‘Lv

of Unified Science

: v :

VOLUME I PART 1

I JH

.

i

f i

EDITED BY

Otto Neurath Rudolf Carnap & Charles W. Morris

Leonard Bloomfield

Niels Bohr

Rudolf Carnap

John Dewey

Victor F. Lenzen

Charles W. Morris

Otto Neurath

Bertrand Russell

Recent years have witnessed a striking
growth of interest in scientific enterprise
and especially in the unity of science. A
new concern throughout the thinking
world for the logic, history, and sociol-
ogy of science led to this Encyclopedia,
begun in 1938 under the editorship of
the late Otto Neurath. The first volume,
containing the first ten of twenty intro-
ductory monographs, is here presented
in two permanently bound parts.

Part 1

ENCYCLOPEDIA AND UNI-
FIED SCIENCE

OTTO NEURATH opens the introduc-
tory monograph with an essay on Uni-
fied Science as Encyclopedic Integration;
JOHN DEWEY discusses Unity of Sci-
ence as a Social Problem; RUDOLF
CARNAP analyzes the Logical Founda-
tions of the Unity of Science; and
CHARLES W. MORRIS, in his essay Sci-
entific Empiricism, attempts to formulate
a general framework for the encyclopedic
enterprise as a whole. There are two short
papers — by NIELS BOHR on Analysis
and Synthesis in Science, and by BER-
TRAND RUSSELL On the Importance
of Logical Form.

FOUNDATIONS OF THE
THEORY OF SIGNS

In the second section CHARLES W.
MORRIS outlines the field of the theory
of signs, showing the place of this dis-
cipline within the unity of science move-
ment. After a general discussion of sign
processes he examines in turn the three

( Continued on back flap)

r- <0

G/tcn^e, V Gtza-Usyu-

i

International Encyclopedia of Unified Science

International Encyclopedia

of

Unified Science

Otto Neurath

Volume I, Nos. 1-5

Edited by

Charles Morris

Rudolf Carnap

The University of Chicago Press

Chicago, Illinois

This edition combines in two cloth-bound vol-
umes the ten numbers of Volume I of the In-
ternational Encyclopedia of Unified Science.

Copyright 1938, 1939, 1946, 1951, 1952, 1955
by The University of Chicago. All rights reserved.
Combined edition published 1955.

Copyright 1955 under the International Copyright Union.
Composed by the University of Chicago Press, Chicago 37, Illinois.
Printed in the United States of America.

CONTENTS

Page

PART 1

Encyclopedia and Unified Science 1

Foundations of the Theory of Signs 77

Foundations of Logic and Mathematics 139

Linguistic Aspects of Science 215

Procedures of Empirical Science 279

PART 2

Principles of the Theory of Probability 3 41

Foundations of Physics 423

Cosmology 505

Foundations of Biology 567

The Conceptual Framework of Psychology 655

Encyclopedia and Unified
Science

Otto Neurath, Niels Bohr, John Dewey,
Bertrand Russell, Rudolf Carnap,

Charles W. Morris

Encyclopedia and Unified Science

Contents:

OTTO NEURATH : Unified Science as Encyclopedic paoe
Integration 1

I. Unity of Science Movement 1

II. Mosaic of Empirical Science 3

III. From Metaphysical Comprehensiveness to Em-

piristic Synthesis 5

IV. Scientific Attitude and Systematization of Em-
pirical Procedure 8

V. Logical Analysis of Scientific Statements 10

VI. Logico-empirical Integration 15

VII. Unified Science and Encyclopedism .... 20

VIII. Structure of the Encyclopedia 23

NIELS BOHR: Analysis and Synthesis in Science 28

JOHN DEWEY : Unity of Science as a Social Prob-
lem 29

I. The Scientific Attitude 29

II. The Social Unity of Science 32

III. Education and the Unity of Science .... 35

BERTRAND RUSSELL: On the Importance of Log-
ical Form 39

VIII

Contents

PAGE

RUDOLF CARNAP: Logical Foundations of the Uni-
ty of Science 42

I. What Is Logical Analysis of Science? ... 42

II. The Main Branches of Science 45

III. Reducibility 49

IV. The Unity of the Language of Science ... 52

V. The Problem of the Unity of Laws .... 60

CHARLES W. MORRIS: Scientific Empiricism . . 63

I. Method in Science 63

II. Generalization of Scientific Method .... 65

III. The Viewpoint of Scientific Empiricism ... 68

IV. Science and Practice 72

V. Scientific Empiricism and Encyclopedism 74

IX

Unified Science as Encyclopedic
Integration

Otto Neurath

I. Unity of Science Movement

Unified science became historically the subject of this En-
cyclopedia as a result of the efforts of the unity of science move-
ment, which includes scientists and persons interested in science
who are conscious of the importance of a universal scientific
attitude.

The new version of the idea of unified science is created by the
confluence of divergent intellectual currents. Empirical work of
scientists was often antagonistic to the logical constructions of
a priori rationalism bred by philosophico-religious systems;
therefore, “empiricalization” and “logicalization” were con-
sidered mostly to be in opposition — the two have now become
synthesized for the first time in history. Certain empiricists be-
gan to appreciate logical analysis and construction as universal
scientific aids; other thinkers, especially interested in the his-
torical importance of scientific imagination — many call it ra-
tionalistic fancy — stress their opposition to all kinds of a priori
reasoning and demand empirical tests for all their theories. The
term ‘logical empiricism’ expresses very well this new type of
synthesis; it may be used synonymously with the term ‘em-
pirical rationalism.’ To further all kinds of scientific synthesis
is one of the most important purposes of the unity of science
movement, which is bringing together scientists in different
fields and in different countries, as well as persons who have
some interest in science or hope that science will help to amelio-
rate personal and social life. The terms ‘unified science’ and
‘unity of science,’ used before this movement came into being,
are becoming more and more widespread in usage. This move-

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Encyclopedia and Unified Science

ment has found in the International Congresses for the Unity
of Science its organized contemporary expression. These con-
gresses, held yearly, show a new field for co-operation . 1 Phys-
icists are familiar with co-operation in the field of physics,
biologists in the field of biology; the same type of scientific co-
operation is shown by these congresses, because the members
of these congresses and all scientists in the movement co-oper-
ate in the field of unified science — physicists with biologists,
biologists with sociologists, and other specialists with logicians
and mathematicians. One important work within the wider
unity of science movement will be this Encyclopedia.

The International Encyclopedia of Unified Science aims to
show how various scientific activities such as observation, ex-
perimentation, and reasoning can be synthesized, and how all
these together help to evolve unified science. These efforts to
synthesize and systematize wherever possible are not directed
at creating the system of science; this Encyclopedia continues
the work of the famous French Encyclopedic in this and other
respects.

About one hundred and ninety years ago D’Alembert wrote a
Discours preliminaire for the French Encyclopedic , a gigantic
work achieved by the co-operation of a great many specialists.
Although a lover of the systematizing scientific mind, D’Alem-
bert objected to the making of a universal system, just as
Condillac opposed such attempts in his TraitS des systemes, in
which he criticized the great rationalistic systems of his day.
D’Alembert’s idea of the procedure of empirical science was
mostly based on Francis Bacon, and the idea of science in gen-
eral on Newton; no comprehensive idea of “logicalization” stim-
ulated the Encyclopedists. One must carefully look at their
work as an important example of organized co-operation. Per-
haps the same kind of scientific tolerance will appear in this
Encyclopedia which appeared in the French Encycloptdie when
D’Alembert, in his Introduction, opposed Rousseau’s aggres-
sions against science and yet expressed his pleasure that Rous-
seau had become a collaborator in the work. As a modern
scientific man, D’Alembert stressed the degree to which all

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Neurath: Unified Science as Encyclopedic Integration

scientific activities depend upon social institutions. Today this
idea is so familiar that this Encyclopedia will give it special at-
tention. Succeeding generations may better be able to assess
how far this present Encyclopedia expresses living activities, old
traditions, and a rising future.

II. Mosaic of Empirical Science

Continual scientific activity throughout the centuries gives
rise step by step to a' distinctive intellectual environment. The
history of this evolution of empirical science and scientific em-
piricism can be regarded as the history of a “mosaic,” the pat-
tern of which has been formed by combining new observations
and new logical constructions of diverse character and origin.
The generations of the “mosaicists” are not only inlaying the
stones but also changing certain stones for others and varying
the whole pattern. Scientific thinkers who were combating one
another in social and intellectual conflicts must nevertheless
often be regarded as the contributors of little stones to the same
part of the whole pattern.

The history of philosophy, on the contrary, can be written
as a history of philosophical systems formed by certain persons
who concentrated upon and focused attention upon particular
groups of ideas. Such a focalization can be more easily corre-
lated to certain social situations than can the mosaic of em-
pirical science to its historical environment. One can speak
about a “republic of scientists” which makes a scientific pattern
but not about a “republic of philosophers.”

Science as a whole can be regarded as a combination of an
enormous number of elements, collected little by little. One ex-
ample may show this: The principle of Huyghens says that
each point on an expanding sphere of light can be regarded as a
center of a new expanding sphere of light. While the followers
of the emission theory could use the idea of a certain periodicity,
the followers of Huyghens at first lacked it ; Euler used the idea
of the periodicity of undulation but objected to the principle of
Huyghens. It happened subsequently that Huyghens’ principle,
together with periodicity and other factors, formed the theory

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Encyclopedia and Unified Science

of light of the nineteenth century. In the same way, in the mod-
em theory of economics one finds a connecting of elements of
diverse origin. Smith and Ricardo did show certain correla-
tions of the market system but did not concentrate their inter-
est in certain decreases of wealth as did Sismondi; in that part
of modern theory which deals with business cycles one finds
more elements of Marxism than of Smithian ideas. Only a com-
plicated comparative scheme could show the amalgamation of
various elements, the common and different features of various
theories. In a similar way one might analyze this Encyclopedia
and show the elements which form the idea of a unified science.

Modem empiricism has grown up in the scientific tradition,
in the activities of daily life, and in philosophico-religious
speculations. The more or less common thought of the Middle
Ages, which was mainly based on a literature written in Latin,
Arabic, Hebrew, or Greek, was also influenced by other peoples
outside the Mediterranean. Indian and Chinese influences were
not very strong, but increased in later times. The social ideas
of Quesnay, Montesquieu, Voltaire, and other thinkers were
directly influenced by Chinese ideas. One also finds such an in-
fluence in philosophers, for instance, in Leibniz. Scholasticism,
roughly speaking, was the native soil of all European thinking.
No great attention was devoted to empirical studies within the
structure of scholasticism, but one sees scientific research and
reasoning arising within this structure. Men who wrote about
physical problems also gave metaphysical explanations or com-
bined empirically tested statements with purely traditional
statements. Certain elements of optics in the works of Roger
Bacon (about 1250 ) or in the works of other Scholastics, such as
Dietrich of Freiberg, are of the same type as the corresponding
explanations of Snellius or Descartes (about 1600 ); neverthe-
less, Roger Bacon discussed the question of the distance be-
tween the western coast of Spain and the eastern coast of Asia
without arguing empirically or making particular proposals for
research. Macrobius had previously analyzed Cicero’s Som-
nium Scipionis and his explanations dealing with the five zones
of the globe, with the antipodes and the continent “Australis”;

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Neurath: Unified Science as Encyclopedic Integration

Roger Bacon did not discuss these matters systematically and
maintained only that the distance was relatively short, using
quotations from Aristotle, from an apocryphal book of the
Bible, and from a well-known saint, thus preparing for the dis-
covery of America by Columbus. The fourteenth-century
Oresme, a mathematician, physicist, and economist of impor-
tance, discussed a great many questions in a modern manner,
but he holds a far from universal scientific attitude, as appears
when his work as a whole is analyzed.

The increasing number of technical inventions and geograph-
ical discoveries, together with the increasing secularization of
politics, produced a new attitude and the necessity to analyze
and combine a great deal of empirical data. Machiavelli, for in-
stance, discussed problems of social organization in an attitude
far from the Scholastic tradition. A long series of historians and
economists, physicists and biologists, did the same in the fol-
lowing centuries and worked out a modern idea of the world.
Alexander von Humboldt has shown in his Kosnios how such
empirical studies can be correlated without the help of philo-
sophico-religious construction. Empiristic interest spread in the
public not only in the field of technical activities but also in
daily life: during the eighteenth century, for instance, micros-
copy was a hobby as photography is now. That physiognomic
and phrenologic studies became popular can be regarded as a
sign of a growing — though only uncritical — empirical interest.

The evolution of a comprehensive empiricism in opposition
to the traditional systems has started with work in decentralized
camps. Scholasticism was based on a well-organized movement,
but not so the young empirical science. In the Encyclopedia of
Unified Science this historical situation and its consequences
will be demonstrated by showing the formation of the mosaic of
scientific activities.

III. From Metaphysical Comprehensiveness to
Empiristic Synthesis

All-embracing vision and thought is an old desire of human-
ity. Gnosticism was not only characterized by realistic ideas

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Encyclopedia and Unified Science

pertaining to astrology and magic but was also full of ideas
dealing with angel-like “emanations” and personified concepts
and qualities. Elements of these “cosmic poems”— if one may
use this term — combined with elements of Platonism, Aris-
totelianism, and other philosophical and religious tendencies,
formed the background of the medieval desire for a compre-
hensive viewpoint — a desire which can also be found in medieval
mysticism and not only in Scholastic intellectual systems. A
bewildering multiplicity of logical processes were performed by
the deductions from and combinations of dogmatic texts of the
Catholic church, statements from the Bible, the Church Fa-
thers, Aristotle, and others. One finds mainly theological ex-
planations (moral theology and other theological disciplines in-
cluded) and medieval cosmology (references to heaven, purga-
tory, hell, and other places for the soul) in the Scholastic sys-
tems. Both Suvmnae of Thomas Aquinas are representative of
these structures. 1 he logical instrument of the Scholastics was
sharpened for nonempiristic purposes. Their interest in logical
combination, in discussion, and in argument can be illustrated
by the subjects of the trivium and the quadrivium : grammar,
logic, rhetoric; arithmetic, music, geometry, and astronomy.

This interest in combining concepts and statements without
empirical testing prepared a certain attitude which appeared in
the following ages as metaphysical construction. The neglect of
testing facts and using observation statements in connection
with all systematized ideas is especially found in the different
idealistic systems. One may use Hegel’s Encyclopedia as an
example of this type. His comprehensive work can be re-
garded as substituting a priori philosophico-religious state-
ments for traditional theological statements, and as joining
these philosophico-religious statements to metaphysically trans-
formed empirical statements. Hegel’s vigor and all-embracing
enthusiasm stimulated such empirical thinkers as Feuerbach,
Marx, and Engels; they received more thoughts that breathe
and words that burn from Hegel than from the books in which
Helvetius, Holbach, and others wrote about the world in an
empirical sense. Neither Hegel nor Schelling encouraged a

6

Neurath: Unified Science as Encyclopedic Integration

scientific attitude and produced logical analyses or particular
theses which could be used directly in the sciences, as did, for
instance, certain ideas of Descartes and Leibniz. Schelling’s
N atnr philosophic influenced chemists and other scientists; the
results were such that Liebig and others had to fight Schel-
lingism in their own scientific camp.

The large and panoramic systems of idealistic philosophy are,
as it were, late branches of a deformed scholasticism. Hegelian-
ism is a typical metaphysical system of our age; Thomism is a
typical Scholastic system which is still living in the Catholic
church and also attracting some persons outside the Catholic
church. Both these systems have not so far shown any disposi-
tion to logicalize empirical science, to form a quasi-addition to
their philosophico-religious structures. Synthesis of empirical
science was not directly supported by metaphysical and the-
ological systems, though certain stimulations came from them.

Collections of interesting subjects, biographical, geographi-
cal, philological, and other data were already known in the
Middle Ages, but in the seventeenth and eighteenth centuries
some enormous encyclopedias of a new type were published (not
so large as the Chinese, however). The French philosophical
opposition organized the great French EncyclopSdie. It was not
a “ faute de mieux encyclopedia” in place of a comprehensive sys-
tem, but an alternative to systems. Since the Encyclopedists
stressed the view that empirical statements were to be used not
only for science (in spite of the fact that one can find not a few
metaphysical and theological explanations in the work itself)
but also for engineering and other technical purposes, collec-
tions of pictures were added to the text. The French Encyclo-
pedic and its empirical attitude were combated by the church
and government. Persons writing papers which seemed able to
stimulate social perturbation had to fear persecution. Diderot
(a polyhistor such as Voltaire but, unlike him, not doing exper-
imental research) had to permit of textual variations by the
publisher who feared the powerful enemies of the EncyclopSdie.
This encyclopedia had no comprehensive unity despite the ex-
pression of a certain empirical attitude; it was organized by

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Encyclopedia and Unified Science

means of a classification of sciences, references, and other de-
vices.

More constructive ideas formed the basis of Comte’s Phi-
losophic positive. Comte, familiar with mathematics and phys-
ics, on the one hand, and very interested in social problems, on
the other (the term ‘sociology’ was coined by him), tried to
combine his ordering and classification of science with historical
interests. Spencer’s gigantic work, the system of Synthetic Phi-
losophy, is also an example of this synthetic tendency which
avoids metaphysical construction (this is not the place to dis-
cuss some metaphysical arguments in Comte’s and Spencer’s
work) but endeavors to substitute an empirical scientific whole
for philosophical speculations. One must stress the fact that
Spencer’s interest was concentrated in biology and sociology,
because it is a usual prejudice that the idea of unified science
and the unity of science movement is especially based on an
interest in physics. Neither the Encyclopedists nor Comte and
Spencer nor similar thinkers made an attempt to organize a
logical synthesis of science.

IV. Scientific Attitude and Systematization of
Empirical Procedure

The parallel increase of the scientific attitude and the system-
atic analysis of scientific procedures prepared the way for logical
empiricism. During the Renaissance modern thinkers began to
be interested in the procedure of empirical science and in the
scientific attitude. Leonardo da Vinci, a universal genius in-
fluenced by the Scholastics, became a man with a comprehen-
sive scientific attitude. He worked in different fields of en-
gineering, was interested in scientific problems, and of course
in all matters connected with painting. He began to feel the
common root of all empirical science and stressed, for instance,
the importance of “generalization” and other scientific aids. He
understood what we call the empirical procedure. The uni-
versality and versatility of Leonardo da Vinci and other think-
ers, scientists, and amateurs in science were stimuli of great

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Neurath: Unified Science as Encyclopedic Integration

importance for the origin of a comprehensive empiricism and a
scientific attitude.

History shows us a great many scientists whose scientific atti-
tude is mot equally maintained in the various fields of thought.
Such a scientist may be very critical in his own domain, for
instance in physics, but of a totally different behavior when he
speaks about “free will,” “social privileges,” or similar tradi-
tional problems. On such occasions some scientists sharply
change their criticism and exactness of arguments and their
style of language. Newton, for instance, was a scientist in whom
theological speculations and scientific empiricism existed partly
side by side, partly in actual connection — he speaks, for in-
stance, about space as sensorium dei. A comprehensive scien-
tific attitude has come into being, within scientific activity,
since the Renaissance, but neither Leonardo da Vinci, nor
Galileo a hundred years after him, analyzed the rules of em-
pirical procedure or the scientific attitude. Francis Bacon was
not very active in science or engineering, but he promoted cer-
tain ideas of empiricism successfully, especially the idea of
“induction” as a scientific aid, in spite of the fact that he gave
very poor examples in his work and did not recognize the
scientific importance of Galileo and other empirical thinkers of
his time.

Modern scientific empiricism attained very late in its de-
velopment a comprehensive work which analyzes empirical pro-
cedure in all scientific fields: John Stuart Mill’s A System of
Logic, Ratiocinative and Inductive, Being a Connected View of the
Principles of Evidence and the Methods of Scientific Investigation.
Mill does not question the fact that astronomy and social sci-
ence, physics and biology, are sciences of the same type. Mill,
who was familiar with the problem of utilitarianism, with polit-
ical economy and governmental practice as a pupil of his fa-
ther, and with Bentham (who is not even now sufficiently ap-
preciated), could effectively use Whewell’s famous History of
the Inductive Sciences from the Earliest to the Present Times.
Mill’s work influenced modern empiricism despite the fact that
many of his particular statements were criticized. This type of

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Encyclopedia and Unified Science

comprehensiveness is also represented by the Principles of Sci-
ence of Stanley Jevons; he was an economist like Mill and one of
the promoters of modern symbolic logic. Karl Pearson’s The
Grammar of Science details the measures and procedures of
modern science. Karl Pearson was very interested in biologi-
cal and sociological questions, as was his teacher, Galton, but he
was also familiar with physics.

But neither Mill nor other thinkers of similar type applied
logical analysis consistently to the various sciences, thus at-
tempting to make science a whole on a “logicalized” basis.
They only achieved a comprehensive understanding of all the
arguments which a scientist needs if he mak£s generalizations
and tests scientific hypotheses. This Encyclopedia will show
modern attempts to reform generalization, classification, test-
ing, and other scientific activities, and to develop them by
means of modern logic.

V. Logical Analysis of Scientific Statements

One science after the other separated from the mother-phi-
losophy; scientists became more capable of solving difficult
scientific problems than were philosophers occupied by a great
many unscientific speculations and the particular problems of
their own systems. One example may demonstrate this situa-
tion. During the eighteenth century people of different inter-
ests discussed the problem of “inertia.” Some were influenced
by Descartes, who used the term ‘motion’ when a body which
is in the neighborhood of a second body is later found in the
neighborhood of a third body. ‘Space’ and ‘groups of bodies’
were for these people more or less the same terms and ‘empty
absolute space’ seemed to be a meaningless term. Others used
the idea on which actual physical calculation was based in these
times. The question arose whether to describe “inertia” in terms
of relative correlations between bodies (a certain body together
with the fixed stars), in terms of absolute space, or by means of
other conceptions. Euler (about 1750) discussed the kinds
of argumentation and was inclined to oppose the opinion
that inertia could depend upon the totality of the fixed stars —

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Neurath: Unified Science as Encyclopedic Integration

an opinion which is similar to modern ideas and which has been
developed step by step by Mach and others. Kant, discussing
this group of problems and analyzing Euler’s paper, did not
feel that this idea about correlation between inertia and the
fixed stars was important enough to be criticized and did not
even mention this, for him, strange construction, being more
interested in his own a priori philosophy and in supporting
Newton’s ideas. The “essayistic” criticism by Hume and simi-
lar thinkers loosened the firmness and coherence of compact
traditional opinions, but Kant, who stimulated some scientists,
formed new barriers of peculiar rigidness by focusing and regu-
lating criticism and skepticism. The essayist-philosopher Nietz-
sche showed how much of an antiscientific attitude can be
found in Kant’s system, which reduces the power of science and
thus opens the doors to metaphysical and philosophico-religious
speculations.

The evolution of non-Euclidean geometry, for instance, which
prepared modern theories of measuring time and space, was
hardly supported by modern philosophers — another example of
the inadequacy of philosophers. One can rather assume that
the ideas of Gauss, Bolyai, Lobachevski, and others were im-
peded by Kantianism : they had to start by opposing all kinds
of apriorism. Not a few philosophers opposed the theory of rela-
tivity. The new intellectual environment was prepared more
by specialists in physics and mathematics, or by certain im-
aginative amateurs in the field of science, and by poets, than by
systematic philosophers. How many people may have been edu-
cated in the field of scientific imagination by Jonathan Swift !

A long series of imaginative analyses started with the ani-
mated statue imagined by Condillac, who had been influenced
in his thought by Locke, Lamettrie, and others. Imagined
statues which received one sense after another are relatives
of Caspar Hauser and all the many children who have been
found in the woods or other isolated places. These imagined
and real beings are the subject matter of an old and rich litera-
ture which helped to prepare a logico-empirical attitude by
means of imaginative analysis. Helmholtz and others, op-

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Encyclopedia and Unified Science

posing Kant’s ideology, imagined (partly for pedagogical pur-
poses) two-dimensional beings on a sphere discussing geometri-
cal problems. About the middle of the nineteenth century Fech-
ner and others fancied dreamlands of different kinds: three-di-
mensional beings of different ages were produced by cutting off
slices from a four-dimensional sausage. Some scientists fear
such imaginative analogies as unreliable guides and demand the
use of more systematic analysis. Actually, the history of all
these imaginations has to be regarded as a part of the history of
the empiristic mosaic. One must add to these significant, im-
aginings that of two-dimensional beings who, traversing a hill,
observe a retardation region, an indifference region, and an ac-
celeration region of a geometrically homogeneous world. One
could imagine a country (“Aeonia”) in which beings could be
repaired as are machines. There could be beings with con-
nected nerves, or powerful beings consisting of a brain, one
muscle, and one sense organ, but using complicated mechanical
devices. Poincare’s problem of similar worlds (our world, re-
duced or enlarged in size) was also analyzed imaginatively
about the middle of the nineteenth century by Eberty ( Die
Gestirne und die Weltgeschichte, newly edited by Gregorius Itel-
son with an introduction by Albert Einstein), who fancied also
a trip throughout the universe quicker than light, in concordance
with certain thoughts of Humboldt and Babbage. Renouvier
( Uchronie , Vutopie dans Vhistoire ) used imagination in the field
of history: how history really happened and how it might
have happened. Lichtenberg, together with Chodowiecki (well
known by his graphic work) and Sonya Kovalevska, the mathe-
matician, elaborated similar fantasies for single individuals.

From these imaginations one can enter into the problem of
behavioristics, logico-empirical analysis, and poetry. One can
ask whether a blind man can make a complete physical descrip-
tion by means of certain devices or how the sensorium of
Siamese twins is formed in the common part of their body, and
similar concrete questions; one can write stimulating imagina-
tive novels as did H. G. Wells, or a book of logico-empirical

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Neurath: Unified Science as Encyclopedic Integration

analysis like Carnap’s Logischer Aufbau der Welt. To what ex-
tent imaginative constructions will be useful in the future may
remain an open question.

All these imaginative analyses and constructions formed a
part of the “essayistic” analysis of scientific argumentation
such as that made by Poincare, Duhem, and others. Their mod-
ern logical analysis of scientific statements, hypotheses, and
theories was prepared for by many thinkers : the physicist Brew-
ster, for instance, said that it is of the greatest importance that
the same value d, characterizing the periodicity of light, could
be found in the undulation theory and in the emission theory.
John F. W. Herschel was anticipating Poincare and Duhem
when he wrote in his Preliminary Discourse on the Study of
Natural Philosophy that one might imagine such a development
of Newton’s theory that it could solve certain problems which
seemed reserved for the undulation theory. This sort of analy-
sis of scientific statements was strongly supported by historico-
critical studies. Mach’s leading books dealing with mechanics,
optics, and theory of heat characterize this tendency. He, by
his paradigmatic analysis of concepts such as space and inertia,
furthered the evolution of Einstein’s theory of relativity. He
did not make new experiments for this particular purpose but
used, of course, the physical knowledge of his time. The fact
that Duhem, Enriques, Mach, and others were active in their
special sciences in logical analysis of scientific statements, and
in the historical analysis of science, suggests the idea that the
“history of the history of science” should be very instructive.
In centuries in which such an analysis of scientific statements
was not evolved as a special activity, historians of science were
often busy with the analysis of scientific statements as a prepa-
ration for their historical analysis. A history of the history of
optics, for instance, from Joseph Priestley {The History and the
Present State of Discoveries Relating to Vision, Light, and Colors
[ 1772 ], written as a part of a universal history of physics) up to
Ernst Mach {Die Prinzipien der physikalischen Optik [ 1921 ])
shows clearly the increase of logicalization.

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Encyclopedia and Unified Science

Corresponding historical sequences may be found in all scien-
tific fields. The systematic analysis of “planned economy,” for
instance, has to be based on various fundamental problems; one
group of problems deals with describing various possibilities:
Thomas More’s Utopia and Francis Bacon’s Nova Atlantis are
of another type of social analysis than are the plans of Ballod-
Atlanticus or Popper-Lynkeus — plans which are forerunners of
the science of socio-economic planning. Such kinds of social im-
agination can be combined with historico-social analyses like
those made by Montesquieu, Stein, Marx, and others. The
structure of economics is logically not so developed as the struc-
ture of physics, and so the history of the history of economics is
not so abundant in logico-empirical analysis as is the history of
the history of optics. Whewell’s description and discussion of
Linnaeus’ problems dealing with classification and systematiza-
tion give a picture of the logico-empirical analysis of sciences
in the time of Whewell. A corresponding description and dis-
cussion of biological ideas from Linnaeus’ classifications up to
Woodger’s formalization of biology should put a scientist in a
position to compare directly the logico-empirical analysis of our
time with the logico-empirical analysis made by a historian of
science a hundred years ago.

A comparison of the argumentation in cosmology, geology,
physics, biology, behavioristics (“psychology”), history, and
social sciences in different ages will be furthered by the unity of
science movement. An increasing number of scientists are busy
with such problems, and one can hope for great success if scien-
tists analyzing various sciences co-operate with men concerned
with the history and logic of science. The inconsistency of the
historically given universal pattern appears immediately. Suc-
cessive editions of the Encyclopedia of Unified Science would
show progress in logical analysis of scientific statements. Per-
haps a special technique will be evolved which is able to describe
systematically all these changes in the different sciences, thus
continuing the work of Ernst Mach, whose centennial will be
celebrated this year.

14

Neurath: Unified Science as Encyclopedic Integration

VI. Logico-empirical Integration

The mosaic pattern of empirical science progressively shows
more marked interconnections than in the times in which em-
pirical studies were relatively isolated. Scientific analysis of the
sciences led to the observation that an increase of logical inter-
correlation between statements of the same science and between
statements of different sciences is a historical fact; one finds
rationalism (as a quality of our experience), as it were, em-
pirically, and may use the term ‘empirical rationalism’ with this
meaning in which Gregorius Itelson proposed it, not merely as
“rationalism based on experience.” Comprehensiveness arises
thus as a scientific need and is no longer a desire for vision only.
The evolving of all such logical connections and the integration
of science is a new aim of science.

Logical empiricism or empirical rationalism can also be re-
garded as a regeneration of certain elements of a priori ration-
alism. The Scientia generalis of Leibniz was the background of
his “panlogism” — if this term is permitted in this sense and can
be regarded as a “secularization of logic.” The driving power of
panlogism in the framework of a priori rationalism depends
partly upon the idea that one can anticipate by means of logical
combinations the progress of empirical science and not simply
fructify its results or give certain suggestions. Leibniz was the
first and last of the great philosophers who planned seriously to
work out a comprehensive calculus adequate for all scientific
progress. He promoted a universal logicalization of the whole of
human thinking by means of a general calculus and a general
terminology. He worked as a scientist and also began to or-
ganize scientific co-operation by means of scientific academies,
but he was far from attempting or executing a universal scien-
tific empiricism— too busy writing his Theodicee, elaborating the
Monadology, and moreover getting entangled in theological dis-
putes and church diplomacy.

His career was closely connected with Scholastic influences.
As a boy he played, as it were, with logical elements such as
“notions” and “subnotions.” Young Leibniz at twenty years of

15

Encyclopedia and Unified Science

age published his Ars combinatorics, which is influenced by
Raimundus Lullus and other Scholastic authors. He was later
influenced by the rationalistic Descartes and other modern
thinkers, but he made clear that one could successfully use
certain ideas of such Scholastic thinkers as Thomas Aquinas.
Leibniz, the grandfather of modern logistic, transformed the
often vague logical ideas of Scholasticism, and took the first
steps toward modern exactness in logic, preparing the way for
a great many modern ideas in the field of mathematics and logic.

He planned to organize a large encyclopedia, together with
an Atlas universalis, in close connection with his Characteristica
universalis. The plan embraced not only scientific disciplines,
including rational grammar, moral science, geo-politics, but also
natural theology. This gigantic plan was intended to form a
logically organized whole. One may say that the Pansophia of
Comemus (who came in touch with Leibniz) together with his
Orbis pictus can be regarded as the parallel to the Encyclopedia
and the Atlas of Leibniz. Both these pairs of works are based on
a philosophico-religious rationalism and correspond in a certain
sense to the medieval pair: Scholastic system and the over-
whelming visual presentation in a medieval Catholic church.

Since Leibniz, like other a priori rationalists, was seeking the
system of science and the logical key for it, one can understand
that such an ideal was strange to empiricists. Most of the logi-
cal studies of Leibniz were not published, and only a few per-
sons, like Lambert, were interested in special logistic problems
Public opinion was against formal logic. Kant and his followers
discredited formal logic and thus petrified the aversion of Gali-
leo and others against logic as an instrument of the traditional
scholasticism. The growing new logic of Boole, De Morgan, and
Grassman was not supported by philosophical thinkers of this
period. Bolzano, for instance, influenced Austrian scientists and
pedagogues about the middle of the nineteenth century by cer-
tain of his ideas, but his important investigations in the logical
faeM (one example: Paradoxien des Unendlichen ) were not
studied and esteemed for a long period.

A universal application of logical analysis and construction

16

Neurath: Unified Science as Encyclopedic Integration

to science in general was prepared not only by the systematiza-
tion of empirical procedure and the systematization of logico-
empirical analysis of scientific statements, but also by the analy-
sis of language from different points of view. A direct route
leads from Scholastic analysis of language, made especially by
nominalists, up to Condillac (Essai sur Vorigine des connais-
sances humaines), who influenced French and English thinkers,
to Bentham, whose multifarious work still lives today, and to
other thinkers interested in language as an aid for our daily life
and for science. The connection between modern logic and em-
piricism did not arise instantly. The importance of logicians of
the second half of the nineteenth century, such as Venn,
Schroeder, Peano, Frege, and others, will be expressly discussed
in the Encyclopedia. A few of the modern logicians, such as
Peirce and, later on, Bertrand Russell, combined the interest in
logic with an interest in empiricism. Traditional idealistic phi-
losophers did not discuss carefully or look with favor upon this
new combination of logicalization and empiricalization. The
fact that Peirce was a logician and simultaneously interested in
empiricism was in turn important for the preparation of modern
scientific empiricism in the United States. In Europe the
Vienna Circle, the Berlin group, related thinkers in England
and Scandinavia, the Centre de synthese in I ranee, and the
Scientia group in Italy are evidence of the interest in this evo-
lution of logic and empiricism. Some thinkers are mainly busy
with logical calculi, such as the members of the Polish School
or the Munster group. One cannot judge at the moment what
elements of these and other circles of thinkers may become most
essential for the future of unified science. The importance of
Riemann’s geometry for modern physics did not appear at once.
What part will metalogic, semantics, and other disciplines play
in the unification of the language of the empirical sciences?

The important opinion arose (the influence of Wittgenstein,
a metaphysician in many respects, has to be mentioned in this
connection) that all statements can be expressed as “scientific
statements” and that one cannot speak of special “philosophic
statements.” Some persons proposed to use the term ‘phi-

17

Encyclopedia and Unified Science

losophize’ for an activity which makes concepts and statements
clear; others proposed to use the term ‘philosophy’ for ‘logic of
science.’ If one takes the thesis seriously that in the field of
knowledge one only has to deal with scientific statements, the
most comprehensive field of statements must be that of unified
science. If one does not care to avoid the term ‘philosopher,’
one may use it for persons engaged in unified science. Such
“philosophers” may be specialists in one discipline and ama-
teurs in others or comprehensive scientific amateurs like Vol-
taire, but not speculative thinkers.

It is common to all these persons that they do not join scien-
tific statements with a second type of specific “philosophical”
formulation; this attitude gives one the feeling that one is acting
within the collective scientific atmosphere and not in the sphere
of individual philosophemes. Voltaire mentioned that opinions
which become common do not bear the names of their creators
(Du Bois-Reymond added that Voltaire’s name and activity are
insufficiently known because “Voltairianism” is a quality of
the age after Voltaire). Nietzsche stressed esteem for “the
unpretentious truths, objecting to the fascinating errors of
metaphysical ages. An evolved civilization likes, according to
Nietzsche, the modest results found by means of exact methods
which are fruitful for the whole future; and such manliness,
simplicity , and temperance will characterize not only an
increasing number but also the whole of humanity in the
future. Moritz Schlick explained in a similar sense that the
evolution of modern critical thinking is founded on an anony-
mous mass of thinkers, especially scientists, and that progress
does not arise from the sensational philosophical systems which
form an endless row, each contradicting the others.

Scientists may now build up systematical bridges from science
to science, analyzing concepts which are used in different sci-
ences, considering all questions dealing with classification, or-
der, etc. Axiomatization of science seems to give an opportu-
nity to make the use of fundamental terms more precise and to
prepare the combination of different sciences; preliminary axi-

18

Neurath: Unified Science as Encyclopedic Integration

omatization has to be founded on a long evolution of science.
We cannot anticipate a “final axiomatization.”

Some difficulties in science, even within a special discipline,
arise frpm the fact that one cannot always decide whether two
scientists (for instance, psychologists) speak about the same or
different problems, or whether they explain the same or differ-
ent opinions, by means of different scientific languages. Unifi-
cation of scientific language is one of the purposes of the unity
of science movement. It is a question to what extent such uni-
fication can be furthered. One can perhaps reduce all scientific
terms to one kind of term by means of a special logical tech-
nique. The thesis of physicalism which will be discussed in this
Encyclopedia (see the following article by Carnap) emphasizes
that it is possible to reduce all terms to well-known terms of our
language of daily life. Another question is to what extent one
can reduce the statements or laws of biology, behavioristics,
or sociology to physicalistic statements or laws. All studies
dealing with languages and scientific terminology are regarded
seriously not only in connection with what one usually calls
logical questions but also in connection with questions of so-
ciology and behavioristics; one may ask, for instance, how prob-
lems discussed by the Dutch group of thinkers interested in
“signifies” are connected with problems of semantics and other
new disciplines. A great many scientists, working in different
fields, are pushing these analyses forward.

Since more and more scientists stress the fact that in the end
one must test all theories by means of the language of daily life,
the correlations between the calculus of theories and the lan-
guage of daily life will be systematically analyzed.

Many people think that logic (or logistic) is, as it were, an
antidote to metaphysical speculations; that is wrong: one can
elaborate a speculative metaphysical system more logico demon-
strata. There is no automatically acting antidote against state-
ments which, though formulated in an empirical language, yet
need scientific criticism. Of which temper one’s mind is, one can
show by presenting one’s work.

19

Encyclopedia and Unif ied Science

VII. Unified Science and Encyclopedism

Science itself is supplying its own integrating glue instead of
aiming at a synthesis on the basis of a “super science” which is
to legislate for the special scientific activities. The historical
tendency of the unity of science movement is toward a unified
science departmentalized into special sciences, and not toward
a speculative juxtaposition of an autonomous philosophy and a
group of scientific disciplines. If one rejects the idea of such a
super science as well as the idea of a pseudo-rationalistic anticipa-
tion of the system of science, what is the maximum of scientific
co-ordination which remains possible? The answer given by the
unity of science movement is: an encyclopedia of unified sci-
ence. An encyclopedia (in contradistinction to an anticipated
system or a system constructed a priori) can be regarded as the
model of man’s knowledge. For, since one cannot compare the
historically given science with “the real science,” the most one
can achieve in integration of scientific work seems to be an
encyclopedia, constructed by scientists in co-operation. It may
happen that one must use in one hypothesis, destined for a par-
ticular purpose, a supposition which contradicts another sup-
position used in another hypothesis, destined for another par-
ticular purpose. One may try to eliminate such contradictions,
but in the historically given science, and so in a real encyclo-
pedia, these and other difficulties always appear.

Encyclopedism may be regarded as a special attitude; one
may also speak of encyclopedism as a program. Encyclopedism
starts with the analysis of certain groups of scientific state-
ments; it may happen that these can be axiomatized and that
this axiomatized group of statements can be combined with
others expressed in a similar form. But such a system of state-
ments must not be regarded as a model of the scientific knowl-
edge of a given age. An encyclopedia and not a system is the
genuine model of science as a whole. An encyclopedic integra-
tion of scientific statements, with all the discrepancies and diffi-
culties which appear, is the maximum of integration which we
can achieve. It is against the principle of encyclopedism to im-

20

Neurath: Unified Science as Encyclopedic Integration

agine that one “could” eliminate all such difficulties. To believe
this is to entertain a variation of Laplace’s famous demon who
was supposed to have a complete knowledge of present facts
sufficient for making complete predictions of the future. Such
is the idea of the system, in contrast to the idea of an encyclo-
pedia; the anticipated completeness of the system is opposed to
the stressed incompleteness of an encyclopedia.

Such encyclopedism is the expression of a certain skepticism
which objects not only to metaphysical speculations but also
to overstatements within the field of empirical sentences. But
how many theories and predictions, especially those which are
used for stimulating people to practical work, often use terms
such as ‘certain,’ ‘always,’ instead of terms such as ‘perhaps,’
‘sometimes’! An empiricist must permit himself, if necessary,
a certain vagueness. Scientism — if one may use this term, in-
troduced by French positivists a hundred years ago — does not
depend upon “exactness” but only upon the permanence of
scientific criticism. New ideas of scientific importance start
mostly with vague and sometimes queer explanations; they
become clearer and clearer, but the theories which follow will
stand in time before the door with all their new vagueness and
queerness. Niels Bohr expressed this historically and peda-
gogically essential fact in his paradoxical manner: the law of
complementarity is valid also for fruitfulness and clearness of
scientific theories. Must one fear by this to encourage vague
speculations? No! Persons who are interested in unscientific
speculation will undertake it under all circumstances. But it is
useful to avoid dogmatism and bumptiousness in scientism and
empirical panlogism. One can love exactness and nevertheless
consciously tolerate a certain amount of vagueness.

How can one combine such a critical and skeptical attitude
with the unparalyzed driving power which is needed to attain
success in social and private life? The wish to eliminate all these
limitations to a comprehensive scientific attitude often leads
men to hypocrisy and cant; one cannot deny that a certain
practical antagonism may arise between the scientific attitude
and human activity, be it in a particular or in all social orders.

21

Encyclopedia and Unified Science

Science forms an essential part of the rich pattern character-
izing modern life, which is becoming more and more uniform.
If one looks at the whole of humanity, one sees a constant in-
crease of the scientific attitude in daily life during the last cen-
turies, in spite of the fact that books, speeches, and propaganda
dealing with metaphysical speculation show us a “to and fro.”
There is no direct correlation between empirical activity in life
and business, on the one hand, and systematically expressed
empiricism, on the other. How much modern engineering and
technical activity, together with all the helpful special sciences,
were evolved, for instance, in Germany during the nineteenth
century and how little comprehensive scientific empiricism !

The empiricalization of daily life is increasing in all countries:
cities in the United States and in Japan, highwavs in IVIexico and
Germany, armies in China and France, universities in Turkey
and Italy — all show us certain common features. A meteorolo-
gist trained in Denmark may become a useful collaborator to a
Canadian polar expedition; English economists can discuss a
Russian analysis of American business cycles; and Russian
economists may object to or accept the opinions of English econ-
omists about the effect of rural collectivization in the Soviet
Union. If fundamental difficulties arise in discussions between
engineers, generals, and scientists of different countries, they
are not based on the fact that the debaters are of different na-
tions, one can show that analogous fundamental differences also
arise within the same nation — generally when one of the de-
baters, or both of them, intends to use “superscientific” or
“isolated” sentences. The fact that the scientific language is
common to all these people is almost concealed by this type of
discussion.

The unified-science attitude based on the simplicity and
straightforwardness of scientific empiricism is concentrating its
attention on generalizations and predictions made by the de-
baters. What people in all the countries expect scientists to do
is always to predict successfully by means of so-called scientific
procedure, one hopes that a surgeon who knows about bones
and veins will make a diagnosis and then perform a satisfactory

22

Neurath: Unified Science as Encyclopedic Integration

operation, that a historian knows much about human history
and can foretell the main results of a newly undertaken excava-
tion, that an economist judging from the first symptoms can
warn the public of an impending slump, that a political leader
can systematically predict social changes which are arising.

One can state all these scientific prognostications in terms of
everyday language — the language which is common to all men
in the world irrespective of the fact that the scientist himself
uses expressions and symbols in preparatory work which are
mostly of an international character. Unified science is there-
fore supported, in general, by the scientific attitude which is
based on the internationality of the use of the language of every-
day life and on the internationality of the use of scientific lan-
guage.

It may happen that people create and prefer certain terms
and formulations not for universal understanding but for stimu-
lating certain emotions, and may decide that in certain cases
an emotional activity is more important than a scientific atti-
tude. It is not the subject of a scientific explanation to support
or oppose such a decision. If one prefers a comprehensive
scientific attitude, this Encyclopedia tries to show him the spec-
trum of scientific thinking. Each scientifically oriented man
knows very well that the elaboration of such an encyclopedia,
like other activities, is influenced by wishing and fearing, but
there is a difference between men who intend to discover such
influences and others who do not. That leads one to a great
many unsolved problems. Incompleteness and open questions
arise in all parts of this work, but encyclopedism maintains,
nevertheless, that the integration of science is an inevitable part
of man’s scientific activities.

VIII. Structure of the Encyclopedia

One may ask: “What program is common to all the col-
laborators of the Encyclopedia?” A program formed of state-
ments accepted by all the collaborators would be narrow and
would be a source of divergences in the near future. This En-
cyclopedia will show that scientists, though working in different

23

Encyclopedia and Unified Science

scientific fields and in different countries, may nevertheless co-
operate as successfully within unified science as when scientists
co-operate within physics or biology. The Encyclopedia will per-
haps be a mainstay of scientific empiricism as well as of the
unity of science movement in the widest sense. The maximum
of co-operation — that is the program ! This co-operation strives
to elaborate the framework of unified science. Encyclopedism
based on logical empiricism was the general historical back-
ground which underlay the proposal of an international en-
cyclopedia of unified science . 2

The general purpose of the International Encyclopedia of
Unified Science is to bring together material pertaining to the
scientific enterprise as a whole. The work will not be a series of
alphabetically arranged articles; rather will it be a series of
monographs with a highly analytical index, which will make it
possible to find the bit of information sought if the Encyclopedia
is to be used as a reference work. Each monograph, sometimes
written by more than one collaborator, is devoted to a particular
group of problems. The collaborators and organizers of this
work are concerned with the analysis and interrelation of cen-
tral scientific ideas, with all problems dealing with the analysis
of sciences, and with the sense in which science forms a unified
encyclopedical whole. The new Encyclopedia so aims to inte-
grate the scientific disciplines, so to unify them, so to dovetail
them together, that advances in one will bring about advances
in the others.

The Encyclopedia is to be constructed like an onion. The
heart of this onion is formed by twenty pamphlets which con-
stitute two introductory volumes. These volumes, entitled
Foundations of the Unity of Science, are complete in themselves
but also serve as the introduction to what will follow.

The first “layer” of the onion which will inclose this “heart,”
consisting of the first two volumes, is planned as a series of
volumes which will deal with the problems of systematization in
special sciences and in unified science — including logic, mathe-
matics, theory of signs, linguistics, history and sociology of
science, classification of sciences, and educational implications

24

Neurath: Unified Science as Encyclopedic Integration

of the scientific attitude. In these volumes scientists with dif-
ferent opinions will be given an opportunity to explain their
individual ideas in their own formulation, since it is a special
aim of this work to stress the gaps in our present knowledge and
the difficulties and discrepancies which are found at present in
various fields of science. “Heart” and “first layer” together will
be a completely self-contained unit. The following “layers”
may deal with more specialized problems; the interests of the
reader and the collaborators in the particular problems will lead
the members of the Committee of Organization and the Ad-
visory Committee to consider various possible lines of develop-
ment. It is hoped that an Atlas can be worked out as an lsotype
Thesaurus showing important facts by means of unified visual
aids . 3 The plan of this Encyclopedia could not be based on a
generally accepted classification of the sciences — indeed, the
collaborators may perhaps find a new way to assemble system-
atically all the special sciences. The organizers and collabora-
tors know very well that certain frontiers of sciences are un-
satisfactory and that certain terms are not sufficiently defined.
The Encyclopedia will eliminate these defects where possible.

This Introduction has aimed to show the historical position of
the Encyclopedia; it is amplified by special articles. Rough out-
lines are augmented by the articles of Charles W. Morris and
Rudolf Carnap, the one explaining how scientific empiricism is
an even more comprehensive movement than logical empiricism,
the other stressing the importance of the logical analysis of
sciences. The other articles in this introductory monograph am-
plify some aspects of the problems connected with unified sci-
ence. Niels Bohr and Bertrand Russell are concerned with the
importance for the sciences of certain phases of the unity of
science movement, while John Dewey stresses the wider social
implications involved in the unification of the forces of science.

Without pursuing utopian ideals, an effort will be made to
have the scientific language of the Encyclopedia as homogeneous
as it is possible to make it at the present. The Encyclopedia will
express the situation of a living being and not of a phantom;
those who read the Encyclopedia should feel that scientists are

25

Encyclopedia and Unified Science

speaking about science as a being of flesh and blood. The col-
laborators will certainly learn from their encyclopedical work.
Suggestions from different sources will stimulate this activity, so
that this Encyclopedia will become a platform for the discussion
of all aspects of the scientific enterprise. In this way the Inter-
national Encyclopedia of Unified Science hopes to avoid becom-
ing a mausoleum or a herbarium, and to remain a living intel-
lectual force growing out of a living need of men, and so in turn
serving humanity.

NOTES

1 PIans were laW at the congress at Charles University in Prague (1934) for a series
of annual congresses devoted to the unity of science. The proceedings of this preliminary
congress were published in the 1935 Erkenntnis (Leipzig: F. Meiner) and "in separate
volume form as Einheit der Wissenschaft (F. Meiner). The First International Congress
for the Unity of Science was held at the Sorbonne, Paris, in 1935, and the proceedings
were published under the title Actes du congres international de philosophic scientifique
(Pans: Hermann & Cie, 1936). The proceedings of the second congress, held at Copen-
hagen in 1936 and devoted to the problem of causality, appeared in the 1937 Erkenntnis
and also as an independent volume (Das Kausalproblem [Leipzig: F. Meiner; Copen-
hagen: Levin & Munksgaard]). The third congress (Paris, 1937) took the form of a
conference devoted to the project of the International Encyclopedia of Unified Science.
Annual congresses are being planned, and preparations are now being made for the
fourth congress, to be held at Girton College, Cambridge, England, July 14-19, 1938,
and for the fifth congress, to be held at Harvard University, September 5-10, 1939.
The general theme of this congress will be the “Logic of Science,” and the publication
of the Foundations of the Unity of Science is so arranged as to provide a background for
the congress.

The International Congresses for the Unity of Science are being administered by an
International Committee composed of the following members:

N. Bohr (Copenhagen)

M. Boll (Paris)

H. Bonnet (Paris)

P. W. Bridgman (Cam-
bridge, Mass.)

E. Brunswik (Vienna,
Berkeley)

R. Carnap (Chicago)

E. Cartan (Paris)

J. Clay (Amsterdam)

M. R. Cohen (Chicago)

J. Dewey (New York
City)

F. Enriques (Rome)

P. Frank (Prague)

M. Fr£chet (Paris)

F. Gonseth (Zurich)

J. Hadamard (Paris)

P. Janet (Paris)

H. S. Jennings (Balti-
more)

J. Joergensen (Copenha-
gen)

E. Kaila (Helsingfors)

T. Kotarbinski (Warsaw)
A. Lalande (Paris)

P. Langevin (Paris)

K. S. Lashley (Cambridge,
Mass.)

C. I. Lewis (Cambridge,
Mass.)

J. Lukasiewicz (Warsaw)

G. Mannoury (Amster-
dam)

R. von Mises (Istanbul)

C. W. Morris (Chicago)

O. Neurath (The Hague)
C. K. Ogden (London)

J. Perrin (Paris)

H. Reichenbach (Istan-
bul)

A. Rey (Paris)

C. Rist (Paris)

L. Rougier (Besangon,
Cairo)

B. Russell (Petersfield)

L. S. Stebbing (London)

J. H. Woodger (London)

26

Neurath: Unified Science as Encyclopedic Integration

2. See Otto Neurath, “An International Encyclopedia of Unified Science” — a paper
read at the First International Congress for the Unity of Science (Paris, 1935), pub-
lished in Actes du congrh international de philosophic scientifique (Paris: Hermann &
Cie, 1936), Part II. This idea was supported and accompanied by important explana-
tions by the members of the Encyclopedia Committee of Organization (Rudolf Carnap,
Philipp Frank, Joergen Joergensen, Charles W. Morris, Otto Neurath, Louis Rougier),
who spoke about the problems, the importance, and the logical basis of this project (see
papers read by Charles W. Morris, Rudolf Carnap, and Philipp Frank at the same
congress). The First International Congress for the Unity of Science approved the plan
and expressed willingness to help in its fulfilment. See also Otto Neurath, “L’Encyclo-
pedie comme ‘modele,’ ” RemCe de synthlse, October, 1936.

3. See Otto Neurath, International Picture Language: The First Rules of ISOTYPE
(London: Kegan Paul, 1936).

27

Analysis and Synthesis in Science

Niels Bohr

Notwithstanding the admittedly practical necessity for most
scientists to concentrate their efforts in special fields of re-
search, science is, according to its aim of enlarging human
understanding, essentially a unity. Although periods of fruitful
exploration of new domains of experience may often naturally
be accompanied by a temporary renunciation of the compre-
hension of our situation, history of science teaches us again and
again how the extension of our knowledge may lead to the
recognition of relations between formerly unconnected groups
of phenomena, the harmonious synthesis of which demands a re-
newed revision of the presuppositions for the unambiguous
application of even our most elementary concepts. This circum-
stance reminds us not only of the unity of all sciences aiming at
a description of the external world but, above all, of the in-
separability of epistemological and psychological analysis. It is
just in the emphasis on this last point, which recent develop-
ment in the most different fields of science has brought to the
foreground, that the program of the present great undertaking
distinguishes itself from that of previous encyclopedic enter-
prises, in which stress was essentially laid on the completeness
of the account of the actual state of knowledge rather than on
the elucidation of scientific methodology. It is therefore to be
hoped that the forthcoming Eticyclopedia will have a deep in-
fluence on the whole attitude of our generation which, in spite
of the ever increasing specialization in science as well as in
technology, has a growing feeling of the mutual dependency of
all human activities. Above all, it may help us to realize that
even in science any arbitrary restriction implies the danger of
prejudices and that our only way of avoiding the extremes of
materialism and mysticism is the never ending endeavor to bal-
ance analysis and synthesis.

28

Unity of Science as a Social Problem

John Dewey

I. The Scientific Attitude

Anyone who attempts to promote the unity of science must
ask himself at least two basic questions: “What is meant by
that whose unity is to be promoted, namely, science?” and
“What sort of unity is feasible or desirable?” The following
pages represent the conclusions the present writer has reached
in reflecting upon these two themes.

With respect to the question as to the meaning of science, a
distinction needs to be made between science as attitude and
method and science as a body of subject matter. I do not mean
that the two can be separated, for a method is a way of dealing
with subject matter and science as a body of knowledge is a
product of a method. Each exists only in connection with the
other. An attitude becomes psychopathic when it is not di-
rected to objects beyond itself. What is meant is, first, that
attitude and method come before the material which is found
in books, journals, and the proceedings of scientific organiza-
tions; and, second, that the attitude is manifested primarily
toward the objects and events of the ordinary world and only
secondarily toward that which is already scientific subject mat-
ter.

Stated in other words, the scientific method is not confined
to those who are called scientists. The body of knowledge and
ideas which is the product of the work of the latter is the fruit
of a method which is followed by the wider body of persons who
deal intelligently and openly with the objects and energies of
the common environment. In its specialized sense, science is an
elaboration, often a highly technical one, of everyday opera-
tions. In spite of the technicality of its language and proce-
dures, its genuine meaning can be understood only if its con-

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nection with attitudes and procedures which are capable of
being used by all persons who act intelligently is borne in mind.

On the level of common sense there are attitudes which are
like those of science in its more specialized sense, while there are
attitudes which are thoroughly unscientific. There are those
who work by routine, by casual cut-and-try methods, those who
are enslaved to dogma and directed by prejudice, just as there
are those who use their hands, eyes, and ears to gain knowledge
of whatever comes their way and use whatever brains they have
to extract meaning from what they observe. Few would rule
engineers from out the scientific domain, and those few would
rest their case upon a highly dubious distinction between some-
thing called “pure” science and something else called “applied”
science.

As Dr. Karl Darrow has said in his Renaissance of Science:

Many of the things which modern science has to tell us are fantastic and
inconceivable indeed, but they have been attested by the same sort of man
with the same sort of training and using the same sort of reasoning as those
who have made it possible to speak over a wire with San Francisco and over
the ether of space to London, to cross the Atlantic in four days by steamer
and in twenty-four hours by aeroplane, to operate a railroad with power trans-
mitted invisibly through rails, and to photograph the bones inside the body
with a light no eye can see and no fire can send forth.

When the achievements of the engineer are disparaged under
the name “applied” science, it is forgotten that the inquiries and
the calculations required to produce these achievements are as
exacting as those which generate the science called “pure.”
Pure science does not apply itself automatically; application
takes place through use of methods which it is arbitrary to dis-
tinguish from those employed in the laboratory or observatory.
And if the engineer is mentioned, it is because, once he is ad-
mitted, we cannot exclude the farmer, the mechanic, and the
chauffeur, as far as these men do what they have to do with in-
telligent choice of means and with intelligent adaptation of
means to ends, instead of in dependence upon routine and
guesswork. On the other hand, it is quite possible for the scien-
tist to be quite unscientific in forming his beliefs outside his

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Dewey: Unity of Science as a Social Problem

special subject, as he does whenever he permits such beliefs to
be dictated by unexamined premisses accepted traditionally or
caught up out of the surrounding social atmosphere.

In short, the scientific attitude as here conceived is a quality
that is manifested in any walk of life. What, then, is it? On its
negative side, it is freedom from control by routine, prejudice,
dogma, unexamined tradition, sheer self-interest. Positively, it
is the will to inquire, to examine, to discriminate, to draw con-
clusions only on the basis of evidence after taking pains to
gather all available evidence. It is the intention to reach be-
liefs, and to test those that are entertained, on the basis of
observed fact, recognizing also that facts are without meaning
save as they point to ideas. It is, in turn, the experimental atti-
tude which recognizes that while ideas are necessary to deal
with facts, yet they are working hypotheses to be tested by the
consequences they produce.

Above all, it is the attitude which is rooted in the problems
that are set and questions that are raised by the conditions of
actuality. The unscientific attitude is that which shuns such
problems, which runs away from them, or covers them up in-
stead of facing them. And experience shows that this evasion
is the counterpart of concern with artificial problems and al-
leged ready-made solutions. For all problems are artificial
which do not grow, even if indirectly, out of the conditions
under which life, including associated living, is carried on. Life
is a process which goes on in connection with an environment
which is complex, physically and culturally. There is no form
of interaction with the physical environment and the human
environment that does not generate problems that can be coped
with only by an objective attitude and an intelligent method.
The home, the school, the shop, the bedside and hospital, pre-
sent such problems as truly as does the laboratory. They usual-
ly present the problems in a more direct and urgent fashion.
This fact is so obvious that it would be trite to mention it were
it not that it shows the potential universality of the scientific
attitude.

The existence of artificial problems is also an undeniable fact

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in human history. The existence of such problems and the ex-
penditure of energy upon the solution of them are the chief
reasons why the potentiality of scientific method is so often
unrealized and frustrated. The word ‘metaphysics’ has many
meanings, all of which are generally supposed to be so highly
technical as to be of no interest to the man in the street. But in
the sense that ‘metaphysical’ means that which is outside of
experience, over and beyond it, all human beings are meta-
physical when they occupy themselves with problems which
do not rise out of experience and for which solutions are sought
outside experience. Men are metaphysical not only in technical
philosophy but in many of their beliefs and habits of thought in
religion, morals, and politics. The waste of energy that results
is serious enough. But this is slight compared with that which
is wrought by artificial problems and solutions in preventing,
deflecting, and distorting the development of the scientific atti-
tude which is the proper career of intelligence.

II. The Social Unity of Science

When we turn from the question of what is meant by science
to the question of what is meant by its unity, we seem, at first
sight, to have shifted ground and to be in another field. The
unity of science is usually referred to in connection with unifica-
tion of the attained results of science. In this field the problem
of attaining the unity of science is that of co-ordinating the
scattered and immense body of specialized findings into a sys-
tematic whole. This problem is a real one and cannot be neg-
lected. But there is also a human, a cultural, meaning of the
unity of science. There is, for instance, the question of unifying
the efforts of all those who exercise in their own affairs the sci-
entific method so that these efforts may gain the force which
comes from united effort. Even when an individual is or tries
to be intelligent in the conduct of his own life-affairs, his efforts
are hampered, often times defeated, by obstructions due not
merely to ignorance but to active opposition to the scientific
attitude on the part of those influenced by prejudice, dogma,
class interest, external authority, nationalistic and racial senti-

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Dewey: Unity of Science as a Social Problem

ment, and similar powerful agencies. Viewed in this light, the
problem of the unity of science constitutes a fundamentally im-
portant social problem.

At the present time the enemies of the scientific attitude are
numerous and organized — much more so than appears at super-
ficial glance. The prestige of science is indeed great, especially
in the field of its external application to industry and war. In
the abstract, few would come out openly and say that they were
opposed to science. But this small number is no measure of the
influence of those who borrow the results of science to advance
by thoroughly unscientific and antiscientific methods private,
class, and national interests. Men may admire science, for ex-
ample, because it gives them the radio to use, and then employ
the radio to create conditions that prevent the development of
the scientific attitude in the most important fields of human ac-
tivity — fields which suffer terribly because of failure to use
scientific method. In particular, science is not welcomed but
rather opposed when it “invades” (a word often used) the field
now pre-empted by religion, morals, and political and economic
institutions.

To bring about unity of the scientific attitude is, then, to
bring those who accept it and who act upon it into active co-
operation with one another. This problem transcends in im-
portance the more technical problem of unification of the re-
sults of the special sciences. It takes precedence over the latter
issue. For it is not too much to say that science, even in its more
specialized sense, now stands at a critical juncture. It must
move forward in order to maintain its achievements. If it
stands still, it will be confined to the field in which it has al-
ready won victories and will see the fruits of its victories ap-
propriated by those who will use them by antiscientific methods
for nonhumane ends.

Accordingly, the great need is for those who are actuated by
the scientific spirit to take counsel legarding the place and func-
tion of science in the total scene of life. It follows that a move-
ment in behalf of the unity of science need not and should not
lay down in advance a platform to be accepted. It is essentially

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a co-operative movement, so that detailed specific common
standpoints and ideas must emerge out of the very processes of
co-operation. To try to formulate them in advance and insist
upon their acceptance by all is both to obstruct co-operation
and to be false to the scientific spirit. The only thing necessary
in the form of agreement is faith in the scientific attitude and
faith in the human and social importance of its maintenance
and expansion.

What has been said does not minimize the difficulties that
arise from the great degree of isolated specialization that now
characterizes science or the importance of overcoming these
difficulties. To a great extent those who now pursue the differ-
ent branches of science speak different languages and are not
readily understood by one another. Translation from one
branch to another is not easy. In consequence, workers tend to
be deprived of the useful intellectual instruments that would be
available in their own special work if there were a freer give and
take.

But the needed work of co-ordination cannot be done me-
chanically or from without. It, too, can be the fruit only of co-
operation among those animated by the scientific spirit. Con-
vergence to a common center will be effected most readily and
most vitally through the reciprocal exchange which attends
genuine co-operative effort. The attempt to secure unity by de-
fining the terms of all the sciences in terms of some one science
is doomed in advance to defeat. In the house which science
might build there are many mansions. The first task, to change
the metaphor, is to build bridges from one science to another.
There are many gaps to be spanned. It seems to me, however,
that the great need is the linkage of the physico-chemical sci-
ences with psychological and social fields of science through the
intermediary of biology. I should probably be expressing my
own view or that of a particular and perhaps small group if I
said that convergence can best be attained by considering how
various sciences may be brought together in common attack
upon practical social problems. But it is wholly within the scope
of the present theme to say that the co-operative endeavor held

34

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Dewey: Unity of Science as a Social Problem

in view by the present movement for the unity of science is
bound gradually to disclose the causes of present gaps and to
indicate where and how bridges may be built across the gulfs
that still separate workers in different fields.

A very short history has been enjoyed by free scientific meth-
od in comparison with the long history enjoyed by forces which
have never felt the influence of science. Ideas that descend from
the prescientific epoch are still with us and are crystallized in
institutions. They are not to be exorcised by reiteration of the
word ‘science.’ Every scientific worker is still subject to
their influence, certainly outside his special field and some-
times even within it. Only constant critical care, exercised m
the spirit of the scientific attitude, can bring about their gradual
elimination. Ultimately, this criticism must be self-criticism.
But the agencies and instrumentalities of self-criticism can be
had only by means of as full and free co-operation with others
as it is possible to secure.

The advance of scientific method has brought with it, where
the influence of the method has been felt, a great increase in
toleration. We are now in a world where there is an accelerated
development of intolerance. Part of the cause for this growth
can be found, I think, in the fact that tolerance so far has been
largely a passive thing. We need a shift from acceptance of
responsibility for passive toleration to active responsibility for
promoting the extension of scientific method. The first step is
to recognize the responsibility for furthering mutual under-
standing and free communication.

III. Education and the Unity of Science

It is perhaps within the scope of my theme to say something
about the connection of the movement for the unity of science
with education. I have already mentioned the fact that scien-
tific method has reached a crisis in its history, due, in final
analysis, to the fact that the ultra-reactionary and the ultra-
radical combine, even while acclaiming the prestige of science
in certain fields, to use the techniques of science to destroy the
scientific attitude. The short history of science in comparison

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Encyclopedia and Unified Science

with the history of institutions that resist its application by the
mere fact of their inertia has also been mentioned. These two
influences combine to render the agencies of education the cru-
cial point in any movement to bring about a greater and more
progressive unity of the scientific spirit.

After a struggle, the various sciences have found a place for
themselves in the educational institutions. But to a large extent
they exist merely side by side with other subjects which have
hardly felt the touch of science. This, however, is far from being
the most depressing feature of the educational situation as
respects the place of science. For it is also true that the spirit
in which the sciences are often taught, and the methods of in-
struction employed in teaching them, have been in large meas-
ure taken over from traditional nonscientific subjects.

I mention certain things which confirm this statement. In the
first place, science has barely affected elementary education.
With a very few exceptions it has not touched the early years of
the elementary school. Aet this is the time when curiosity is
most awake, the interest in observation the least dulled, and
desire for new experiences most active. It is also the period in
which the fundamental attitudes are formed which control, sub-
consciously if not consciously, later attitudes and methods.

In the second place, scientific subjects are taught very largely
as bodies of subject matter rather than as a method of universal
attack and approach. There may be laboratories and labora-
tory exercises and yet this statement remain true. For they
may be employed primarily in order that pupils acquire a cer-
tain body of information. The resulting body of information
about facts and laws has a different content from that provided
in other studies. But as long as the ideal is information, the
sciences taught are still under the dominion of ideas and prac-
tices that have a prescientific origin and history. Laboratory
exercises and class demonstrations may be a part of a regular
routine of instruction, and yet accomplish little in developing
the scientific habit of mind. Indeed, except in a chosen few the
mere weight of information may be a load carried in the memo-
ry, not a resource for further observation and thought.

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Dewey: Unity of Science as a Social Problem

In the third place, apart from some institutions of research
and graduate departments of universities which attract rela-
tively a small number, most money and energy go into institu-
tions in which persons are prepared for special professional
pursuits. This fact is not itself objectionable, as I have already
indicated in speaking of “applied” and pure science. But this
technical education, as it is at present conducted, is directed to
narrow ends rather that to the wide and liberal end of develop-
ing interest and ability to use the scientific method in all fields
of human betterment. It is quite possible, unfortunately, for a
person to have the advantage of this special training and yet re-
main indifferent to the application of the scientific attitude in
fields that lie outside his own specialized calling.

The final point is a corollary. Something called by the name
of “science” gets shut off in a segregated territory of its own.
There are powerful special interests which strive in any case to
keep science isolated so that the common life may be immune
from its influence. Those w T ho have these special interests fear
the impact of scientific method upon social issues.

They fear this impact even if they have not formulated the
nature and ground of their fear. But there are influences within
the status of science itself in the educational system which pro-
mote its isolation. If the schools are used for the purpose of in-
stilling belief in certain dogmas — a use in which something
called “education” becomes simply an organ of propaganda—
and this use continues to grow, it will be in some measure be-
cause science has not been conceived and practiced as the sole
universal method of dealing intellectually with all problems.
The movement to unify workers in different fields of science
is itself an educative movement for those w T ho take part in it.
It is also a precondition of effort to give the scientific attitude
that place in educational institutions which will create an ever
increasing number of persons who habitually adopt the scientific
attitude in meeting the problems that confront them.

I said that I thought that reference to education belonged
within the scope of the present theme. On the one hand, the
future of the scientific attitude as a socially unified force de-

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pends more upon the education of children and youth than
upon any other single force. On the other hand, the teaching of
science can hardly take the place which belongs to it, as an atti-
tude of universal application, unless those who are already
animated by the scientific attitude and concerned for its ex-
pansion actively co-operate. The first condition to be satisfied
is that such persons bestir themselves to become aware of what
the scientific attitude is and what it is about so as to become
diligently militant in demonstrating its rightful claims.

Ihe import of what has been said is that the scientific attitude
and method are at bottom but the the method of free and ef-
fective intelligence. The special sciences reveal what this meth-
od is and means, and what it is capable of. It is neither feasible
nor desirable that all human beings should become practitioners
of a special science. But is intensely desirable and under certain
conditions practicable that all human beings become scientific in
their attitudes: genuinely intelligent in their ways of thinking
and acting. It is practicable because all normal persons have the
potential germs which make this result possible. It is desirable
because this attitude forms the sole ultimate alternative to
prejudice, dogma, authority, and coercive force exercised in
behalf of some special interest. Those who are concerned with
science in its more technical meaning are obviously those who
should take the lead by co-operation with one another in bring-
ing home to all the inherent universality of scientific method.

38

On the Importance of Logical Form

Bertrand Russell

The instrument of mathematical logic, which has begun to
be appreciated during the present century, possesses two rather
different kinds of utility— one in pure mathematics, the other
in the various empirical sciences. Of the former I shall say noth-
ing, since the ground is familiar; but on the latter there are some
things to be said that bear on the importance of a modern
encyclopedia.

In the empirical sciences it is not so much in relation to in-
ference that mathematical logic is useful as in relation to analy-
sis and the apprehension of identity and difference of form.
Where identity of form is of the traditional mathematical kind,
its importance has long been realized. The kinetic theory of
gases has been applied to the stellar universe, which, to the non-
mathematical mind, appears very different from a gas. . A
British mathematical professor at Tokyo was led by his location
to study earthquakes, and made useful applications of his re-
sults to the vibrations of the footplates of locomotives. But,
where identity of form is not of the sort that can be expressed
without logical symbols, men of science have been less quick to
recognize it; while the general public, through logical incom-
petence, has been led into grave practical errors. During the
Black Death the inhabitants of Siena attributed the calamity
to their presumption in planning a much enlarged cathedral,
oblivious of the fact that the mortality was just as great else-
where. Similarly, in 1931, the population of every country at-
tributed the depression to the sins of its own government; this
caused a movement to the Left where there was a Right go\-
ernment, and to the Right where there was a Left government.
Only a few impotent intellectuals observed that the phenome-
non to be explained was world- wide, not local.

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The distinction between macroscopic and microscopic phys-
ics, which has become important since the rise of quantum
theory, suggests possibilities as regards scientific method in
other fields. Although, as a mathematical ideal, macroscopic
physics may be supposed deducible from the behavior of the
individual atoms, it was in fact discovered first, and its laws
remain valid, for most practical purposes, in spite of the dis-
coveries of the quantum physicists. This suggests the possibility
of a social science not deduced from the laws of individual be-
havior but based upon laws which are only valid for large num-
bers. The theory of evolution, in biology, is the most striking
example. Economics, in so far as it is a science, is another. Vital
statistics afford another field for the observation of statistical
behavior; it might be thought, for instance, that there is an
inverse correlation between increase and density of the popula-
tion, but Australia, though confirming this as regards rabbits,
negatives it as regards human beings.

Logical method has important applications to psychology.
Suppose, for example, that, in order to deal with dual' and mul-
tiple personality, we desire a definition of ‘person’ not derived
from bodily continuity. We may observe that dual personality
is connected with amnesia. We may define a relation M be-
tween two experiences, consisting in the fact that one is, in
whole or part, a recollection of the other, or the other of the one.
If N is the ancestral relation of M, all the experiences which
have to a given experience the relation N may be defined as the
person to whom the given experience belongs; for the student
of dual and multiple personality this is probably the most con-
venient definition.

I said that mathematical logic has less importance in relation
to scientific inference than in relation to analysis, but this state-
ment needs qualification. Outside mathematics, the important
inferences are not deductive, i.e., they are not such as mathe-
matical logic makes. But logic can state their character with a
precision which was formerly impossible. Much has been done,
for example, by Carnap, in analyzing the kind of inference upon
which scientific laws are based. Since all inferences of this kind

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Russell: On the Importance of Logical Form

are probable, not demonstrative, the study of probability, as
Reichenbach insists, is of fundamental importance in scientific
method.

The importance of logical form may be illustrated by what
may be called the principle of the dictionary: Given two sets
of propositions such that, by a suitable dictionary, any proposi-
tion of either set can be translated into a proposition of the other
set, there is no effective difference between the two sets. Sup-
pose — to take a hypothesis that I neither affirm nor deny— -that
all scientific propositions can be tested in terms of physics, and
can also be stated on Berkeleian principles, in terms of psy-
chology; then the question as to which of these forms of state-
ment is the more correct has no meaning, since both or neither
must be correct. Such dictionaries, which can, as a rule, only
be constructed by the help of modern logic, suffice to dispose of
large numbers of metaphysical questions, and thus facilitate
concentration upon genuine scientific problems.

Let us take another example of the principle of the dictionary.
The general principle of relativity showed that, in expressing
the laws of macroscopic physics, we can transform our co-
ordinates in any way we choose, so long as topological relations
in space-time are preserved as topological relations among co-
ordinates. It follows that the laws of macroscopic physics are
topological laws, and that the introduction of number through
co-ordinates is only a practical convenience, the laws being such
as can, in theory, be expressed without the use of number. The
old view that measurement is of the essence of science would
therefore seem to be erroneous.

The unity of science, which is sometimes lost to view through
immersion in specialist problems, is essentially a unity of meth-
od, and the method is one upon which modern logic throws
much new light. It may be hoped that the Encyclopedia will do
much to bring about an awareness of this unity.

41

Logical Foundations of the Unity
of Science

Rudolf Carnap

I. What Is Logical Analysis of Science?

The task of analyzing science may be approached from
various angles. The analysis of the subject matter of the sci-
ences is carried out by science itself. Biology, for example,
analyzes organisms and processes in organisms, and in a similar
way every branch of science analyzes its subject matter. Mostly,
however, by analysis of science’ or ‘theory of science’ is meant
an investigation which differs from the branch of science
to which it is applied. We may, for instance, think of an in-
vestigation of scientific activity. We may study the historical
development of this activity. Or we may try to find out in
which way scientific work depends upon the individual condi-
tions of the men working in science, and upon the status of the
society surrounding them. Or we may describe procedures and
appliances used in scientific work. These investigations of sci-
entific activity may be called history, psychology, sociology,
and methodology of science. The subject matter of such studies
is science as a body of actions carried out by certain persons
under certain circumstances. Theory of science in this sense
will be dealt with at various other places in this Encyclopedia;
it is certainly an essential part of the foundation of science.

We come to a theory of science in another sense if we study
not the actions of scientists but their results, namely, science
as a body of ordered knowledge. Here, by ‘results’ we do not
mean beliefs, images, etc., and the behavior influenced by them.
That would lead us again to psychology of science. We mean by
results’ certain linguistic expressions, viz., the statements as-
serted by scientists. The task of the theory of science in this

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Carnap: Logical Foundations of the Unity of Science

sense will be to analyze such statements, study their kinds and
relations, and analyze terms as components of those statements
and theories as ordered systems of those statements. A state-
ment is a kind of sequence of spoken sounds, written marks,
or the like, produced by human beings for specific purposes.
But it is possible to abstract in an analysis of the statements of
science from the persons asserting the statements and from the
psychological and sociological conditions of such assertions.
The analysis of the linguistic expressions of science under such
an abstraction is logic of science.

Within the logic of science we may distinguish between two
chief parts. The investigation may be restricted to the forms of
the linguistic expressions involved, i.e., to the way in which
they are constructed out of elementary parts (e.g., words) with-
out referring to anything outside of language. Or the investiga-
tion goes beyond this boundary and studies linguistic expres-
sions in their relation to objects outside of language. A study
restricted in the first-mentioned way is called formal; the field
of such formal studies is called formal logic or logical syntax.
Such 'a formal or syntactical analysis of the language of science
as a whole or in its various branches will lead to results of the
following kinds. A certain term (e.g., a word) is defined within
a certain theory on the basis of certain other terms, or it is
definable in such a way. A certain term, although not definable
by certain other terms, is reducible to them (in a sense to be
explained later). A certain statement is a logical consequence
of (or logically deducible from) certain other statements; and
a deduction of it, given within a certain theory, is, or is not,
logically correct . A certain statement is incompatible with cer-
tain other statements, i.e., its negation is a logical consequence
of them. A certain statement is independent of certain other
statements, i.e., neither a logical consequence of them nor
incompatible with them. A certain theory is inconsistent, i.e.,
some of its statements are incompatible with the other ones.
The last sections of this essay will deal with the question of the
unity of science from the logical point of view, studying the
logical relations between the terms of the chief branches of

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science and between the laws stated in these branches; thus it
will give an example of a syntactical analysis of the language
of science.

In the second part of the logic of science, a given language
and the expressions in it are analyzed in another way. Here
also, as in logical syntax, abstraction is made from the psycho-
logical and sociological side of the language. This investigation,
however, is not restricted to formal analysis but takes into
consideration one important relation between linguistic expres-
sions and other objects — that of designation. An investigation
of this kind is called semantics . Results of a semantical analysis
of the language of science may, for instance, have the following
forms. A certain term designates a certain particular object
(e.g., the sun), or a certain property of things (e.g., iron), or
a certain relation between things (e.g., fathership), or a certain
physical function (e.g., temperature); two terms in different
branches of science (e.g., ‘homo sapiens’ in biology and ‘person’
in economics, or, in another way, ‘man’ in both cases) designate
(or: do not designate) the same. What is designated by a cer-
tain expression may be called its designatum. Two expressions
designating the same are called synonymous. The term ‘true,’
as it is used in science and in everyday life, can also be defined
within semantics. We see that the chief subject matter of a
semantical analysis of the language of science are such proper-
ties and relations of expressions, and especially of statements,
as are based on the relation of designation. (Where we say ‘the
designatum of an expression,’ the customary phrase is ‘the
meaning of an expression.’ It seems, however, preferable to
avoid the word ‘meaning’ wherever possible because of its
ambiguity, i.e., the multiplicity of its designata. Above all, it
is important to distinguish between the semantical and the psy-
chological use of the word ‘meaning.’)

It is a question of terminological convention whether to use
the term logic in the wider sense, including the semantical
analysis of the designata of expressions, or in the narrower sense
of logical syntax, restricted to formal analysis, abstracting from
designation. And accordingly we may distinguish between logic

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of science in the narrower sense, as the syntax of the language
of science, and logic of science in the wider sense, comprehending
both syntax and semantics.

II. The Main Branches of Science

We use the word ‘science’ here in its widest sense, including
all theoretical knowledge, no matter whether in the field of
natural sciences or in the field of the social sciences and the
so-called humanities, and no matter whether it is knowledge
found by the application of special scientific procedures, or
knowledge based on common sense in everyday life. In the same
way the term ‘language of science’ is meant here to refer to the
language which contains all statements (i.e., theoretical sen-
tences as distinguished from emotional expressions, commands,
lyrics, etc.) used for scientific purposes or in everyday life.
What usually is called science is merely a more systematic con-
tinuation of those activities which we carry out in everyday life
in order to know something.

The first distinction which we have to make is that between
formal science and empirical science. Formal science consists of
the analytic statements established by logic and mathematics;
empirical science consists of the synthetic statements estab-
lished in the different fields of factual knowledge. The relation
of formal to empirical science will be dealt with at another
place; here we have to do with empirical science, its language,
and the problem of its unity.

Let us take ‘physics’ as a common name for the nonbiological
field of science, comprehending both systematic and historical
investigations within this field, thus including chemistry, min-
eralogy, astronomy, geology (which is historical), meteorology,
etc. How, then, are we to draw the boundary line between
physics and biology ? It is obvious that the distinction between
these two branches has to be based on the distinction between
two kinds of things which we find in nature: organisms and
nonorganisms. Let us take this latter distinction as granted; it
is the task of biologists to lay down a suitable definition for the
term ‘organism,’ in other words, to tell us the features of a

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thing which we take as characteristic for its being an organism.
How, then, are we to define ‘biology’ on the basis of organism’?
We could perhaps think of trying to do it in this way: biology
is the branch of science which investigates organisms and the
processes occurring in organisms, and physics is the study of
nonorganisms. But these definitions would not draw the dis-
tinction as it is usually intended. A law stated in phvsics is
intended to be valid universally, without any restriction. For
example, the law stating the electrostatic force as a function
of electric charges and their distance, or the law determining
the pressure of a gas as a function of temperature, or the law
determining the angle of refraction as a function of the coeffi-
cients of refraction of the two media involved, are intended to
apply to the processes in organisms no less than to those in in-
organic nature. The biologist has to know these laws of physics
in studying the processes in organisms. He needs them for the
explanation of these processes. But since they do not suffice,
he adds some other laws, not known by the physicist, viz.,
the specifically biological laws. Biology presupposes physics,
but not vice versa.

These reflections lead us to the following definitions. Let us
call those terms which we need— in addition to logico-mathe-
matical terms for the description of processes in inorganic
nature 'physical terms, no matter whether, in a given instance,
they are applied to such processes or to processes in organisms.
That sublanguage of the language of science, which contains

besides logico-mathematical terms— all and only physical
terms, may be called physical language. The system of those
statements which are formulated in the physical language and
are acknowledged by a certain group at a certain time is called
the physics of that group at that time. Such of these state-
ments as have a specific universal form are called physical laws.
The physical laws are needed for the explanation of processes
in inorganic nature; but, as mentioned before, they apply to
processes in organisms also.

The whole of the rest of science may be called biology (in
the wider sense). It seems desirable, at least for practical pur-

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poses, e.g., for the division of labor in research work, to sub-
divide this wide field. But it seems questionable whether any
distinctions can be found here which, although not of a funda-
mental nature, are at least clear to about the same degree as
the distinction between physics and biology. At present, it is
scarcely possible to predict which subdivisions will be made in
the future. The traditional distinction between bodily (or ma-
terial) and mental (or psychical) processes had its origin in the
old magical and later metaphysical mind-body dualism. The
distinction as a practical device for the classification of branches
of science still plays an important role, even for those scientists
who reject that metaphysical dualism; and it will probably
continue to do so for some time in the future. But when the
aftereffect of such prescientific issues upon science becomes
weaker and weaker, it may be that new boundary lines for
subdivisions will turn out to be more satisfactory.

One possibility of dividing biology in the wider sense into two
fields is such that the first corresponds roughly to what is
usually called biology, and the second comprehends among
other parts those which usually are called psychology and social
science. The second field deals with the behavior of individual
organisms and groups of organisms within their environment,
with the dispositions to such behavior, with such features of
processes in organisms as are relevant to the behavior, and with
certain features of the environment which are characteristic of
and relevant to the behavior, e.g., objects observed and work
done by organisms.

The first of the two fields of biology in the wider sense may
be called biology in the narrower sense, or, for the following
discussions, simply biology. This use of the term ‘biology’ seems
justified by the fact that, in terms of the customary classifica-
tion, this part contains most of what is usually called biology,
namely, general biology, botany, and the greater part of zo-
ology. The terms which are used in this field in addition to
logico-mathematical and physical terms may be called biological
terms in the narrower sense, or simply biological terms. Since
many statements of biology contain physical terms besides bio-

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logical ones, the biological language cannot be restricted to bio-
logical terms; it contains the physical language as a sublanguage
and, in addition, the biological terms. Statements and laws be-
longing to this language but not to physical language will be
called biological statements and biological laws.

The distinction between the two fields of biology in the wider
sense has been indicated only in a very vague way. At the
present time it is not yet clear as to how the boundary line may
best be drawn. Which processes in an organism are to be as-
signed to the second field? Perhaps the connection of a process
with the processes in the nervous system might be taken as
characteristic, or, to restrict it more, the connection with speak-
ing activities, or, more generally, with activities involving signs.
xVnother way of characterization might come from the other
direction, from outside, namely, selecting the processes in an
organism from the point of view of their relevance to achieve-
ments in the environment (see Bruns wik and Ness). There is
no name in common use for this second field. (The term ‘men-
tal sciences’ suggests too narrow a field and is connected too
closely with the metaphysical dualism mentioned before.) The
term ‘behavioristics’ has been proposed. If it is used, it must be
made clear that the word ‘behavior’ has here a greater extension
than it had with the earlier behaviorists. Here it is intended
to designate not only the overt behavior which can be observed
from outside but also internal behavior (i.e., processes within
the organism) ; further, dispositions to behavior which may not
be manifest in a special case; and, finally, certain effects upon
the environment. Within this second field we may distinguish
roughly between two parts dealing with individual organisms
and with groups of organisms. But it seems doubtful whether
any sharp line can be drawn between these two parts. Com-
pared with the customary classification of science, the first part
would include chiefly psychology, but also some parts of physi-
ology and the humanities. The second part would chiefly in-
clude social science and, further, the greater part of the humani-
ties and history, but it has not only to deal with groups of
human beings but also to deal with groups of other organisms.

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For the following discussion, the terms ‘psychology’ and ‘social
science’ will he used as names of the two parts because of lack
of better terms. It is clear that both the question of boundary
lines and the question of suitable terms for the sections is still
in need of much more discussion.

III. Reducibility

The question of the unity of science is meant here as a prob-
lem of the logic of science, not of ontology. We do not ask:
“Is the world one?” “Are all events fundamentally of one
kind?” “Are the so-called mental processes really physical
processes or not?” “Are the so-called physical processes really
spiritual or not?” It seems doubtful whether we can find any
theoretical content in such philosophical questions as discussed
by monism, dualism, and pluralism. In any case, when we ask
whether there is a unity in science, we mean this as a question
of logic, concerning the logical relationships between the terms
and the laws of the various branches of science. Since it be-
longs to the logic of science, the question concerns scientists and
logicians alike.

Let us first deal with the question of terms. (Instead of the
word ‘term’ the word ‘concept’ could be taken, which is more
frequently used by logicians. But the word ‘term’ is more clear,
since it shows that we mean signs, e.g., words, expressions con-
sisting of words, artificial symbols, etc., of course with the
meaning they have in the language in question. We do not
mean ‘concept’ in its psychological sense, i.e., images or
thoughts somehow connected with a word; that would not be-
long to logic.) We know the meaning (designatum) of a term
if we know under what conditions we are permitted to apply
it in a concrete case and under what conditions not. Such a
knowledge of the conditions of application can be of two differ-
ent kinds. In some cases we may have a merely practical knowl-
edge, i.e., we are able to use the term in question correctly with-
out giving a theoretical account of the rules for its use. In
other cases we may be able to give an explicit formulation of the
conditions for the application of the term. If now a certain

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term x is such that the conditions for its application (as used in
the language of science) can be formulated with the help of the
terms y, z, etc., we call such a formulation a reduction statement
for x in terms of y, z, etc., and we call x reducible to y, z, etc.
There may be several sets of conditions for the application of
x; hence x may be reducible to y, z, etc., and also to u, v, etc.,
and perhaps to other sets. There may even be cases of mutual
reducibility, e.g., each term of the set x u x 2 , etc., is reducible to
Vi, 2/2, etc.; and, on the other hand, each term of the set y u y 2 ,
etc., is reducible to x u x 2 , etc.

A definition is the simplest form of a reduction statement.
For the formulation of examples, let us use ‘ = ’ (called the sym-
bol of equivalence) as abbreviation for ‘if and only if.’ Example
of a definition for ‘ox’ : ‘x is an ox = x is a quadruped and horned
and cloven-footed and ruminant, etc.’ This is also a reduction
statement because it states the conditions for the application
of the term ‘ox,’ saying that this term can be applied to a thing
if and only if that thing is a quadruped and horned, etc. By
that definition the term ‘ox’ is shown to be reducible to — more-
over definable by— the set of terms ‘quadruped,’ ‘homed,’ etc.

A reduction statement sometimes cannot be formulated in
the simple form of a definition, i.e., of an equivalence statement,
but only ip the somewhat more complex form

then: . . . . = ’ Thus a reduction statement is

either a simple (i.e., explicit) definition or, so to speak, a condi-
tional definition. (The term ‘reduction statement’ is generally
used in the narrower sense, referring to the second, conditional
form.) For instance, the following statement is a reduction
statement for the term ‘electric charge’ (taken here for the sake
of simplicity as a nonquantitative term), i.e., for the statement
form ‘the body x has an electric charge at the time t’ : ‘If a light
body y is placed near x at t, then: x has an electric charge at t
— V ^ attracted by x at t.’ A general way of procedure which
enables us to find out whether or not a certain term can be
applied in concrete cases may be called a method of determina-
tion for the term in question. The method of determination
for a quantitative term (e.g., ‘temperature’) is the method of

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measurement for that term. Whenever we know an experimen-
tal method of determination for a term, we are in a position
to formulate a reduction statement for it. To know an experi-
mental method of determination for a term, say ‘Q 3 ,’ means to
know two things. First, we must know an experimental situa-
tion which we have to create, say the state Qi, e.g., the arrange-
ment of measuring apparatuses and of suitable conditions for
their use. Second, we must know the possible experimental re-
sult, say Q 2 , which, if it occurs, will confirm the presence of the
property Q 3 . In the simplest case — let us leave aside the more
complex cases — Q 2 is also such that its nonoccurrence shows
that the thing in question does not have the property Q 3 . Then
a reduction statement for ‘Q 3 ,’ i.e., for the statement form ‘the
thing (or space-time-point) x is Q 3 (i.e., has the property Q 3 )
at the time t ,’ can be formulated in this way: ‘If x is Qi (i.e., x
and the surroundings of x are in the state QO at time t, then:
x is Q 3 at t = x is Q 2 at On the basis of this reduction state-
ment, the term ‘Q 3 ’ is reducible to ‘Qi,’ ‘Q 2 ,’ and spatio-temporal
terms. Whenever a term ‘Q 3 ’ expresses the disposition of a
thing to behave in a certain way (Q 2 ) to certain conditions
(Qi), we have a reduction statement of the form given above.
If there is a connection of such a kind between Qi, Q 2 , and Q 3 ,
then in biology and psychology in certain cases the following
terminology is applied : ‘To the stimulus Qi we find the reaction
Q 2 as a symptom for Q 3 .’ But the situation is not essentially
different from the analogous one in physics, where we usually
do not apply that terminology.

Sometimes we know several methods of determination for a
certain term. For example, we can determine the presence
of an electric current by observing either the heat produced in
the conductor, or the deviation of a magnetic needle, or the
quantity of a substance separated from an electrolyte, etc. Thus
the term ‘electric current’ is reducible to each of many sets of
other terms. Since not only can an electric current be measured
by measuring a temperature but also, conversely, a temperature
can be measured by measuring the electric current produced
by a thermo-electric element, there is mutual reducibility be-

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tween the terms of the theory of electricity, on the one hand,
and those of the theory of heat, on the other. The same holds
for the terms of the theory of electricity and those of the theory
of magnetism.

Let us suppose that the persons of a certain group have a
certain set of terms in common, either on account of a merely
practical agreement about the conditions of their application
or with an explicit stipulation of such conditions for a part of
the terms. Then a reduction statement reducing a new term
to the terms of that original set may be used as a way of
introducing the new term into the language of the group. This
way of introduction assures conformity as to the use of the new
term. If a certain language (e.g., a sublanguage of the language
of science, covering a certain branch of science) is such that
every term of it is reducible to a certain set of terms, then this
language can be constructed on the basis of that set by intro-
ducing one new term after the other by reduction statements. In
this case we call the basic set of terms a sufficient reduction basis
for that language.

IV. The Unity of the Language of Science

Now we will analyze the logical relations among the terms
of different parts of the language of science with respect to
reducibility. We have indicated a division of the whole lan-
guage of science into some parts. Now we may make another
division cutting across the first, by distinguishing in a rough
way, without any claims to exactness, between those terms
which we use on a prescientific level in our everyday language,
and for whose application no scientific procedure is necessary,
and scientific terms in the narrower sense. That sublanguage
which is the common part of this prescientific language and the
physical language may be called physical thing-language or
briefly thing-language. It is this language that we use in speak-
ing about the properties of the observable (inorganic) things
surrounding us. Terms like ‘hot’ and ‘cold’ may be regarded as
belonging to the thing-language, but not ‘temperature’ because
its determination requires the application of a technical instru-

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ment; further, ‘heavy’ and ‘light’ (but not ‘weight’); ‘red,’
‘blue,’ etc.; ‘large,’ ‘small,’ ‘thick,’ ‘thin,’ etc.

The terms so far mentioned designate what we may call
observable properties, i.e., such as can be determined by a
direct observation. We will call them observable thing-predicates.
Besides such terms the thing-language contains other ones, e.g.,
those expressing the disposition of a thing to a certain behavior
under certain conditions, e.g., ‘elastic,’ ‘soluble,’ ‘flexible,’
‘transparent,’ ‘fragile,’ ‘plastic,’ etc. These terms — they might
be called disposition-predicates — are reducible to observable
thing-predicates because we can describe the experimental con-
ditions and the reactions characteristic of such disposition-pred-
icates in terms of observable thing-predicates. Example of a
reduction statement for ‘elastic’: ‘If the body x is stretched
and then released at the time t, then: x is elastic at the time
t =3 x contracts at t' where the terms ‘stretched,’ ‘released,’
and ‘contracting’ can be defined by observable thing-predicates.
If these predicates are taken as a basis, we can moreover intro-
duce, by iterated application of definition and (conditional) re-
duction, every other term of the thing-language, e.g., designa-
tions of substances, e.g., ‘stone,’ ‘water,’ ‘sugar,’ or of processes,
e.g., ‘rain,’ ‘fire,’ etc. For every term of that language is such
that we can apply it either on the basis of direct observation
or with the help of an experiment for which we know the condi-
tions and the possible result determining the application of the
term in question.

Now we can easily see that every term of the physical lan-
guage is reducible to those of the thing-language and hence
finally to observable thing-predicates. On the scientific level,
we have the quantitative coefficient of elasticity instead of the
qualitative term ‘elastic’ of the thing-language; we have the
quantitative term ‘temperature’ instead of the qualitative ones
‘hot’ and ‘cold’; and we have all the terms by means of which
physicists describe the temporary or permanent states of things
or processes. For any such term the physicist knows at least
one method of determination. Physicists would not admit into
their language any term for which no method of determination

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by observations were given. The formulation of such a method,
i.e., the description of the experimental arrangement to be car-
ried out and of the possible result determining the application
of the term in question, is a reduction statement for that term.
Sometimes the term will not be directly reduced by the reduc-
tion statement to thing-predicates, but first to other scientific
terms, and these by their reduction statements again to other
scientific terms, etc.; but such a reduction chain must in any
case finally lead to predicates of the thing-language and, more-
over, to observable thing-predicates because otherwise there
would be no way of determining whether or not the physical
term in question can be applied in special cases, on the basis of
given observation statements.

If we come to biology (this term now always understood in
the narrower sense), we find again the same situation. For
any biological term the biologist who introduces or uses it must
know empirical criteria for its application. This applies, of
course, only to biological terms in the sense explained before,
including all terms used in scientific biology proper, but not to
certain terms used sometimes in the philosophy of biology — ‘a
whole,’ ‘entelechy,’ etc. It may happen that for the description
of the criterion, i.e., the method of determination of a term,
other biological terms are needed. In this case the term in ques-
tion is first reducible to them. But at least indirectly it must
be reducible to terms of the thing-language and finally to ob-
servable thing-predicates, because the determination of the term
in question in a concrete case must finally be based upon ob-
servations of concrete things, i.e., upon observation statements
formulated in the thing-language.

Let us take as an example the term ‘muscle.’ Certainly bi-
ologists know the conditions for a part of an organism to be a
muscle; otherwise the term could not be used in concrete cases.
The problem is : Which other terms are needed for the formula-
tion of those conditions? It will be necessary to describe the
functions within the organism which are characteristic of mus-
cles, in other words, to formulate certain laws connecting the
processes in muscles with those in their environment, or, again

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in still other words, to describe the reactions to certain stimuli
characteristic of muscles. Both the processes in the environ-
ment and those in the muscle (in the customary terminology:
stimuli and reactions) must be described in such a way that
we can determine them by observations. Hence the term ‘mus-
cle,’ although not definable in terms of the thing-language,
is reducible to them. Similar considerations easily show the
reducibility of any other biological term— whether it be a desig-
nation of a kind of organism, or of a kind of part of organisms,
or of a kind of process in organisms.

The result found so far may be formulated in this way: The
terms of the thing-language, and even the narrower class of
the observable thing-predicates, supply a sufficient basis for the
languages both of physics and of biology. (There are, by the
way, many reduction bases for these languages, each of which
is much more restricted than the classes mentioned.) Now the
question may be raised whether a basis of the kind mentioned
is sufficient even for the whole language of science. The affirma-
tive answer to this question is sometimes called physicalism
(because it was first formulated not with respect to the thing-
language but to the wider physical language as a sufficient
basis). If the thesis of physicalism is applied to biology only,
it scarcely meets any serious objections. The situation is some-
what changed, however, when it is applied to psychology and
social science (individual and social behavioristics) . Since many
of the objections raised against it are based on misinterpreta-
tions, it is necessary to make clear what the thesis is intended
to assert and what not.

The question of the reducibility of the terms of psychology
to those of the biological language and thereby to those of the
thing-language is closely connected with the problem of the
various methods used in psychology. As chief examples of meth-
ods used in this field in its present state, the physiological, the
behavioristic, and the introspective methods may be considered.
The physiological approach consists in an investigation of the
functions of certain organs in the organism, above all, of the
nervous system. Here, the terms used are either those of biology

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or those so closely related to them that there will scarcely be
any doubt with respect to their reducibility to the terms of the
biological language and the thing-language. For the behavioris-
tic approach different ways are possible. The investigation may
be restricted to the external behavior of an organism, i.e., to
such movements, sounds, etc., as can be observed by other
organisms in the neighborhood of the first. Or processes within
the organism may also be taken into account so that this
approach overlaps with the physiological one. Or, finally, ob-
jects in the environment of the organism, either observed or
worked on or produced by it, may also be studied. Now it is
easy to see that a term for whose- determination a behavioristic
method — of one of the kinds mentioned or of a related kind —
is known, is reducible to the terms of the biological language,
including the thing-language. As we have seen before, the for-
mulation of the method of determination for a term is a reduc-
tion statement for that term, either in the form of a simple
definition or in the conditional form. By that statement the
term is shown to be reducible to the terms applied in describing
the method, namely, the experimental arrangement and the
characteristic result. Now, conditions and results consist in the
behavioristic method either of physiological processes in the
organism or of observable processes in the organism and in its
environment. Hence they can be described in terms of the
biological language. If we have to do with a behavioristic ap-
proach in its pure form, i.e., leaving aside physiological investi-
gations, then the description of the conditions and results char-
acteristic for a term can in most cases be given directly in terms
of the thing-language. Hence the behavioristic reduction of
psychological terms is often simpler than the physiological re-
duction of the same term.

Let us take as an example the term ‘angry.’ If for anger we
knew a sufficient and necessary criterion to be found by a
physiological analysis of the nervous system or other organs,
then we could define ‘angry’ in terms of the biological language.
The same holds if we knew such a criterion to be determined by
the observation of the overt, external behavior. But a physio-

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logical criterion is not yet known. And the peripheral symptoms
known are presumably not necessary criteria because it might
be that a person of strong self-control is able to suppress these
symptoms. If this is the case, the term ‘angry’ is, at least at
the present time, not definable in terms of the biological lan-
guage. But, nevertheless, it is reducible to such terms. It is
sufficient for the formulation of a reduction sentence to know
a behavioristic procedure which enables us — if not always, at
least under suitable circumstances — to determine whether the
organism in question is angry or not. And we know indeed
such procedures; otherwise we should never be able to apply
the term ‘angry’ to another person on the basis of our observa-
tions of his behavior, as we constantly do in everyday life and
in scientific investigation. A reduction of the term ‘angry’ or
similar terms by the formulation of such procedures is indeed
less useful than a definition would be, because a definition
supplies a complete (i.e., unconditional) criterion for the term
in question, while a reduction statement of the conditional form
gives only an incomplete one. But a criterion, conditional or
not, is all we need for ascertaining reducibility. Thus the result
is the following: If for any psychological term we know either
a physiological or a behavioristic method of determination, then
that term is reducible to those terms of the thing-language.

In psychology, as we find it today, there is, besides the physio-
logical and the behavioristic approach, the so-called introspec-
tive method. The questions as to its validity, limits, and neces-
sity are still more unclear and in need of further discussion than
the analogous questions with respect to the two other methods.
Much of what has been said about it, especially by philosophers,
may be looked at with some suspicion. But the facts themselves
to which the term ‘introspection’ is meant to refer will scarcely
be denied by anybody, e.g., the fact that a person sometimes
knows that he is angry without applying any of those pro-
cedures which another person would have to apply, i.e., with-
out looking with the help of a physiological instrument at his
nervous system or looking at the play of his facial muscles. The
problems of the practical reliability and theoretical validity of

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the introspective method may here be left aside. For the dis-
cussion of reducibility an answer to these problems is not
needed. It will suffice to show that in every case, no matter
whether the introspective method is applicable or not, the be-
havioristic method can be applied at any rate. But we must
be careful in the interpretation of this assertion. It is not meant
as saying: ‘Every psychological process can be ascertained by
the behavioristic method.’ Here we have to do not with the
single processes themselves (e.g., Peter’s anger yesterday morn-
ing) but with kinds of processes (e.g., anger). If Robinson Cru-
soe is angry and then dies before anybody comes to his island,
nobody except himself ever knows of this single occurrence of
anger. But anger of the same kind, occurring with other per-
sons, may be studied and ascertained by a behavioristic method,
if circumstances are favorable. (Analogy: if an electrically
charged raindrop falls into the ocean without an observer or
suitable recording instrument in the neighborhood, nobody will
ever know of that charge. But a charge of the same kind can
be found out under suitable circumstances by certain observa-
tions.) Further, in order to come to a correct formulation of the
thesis, we have to apply it not to the kinds of processes (e.g.,
anger) but rather to the terms designating such kinds of proc-
esses (e.g., ‘anger’). The difference might seem trivial but is, in
fact, essential. We do not at all enter a discussion about the
question whether or not there are kinds of events which can
never have any behavioristic symptoms, and hence are know-
able only by introspection. We have to do with psychological
terms not with kinds of events. For any such term, say, ‘Q,’
the psychological language contains a statement form applying
that term, e.g., ‘The person .... is at the time .... in the
state Q.’ Then the utterance by speaking or writing of the
statement ‘I am now (or: I was yesterday) in the state Q,’ is
(under suitable circumstances, e.g., as to reliability, etc.) an
observable symptom for the state Q. Hence there cannot be a
term in the psychological language, taken as an intersubjective
language for mutual communication, which designates a kind
of state or event without any behavioristic symptom. There-

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fore, there is a behavioristic method of determination for any
term of the psychological language. Hence every such term is
reducible to those of the thing-language.

The logical nature of the psychological terms becomes clear
by an analogy with those physical terms which are introduced
by reduction statements of the conditional form. Terms of both
kinds designate a state characterized by the disposition to cer-
tain reactions. In both cases the state is not the same as those
reactions. Anger is not the same as the movements by which
an angry organism reacts to the conditions in his environment,
just as the state of being electrically charged is not the same as
the process of attracting other bodies. In both cases that state
sometimes occurs without these events which are observable
from outside; they are consequences of the state according to
certain laws and may therefore under suitable circumstances
be taken as symptoms for it ; but they are not identical with it.

The last field to be dealt with is social science (in the wide
sense indicated before; also called social behavioristics) . Here
we need no detailed analysis because it is easy to see that every
term of this field is reducible to terms of the other fields. The
result of any investigation of a group of men or other organisms
can be described in terms of the members, their relations to
one another and to their environment. Therefore, the condi-
tions for the application of any term can be formulated in
terms of psychology, biology, and physics, including the thing-
language. Many terms can even be defined on that basis, and
the rest is certainly reducible to it.

It is true that some terms which are used in psychology are
such that they designate a certain behavior (or disposition to
behavior) within a group of a certain kind or a certain attitude
toward a group, e.g., ‘desirous of ruling,’ ‘shy,’ and others. It
may be that for the definition or reduction of a term of this
kind some terms of social science describing the group involved
are needed. This shows that there is not a clear-cut line between
psychology and social science and that in some cases it is not
clear whether a term is better assigned to one or to the other
field. But such terms are also certainly reducible to those of the

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thing-language because every term referring to a group of
organisms is reducible to terms referring to individual organ-
isms.

The result of our analysis is that the class of observable
thing-predicates is a sufficient reduction basis for the whole of
the language of science, including the cognitive part of the
everyday language.

V. The Problem of the Unity of Laws

The relations between the terms of the various branches of
science have been considered. There remains the task of ana-
lyzing the relations between the laws. According to our previ-
ous consideration, a biological law contains only terms which
are reducible to physical terms. Hence there is a common lan-
guage to which both the biological and the physical laws be-
long so that they can be logically compared and connected. We
can ask whether or not a certain biological law is compatible
with the system of physical laws, and whether or not it is
derivable from them. But the answer to these questions cannot
be inferred from the reducibility of the terms. At the present
state of the development of science, it is certainly not possible
to derive the biological laws from the physical ones. Some phi-
losophers believe that such a derivation is forever impossible
because of the very nature of the two fields. But the proofs
attempted so far for this thesis are certainly insufficient. This
question is, it seems, the scientific kernel of the problem of
vitalism; some recent discussions of this problem are, however,
entangled with rather questionable metaphysical issues. The
question of derivability itself is, of course, a very serious scien-
tific problem. But it will scarcely be possible to find a solution
for it before many more results of experimental investigation
are available than we have today. In the meantime the efforts
toward derivation of more and more biological laws from physi-
cal laws — in the customary formulation: explanation of more
and more processes in organisms with the help of physics and
chemistry — will be, as it has been, a very fruitful tendency in
biological research.

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Carnap: Logical Foundations of the Unity of Science

As we have seen before, the fields of psychology and social
science are very closely connected with each other. A clear divi-
sion of the laws of these fields is perhaps still less possible than
a division of the terms. If the laws are classified in some way
or other, it will be seen that sometimes a psychological law is
derivable from those of social science, and sometimes a law of
social science from those of psychology. (An example of the
first kind is the explanation of the behavior of adults — e.g., in
the theories of A. Adler and Freud — by their position within
the family or a larger group during childhood; an example of
the second kind is the obvious explanation of an increase of
the price of a commodity by the reactions of buyers and sellers
in the case of a diminished supply.) It is obvious that, at the
present time, laws of psychology and social science cannot be
derived from those of biology and physics. On the other hand,
no scientific reason is known for the assumption that such a
derivation should be in principle and forever impossible.

Thus there is at present no unity of laws. The construction
of one homogeneous system of laws for the whole of science is
an aim for the future development of science. This aim cannot
be shown to be unattainable. But we do not, of course, know
whether it will ever be reached.

On the other hand, there is a unity of language in science, viz.,
a common reduction basis for the terms of all branches of sci-
ence, this basis consisting of a very narrow and homogeneous
class of terms of the physical thing-language. This unity of
terms is indeed less far-reaching and effective than the unity
of laws would be, but it is a necessary preliminary condition
for the unity of laws. We can endeavor to develop science more
and more in the direction of a unified system of laws only be-
cause we have already at present a unified language. And, in
addition, the fact that we have this unity of language is of the
greatest practical importance. The practical use of laws con-
sists in making predictions with their help. The important fact
is that very often a prediction cannot be based on our knowl-
edge of only one branch of science. For instance, the construc-
tion of automobiles will be influenced by a prediction of the

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presumable number of sales. This number depends upon the
satisfaction of the buyers and the economic situation. Hence
we have to combine knowledge about the function of the motor,
the effect of gases and vibration on the human organism, the
ability of persons to learn a certain technique, their willingness
to spend so much money for so much service, the development
of the general economic situation, etc. This knowledge con-
cerns particular facts and general laws belonging to all the four
branches, partly scientific and partly common-sense knowledge.
For very many decisions, both in individual and in social life,
we need such a prediction based upon a combined knowledge of
concrete facts and general laws belonging to different branches of
science. If now the terms of different branches had no logical
connection between one another, such as is supplied by the
homogeneous reduction basis, but were of fundamentally differ-
ent character, as some philosophers believe, then it would not
be possible to connect singular statements and laws of different
fields in such a way as to derive predictions from them. There-
fore, the unity of the language of science is the basis for the
practical application of theoretical knowledge.

Selected Bibliography

I. LOGICAL ANALYSIS

Carnap, R. Philosophy and Logical Syntax. London, 1935. (Elementary.)
. Logical Syntax of Language. London, 1937. (Technical.)

II. REDUCIBILITY

Carnap, R. “Testability and Meaning,” Philosophy of Science, Vols. Ill
(1936) and IV (1937).

III. THE UNITY OF THE LANGUAGE OF SCIENCE; PHYSICALISM

Papers by Neurath and Carnap, Erlcenntnis Vol. II (1932); ibid., Vol. Ill
(1933). Translation of one of these papers: Carnap, The Unity of Science.
London, 1934. Concerning psychology : papers by Schlick, Hempel, and
Carnap, Revue de synthese, Vol. X (1935).

62

Scientific Empiricism

Charles W. Morris

The mind of Leibniz — which was too comprehensive for any single
individual of our time— seemed to have diffused itself over the various
sections. This struck me particularly in connection with the project of a
scientifically philosophical Encyclopedia, advocated by Dr. Otto Neurath.
Leibniz, if he were alive, would no doubt write the whole of it, but in our
day different sections of it will have to be undertaken by different men. It
must, however, be said that in one point of great importance the modern
movement surpasses anything imagined by Leibniz or his contemporaries : I
mean the combination of empiricism with mathematical method. In science,
this combination has existed since the time of Galileo; but in philosophy,
until our time, those who were influenced by mathematical method were
anti-empirical, and the empiricists had little knowledge of mathematics.
Modern science arose from the marriage of mathematics and empiricism;
three centuries later the same union is giving birth to a second child, scientific
philosophy, which is perhaps destined to as great a career. For it alone can
provide the intellectual temper in which it is possible to find a cure for the
diseases of the modern world.

These words of Bertrand Russell appeared in 1936 in the first
volume of the Ades du congres international de philosophic sci-
entifique and form part of the statement in which he registered
his impression of the meeting recorded in these proceedings —
the First International Congress for the Unity of Science
(Paris, 1935). They suggest in miniature the context in which
the present Encyclopedia originates and its possible importance.
For that reason they merit commentary.

I. Method in Science

The development of the method of experimental science is
possibly the most significant intellectual contribution of West-
ern civilization. The development, like most developments, is
a long one, and there is no one moment of absolute origin or
culmination. Certainly, neither experimentation nor mathemat-

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ics had to wait for birth until the flowering of Western science.
Nevertheless, in this flowering something of undeniable impor-
tance took place: the incorporation of mathematics and ex-
perimentation within a single method. Previously they had
functioned as rival methods: mathematics as one way to get
knowledge of nature, and experimental observation as another.
Those scientists who advocated the former were one species of
rationalists, and the advocates of the latter were one species
of “empiricists”— even the philosophical opposition of rational-
ism and empiricism was at bottom a reflection of what were to
be taken as different scientific methods for knowing nature.
Gradually, and in a way that need not at this point be traced
in detail, mathematics came to lose the status of an independent
method for the study of nature, while at the same time it sup-
planted the classical logic as a tool of analysis and as the
structural basis for the scientific edifice. The important result
was a double shift from a metaphysical to a methodological
rationalism, and from a loose-jointed empiricism to an em-
piricism which utilized the techniques and the form of mathe-
matics. Rationalism and empiricism in this way ceased to be
rival methods for knowing nature and became complementary
components of experimental science with its one observational-
hypothetical-deductive-experimental method.

Not only did this method find place for the rational and
empirical factors in the knowledge process, but the emphasis
upon experimentation as against mere observation meant the
breakdown of the radical opposition of theory and practice,
for not only is experimentation itself a kind of practice, but it
is of such a kind as to open up the possibility of a novel and
systematic control of many kinds of natural processes.

The direction of this double movement within science (the
incorporation of the mathematical method within the empiricist
temper and the breakdown through experimentation of the
dichotomy between theory and practice) is discernible in the
Hellenistic period and the late Middle Ages, becomes clearly
evident in Galileo, and reaches a definite expression in Newton.
By the late seventeenth century the great scientists, whatever

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their philosophical differences, had found a place within scien-
tific method for careful and systematic observation, mathe-
matical theory, and experimental practice. Since that time no
fundamental change in the conception of scientific method has
taken place, and science has reaped a rich harvest from its atti-
tude of mathematical experimental empiricism.

II. Generalization of Scientific Method

The attainment of a similar attitude in philosophy came more
slowly and has not even yet received wide agreement. As
Charles S. Peirce remarked, metaphysics has always been the
ape of mathematics, and much that passes for metaphysics
rests upon views of mathematics now largely discarded by the
mathematicians or upon conceptions of philosophic method
once made plausible by the then current conceptions of mathe-
matics. But if philosophic rationalists were slow to see the sig-
nificance of the changed state which mathematics had under-
gone in science, the philosophical empiricists were equally blind
to the significance of this change. The history of empiricism is
not an appropriate theme for an introductory article, but it can
safely be said that the main energy of empiricists was spent in
opposing the a priori rationalists rather than in contributing
to or positively assessing the developing sciences. In fact, these
philosophical empiricists at many points became entangled
themselves in the speculative nets of their opponents, as is evi-
dent in their acceptance of the same superficial subjectivistic,
individualistic, and atomistic conception of experience which
the rationalists had proudly exhibited, ostensibly on the basis
of science but actually because it seemed to show the limitations
of the appeal to experience and the consequent need for other
foundations for knowledge — hence the protracted and exhaust-
ing struggle of past empiricisms against the phantoms of
solipsism and idealism.

The philosophical empiricists, in the main, had connection
with the biological sciences rather than with mathematics or
the physical sciences, and the resulting fact that they were
unable satisfactorily to account for mathematics, or to make

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plausible the rational systematization which the physical
sciences were in fact attaining, gave strong advantages to the
impressive speculative rationalism of their opponents. The
traditional empiricism, in addition to freeing itself from in-
adequate views of experience, had in fact to be supplemented
in two interconnected respects before it could claim to be the
philosophical equivalent of the method which science had
achieved: it had to be able to assimilate and utilize the logical
and mathematical tools of rationalism, and it had to be able
to account for the intellectual significance of practice. The
first expansion was made by logical empiricism; the second by
pragmatism.

The union of formal logic and empiricism is linked with the
development of symbolic or mathematical logic — another theme
which at the proper place is to receive its separate treatment.
This modern version of formal logic developed in the hands of
philosophic rationalists who were themselves mathematicians.
It arose out of the cross-fertilization of the medieval approach
to logic in terms of a general theory of signs and the methods of
modern mathematics, a union which is first significantly made
by Leibniz 1 the great Leibniz whose ideals of a universal scien-
tific language, a generalized mathematical science, a calculus
applicable to all reasoning, and an encyclopedia showing the
logical relation between the concepts of all sciences receive a
contemporary form of expression in this Encyclopedia. The de-
velopment of this logic in the period since Leibniz has made
available a logic adequate to the relational structure of science
and mathematics, and has made possible new and powerful
techniques of analysis. The point to be noted in the present
connection is that in this development logic, like the mathe-
matics which it generalized, came to be regarded as an instru-
ment free from any speculative accretions, and in particular
from the metaphysical (or a priori) rationalism which nourished
it, and thus became available for use by empiricists — another
instance of the fact mentioned in connection with the develop-
ment of science that the formal disciplines become compatible
with empiricism when they pass from the status of rival meth-

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Morris: Scientific Empiricism

ods for the knowledge of nature to that of being formal lin-
guistic structures available to the natural sciences as methodo-
logical tools. Logic thus rests, as Peirce 2 maintained, on a gen-
eral theory of signs, formal logic tracing the relations between
signs within a language. So conceived, logic deals with the lan-
guage in which statements about nature are made, and does not
itself make statements about the nonlinguistic world. Hence it
is not the rival of the empirical knowledge of nature and be-
comes assimilable to the temper and program of empiricism . 3
The clear development of this view and the utilization of the
methods of the new logic within the framework of empiricism
are the significant achievements of logical empiricism as rep-
resented by the Vienna Circle (Carnap, Frank, Hahn, Neurath,
Schlick) and by Russell, Wittgenstein, and Reichenbach.

This union of empiricism and methodological rationalism
requires completion by one further step. Languages are devel-
oped and used by living beings operating in a world of objects,
and show the influence of both the users and the objects. If, as
symbolic logic maintains, there are linguistic forms whose valid-
ity is not dependent upon nonlinguistic objects, then their
validity must be dependent upon the rules of the language in
question, and such rules represent habits actually found in
operation or set up by deliberate convention. The introduction
of such terms as ‘convention,’ ‘decision,’ ‘procedure,’ and rule
involves reference to the users of signs in addition to empirical
and formal factors. It has been the function of pragmatism to
make explicit the instrumental significance of ideas in general
and of scientific results and procedure in particular. Thus
Dewey interprets even logical rules as empirical generalizations
embodying methods of inquiry which have proved particularly
successful for the purpose of inference and which have therefore
been transformed by the users into principles accepted for the
time being as stipulations for the carrying-on of future inquiry.
The introduction of pragmatic considerations avoids the ex-
tremes both of empiricism and of conventionalism in logical
theory while yet doing justice to both. At the same time, in
making explicit the instrumental significance of ideas, it was

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necessary to determine the scientific usage of such terms as
‘idea,’ ‘meaning,’ ‘self,’ and ‘experience’ in the light of post-
Darwinian biology and sociology ; 4 and in doing this, in addition
to gaining results of intrinsic significance, pragmatism helped
free empiricism itself from certain pitfalls which it had en-
countered during its former development. The emphasis upon
the relational and the functional which the biological emphasis
brought with it called attention in general to the previously
neglected relational and functional aspects of experience, and
the realization of the social context in which mind and knowl-
edge arise and operate made manifest the artificiality of the
subjectivistic and individualistic concept of experience with
which English empiricists had often operated. Pragmatism ac-
cordingly not merely brought to the forefront the pragmatical
factor which complements and completes the formal and the
empirical factors, but it helped to enrich the empiricist tradition
through its conception of radical empiricism.

III. The Viewpoint of Scientific Empiricism

The resulting comprehensive point of view, embracing at
once radical empiricism, methodological rationalism, and criti-
cal pragmatism, may appropriately be called scientific empiri-
cism. It is the generalized analogue of the point of view which
has been effective in science for some centuries. It is willing and
able to admit into the scope of its considerations everything
involved in the scientific enterprise as such, together with the
implications of this enterprise for other human interests. It is
an empiricism genuinely oriented around the methods and the
results of science and not dependent upon some questionable
psychological theory as to the “mental” nature of experience.
It is an empiricism which, because of this orientation and the
use of powerful tools of logical analysis, has become positive
in temper and co-operative in attitude and is no longer con-
demned to the negative skeptical task of showing defects in the
methods and results of its opponents.

Such a point of view, characteristic in the main of this En-
cyclopedia (though, of course, not binding on its contributors)

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Morris: Scientific Empiricism

signalizes the widest possible generalization of scientific method.
The field of application of this point of view is science itself.
In analogy with certain other uses of the prefix meta which
are current today, we may introduce the term ‘metascience’ as
a synonym for ‘the science of science.

The attempt to make the scientific enterprise as a whole an
object of scientific investigation— i.e., to develop metascienc^-
requires consideration of the three factors involved in this en-
terprise. Since these factors correspond to the three components
of scientific empiricism, this point of view proves to be ap-
propriate to the task. In its most tangible form science exists
as a body of written characters and spoken words. It is possible
to investigate this linguistic residue of the scientists’ activity
purely formally, without reference to the relation of these marks
and sounds to other objects or to the activity of which they are
the residues. There is no mystery as to how such abstraction
is possible: from a linguistic point of view, to abstract from
some of the properties or relations of an object is simply not to
talk about them. This type of investigation may, in one sense
of the term, be called logical analysis; because of the ambiguity
of the term ‘logical,’ it may be preferable to call it, in the spirit
of Carnap, the syntactical investigation of the language of
science. Such investigation studies the structure of scientific
language: the relation between the terms and the sentences of
the same and different sciences. The degree of unity or disunity
of science reveals itself here in the degree to which the sciences
have or can have a common linguistic structure.

But the signs which constitute scientific treatises have, to
some extent at least, a correlation with objects, and the investi-
gation of all aspects of this relation constitutes a second task of
metascience. Here belong all the problems as to the nature of
this correlation and the analysis of the specific situations under
which scientific terms and sentences are applicable. This may
be called, in the spirit of the Polish logicians, the semantical
investigation of the language of science. The unity of science is
here no longer a purely formal unity, for the unity or disunity of
the scientific language corresponds to some extent to the seman-

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tical relation or lack of relation of the various terms of the sci-
ences — and so to the relations of objects.

A third concern of metascience arises from the fact that the
signs which constitute the language of science are parts and
products of the activity of scientists. The study of the relation
of signs to scientists may be called, in the spirit of pragmatism,
the pragmatical investigation of the language of science. Here
belong the problems as to how the scientist operates, the connec-
tion of science as a social institution with other social institu-
tions, and the relation of scientific activity to other activities.
The question as to the unity of science is now the question as to
the unity of procedures, purposes, and effects of the various
sciences.

Science, as a body of signs with certain specific relations to
one another, to objects, and to practice, is at once a language,
a knowledge of objects, and a type of activity; the interrelated
study of the syntactics, semantics, and pragmatics of the lan-
guage of science in turn constitutes metascience — the science of
science. Discussion of the specific signs of science must be car-
ried on in terms of some theory of signs, and so semiotic, as the
science of signs, occupies an important place in the program —
indeed, the study of the actual language of science is an instance
of applied semiotic. Since the institution of science is a social
institution, certain of its features, as Auguste Comte realized,
reveal themselves especially well in historical perspective, so
that the history of science is of abiding importance within the
study of science.

The elaboration of the syntactics, semantics, and pragmatics
of science may rightly be regarded as the natural extension and
completion of the scientific enterprise itself. It is believed that
this program will appeal to scientists and will prove of im-
portance in the development and assessment of science. It is
inevitable that, in seeking for its greatest unification, science
will make itself an object of scientific investigation. The ful-
filment of these related tasks cannot be left to chance if science
is to grow to full stature: they must be taken in hand by those

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familiar with the results and spirit of science — that is, scientists;
they must be encouraged by scientific institutions, foundations,
and associations; they must be incorporated in educational
programs for the training of scientists. It is true that the indi-
vidual experimenter may not be directly helped in the carrying-
out of a particular experiment — though even here he can be
justly held as responsible for the careful use of his linguistic
tools as he is held for the careful handling of balance, micro-
scope, or telescope. But science has never been content with
isolated facts. It has ever pressed on to larger systematizations.
This process will continue as long as scientific progress con-
tinues. And in this striving for the widest systematization, sci-
ence inevitably has to study itself and incorporate the results of
such study in its systematization. The study of science is not
an intellectual luxury for scientists; it is a movement within
science itself.

But if the study of science which is here contemplated is
science (and so not a domain over and above science), it may
equally well be regarded as philosophy. For the three-faceted
point of view of scientific empiricism, and of metascience which
results from its application to science, can be regarded as em-
bracing the contemporary empiricist equivalents of the tradi-
tional fields of philosophy (logic, metaphysics, and theory of
value). Logic is grounded on semiotic; metaphysics is replaced
by sign analysis and unified science; and axiology becomes the
scientific study of values and judgments of values. Within the
general orientation of scientific empiricism, science and phi-
losophy relinquish all claims to the possession of distinct meth-
ods or subject matters and merge their efforts within a common
task: the erection, the analysis, and the assessment of unified
science. Such community of effort is an ancient ideal and in
part an ancient practice; what is novel today is the scale upon
which fruitful co-operation is possible. Science rounds itself out
humanly and scientifically in this process, and the relevance and
significance which philosophy sometimes had are again re-
gained.

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IV. Science and Practice

What has been called the syntactics and the semantics of the
language of science will receive immediate and extended treat-
ment in the pages of this Encyclopedia. But since this will not
be true to the same degree in the case of the pragmatics of the
language of science, it is well to pause at this point in order both
to see certain implications of the standpoint of scientific em-
piricism and to avoid certain possible misunderstandings.

The very statement that science walks on three legs of theory,
observation, and practice will call out opposition in a certain
type of mind — the statement is more frequent with ‘practice’
omitted. The word ‘practice’ is admittedly equivocal. The ac-
tivity which gives rise to the sentences of science is, like any
other systematic activity, a practice proceeding in terms of
rules or canons. Further, the confirmation of every proposition
always involves some instrument, whether this be simply the
scientist himself or in addition such instruments as those in-
volved in experimentation — and methodologically there is no
important distinction between the two cases. In this (theoreti-
cally the most important) sense, all empirical science involves
experimentation, and experimentation is an activity, a practice.
In the third place, science is part of the practice of the com-
munity in which it is an institution, ministering — however in-
directly — to the needs of the community and being affected —
and very directly — in its development by the community of
social institutions of which it is a part.

It is clear that any adequate account of science must take
account of these psychological, methodological, and sociological
aspects of scientific practice. The present work recognizes that
fact. But it should also be clear that ‘practice’ in all three senses
of the word is not an unessential factor added to the theoretical
and empirical aspects of the scientific enterprise but an equally
essential factor, since, at the minimum, ‘confirmation’ is a con-
cept which contains irreducible pragmatical features. If this is
so, it would be well for scientists to become fully aware of this
factor of practice and, in becoming aware of it, to assume the
entailed responsibilities.

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Morris: Scientific Empiricism

The same point may be given an alternative formulation in
terms of the notion of value. It is often said that science gives
only “facts” and has nothing to do with “values.” There is an
element of truth in such a statement, since the pragmatical
factor in language cannot be reduced at a given moment to the
empirical, and since life is more than knowledge. But this is
hardly the usual import of this statement, which is often made
against a background which involves a sharp distinction be-
tween the natural sciences and the socio-humanistic sciences.
The detailed study of the actual relations must be left for other
writers. Nevertheless, it seems clear that, while a program
which stresses the unity of science can admit of whatever diver-
sity is in fact found in the various sciences (for unity does not
exclude differentiation), it must naturally be skeptical of any
such wholesale cleavage. Later treatments of these matters will
maintain in harmony with the empiricist attitude that there is
no unbridgeable gap within science between the procedures and
subject matters of the natural sciences and the socio-humanistic
sciences. All knowledge forms in principle one unified w T hole,
and there exists no system of knowledge (such as metaphysics,
aesthetics, ethics, religion) alongside of or superior to unified
science.

This statement should not be misunderstood. Science is an
activity eventuating in a certain sort of product having certain
effects; but science is not the only activity of that sort. Art,
morality, piety, play, work, and war are also activities with
characteristic products and effects, and they must in no sense be
confused with the sciences of such activities (aesthetics, ethics,
science of religion, etc.). As activities they are co-ordinate with
science considered as an activity, but the sciences of these ac-
tivities fall within the field of unified science. In the first case,
they are alternatives to the scientific attitude; in the second
case, they are part of science. But an alternative is not neces-
sarily a rival. Indeed, once science is distinguished in terms of
its specific goal (reliable knowledge), then science not only does
not, in general, clash with other activities but may itself further
any other activity in so far as it can be furthered by knowledge.

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In this respect science is the most practical of human activities.
It is in opposition only to such activities as claim to usurp its
own cognitive goal or which wither and die when the light of
scientific investigation is turned upon them; it is at once both co-
ordinate with and instrumental to the realization of the pur-
poses of most other activities.

It is because of this relation of scientific activity to other ac-
tivities that the scientific habit of mind and scientific results
are of such potential promise in society at large and education
in particular . 6 For this habit of mind is the best guaranty of an
objective consideration of the multiplicity of factors which enter
into the complex problems of contemporary man.

V. Scientific Empiricism and Encyclopedism

The standpoint of scientific empiricism is thus ample enough
to embrace and to integrate the various factors which must be
taken into account in an Encyclopedia devoted to unified science
— i.e., to the scientific study of the scientific enterprise in its
totality. The theory of signs gives the general background for
the consideration of the language of science. The investigation
of this language breaks up into the distinguishable but interre-
lated investigations of the syntactics, semantics, and pragmatics
of the language of science. In this way, and on a comprehensive
scale, science is made an object of scientific investigation; meta-
science appears both as a tool for, and as an element within uni-
fied science. The attitude of scientific empiricism is simul-
taneously congenial to the temper of the rationalist, the empiri-
cist, and the pragmatist, and provides the corrective to the one-
sidedness of these attitudes when held in isolation. It is from
the standpoint of method the complement of encyclopedism,
since while it accepts the encyclopedia as the necessary form of
human knowledge it yet recognizes that science strives for the
greatest degree of systematization compatible with its con-
tinual growth.

This Encyclopedia, reflecting this inclusive standpoint, right-
fully sounds the roll call of those distinguished logicians, scien-
tists, and empiricists whom the traditional history of ideas has

74

Morris: Scientific Empiricism

so shamefully neglected. But basically it aims to present
through extensive co-operation the existing status and the un-
realized possibilities for the integration of science. Its existence
signalizes the union of scientific and philosophic traditions in a
common task. The Encyclopedia presents a contemporary ver-
sion of the ancient encyclopedic ideal of Aristotle, the Scholas-
tics, Leibniz, the Encyclopedists, and Comte. It wishes to give
satisfaction to the pervasive human interest in intellectual uni-
ty, but its common point of view permits divergences and dif-
ferences in emphasis and does not blur the fact that an in-
separable feature of the institution of science is constant growth.
It aims to provide a basis for co-operative activity and not a
panacea.

NOTES

1. See the splendid work of Louis Couturat, La Logique de Leibniz (Paris, 1901).

2. See especially Collected Papers (Cambridge, Mass., 1932), Vol. II.

3. This point was clearly elaborated by H. Hahn (see Erkenntnis, II [1931], 135-41;
Logik, Mathematik und Naturerkennen [Vienna, 1933]). It may be remarked that from
this point of view Leibniz’ plans for a universal mathematics, a calculus of reasoning, a
general characteristic, and a unified science expressed in the form of an encyclopedia all
remain valid when interpreted as logical rather than as metaphysical doctrines. Leib-
niz’ rationalistic metaphysics, which came from the simple conversion of formal logic
into a metaphysics through the neglect of the criterion of the empirically meaningful,
is, in terms of the present conception of the relation of logic to empiricism, no longer
the necessary cosmological corollary of his logical doctrines.

4. The writings of George H. Mead are of importance in this connection, especially
Mind, Self, and Society (Chicago, 1934).

5. This term and certain of the more general features of the point of view are charac-
terized in the author’s pamphlet. Logical Positivism, Pragmatism, and Scientific Empiri-
cism (Paris: Hermann & Cie, 1937).

6. John Dewey, in particular, has devoted his life to the formulation and assess-
ment of the social, cultural, and educational implications of the scientific habit of
mind. See Philosophy and Civilization (New York, 1931) and his forthcoming work,
Logic: The Theory of Inquiry.

75

Foundations of the Theory
of Signs

Charles W. Morris

Foundations of the Theory of Signs

Contents:

I. Introduction paqe

1. Semiotic and Science 79

II. Semiosis and Semiotic

2. The Nature of a Sign 81

3. Dimensions and Levels of Semiosis .... 84

4. Language 88

III. Syntactics

5. The Formal Conception of Language ... 91

6. Linguistic Structure 94

IV. Semantics

7. The Semantical Dimension of Semiosis 99

8. Linguistic and Nonlinguistic Structures 104

V. Pragmatics

9. The Pragmatical Dimension of Semiosis 107

10. Individual and Social Factors in Semiosis . 112

11. Pragmatic Use and Abuse of Signs .... 116

VI. The Unity of Semiotic

12. Meaning 121

13. Universals and Universality 126

14. Interrelation of the Semiotical Sciences . 130

VII. Problems and Applications

15. Unification of the Semiotical Sciences 132

16. Semiotic as Organon of the Sciences . 134

17. Humanistic Implications of Semiotic 135

Selected Bibliography 137

78

Foundations of the Theory of Signs

Charles W. Morris

Nemo autem vereri debet ne eharacterum contemplatio nos a rebus
abducat, imo contra ad intima rerum ducet. — Gottfried Leibniz

I. Introduction

1. Semiotic and Science

Men are the dominant sign-using animals. Animals other
than man do, of course, respond to certain things as signs of
something else, but such signs do not attain the complexity and
elaboration which is found in human speech, writing, art, testing
devices, medical diagnosis, and signaling instruments. Science
and signs are inseparately interconnected, since science both
presents men with more reliable signs and embodies its results
in systems of signs. Human civilization is dependent upon
signs and systems of signs, and the human mind is inseparable
from the functioning of signs — if indeed mentality is not to be
identified with such functioning.

It is doubtful if signs have ever before been so vigorously
studied by so many persons and from so many points of view.
The army of investigators includes linguists, logicians, philoso-
phers, psychologists, biologists, anthropologists, psychopathol-
ogists, aestheticians, and sociologists. There is lacking, how-
ever, a theoretical structure simple in outline and yet compre-
hensive enough to embrace the results obtained from different
points of view and to unite them into a unified and consistent
whole. It is the purpose of the present study to suggest this
unifying point of view and to sketch the contours of the science
of signs. This can be done only in a fragmentary fashion, partly
because of the limitations of space, partly because of the un-
developed state of the science itself, but mainly because of the

79

Foundations of the Theory of Signs

purpose which such a study aims to serve by its inclusion in
this Encyclopedia.

Semiotic has a double relation to the sciences: it is both a
science among the sciences and an instrument of the sciences.
The significance of semiotic as a science lies in the fact that it
is a step in the unification of science, since it supplies the founda-
tions for any special science of signs, such as linguistics, logic,
mathematics, rhetoric, and (to some extent at least) aesthetics.
The concept of sign may prove to be of importance in the uni-
fication of the social, psychological, and humanistic sciences in
so far as these are distinguished from the physical and biological
sciences. And since it will be shown that signs are simply the
objects studied by the biological and physical sciences related
in certain complex functional processes, any such unification
of the formal sciences on the one hand, and the social, psycho-
logical, and humanistic sciences on the other, would provide
relevant material for the unification of these two sets of sciences
with the physical and biological sciences. Semiotic may thus
be of importance in a program for the unification of science,
though the exact nature and extent of this importance is yet
to be determined.

But if semiotic is a science co-ordinate with the other sci-
ences, studying things or the properties of things in their func-
tion of serving as signs, it is also the instrument of all sciences,
since every science makes use of and expresses its results in
terms of signs. Hence metascience (the science of science) must
use semiotic as an organon. It was noticed in the essay “Sci-
entific Empiricism” (Vol. I, No. 1) that it is possible to include
without remainder the study of science under the study of the
language of science, since the study of that language involves
not merely the study of its formal structure but its relation
to objects designated and to the persons who use it. From this
point of view the entire Encyclopedia, as a scientific study of
science, is a study of the language of science. But since nothing
can be studied without signs denoting the objects in the field
to be studied, a study of the language of science must make use
of signs referring to signs — and semiotic must supply the rele-

80

The Nature of a Sign

vant signs and principles for carrying on this study. Semiotic
supplies a general language applicable to any special language
or sign, and so applicable to the language of science and specific
signs which are used in science.

The interest in presenting semiotic as a science and as part
of the unification of science must here be restricted by the prac-
tical motive of carrying the analysis only so far and in such
directions as to supply a tool for the work of the Encyclopedia,

1. e., to supply a language in which to talk about, and in so
doing to improve, the language of science. Other studies would
be necessary to show concretely the results of sign analysis
applied to special sciences and the general significance for the
unification of science of this type of analysis. But even without
detailed documentation it has become clear to many persons
today that man — including scientific man — must free himself
from the web of words which he has spun and that language —
including scientific language — is greatly in need of purification,
simplification, and systematization. The theory of signs is a
useful instrument for such debabelization.

II. Semiosis and Semiotic

2. The Nature of a Sign

The process in which something functions as a sign may be
called semiosis. This process, in a tradition which goes back
to the Greeks, has commonly been regarded as involving three
(or four) factors: that which acts as a sign, that which the
sign refers to, and that effect on some interpreter in virtue of
which the thing in question is a sign to that interpreter. These
three components in semiosis may be called, respectively, the
sign vehicle, the designatum, and the interpretant; the interpreter
may be included as a fourth factor. These terms make explicit
the factors left undesignated in the common statement that a
sign refers to something for someone.

A dog responds by the type of behavior (7) involved in the
hunting of chipmunks (D) to a certain sound (S); a traveler
prepares himself to deal appropriately (7) with the geographical
region (7?) in virtue of the letter (S) received from a friend. In

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Foundations of the Theory of Signs

such cases S is the sign vehicle (anti a sign in virtue of its func-
tioning), D the designatum, and I the interpretant of the in-
terpreter. The most effective characterization of a sign is the
following: S is a sign of D for I to the degree that I takes
account of D in virtue of the presence of S. Thus in semiosis
something takes account of something else mediately, i.e., by
means of a third something. Semiosis is accordingly a mediated-
taking-account-of. The mediators are sign vehicles; the takings-
acceunt-of are interpretants; the agents of the process are in-
terpreters; what is taken account of are designata. There are
several comments to be made about this formulation.

It should be clear that the terms ‘sign,’ ‘designatum,’ ‘inter-
pretant,’ and ‘interpreter’ involve one another, since they are
simply ways of referring to aspects of the process of semiosis.
Objects need not be referred to by signs, but there are no
designata unless there is such reference; something is a sign only
because it is interpreted as a sign of something by some inter-
preter; a taking-account-of-something is an interpretant only
in so far as it is evoked by something functioning as a sign; an
object is an interpreter only as it mediately takes account of
something. The properties of being a sign, a designatum, an
interpreter, or an interpretant are relational properties which
things take on by participating in the functional process of
semiosis. Semiotic, then, is not concerned with the study of a
particular kind of object, but with ordinary objects in so far
(and only in so far) as they participate in semiosis. The im-
portance of this point will become progressively clearer.

Signs which refer to the same object need not have the same
designata, since that which is taken account of in the object
may differ for various interpreters. A sign of an object may, at
one theoretical extreme, simply turn the interpreter of the sign
upon the object, while at the other extreme it would allow the
interpreter to take account of all the characteristics of the ob-
ject in question in the absence of the object itself. There is thus
a potential sign continuum in which with respect to every ob-
ject or situation all degrees of semiosis may be expressed, and
the question as to what the designatum of a sign is in any given

82

The Nature of a Sign

situation is the question of what characteristics of the object
or situation are actually taken account of in virtue of the pres-
ence of the sign vehicle alone.

A sign must have a designatum; yet obviously every sign
does not, in fact, refer to an actual existent object. The diffi-
culties which these statements may occasion are only apparent
difficulties and need no introduction of a metaphysical realm
of “subsistence” for their solution. Since ‘designatum’ is a
semiotical term, there cannot be designata without semiosis —
but there can be objects without there being semiosis. The des-
ignatum of a sign is the kind of object which the sign applies to,
i.e., the objects with the properties which the interpreter takes
account of through the presence of the sign vehicle. And the
taking-account-of may occur without there actually being ob-
jects or situations with the characteristics taken account of.
This is true even in the case of pointing: one can for certain pur-
poses point without pointing to anything. No contradiction
arises in saying that every sign has a designatum but not every
sign refers to an actual existent. Where what is referred to
actually exists as referred to the object of reference is a de-
notatum. It thus becomes clear that, while every sign has a
designatum, not every sign has a denotatum. A designatum is
not a thing, but a kind of object or class of objects — and a class
may have many members, or one member, or no members. The
denotata are the members of the class. This distinction makes
explicable the fact that one may reach in the icebox for an
apple that is not there and make preparations for living on an
island that may never have existed or has long since disap-
peared beneath the sea.

As a last comment on the definition of sign, it should be
noted that the general theory of signs need not commit itself
to any specific theory of what is involved in taking account of
something through the use of a sign. Indeed, it may be possible
to take ‘mediated-taking-account-of as the single primitive
term for the axiomatic development of semiotic. Nevertheless,
the account which has been given lends itself to treatment
from the point of view of behavioristics, and this point of view

83

Foundations of the Theory of Signs

will be adopted in what follows. This interpretation of the
definition of sign is not, however, necessary. It is adopted here
because such a point of view has in some form or other (though
not in the form of Watsonian behaviorism) become widespread
among psychologists, and because many of the difficulties which
the history of semiotic reveals seem to be due to the fact that
through most of its history semiotic linked itself with the faculty
and introspective psychologies. From the point of view of be-
havioristics, to take account of D by the presence of S involves
responding to D in virtue of a response to S. As will be made
clear later, it is not necessary to deny “private experiences”
of the process of semiosis or of other processes, but it is necessary
from the standpoint of behavioristics to deny that such ex-
periences are of central importance or that the fact of their
existence makes the objective study of semiosis (and hence
of sign, designatum, and interpretant) impossible or even in-
complete.

3. Dimensions and Levels of Semiosis

In terms of the three correlates (sign vehicle, designatum,
interpreter) of the triadic relation of semiosis, a number of
other dyadic relations may be abstracted for study. One may
study the relations of signs to the objects to which the signs
are applicable. This relation will be called the semantical dimen-
sion of semiosis, symbolized by the sign ‘Z) sem ’ ; the study of
this dimension will be called semantics. Or the subject of study
may be the relation of signs to interpreters. This relation will
be called the pragmatical dimension of semiosis, symbolized by
‘D p ,’ and the study of this dimension will be named pragmatics.

One important relation of signs has not yet been introduced :
the formal relation of signs to one another. This relationship
was not, in the preceding account, explicitly incorporated in
the definition of ‘sign,’ since current usage would not seem to
eliminate the possibility of applying the term ‘sign’ to some-
thing which was not a member of a system of signs — such
possibilities are suggested by the sign aspects of perception and
by various apparently isolated mnemonic and signaling devices.

84

Dimensions and Levels of Semiosis

Nevertheless, the interpretation of these cases is not perfectly
clear, and it is very difficult to be sure that there is such a
thing as an isolated sign. Certainly, potentially, if not actually,
every sign has relations to other signs, for what it is that the
sign prepares the interpreter to take account of can only be
stated in terms of other signs. It is true that this statement
need not be made, but it is always in principle capable of being
made, and when made relates the sign in question to other signs.
Since most signs are clearly related to other signs, since many
apparent cases of isolated signs prove on analysis not to be such,
and since all signs are potentially if not actually related to
other signs, it is well to make a third dimension of semiosis
co-ordinate with the other two which have been mentioned.
This third dimension will be called the syntactical dimension of
semiosis, symbolized by ‘D ty J and the study of this dimension
will be named syntactics.

It will be convenient to have special terms to designate cer-
tain of the relations of signs to signs, to objects, and to inter-
preters. ‘ Implicates' will be restricted to D syn , ‘designates’ and
‘ denotes ’ to D, tm , and ‘expresses’ to D p . The word ‘table’ impli-
cates (but does not designate) ‘furniture with a horizontal top on
which things may be placed,’ designates a certain kind of object
(furniture with a horizontal top on which things may be placed),
denotes the objects to which it is applicable, and expresses its
interpreter. In any given case certain of the dimensions may
actually or practically vanish: a sign may not have syntactical
relations to other signs and so its actual implication becomes
null; or it may have implication and yet denote no object; or
it may have implication and yet no actual interpreter and so no
expression — as in the case of a word in a dead language. Even
in such possible cases the terms chosen are convenient to refer
to the fact that certain of the possible relations remain un-
realized.

It is very important to distinguish between the relations
which a given sign sustains and the signs used in talking about
such relations — the full recognition of this is perhaps the most
important general practical application of semiotic. The func-

85

Foundations of the T heory of Signs

tioning of signs is, in general, a way in which certain existences
take account of other existences through an intermediate class
of existences. But there are levels of this process which must
be carefully distinguished if the greatest confusion is not to
result. Semiotic as the science of semiosis is as distinct from
semiosis as is any science from its subject matter. If x so func-
tions that y takes account of 2 through x, then we may say that
x is a sign, and that x designates 2, etc.; but here ‘sign,’ and
‘designates’ are signs in a higher order of semiosis referring to
the original and lower-level process of semiosis. What is now
designated is a certain relation of x and 2 and not 2 alone; x
is designated, 2 is designated, and a relation is designated such
that x becomes a sign and 2 a designatum. Designation may
therefore occur at various levels, and correspondingly there are
various levels of designata; ‘designation’ reveals itself to be a
sign within semiotic (and specifically within semantics), since
it is a sign used in referring to signs.

Semiotic as a science makes use of special signs to state facts
about signs; it is a language to talk about signs. Semiotic has
the three subordinate branches of syntactics, semantics, and
pragmatics, dealing, respectively, with the syntactical, the
semantical, and the pragmatical dimensions of semiosis. Each
of these subordinate sciences will need its own special terms;
as previously used ‘implicates’ is a term of syntactics, ‘desig-
nates’ and ‘denotes’ are terms of semantics, and ‘expresses’ is
a term of pragmatics. And since the various dimensions are
only aspects of a unitary process, there will be certain relations
between the terms in the various branches, and distinctive signs
will be necessary to characterize these relations and so the
process of semiosis as a whole. ‘Sign’ itself is a strictly semi-
otical term, not being definable either within syntactics, seman-
tics, or pragmatics alone; only in the wider use of ‘semiotical’
can it be said that all the terms in these disciplines are semi-
otical terms.

It is possible to attempt to systematize the entire set of terms
and propositions dealing with signs. In principle, semiotic could

86

Dimensions and Levels of Semiosis

be presented as a deductive system, with undefined terms and
primitive sentences which allow the deduction of other sentences
as theorems. But though this is the form of presentation to
which science strives, and though the fact that semiotic deals
exclusively with relations makes it peculiarly fit for treatment
by the new logic of relations, yet it is neither advisable nor
possible in the present monograph to attempt this type of ex-
position. It is true that much has been accomplished in the
general analysis of sign relations by the formalists, the em-
piricists, and the pragmatists, but the results which have been
attained seem to be but a small part of what may be expected;
the preliminary systematization in the component fields has
hardly begun. For such reasons, as well as because of the intro-
ductory function of this monograph, it has not seemed advisable
to attempt a formalization of semiotic which goes much beyond
the existing status of the subject, and which might obscure the
role which semiotic is fitted to play in the erection of unified
science.

Such a development remains, however, as the goal. Were it
obtained it would constitute what might be called pure semiotic,
with the component branches of pure syntactics, pure seman-
tics, and pure pragmatics. Here would be elaborated in sys-
tematic form the metalanguage in terms of which all sign situa-
tions would be discussed. The application of this language to
concrete instances of signs might then be called descriptive
semiotic (or syntactics, semantics, or pragmatics as the case
may be). In this sense the present Encyclopedia, in so far as it
deals with the language of science, is an especially important
case of descriptive semiotic, the treatment of the structure of
that language falling under descriptive syntactics, the treat-
ment of the relation of that language to existential situations
falling under descriptive semantics, and the consideration of
the relation of that language to its builders and users being an
instance of descriptive pragmatics. The Encyclopedia as a
whole, from the point of view expressed in this monograph,
falls within the province of pure and descriptive semiotic.

87

Foundations of the Theory of Signs
4. Language

The preceding account is applicable to all signs, however
simple or complex. Hence it is applicable to languages as a
particular kind of sign system. The term ‘language,’ in com-
mon with most terms which have to do with signs, is ambigu-
ous, since its characterization may be given in terms of the
various dimensions. Thus the formalist is inclined to consider
any axiomatic system as a language, regardless of whether there
are any things which it denotes, or whether the system is ac-
tually used by any group of interpreters; the empiricist is in-
clined to stress the necessity of the relation of signs to objects
which they denote and whose properties they truly state; the
pragmatist is inclined to regard a language as a type of com-
municative activity, social in origin and nature, by which
members of a social group are able to meet more satisfactorily
their individual and common needs. The advantage of the
three-dimensional analysis is that the validity of all these points
of view can be recognized, since they refer to three aspects of
one and the same phenomenon; where convenient the type of
consideration (and hence of abstraction) can be indicated by
‘L syn ,’ ‘L, em ,’ or ‘ L p .’ It has already been noted that a sign may
not denote any actual objects (i.e., have no denotatum) or may
not have an actual interpreter. Similarly, there may be lan-
guages, as a kind of sign complex, which at a given time are ap-
plied to nothing, and which have a single interpreter or even no
interpreter, just as an unoccupied building may be called a house.
It is not possible, however, to have a language if the set of signs
have no syntactical dimension, for it is not customary to call a
single sign a language. Even this case is instructive, for in
terms of the view expressed (namely, that potentially every
sign has syntactical relations to those signs which would state
its designatum, that is, the kind of situation to which it is
applicable) even an isolated sign is potentially a linguistic sign.
It could also be said that an isolated sign has certain relations
to itself, and so a syntactical dimension, or that having a null
syntactical dimension is only a special case of having a syn-

88

Language

tactical dimension. These possibilities are important in showing
the degree of independence of the various dimensions and con-
sequently of L sya , L xm , and L v . They also show that there is
no absolute cleft between single signs, sentential signs, and lan-
guages — a point which Peirce especially stressed.

A language, then, as a system of interconnected signs, has a
syntactical structure of such a sort that among its permissible
sign combinations some can function as statements, and sign
vehicles of such a sort that they can be common to a number
of interpreters. The syntactical, semantical, and pragmatical
features of this characterization of language will become clearer
when the respective branches of semiotic are considered. It will
also become clear that just as an individual sign is completely
characterized by giving its relation to other signs, objects, and
its users, so a language is completely characterized by giving
what will later be called the syntactical, semantical, and prag-
matical rules governing the sign vehicles. For the moment it
should be noted that the present characterization of language
is a strictly semiotical one, involving reference to all three
dimensions ; much confusion will be avoided if it is recognized
that the word ‘language’ is often used to designate some aspect
of what is language in the full sense. The simple formula, L =
L syn + L sem + L v , helps to clarify the situation.

Languages may be of various degrees of richness in the com-
plexity of their structure, the range of things they designate,
and the purposes for which they are adequate. Such natural
languages as English, French, German, etc., are in these re-
spects the richest languages and have been called universal
languages, since in them everything can be represented. This
very richness may, however, be a disadvantage for the realiza-
tion of certain purposes. In the universal languages it is often
very difficult to know within which dimension a certain sign
is predominantly functioning, and the various levels of sym-
bolic reference are not clearly indicated. Such languages are
therefore ambiguous and give rise to explicit contradictions —
facts which in some connections (but not in all!) are disad-
vantageous. The very devices which aid scientific clarity may

89

Foundations of the Theory of Signs

weaken the potentialities for the aesthetic use of signs, and
vice versa. Because of such considerations it is not surprising
that men have developed certain special and restricted lan-
guages for the better accomplishment of certain purposes:
mathematics and formal logic for the exhibition of syntactical
structure, empirical science for more accurate description and
prediction of natural processes, the fine and applied arts for
the indication and control of what men have cherished. The
everyday language is especially weak in devices to talk about
language, and it is the task of semiotic to supply a language to
meet this need. For the accomplishment of their own ends these
special languages may stress certain of the dimensions of sign-
functioning more than others; nevertheless, the other dimen-
sions are seldom if ever completely absent, and such languages
may be regarded as special cases falling under the full semiotical
characterization of language which has been suggested.

The general origin of systems of interconnected signs is not
difficult to explain. Sign vehicles as natural existences share in
the connectedness of extraorganic and intraorganic processes.
Spoken and sung words are literally parts of organic responses,
while writing, painting, music, and signals are the immediate
products of behavior. In the case of signs drawn from materials
other than behavior or the products of behavior — as in the sign
factors in perception — the signs become interconnected because
the sign vehicles are interconnected. Thunder becomes a sign
of lightning and lightning a sign of danger just because thun-
der and lightning and danger are, in fact, interconnected in
specific ways. If w expects x on the presence of y, and z on
the presence of x, the interconnectedness of the two expecta-
tions makes it very natural for w to expect z on the presence of
y. From the interconnectedness of events on the one hand,
and the interconnectedness of actions on the other, signs be-
come interconnected, and language as a system of signs arises.
That the syntactical structure of language is, in general, a
function both of objective events and of behavior, and not of
either alone, is a thesis which may be called the dual control of
linguistic structure. This thesis will receive elaboration later.

90

The Formal Conception of Language

but it should be already evident that it gives a way of avoiding
the extremes of both conventionalism and the traditional em-
piricism in accounting for linguistic structure. For the reasons
given, sets of signs tend to become systems of signs; this is as
true in the case of perceptual signs, gestures, musical tones, and
painting as it is in the case of speech and writing. In some cases
the systematization is relatively loose and variable and may
include subsystems of various degrees of organization and inter-
connectedness; in others it is relatively close and stable, as in
the case of mathematical and scientific languages. Given such
sign structures, it is possible to subject them to a three-dimen-
sional analysis, investigating their structure, their relation to
what they denote, and their relations to their interpreters.
This will now be done in general terms, discussing in turn
the syntactics, semantics, and pragmatics of language, but
keeping in mind throughout the relation of each dimension,
and so each field of semiotic, to the others. Later, after making
use of the abstractions involved in this treatment, we will
specifically stress the unity of semiotic.

III. Syntactics

5. The Formal Conception of Language

Syntactics, as the study of the syntactical relations of signs
to one another in abstraction from the relations of signs to
objects or to interpreters, is the best developed of all the
branches of semiotic. A great deal of the work in linguistics
proper has been done from this point of view, though often
unconsciously and with many confusions. Logicians have from
the earliest times been concerned with inference, and this,
though historically overlaid with many other considerations,
involves the study of the relations between certain combina-
tions of signs within a language. Especially important has been
the early presentation by the Greeks of mathematics in the
form of a deductive or axiomatic system; this has kept con-
stantly before men’s attention the pattern of a closely knit
system of signs such that by means of operations upon certain
initial sets all the other sets of signs are obtained. Such formal

91

Foundations of the T heory of Signs

systems presented the material whose considerations made in-
evitable the development of syntactics. It was in Leibniz the
mathematician that linguistic, logical, and mathematical con-
siderations jointly led to the conception of a general formal art
( speciosa general is) which included the general characteristic art
( ars characteristica ) , essentially a theory and art of so forming
signs that all consequences of the corresponding “ideas” could
be drawn by a consideration of the signs alone, and the general
combinatory art ( ars combinatoria ) , a general calculus giving a
universal formal method of drawing the consequences from
signs. This unification and generalization of mathematical form
and method has received since Leibniz’ time a remarkable ex-
tension in symbolic logic, through the efforts of Boole, Frege,
Peano, Peirce, Russell, Whitehead, and others, while the theory
of such syntactical relations has received its most elaborate
contemporary development in the logical syntax of Carnap.
For present purposes only the most general aspect of this point
of view need be mentioned, especially since Carnap treats this
question in Volume I, Numbers 1 and 3.

Logical syntax deliberately neglects what has here been called
the semantical and the pragmatical dimensions of semiosis to
concentrate upon the logico-grammatical structure of language,
i.e., upon the syntactical dimension of semiosis. In this type
of consideration a “language” (i.e., L sya ) becomes any set of
things related in accordance with two classes of rules: forma-
tion rules, which determine permissible independent combina-
tions of members of the set (such combinations being called
sentences), and transformation rules, which determine the sen-
tences which can be obtained from other sentences. These may
be brought together under the term ‘ syntactical rule .’ Syntactics
is, then, the consideration of signs and sign combinations in
so far as they are subject to syntactical rules. It is not inter-
ested in the individual properties of the sign vehicles or in any
of their relations except syntactical ones, i.e., relations de-
termined by syntactical rules.

Investigated from this point of view, languages have proved
to be unexpectedly complex, and the point of view unexpectedly

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The Formal Conception of Language

fruitful. It has been possible accurately to characterize primi-
tive, analytic, contradictory, and synthetic sentences, as well
as demonstration and derivation. Without deserting the formal
point of view, it has proved possible to distinguish logical and
descriptive signs, to define synonymous signs and equipollent
sentences, to characterize the content of a sentence, to deal
with the logical paradoxes, to classify certain types of expres-
sions, and to clarify the modal expressions of necessity, pos-
sibility, and impossibility. These and many other results have
been partially systematized in the form of a language, and most
of the terms of logical syntax may be defined in terms of the
notion of consequence. The result is that there is today avail-
able a more precise language for talking about the formal di-
mension of languages than has ever before existed. Logical syn-
tax has given results of high intrinsic interest and furnished a
powerful analytical tool; it will be used extensively in the
analysis of the language of science in this Encyclopedia.

Our present interest, however, is solely with the relation of
logical syntax to semiotic. It is evident that it falls under syn-
tactics; it has indeed suggested this name. All the results of
logical syntax are assimilable by syntactics. Further, it is with-
out doubt the most highly developed part of syntactics, and
so of semiotic. In its spirit and method it has much to con-
tribute to semantics and pragmatics, and there is evidence that
its influence is at work in these fields.

Many of its specific results have analogues in the other
branches of semiotic. As an illustration let us use the term
‘ thing -sentence ,’ to designate any sentence whose designatum
does not include signs ; such a sentence is about things and may
be studied by semiotic. On this usage none of the sentences of
the semiotical languages are thing-sentences. Now Carnap has
made clear the fact that many sentences which are apparently
thing-sentences, and so about objects which are not signs, turn
out under analysis to be pseudo thing-sentences which must be
interpreted as syntactical statements about language. But in
analogy to these quasi-syntactical sentences there are corre-
sponding quasi-semantical and quasi-pragmatical sentences

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Foundations of the T heory of Signs

which appear to be thing-sentences but which must be inter-
preted in terms of the relation of signs to designata or the
relation of signs to interpreters.

Syntactics is in some respects easier to develop than its co-
ordinate fields, since it is somewhat easier, especially in the case
of written signs, to study the relations of signs to one another
as determined by rule than it is to characterize the existential
situations under which certain signs are employed or what goes
on in the interpreter when a sign is functioning. For this reason
the isolation of certain distinctions by syntactical investigation
gives a clue for seeking their analogues in semantical and prag-
matical investigations.

In spite of the importance thus ascribed to logical syntax, it
cannot be equated with syntactics as a whole. For it (as the
term ‘sentence’ shows) has limited its investigation of syntacti-
cal structure to the type of sign combinations which are domi-
nant in science, namely, those combinations which from a
semantical point of view are called statements, or those com-
binations used in the transformation of such combinations.
Thus on Carnap’s usage commands are not sentences, and many
lines of verse would not be sentences. ‘Sentence’ is not, there-
fore, a term which in his usage applies to every independent sign
combination permitted by the formation rules of a language —
and yet clearly syntactics in the wide sense must deal with all
such combinations. There are, then, syntactical problems in
the fields of perceptual signs, aesthetic signs, the practical use
of signs, and general linguistics which have not been treated
within the framework of what today is regarded as logical syn-
tax and yet which form part of syntactics as this is here con-
ceived.

6. Linguistic Structure

Let us now consider more carefully linguistic structure, in-
voking semantics and pragmatics where they may be of help
in clarifying the syntactical dimension of semiosis.

Given a plurality of signs used by the same interpreter, there
is always the possibility of certain syntactical relations between

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Linguistic Structure

the signs. If there are two signs, Si and S 2 , so used that S t
(say ‘animal’) is applied to every object to which S 2 (say ‘man’)
is applied, but not conversely, then in virtue of this usage the
semiosis involved in the functioning of Si is included in that of
S 2 ; an interpreter will respond to an object denoted by ‘man’
with the responses he would make to an object denoted by
‘animal,’ but in addition there are certain responses which
would not be made to any animal to which ‘man’ was not
applicable and which would not be made to an animal to which
certain other terms (such as ‘amoeba’) were applicable. In this
way terms gain relations among themselves corresponding to
the relations of the responses of which the sign vehicles are a
part, and these modes of usage are the pragmatical background
of the formation and transformation rules. The syntactical
structure of a language is the interrelationship of signs caused
by the interrelationship of the responses of which the sign
vehicles are products or parts. The formalist substitutes for
such responses their formulation in signs; when he begins with
an arbitrary set of rules, he is stipulating the interrelationship
of responses which possible interpreters must have before they
can be said to be using the language under consideration.

In so far as a single sign (such as a particular act of pointing)
can denote only a single object, it has the status of an index;
if it can denote a plurality of things (such as the term ‘man’),
then it is combinable in various ways with signs which explicate
or restrict the range of its application; if it can denote every-
thing (such as the term ‘something’), then it has relations with
every sign, and so has universal implication, that is to say, it
is implicated by every sign within the language. These three
kinds of signs will be called, respectively, indexical signs, char-
acterizing signs, and universal signs.

Signs may thus differ in the degree to which they determine
definite expectations. To say ‘something is being referred to’
does not give rise to definite expectations, does not allow taking
account of what is being referred to; to use ‘animal’ with no
further specification awakens certain sets of response, but they
are not particularized sufficiently to deal adequately with a

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Foundations of the Theory of Signs

specific animal; it is an improvement in the situation to use
‘man,’ as is evident in the contrast between knowing that an
animal is coming and that a man is coming; finally, the use of
‘this’ in an actual situation with the supplementary help of
bodily orientation directs behavior upon a specific object but
gives a minimum of expectations concerning the character of
what is denoted. Universal signs may have a certain importance
in allowing one to talk in general of the designata of signs with-
out having to specify the sign or designatum; the difficulty of
attempting to avoid such terms as ‘object,’ ‘entity,’ and ‘some-
thing’ shows the value of such terms for certain purposes. More
important, however, is the combination of indexical and char-
acterizing signs (as in ‘that horse runs’) since such a combina-
tion gives the definiteness of reference of the indexical sign plus
the determinateness of the expectation involved in the char-
acterizing sign. It is the complex forms of such combinations
that are dealt with formally in the sentences of logical and
mathematical systems, and to which (considered semantically)
the predicates of truth and falsity apply. This importance is
reflected in the fact that all formal systems show a differentia-
tion of two kinds of signs corresponding to indexical and char-
acterizing signs. Further, the fact that the determinateness of
expectation can be increased by the use of additional signs is
reflected in the fact that linguistic structures provide a frame-
work which permits of degrees of specification and makes clear
the sign relations involved.

To use terms suggested by M. J. Andrade, it may be said
that every sentence contains a dominant sign and certain speci-
fiers, these terms being relative to each other, since what is a
dominant sign with respect to certain specifiers may itself be
a specifier with respect to a more general dominant sign — thus
‘white’ may make the reference to horses more specific, while
‘horse’ may itself be a specifier with respect to ‘animal.’ Since
an adequate taking-account-of-something demands an indica-
tion of both its location and (relevant) properties, and since the
relevant degree of specification is obtained by a combination
of characterizing signs, a sentence capable of truth and falsity

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Linguistic Structure

involves indexical signs, a dominant characterizing sign with
possibly characterizing specifiers, and some signs to show the
relation of the indexical and characterizing signs to one another
and to the members of their own class. Hence the general
formula of such a sentence:

Dominant characterizing sign [characterizing specifiers (indexical signs)]

In such a sentence as ‘That white horse runs slowly,’ spoken
in an actual situation with indexical gestures, ‘runs’ may be
taken as the dominant sign, and ‘slowly’ as a characterizing
specifier specifies ‘runs’; ‘horse’ similarly specifies the possible
cases of ‘runs slowly,’ ‘white’ carries the specification further,
and ‘that’ in combination with the indexical gesture serves as
an indexical sign to locate the object to which the dominant
sign as now specified is to be applied. The conditions of utter-
ance might show that ‘horse’ or some other sign is to be taken
as the dominant sign, so that pragmatical considerations de-
termine what, in fact, is the dominant sign. The dominant sign
may even be more general than any which have been mentioned :
it may be a sign to show that what follows is a declaration or
a belief held with a certain degree of conviction. Instead of
the use of the indexical sign in an actual situation, characteriz-
ing signs might be so used as to inform the hearer how to supply
the indexical sign: ‘Find the horse such that . . . . ; it is that
horse to which reference is being made’; or ‘Take any horse; then
that horse ’ In case a set of objects is referred to, the refer-

ence may be to all of the set, to a portion, or to some specified
member or members; terms such as ‘all,’ ‘some,’ ‘three,’ to-
gether with indexical signs and descriptions, perform this func-
tion of indicating which of the possible denotata of a character-
izing sign are referred to. There need not be only a single indexi-
cal sign; in such a sentence as ‘ A gave B to C,’ there are three
correlates of the triadic relation to be specified by indexical
signs, either used alone or in connection with other devices.

The sign ‘to’ in the sentence ‘A gave B to C’ serves as an
occasion for stressing an important point: to have intelligible
sign combinations it is necessary to have special signs within

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Foundations of the T heory of Signs

the language in question to indicate the relation of other signs,
and such signs, being in the language in question, must be
distinguished from those signs in the language of syntactics
which designate these relations. In the English examples which
have been given, the ‘s’ in ‘runs,’ the ‘ly’ in ‘slowly,’ the position
of ‘that’ and ‘white’ with reference to the position of ‘horse,’
the positions of ‘A’ and ‘B’ before and after the dominant sign
‘gives,’ the position of ‘to’ before ‘C’ all furnish indications as
to which sign specifies which other sign, or which indexical sign
denotes which correlate of the relation, or which signs are in-
dexical signs and which are characterizing signs. Pauses, speech
melodies, and emphasis help to perform such functions in spoken
language; punctuation marks, accents, parentheses, italics, size
of letter, etc., are similar aids in written and printed languages.
Such signs within the language perform primarily a pragmatical
function, but the term ‘parenthesis’ and its implicates occur
in the metalanguage. The metalanguage must not be confused
with a language to which it refers, and in the language itself
a distinction must be made between those signs whose designata
fall outside the language and those signs which indicate the
relation of other signs.

All the distinctions which have been recognized as involved
in the functioning of language in the full semiotical sense are
reflected in the features of language which syntactics has thus
far studied. Syntactics recognizes classes of signs, such as indi-
vidual constants and variables, and predicate constants and
variables, which are the formal correlates of various kinds of
indexical and characterizing signs; the operators correspond to
class specifiers; dots, parentheses, and brackets are devices with-
in the language for indicating certain relations between the
signs; terms such as ‘sentence,’ ‘consequence,’ and ‘analytic’
are syntactical terms for designating certain kinds of sign com-
binations and relations between signs; sentential (or “propo-
sitional”) functions correspond to sign combinations lacking
certain indexical specifiers necessary for complete sentences
(“propositions”); the formation and transformation rules cor-
respond to the way in which signs are combined or derived from

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The Semantical Dimension of Semiosis

one another by actual or possible users of the language. In this
way the formalized languages studied in contemporary logic
and mathematics clearly reveal themselves to be the formal
structure of actual and possible languages of the type used in
making statements about things; at point after point they re-
flect the significant features of language in actual use. The de-
liberate neglect by the formalist of other features of language,
and the ways in which language changes, is an aid in isolating a
particular object of interest; linguistic structure. The formal
logician differs from the grammarian only in his greater interest
in the types of sentences and transformation rules operative in
the language of science. The logician’s interest needs to be sup-
plemented by the grammarian’s type of interest and by atten-
tion to sign combinations and transformations in fields other
than science if the whole domain of syntactics is to be adequate-
ly explored.

IV. Semantics

7. The Semantical Dimension of Semiosis

Semantics deals with the relation of signs to their designata
and so to the objects which they may or do denote. As in the
case of the other disciplines dealing with signs, a distinction may
be made between its pure and descriptive aspects, pure seman-
tics giving the terms and the theory necessary to talk about the
semantical dimension of semiosis, descriptive semantics being
concerned with actual instances of this dimension. The latter
type of consideration has historically taken precedence over
the former; for centuries linguists have been concerned with
the study of the conditions under which specific words were
employed, philosophical grammarians have tried to find the
correlates in nature of linguistic structures and the differentia-
tion of parts of speech, philosophical empiricists have studied
in more general terms the conditions under which a sign can be
said to have a denotatum (often in order to show that the terms
of their metaphysical opponents did not meet these conditions),
discussions of the term ‘truth’ have always involved the ques-
tion of the relation of signs to things — and yet, in spite of the

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Foundations of the Theory of Signs

length of this history, relatively little has been done in the way
of controlled experimentation or in the elaboration of a suitable
language to talk about this dimension. The experimental ap-
proach made possible by behavioristics offers great promise in
determining the actual conditions under which certain signs are
employed; the development of the language of semantics has
been furthered by recent discussions of the relation of formal
linguistic structures to their “interpretations,” by attempts
(such as those of Carnap and Reichenbach) to formulate more
sharply the doctrine of empiricism, and by the efforts of the
Polish logicians (notably Tarski) to define formally in a sys-
tematic fashion certain terms of central importance within
semantics. Nevertheless, semantics has not yet attained a clar-
ity and systematization comparable to that obtained by certain
portions of syntactics.

Upon consideration, this situation is not surprising, for a
rigorous development of semantics presupposes a relatively
highly developed syntactics. To speak of the relation of signs
to the objects they designate presupposes, in order to refer both
to signs and to objects, the language of syntactics and the
thing-language. This reliance upon syntactics is particularly
evident in discussing languages, for here a theory of formal lin-
guistic structure is indispensable. For example, the constantly
recurring question as to whether the structure of language is the
structure of nature cannot properly be discussed until the terms
‘structure’ and ‘structure of a language’ are clear; the unsatis-
factoriness of historical discussions of this question are cer-
tainly in part due to the lack of such preliminary clarification
as syntactics has today supplied.

A sign combination such as ‘ ‘Fido’ designates A’ is an in-
stance of a sentence in the language of semantics. Here ‘ ‘Fido’ ’
denotes Fido’ (i.e., the sign or the sign vehicle and not a non-
linguistic object), while ‘A’ is an indexical sign of some object
(it might be the word ‘that’ used in connection with some
directive gesture). ‘ ‘Fido’ ’ is thus a term in the metalanguage
denoting the sign ‘Fido’ in the object language; ‘A’ is a term in
the thing-language denoting a thing. ‘Designates’ is a semanti-

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The Semantical Dimension of Semiosis

cal term, since it is a characterizing sign designating a relation
between a sign and an object. Semantics presupposes syntac-
tics but abstracts from pragmatics; whether dealing with simple
signs or complex ones (such as a whole mathematical system),
semantics limits itself to the semantical dimension of semiosis.

In considering this dimension, the most important addition
to the preceding account lies in the term ‘ semantical rule.’ Un-
like the formation and transformation rules, which deal with
certain sign combinations and their relations, ‘semantical rule’
designates within semiotic a rule which determines under which
conditions a sign is applicable to an object or situation; such
rules correlate signs and situations denotable by the signs.
A sign denotes whatever conforms to the conditions laid down
in the semantical rule, while the rule itself states the conditions
of designation and so determines the designatum (the class or
kind of denotata). The importance of such rules has been
stressed by Reichenbach as definitions of co-ordination, and
by Ajdukiewicz as empirical rules of meaning; the latter
insists that such rules are necessary to characterize uniquely a
language, since with different semantical rules two persons
might share the same formal linguistic structure and yet be
unable to understand each other. Thus, in addition to the syn-
tactical rules, the characterization of a language requires the
statement of the semantical rules governing the sign vehicles
singly and in combination (it will later become clear that the full
semiotical characterization of a language demands in addition
the statement of what will be called pragmatical rules).

Rules for the use of sign vehicles are not ordinarily formulated
by the users of a language, or are only partially formulated;
they exist rather as habits of behavior, so that only certain
sign combinations in fact occur, only certain sign combinations
are derived from others, and only certain signs are applied to
certain situations. The explicit formulation of rules for a given
language requires a higher order of symbolization and is a task
of descriptive semiotic; it would be a very difficult task to for-
mulate, for instance, the rules of English usage, as may be seen
if one even tries to formulate the conditions under which the

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Foundations of the Theory of Signs

words ‘this’ and ‘that’ are used. It is natural, therefore, that
attention has been chiefly devoted to fragments of the common
languages and to languages which have been deliberately con-
structed.

A sign has a semantical dimension in so far as there are
semantical rules (whether formulated or not is irrelevant) which
determine its applicability to certain situations under certain
conditions. If this usage is stated in terms of other signs, the
general formula is as follows: The sign vehicle V designates the
conditions a, h, c . . . . under which it is applicable. The state-
ment of those conditions gives the semantical rule for ‘x.’ When
any object or situation fulfils the required conditions, then it is
denoted by ‘x.’ The sign vehicle itself is simply one object,
and its denotation of other objects resides solely in the fact that
there are rules of usage which correlate the two sets of objects.

The semantical rule for an indexical sign such as pointing is
simple: the sign designates at any instant what is pointed at.
In general, an indexical sign designates what it directs attention
to. An indexical sign does not characterize what it denotes
(except to indicate roughly the space-time co-ordinates) and
need not be similar to what it denotes. A characterizing sign
characterizes that which it can denote. Such a sign may do this
by exhibiting in itself the properties an object must have to be
denoted by it, and in this case the characterizing sign is an
icon; if this is not so, the characterizing sign may be called a
symbol. A photograph, a star chart, a model, a chemical dia-
gram are icons, while the word ‘photograph,’ the names of the
stars and of chemical elements are symbols. A “concept” may
be regarded as a semantical rule determining the use of char-
acterizing signs. The semantical rule for the use of icons is that
they denote those objects which have the characteristics which
they themselves have — or more usually a certain specified set
of their characteristics. The semantical rule for the use of sym-
bols must be stated in terms of other symbols whose rules or
usages are not in question, or by pointing out specific objects
which serve as models (and so as icons), the symbol in question
then being employed to denote objects similar to the models.

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The Semantical Dimension of Semiosis

It is the fact that the semantical rule of usage for a symbol can
be stated in terms of other symbols which makes possible (to
use Carnap’s term) the reduction of one scientific term to others
(or, better, the construction of one term upon others) and thus
the systematization of the language of science. It is because in-
dexical signs are indispensable (for symbols ultimately involve
icons, and icons indices) that such a program of systematization
as physicalism proposes is forced to terminate the process of
reduction by the acceptance of certain signs as primitive terms
whose semantical rules of usage, determining their applicability
to things indicated by indices, must be taken for granted but
cannot, within that particular systematization, be stated.

The semantical rule for the use of a sentence involves refer-
ence to the semantical rules of the component sign vehicles. A
sentence is a complex sign to the effect that the designatum of
the indexical component is also a designatum of the component
which is a characterizing sign. The designatum of a sentence is
thus the designatum-of-an-indexical-sign-as-the-designatum-of-
a-characterizing-sign ; when the situation conforms to the se-
mantical rule of a sentence, the situation is a denotatum of that
sentence (and the sentence may then be said to be true of that
situation).

The difference between indices, icons, and symbols (sentences
being compounds of other signs) is accounted for by different
kinds of semantical rules. Things may be regarded as the desig-
nata of indexical signs, properties as the designata of one-place
characterizing signs, relations as the designata of two- (or more)
place characterizing signs, facts or state of affairs as designata
of sentences, and entities or beings as the designata of all signs
whatsoever.

It is because a sign may have a rule of usage to determine
what it can denote without actually being so used that there
can be signs which in fact denote nothing or have null denota-
tion. It was previously noted that the very notion of sign in-
volves that of designatum, but not that there be actually ex-
isting objects which are denoted. The designatum of a sign is
such things which the sign can denote, i.e., such objects or situa-

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Foundations of the Theory of Signs

tions which according to the semantical rule of usage could be
correlated to the sign vehicle by the semantical relation of
denotation. It is now clear, as formerly it could not be, that the
statement of what would constitute a designatum of a certain
sign must itself make use of terms with syntactical relations,
since the semantical rule of usage states what the sign in question
signifies by using the sign in relation to other signs. ‘Desig-
natum’ is clearly a semiotical term, while the question as to
whether there are objects of such and such a kind is a question
to be answered by considerations which go beyond semiotic.
The failure to keep separate the statements of semiotic from
thing-sentences has led to many pseudo thing-sentences. To
say that there is a “realm of subsistence” in addition to, but on
a par with, the realm of existences, since “When we think, we
must think about something,” is a quasi-semantical statement:
it seems to speak about the world in the same way that physics
does, but actually the statement is an ambiguous form of a
semantical sentence, namely, the sentence that for every sign
that can denote something a semantical rule of usage can be
formulated which will state the conditions under which the sign
is applicable. This statement, analytically correct within se-
mantics, does not in any sense imply that there are objects
denoted by every such sign — objects which are “subsistential”
when not existential.

8. Linguistic and Nonlinguistic Structures

One of the oldest and most persistent theories is that lan-
guages mirror (correspond with, reflect, are isomorphic with)
the realm of nonlinguistic objects. In the classical tradition it
was often held that this mirroring was threefold: thought re-
flected the properties of objects; and spoken language, com-
posed of sounds which had been given a representative function
by mind, in turn reflected the kinds and relations of mental
phenomena and so the realm of nonmental objects.

It goes without saying that such a persistent tradition as lies
behind the doctrine in question must have something to com-
mend it; it is, nevertheless, significant that this tradition has

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Linguistic and Nonlinguistic Structures

progressively weakened and has even been repudiated by some
of its most vigorous former champions. What light can the gen-
eral semiotical point of view throw on the situation? In at-
tempting to answer this question, it will be seen that the heart
of the matter lies in the fact that the only relevant correlation
which exists between signs and other objects is that established
by semantical rules.

It seems plausible that the excesses and difficulties of the at-
tempt to find a complete semantical correlation between linguis-
tic signs and other objects lies in the neglect or oversimplifica-
tion of the syntactical and pragmatical dimensions of semiosis.
It has been noted that the very possibility of language requires
that there be some special signs to indicate the syntactical rela-
tions of other signs in the language. Examples of such signs are
pauses, intonations, order of signs, prepositions, affixes, suf-
fixes, etc. Such signs function predominantly in the syntactical
and pragmatical dimensions; in so far as they have a semantical
dimension, they denote sign vehicles and not nonlinguistic ob-
jects. It need not be denied that such signs might help to estab-
lish some kind of isomorphism between the remaining signs and
nonlinguistic objects, for such isomorphism might be much
more complicated than the relation of a model to that of which
it is a model. Spatial relations of signs might not correspond to
spatial relations between things, but there might be a corre-
lating relation such that for every spatial relation between signs
there holds some other relation between the objects denoted by
the signs. Such possibilities are open to investigation and
should be specifically explored; if they do not hold for all signs,
they may hold for certain of them, namely, for such as have
semantical rules correlating them with nonlinguistic situations.
Nevertheless, the defenders of isomorphism have not showm
that such is the case, or that such must be the case if language
is to be possible.

The unconvincingness of the general theory increases if notice
is taken of such signs as ‘all,’ ‘some,’ ‘the,’ ‘not,’ ‘point at in-
finity,’ ‘-1.’ The first three terms indicate how much of the
class determined by some characterizing sign is to be taken ac-

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Foundations of the Theory of Signs

count of. The term ‘not’ is primarily of practical importance,
since it allows reference to something other than what is speci-
fically referred to without specifying what the other is. So
clarified semantically, the practical importance of the term is
obvious, but it is not theoretically necessary in a language, and
certainly no existential negative facts’’ need be invoked to
correspond to it. The mathematical terms mentioned are com-
monly regarded as signs added to the language so that certain
operations, otherwise impossible in certain cases, are always
possible, and certain formulas, otherwise needing qualification,
can be stated in their full generality.

There are also many signs in a common language which indi-
cate the reaction of the user of the signs to the situation being
described (as the ‘fortunately’ in ‘Fortunately, he came’), or
even to the signs he is himself using in the description (as in
expressing his degree of confidence in a statement) . Such terms
within discourse have a semantical dimension only at a higher
level of semiosis, since the pragmatical dimension of a process
of semiosis is not denoted in that process but only in one of a
higher level. As in the case of the predominantly syntactical
features of a language, the predominantly pragmatical features
should not be confounded with those elements correlated by
means of semantical rules with the nonlinguistic objects which
are being denoted, dhe traditional versions of isomorphism
failed to distinguish the various dimensions of semiosis and the
various levels of languages and designata. To what extent some
qualified version of the thesis may be held can only be deter-
mined after it is formulated. But it is clear that, when a lan-
guage as a whole is considered, its syntactical structure is a
function of both pragmatic and empirical considerations and is
not a bare mirroring of nature considered in abstraction from
the users of the language.

The main point of the discussion is not to deny that all the
signs in a language may have designata and so a semantical
dimension but rather to call attention to the fact that the
designata of signs in a given discourse (and so the objects de-
noted, if there are such) do not stand at the same level: the

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The Pragmatical Dimension of Semiosis

designata of some signs must be sought at the level of semiotic
rather than at the level of the thing-language itself; in the given
discourse such signs simply indicate (but do not designate)
relations of the other signs to one another or to the interpreter —
in Scholastic terms they bring something of material and simple
supposition into the functioning of terms in personal supposi-
tion. The strata of signs are as complex and as difficult to un-
ravel as geological strata; the scientific and psychological ef-
fects of unraveling them may be as great in the former case as
it has been in the latter.

So much for a bare indication of the field of semantics. The
precise analysis of semantical terms, their formal systematiza-
tion, and the question of the applicability of semantics to do-
mains other than the language of science (for instance, to
aesthetic signs) obviously are not possible in an introductory
account. If pragmatical factors have appeared frequently in
pages belonging to semantics, it is because the current recogni-
tion that syntactics must be supplemented by semantics has
not been so commonly extended to the recognition that se-
mantics must in turn be supplemented by pragmatics. It is true
that syntactics and semantics, singly and jointly, are capable
of a relatively high degree of autonomy. But syntactical and
semantical rules are only the verbal formulations within se-
miotic of what in any concrete case of semiosis are habits of
sign usage by actual users of signs. ‘Rules of sign usage,’ like
‘sign’ itself, is a semiotical term and cannot be stated syntacti-
cally or semantically.

V. Pragmatics

9. The Pragmatical Dimension of Semiosis

The term ‘pragmatics’ has obviously been coined with refer-
ence to the term ‘pragmatism.’ It is a plausible view that the
permanent significance of pragmatism lies in the fact that it has
directed attention more closely to the relation of signs to their
users than had previously been done and has assessed more pro-
foundly than ever before the relevance of this relation in under-
standing intellectual activities. The term ‘pragmatics’ helps to

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Foundations of the Theory of Signs

signalize the significance of the achievements of Peirce, James,
Dewey, and Mead within the field of semiotic. At the same
time, pragmatics as a specifically semiotical term must receive
its own formulation. By ‘pragmatics’ is designated the science
of the relation of signs to their interpreters. ‘Pragmatics’ must
then be distinguished from ‘pragmatism,’ and ‘pragmatical’
from ‘pragmatic.’ Since most, if not all, signs have as their in-
terpreters living organisms, it is a sufficiently accurate char-
acterization of pragmatics to say that it deals with the biotic
aspects of semiosis, that is, with all the psychological, biologi-
cal, and sociological phenomena which occur in the functioning
of signs. Pragmatics, too, has its pure and descriptive aspects;
the first arises out of the attempt to develop a language in which
to talk about the pragmatical dimension of semiosis; the latter
is concerned with the application of this language to specific
cases.

Historically, rhetoric may be regarded as an early and re-
stricted form of pragmatics, and the pragmatical aspect of sci-
ence has been a recurrent theme among the expositors and in-
terpreters of experimental science. Reference to interpreter and
interpretation is common in the classical definition of signs.
Aristotle, in the De inter pretatione, speaks of words as conven-
tional signs of thoughts which all men have in common. His
words contain the basis of the theory which became tradition-
al: The interpreter of the sign is the mind; the interpretant is
a thought or concept; these thoughts or concepts are common to
all men and arise from the apprehension by mind of objects and
their properties; uttered words are then given by the mind the
function of directly representing these concepts and indirectly
the corresponding things; the sounds chosen for this purpose are
arbitrary and vary from social group to social group; the rela-
tions between the sounds are not arbitrary but correspond to the
relations of concepts and so of things. In this way throughout
much of its history the theory of signs was linked with a par-
ticular theory of thought and mind, so much so that logic, which
has always been affected by current theories of signs, was often
conceived as dealing with concepts— a view made precise in the

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The Pragmatical Dimension of S emiosis

Scholastic doctrine of logical terms as terms of second inten-
tion. Even Leibniz’ insistence upon the empirical study of the
sign vehicle as determined by rule was not a repudiation of the
dominant tradition but merely an insistence that in this way
a new and better technique could be obtained for analyzing con-
cepts than by the attempt to inspect thought directly.

In the course of time most of the tenets of this traditional
version of pragmatics were questioned, and today they would
be accepted only with serious qualifications. The change in
point of view has been most rapid as a result of the implications
for psychology of the Darwinian biology — implications which
received an early interpretation in pragmatism. Charles S.
Peirce, whose work is second to none in the history of semiotic,
came to the conclusion that in the end the interpretant of a
symbol must reside in a habit and not in the immediate physio-
logical reaction which the sign vehicle evoked or in the at-
tendant images or emotions — a doctrine which prepared the
way for the contemporary emphasis on rules of usage. William
James stressed the view that a concept was not an entity but a
way in which certain perceptual data functioned representa-
tively and that such “mental” functioning, instead of being a
bare contemplation of the world, is a highly selective process in
which the organism gets indications as to how to act with
reference to the world in order to satisfy its needs or interests.
George H. Mead was especially concerned with the behavior
involved in the functioning of linguistic signs and with the so-
cial context in which such signs arise and function. His work is
the most important study from the point of view of pragmatism
of these aspects of semiosis. John Dewey’s instrumentalism is
the generalized version of the pragmatists’ emphasis upon the
instrumental functioning of signs or “ideas.”

If from pragmatism is abstracted the features of particular
interest to pragmatics, the result may be formulated somewhat
as follows : The interpreter of a sign is an organism ; the inter-
pretant is the habit of the organism to respond, because of the
sign vehicle, to absent objects which are relevant to a present
problematic situation as if they were present. In virtue of

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Foundations of the Theory of Signs

semiosis an organism takes account of relevant properties of
absent objects, or unobserved properties of objects which are
present, and in this lies the general instrumental significance
of ideas. Given the sign vehicle as an object of response, the
organism expects a situation of such and such a kind and, on
the basis of this expectation, can partially prepare itself in ad-
vance for what may develop. The response to things through
the intermediacy of signs is thus biologically a continuation of
the same process in which the distance senses have taken pre-
cedence over the contact senses in the control of conduct in
higher animal forms; such animals through sight, hearing, and
smell are already responding to distant parts of the environ-
ment through certain properties of objects functioning as signs
of other properties. This process of taking account of a con-
stantly more remote environment is simply continued in the
complex processes of semiosis made possible by language, the
object taken account of no longer needing to be perceptually
present.

With this orientation, certain of the terms which have pre-
viously been used appear in a new light. The relation of a
sign vehicle to its designatum is the actual taking-account in the
conduct of the interpreter of a class of things in virtue of the
response to the sign vehicle, and what are so taken account of
are designata. The semantical rule has as its correlate in the
pragmatical dimension the habit of the interpreter to use the
sign vehicle under certain circumstances and, conversely, to
expect such and such to be the case when the sign is used. The
formation and transformation rules correspond to the actual
sign combinations and transitions which the interpreter uses, or
to stipulations for the use of signs which he lays down for him-
self in the same way in which he attempts to control deliberately
other modes of behavior with reference to persons and things.
Considered fron the point of view of pragmatics, a linguistic
structure is a system of behavior: corresponding to analyti-
cal sentences are the relations between sign responses to the
more inclusive sign responses of which they are segments; cor-
responding to synthetical sentences are those relations between

no

The Pragmatical Dimension of Semiosis

sign responses which are not relations of part to whole. The
indexical signs (or their substitutes) in a sign combination direct
the attention of the interpreter to parts of the environment; the
dominant characterizing sign determines some general response
(expectation) to these parts; the characterizing specifiers de-
limit the general expectation, the degree of specification and the
choice of the dominant sign being determined with respect to
the problem at hand. If the indexical and characterizing func-
tions are both performed, the interpreter is judging and the sign
combination is a judgment (corresponding to the sentence of
syntactics and the statement or proposition of semantics). To
the degree that what is expected is found as expected the sign is
confirmed; expectations are, in general, only partially con-
firmed; there may be, in addition, various degrees of indirect
confirmation that what is indexically referred to has the proper-
ties it was expected to have. In general, from the point of view
of behavior, signs are “true” in so far as they correctly deter-
mine the expectations of their users, and so release more fully
the behavior which is implicitly aroused in the expectation or
interpretation.

Such statements go somewhat beyond pragmatics proper into
the strictly semiotical question as to the interrelation of the
dimensions— a topic yet to be specifically discussed. Pragmatics
itself would attempt to develop terms appropriate to the study
of the relation of signs to their users and to order systematically
the results which come from the study of this dimension of
semiosis. Such terms as ‘interpreter,’ ‘interpretant,’ ‘conven-
tion’ (when applied to signs), ‘taking-account-of’ (when a func-
tion of signs), ‘verification,’ and ‘understands’ are terms of prag-
matics, while many strictly semiotical terms such as ‘sign,’
‘language,’ ‘truth,’ and ‘knowledge’ have important pragmati-
cal components. In a systematic presentation of semiotic, prag-
matics presupposes both syntactics and semantics, as the latter
in turn presupposes the former, for to discuss adequately the
relation of signs to their interpreters requires knowledge of the
relation of signs to one another and to those things to which
they refer their interpreters. The unique elements within prag-

1 1 1

Foundations of the Theory of Signs

matics would be found in those terms which, while not strictly
semiotical, cannot be defined in syntactics or semantics; in the
clarification of the pragmatical aspect of various semiotical
terms; and in the statement of what psychologically, biological-
ly, and sociologically is involved in the occurrence of signs.
Attention may now be turned to some aspects of this latter
problem.

10. Individual and Social Factors in Semiosis

The topic in question may be approached, and a possible
objection forestalled, by asking why there is any need of adding
pragmatics to semantics; since semantics deals with the relation
of signs to objects, and since interpreters and their responses are
natural objects studied by the empirical sciences, it would seem
as if the relation of signs to interpreters fell within semantics.
The confusion here arises from the failure to distinguish levels
of symbolization and to separate— in the use of ‘object’—
semiotical from nonsemiotical terms. Everything that is desig-
natable is subject matter for a (in principle) unified science,
and in this sense all the semiotical sciences are parts of unified
science. When descriptive statements are made about any di-
mension of semiosis, the statements are in the semantical dimen-
sion of a higher level of semiosis and so are not necessarily of the
same dimension that is being studied. Statements in pragmatics
about the pragmatical dimension of specific signs are function-
ing predominantly in the semantical dimension. The fact that
the pragmatical dimension becomes a designatum for a higher-
level process of description does not signify that the interpretant
of a sign at any given level is a designatum of that par-
ticular sign. The interpretant of a sign is the habit in virtue
of which the sign vehicle can be said to designate certain kinds
of objects or situations; as the method of determining the set
of objects the sign in question designates, it is not itself a mem-
ber of that set. Even the language of a unified science which
would contain an account of the pragmatical dimension would
not at the moment of use denote its own pragmatical dimen-
sion, though at a higher level of usage the account given of the

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Individual and Social Factors in Semiosis

pragmatical dimension may be found applicable to the prag-
matical dimension of the lower level. Since the pragmatical di-
mension is involved in the very existence of the relation of
designation, it cannot itself be put within the semantical dimen-
sion. Semantics does not deal with all the relations of signs to
objects but, as a semiotical science, deals with the relation of
signs to their designata; pragmatics, dealing with another rela-
tion of signs, cannot be put within semantics alone or in com-
bination with syntactics. This conclusion is completely inde-
pendent of the relation of physical and biological existences; the
distinction of the semantical and pragmatical dimensions is a
semiotical distinction and has nothing to do with the relation of
biology and physics.

The point can perhaps be made sharper if we introduce the
term ‘ pragmatical rule.’ Syntactical rules determine the sign rela-
tions between sign vehicles; semantical rules correlate sign ve-
hicles with other objects; pragmatical rules state the conditions
in the interpreters under which the sign vehicle is a sign. Any
rule when actually in use operates as a type of behavior, and in
this sense there is a pragmatical component in all rules. But in
some languages there are sign vehicles governed by rules over
and above any syntactical or semantical rules which may govern
those sign vehicles, and such rules are pragmatical rules. Inter-
jections such as ‘Oh!,’ commands such as ‘Come here!,’ value
terms such as ‘fortunately,’ expressions such as ‘Good morn-
ing!,’ and various rhetorical and poetical devices occur only
under certain definite conditions in the users of the language;
they may be said to express such conditions, but they do not
denote them at the level of semiosis in which they are actual-
ly employed in common discourse. The statement of the condi-
tions under which terms are used, in so far as these cannot be
formulated in terms of syntactical and semantical rules, consti-
tutes the pragmatical rules for the terms in question.

The full characterization of a language may now be given:
A language in the full semiotical sense of the term is any inter-
subjective set of sign vehicles whose usage is determined by syntac-
tical, semantical, and pragmatical rules.

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Foundations of the Theory of Signs

Interpretation becomes especially complex, and the individ-
ual and social results especially important, in the case of linguis-
tic signs. In terms of pragmatics, a linguistic sign is used in
combination with other signs by the members of a social group;
a language is a social system of signs mediating the responses of
members of a community to one another and to their environ-
ment. To understand a language is to employ only those sign
combinations and transformations not prohibited by the usages
of the social group in question, to denote objects and situations
as do the members of this group, to have the expectations which
the others have when certain sign vehicles are employed, and to
express one’s own states as others do— in short, to understand a
language or to use it correctly is to follow the rules of usage

(syntactical, semantical, and pragmatical) current in the given
social community.

There is a further stipulation often made in connection with
the linguistic sign: it must be capable of voluntary use for the
function of communicating. Such terms as Voluntary’ and
communication’ need more extended analysis than is here pos-
sible, but Mead’s account, in Mind, Self, and Society, of the
inguistic sign (which he calls the significant symbol) seems to
cover the point intended in this stipulation. According to
Mead, the primary phenomenon out of which language in the
full human sense emerges is the gesture, especially the vocal
gesture. The gesture sign (such as a dog’s snarl) differs from
such a nongestural sign as thunder in the fact that the sign
vehicle is an early phase of a social act and the designatum a
later phase of this act (in this case the attack by the dog). Here
one organism prepares itself for what another organism-the
dog is to do by responding to certain acts of the latter or-
ganism as signs; m the case in question the snarl is the sign, the
attack is the designatum, the animal being attacked is the inter-
preter, and the preparatory response of the interpreter is the
interpretant. The utility of such gesture signs is limited by the
fact that the sign is not a sign to the producer as it is to the
receiver: the dog which snarls does not respond to his snarl as

Individual and Social Factors in Semiosis

does his opponent; the sign is not held in common and so is
not a linguistic sign.

On the other hand, the important characteristic of the vocal
gesture lies precisely in the fact that the emitter of the sound
himself hears the sound just as others do. When such sounds
become connected with social acts (such as a fight, a game, a
festival), the various participants in the act have through this
common sign, and in spite of their differentiated functions with-
in the act, a common designatum. Each participant in the com-
mon activity stimulates himself by his vocal gestures as he
stimulates others. Couple this with what Mead termed the tem-
poral dimension of the nervous system (namely, an earlier but
more slowly aroused activity may initiate a later and more
rapid activity which in turn furthers or checks the complete
arousal of the first activity), and one obtains a possible explana-
tion of how linguistic signs serve for voluntary communication.
To use one of Mead’s frequent examples, we may consider the
situation of a person noticing smoke in a crowded theater.
Smoke is a nongestural sign of fire, and its perception calls out
to some degree responses appropriate to fire. But further, the
spoken word ‘fire,’ as a response which is connected with a whole
set of responses to fire, tends to be uttered. Since this is a lin-
guistic sign, the utterer begins to respond toward this tendency
toward utterance as other members of his social group would
respond — to run toward an exit, to push, and perhaps trample
over, others blocking the way, etc. But the individual, in virtue
of certain fundamental attitudes, will respond either favorably
or unfavorably to these tendencies and will thus check or
further the tendency to say ‘Fire!’

In such a case it is said that the man “knew what he was
about,” that he “deliberately used (or did not use) a certain
sign to communicate to others,” that he “took account of
others.” Mead would generalize from such common usages:
from his point of view “to have a mind” or “to be conscious of
something” was equivalent to “using linguistic signs.” It is
through such signs that the individual is able to act in the light
of consequences to himself and to others, and so to gain a certain

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Foundations of the Theory of Signs

amount of control over his own behavior; the presentation of
possible consequences of action through the production of lin-
guistic signs becomes a factor in the release or inhibition of the
action which has (or seems to have) such consequences. It is in
such processes that the term ‘choice’ gains its clarification— and
also whatever distinction is to be made between senders and
receivers of linguistic signs. Since the linguistic sign is socially
conditioned, Mead, from the standpoint of his social behavior-
ism, regarded the individual mind and self-conscious self as
appearing in a social process when objective gestural communi-
cation becomes internalized in the individual through the func-
tioning of vocal gestures. Thus it is through the achievements of
the community, made available to the individual by his par-
ticipation in the common language, that the individual is able
to gain a self and mind and to utilize those achievements in the
furtherance of his interests. The community benefits at the
same time in that its members are now able to control their
behavior in the light of the consequences of this behavior to
others and to make available to the whole community their own
experiences and achievements. At these complex levels of se-
miosis, the sign reveals itself as the main agency in the develop-
ment of individual freedom and social integration.

11. Pragmatic Use and Abuse of Signs

When a sign produced or used by an interpreter is employed
as a means of gaining information about the interpreter, the
point of view taken is that of a higher process of semiosis,
namely, that of descriptive pragmatics. Psychoanalysis among
the psychologies, pragmatism among the philosophies, and now
the sociology of knowledge among the social sciences have made
this way of looking at signs a common possession of educated
persons. Newspaper statements, political creeds, and philosoph-
ical systems are increasingly being looked at in terms of the
interests which are expressed and served by the production and
use of the signs in question. The psychoanalyist is interested in
dreams for the light they throw upon the dreamer; the so-
ciologist of knowledge is interested in the social conditions under

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Pragmatic Use and Abuse of Signs

which doctrines and systems of doctrine are current. In neither
case is the interest in the question whether the dreams or doc-
trines are true in the semantical sense of the term, i.e., whether
there are situations which the dreams and the doctrines may be
said to denote. Such studies, together with many others, have
confirmed over a wide range the general thesis of pragmatism
as to the instrumental character of ideas.

Any sign whatever may be looked at in terms of the psycho-
logical, biological, and sociological conditions of its usage. The
sign expresses but does not denote its own interpretant; only at
a higher level is the relation of the sign to the interpreter itself
made a matter for designation. When this is done and a cor-
relation found, the sign becomes of individual and social diag-
nostic value, and so a new sign at a higher level of semiosis.
Signs as well as things not signs can become diagnostic signs:
the fact that a patient has a fever shows certain things about his
condition; equally well the fact that a certain sign is used by
someone expresses that person’s condition, for the interpretant
of the sign is part of the conduct of the individual. In such
cases the same sign vehicle may be functioning as two signs,
interpreted by the patient as referring to its denotata and by
the diagnostician as referring to the interpretant involved in
the patient’s sign.

Not only may all signs be regarded in terms of pragmatics,
but it is also perfectly legitimate for certain purposes to use
signs simply in order to produce certain processes of interpreta-
tion, regardless of whether there are objects denoted by the
signs or even whether the sign combinations are formally pos-
sible in terms of the formation and transformation rules of the
language in which the sign vehicles in question are normally
used. Some logicians seem to have a generalized fear of con-
tradictions, forgetting that, while contradictions frustrate the
normal uses of deduction, they may be perfectly compatible
with other interests. Even linguistic signs have many other uses
than that of communicating confirmable propositions: they
may be used in many ways to control the behavior of one’s self
or of other users of the sign by the production of certain inter-

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Foundations of the T heory of Signs

pretants. Commands, questions, entreaties, and exhortations
are of this sort, and to a large degree the signs used in the
literary, pictorial, and plastic arts. For aesthetic and practical
purposes the effective use of signs may require rather extensive
variations from the use of the same sign vehicles most effective
for the purposes of science. Scientists and logicians may be ex-
cused if they judge signs in terms of their own purposes, but the
semiotician must be interested in all the dimensions and all the
uses of signs; the syntactics, semantics, and pragmatics of the
signs used in literature, the arts, morality, religion, and in value
judgments generally are as much his concern as studies of the
signs used in science. In one case as in the other the usage of
the sign vehicle varies with the purpose to be served.

If semiotic must defend the legitimacy for certain purposes
of a concern for the effect of the sign on those who will interpret
it, it must equally set itself the task of unmasking confusion of
these various purposes which signs serve, whether the confusion
be unintentional or deliberate. Just as properly syntactical or
semantical statements may masquerade in a form which causes
them to appear as statements about nonlinguistic objects, so may
pragmatical statements thus masquerade; they then become,
as quasi-pragmatical statements, one particular form of pseudo
thing-sentences. In the clearly dishonest cases a purpose is ac-
complished by giving the signs employed the characteristics of
statements with syntactical or semantical dimensions, so that
they seem to be rationally demonstrated or empirically sup-
ported when in fact they are neither. An intellectual intuition,
superior to scientific method, may be invoked to bolster up the
validity of what is apparently affirmed. The masquerading may
not be of one dimension in terms of others but within the prag-
matical dimension itself; a purpose that cannot fully stand the
light of scrutiny expresses itself in a form suitable to other pur-
poses: aggressive acts of individuals and social groups often
drape themselves in the mantle of morality, and the declared
purpose is often not the real one. A peculiarly intellectualistic
justification of dishonesty in the use of signs is to deny that
truth has any other component than the pragmatical, so that

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Pragmatic Use and Abuse of Signs

any sign which furthers the interest of the user is said to be true.
In terms of the preceding analysis it should be clear that ‘truth’
as commonly used is a semiotical term and cannot be used in
terms of any one dimension unless this usage is explicitly
adopted. Those who like to believe that ‘truth’ is a strictly
pragmatical term often refer to the pragmatists in support of
their view and naturally fail to note (or to state) that prag-
matism as a continuation of empiricism is a generalization of
scientific method for philosophical purposes and could not hold
that the factors in the common usage of the term ‘truth’ to
which attention was being drawn rendered nonexistent previ-
ously recognized factors. Certain of James’s statements taken
in isolation might seem to justify this perversion of pragmatism,
but no one can seriously study James without seeing that his
doctrine of truth was in principle semiotical: he clearly recog-
nized the need of bringing in formal, empirical, and pragmatic
factors; his main difficulty was in integrating these factors,
since he lacked the base which a developed theory of signs pro-
vides. Dewey has specifically denied the imputed identification
of truth and utility. Pragmatism has insisted upon the prag-
matical and pragmatic aspects of truth; the perversion of this
doctrine into the view that truth has only such aspects is an
interesting case of how the results of a scientific analysis may
be distorted to lend credibility to quasi-pragmatical statements.

Pseudo thing-sentences of the quasi-pragmatical type are not
for the most part deliberate deception of others by the use of
signs but cases of unconscious self-deception. Thus a philoso-
pher with certain imperious needs may from a relatively small
empirical base construct an elaborate sign system, perhaps in
mathematical form, and yet the great majority of terms may be
without semantical rules of usage; the impression that the sys-
tem is about the world, and perhaps superior in truth to science,
comes from the confusion of analytic and synthetic sentences
and from the illusion that the congenial attitudes evoked by the
signs constitute semantical rules. A somewhat similar manifes-
tation is found in mythology, but without the evident influence
of scientific types of expression.

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Foundations of the Theory of Signs

A particularly interesting aberration of the semiotical proc-
esses takes place in certain phenomena studied by psycho-
pathology. Signs normally take the place of objects they desig-
nate only to a limited extent; but if for various reasons inter-
ests cannot be satisfied in the objects themselves, the signs come
more and more to take the place of the object. In the aesthetic
sign this development is already evident, but the interpreter
does not actually confuse the sign with the object it designates:
the described or painted man is called a man, to be sure, but
with more or less clear recognition of the sign status — it is only
a painted or described man. In the magical use of signs the dis-
tinction is less clearly made; operations on the sign vehicle take
the place of operations on the more elusive object. In certain
kinds of insanity the distinction between the designatum and
denotata vanishes; the troublesome world of existences is
pushed aside, and the frustrated interests get what satisfaction
they can in the domain of signs, oblivious in varying degrees to
the restrictions of consistency and verifiability imposed by the
syntactical and semantical dimensions. The field of psychopa-
thology offers great opportunities for applications of, and con-
tributions to, semiotic. A number of workers in this field have
already recognized the key place which the concept of sign
holds. If, following the lead of the pragmatist, mental phe-
nomena be equated with sign responses, consciousness with ref-
erence by signs, and rational (or “free”) behavior with the con-
trol of conduct in terms of foreseen consequences made available
by signs, then psychology and the social sciences may recognize
what is distinctive in their tasks and at the same time see their
place within a unified science. Indeed, it does not seem fantastic
to believe that the concept of sign may prove as fundamental to
the sciences of man as the concept of atom has been for the
physical sciences or the concept of cell for the biological sci-
ences.

120

Meaning

VI. The Unity of Semiotic

12. Meaning

We have been studying certain features of the phenomenon
of sign-functioning by making use of the abstractions involved
in distinguishing syntactics, semantics, and pragmatics — just as
biologists study anatomy, ecology, and physiology. While we
have recognized explicitly the abstractions involved and have
constantly correlated the three subdisciplines of semiotic, we
must now draw even more explicitly the unity of semiotic into
the focus of attention.

In a wide sense, any term of syntactics, semantics, or prag-
matics is a semiotical term ; in a narrow sense, only those terms
are semiotical which cannot be defined in any of the component
fields alone. In the strict sense ‘sign,’ ‘language,’ ‘semiotic,’ ‘se-
miosis,’ ‘syntactics,’ ‘truth,’ ‘knowledge,’ etc., are semiotical
terms. What of the term ‘meaning’? In the preceding discus-
sion the term ‘meaning’ has been deliberately avoided. In gen-
eral it is well to avoid this term in discussions of signs; the-
oretically, it can be dispensed with entirely and should not be
incorporated into the language of semiotic. But since the term
has had such a notorious history, and since in its consideration
certain important implications of the present account can be
made clear, the present section is devoted to its discussion.

The confusion regarding the “meaning of ‘meaning’ ” lies in
part in the failure to distinguish with sufficient clarity the di-
mension of semiosis which is under consideration, a situation
which also obtains in the confusions as to the terms ‘truth’ and
‘logic.’ In some cases ‘meaning’ refers to designata, in other
cases to denotata, at times to the interpretant, in some cases to
what a sign implicates, in some usages to the process of semiosis
as such, and often to significance or value. Similar confusions
are found in the common usages of ‘designates,’ ‘signifies,’ ‘indi-
cates,’ ‘expresses,’ and in various attempts by linguists to define
such terms as ‘sentence,’ ‘word,’ and ‘part of speech.’ The most
charitable interpretation of such confusions is to suggest that
for the major purposes which the everyday languages serve it

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has not been necessary to denote with precision the various
factors in semiosis — the process is merely referred to in a vague
way by the term ‘meaning.’ When, however, such vague usages
are taken over into domains where an understanding of semiosis
is important, then confusion results. It then becomes necessary
to either abandon the term ‘meaning’ or to devise ways to make
clear the usage in question. Semiotic does not rest upon a theory
of “meaning”; the term ‘meaning’ is rather to be clarified in
terms of semiotic.

Another factor in the confusion is a psychological-linguistic
one : men in general find it difficult to think clearly about com-
plex functional and relational processes, a situation reflected in
the prevalence of certain linguistic forms. Action centers
around handling things with properties, and the fact that these
things and properties appear only in complex contexts is a
much later and more difficult realization. Hence the naturalness
of what Whitehead has called the fallacy of simple location. In
the present case this takes the form of looking for meanings as
one would look for marbles: a meaning is considered as one
thing among other things, a definite something definitely lo-
cated somewhere. This may be sought for in the designatum,
which thus becomes transformed in certain varieties of- “real-
ism” into a special kind of object— a “Platonic idea” inhabiting
the “realm of subsistence,” perhaps grasped by a special faculty
for intuiting “essences”; or it may be sought for in the inter-
pretant, which then becomes transformed in conceptualism
into a concept or idea inhabiting a special domain of mental
entities whose relation to the “psychical states” of individual
interpreters becomes very difficult to state; or in desperation
caused by contemplation of the previous alternatives it may be
sought in the sign vehicle — though historically few if any “nom-
inalists” have held this position. As a matter of fact, none of
these positions has proved satisfactory and none of them is
demanded. As semiotical terms, neither ‘sign vehicle,’ ‘desig-
natum,’ nor ‘interpretant’ can be defined without reference
to one another; hence they do not stand for isolated existences
but for things or properties of things in certain specifiable func-

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Meaning

tional relations to other things or properties. A “psychical
state,” or even a response, is not as such an interpretant but
becomes such only in so far as it is a “taking-account-of-some-
thing” evoked by a sign vehicle. No object is as such a deno-
tatum but becomes one in so far as it is a member of the class
of objects designatable by some sign vehicle in virtue of the
semantical rule for that sign vehicle. Nothing is intrinsically a
sign or a sign vehicle but becomes such only in so far as it per-
mits something to take account of something through its media-
tion. Meanings are not to be located as existences at any place
in the process of semiosis but are to be characterized in terms of
this process as a whole. ‘Meaning’ is a semiotical term and not
a term in the thing-language ; to say that there are meanings in
nature is not to affirm that there is class of entities on a par with
trees, rocks, organisms, and colors, but that such objects and
properties function within processes of semiosis.

This formulation also avoids another persistent stumbling
block, namely, the belief that meaning is in principle personal,
private, or subjective. Such a view historically owes much to
the assimilation of the conceptualistic position within an as-
sociational psychology which itself uncritically accepted the
current metaphysical view of the subjectivity of experience.
Persons such as Ockham and Locke were well aware of the im-
portance of habit in the functioning of signs, but as the associa-
tional psychology came more and more to reduce mental phe-
nomena to combinations of “psychical states,” and to conceive
these states as within the individual’s “mind” and only acces-
sible to that mind, meaning itself came to be considered in the
same terms. Meanings were inaccessible to observation from
without, but individuals somehow managed to communicate
these private mental states by the use of sounds, writing, and
other signs.

The notion of the subjectivity of experience cannot be here
analyzed with the detail the problem merits. It is believed,
however, that such an analysis would show that ‘experience’
itself is a relational term masquerading as a thing-name. x is an
experience if and only if there is some y (the experiencer) which

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Foundations of the Theory of Signs

stands in the experience relation to x. If E is an abbreviation for
‘experience relation,’ then the class of y s such that y stands in
the relation of E to something or other is the class of experi-
encers, and the x’s to which something or other stands in the
relation E constitute the class of experiences. An experience is
not, then, a special class of objects on a par with other objects,
but objects in a certain relation. The relation E will not here be
exhaustively characterized (that is a central task for em-
piricism) , but as a first approximation it can be said that to ex-
perience something is to take account of its properties by ap-
propriate conduct; the experience is direct to the degree that this
is done by direct response to the something in question, and
indirect to the degree that it is done through the intermediacy of
signs. For y x to experience x x it is sufficient that y x Ex x holds;
there is conscious experience if y x Ex x is an experience (e.g., if
y x E[y x Ex x ] holds), otherwise the experience is unconscious. An
experience x x is de facto subjective with respect to y x if y x is the
only one who stands in the relation E to x x ; an experience x x is
intrinsically subjective with respect to y x , relative to a certain
state of knowledge, if the known laws of nature permit the
deduction that no other y can stand in this relation to x x . An
experience is de facto inter subjective if it is not de facto subjective,
and it is potentially inter subjective if it is not intrinsically sub-
jective. It should be noted that with such usages a person may
not be able directly to experience aspects of himself that others
can directly experience, so that the line between subjective and
intersubjective experience in no sense coincides with the dis-
tinction between experiencers and external objects.

What bearing does this (tentative and preliminary) analysis
have on the question of meaning? It may be admitted, if the
facts warrant it, that there are certain experiences which are
de facto subjective as far as direct experience is concerned and
that this may even be true of the direct experience of the process
of semiosis; there would be nothing surprising in the conclusion
that, if I am the interpreter of a particular sign, there are then
aspects of the process of interpretation which I can directly ex-
perience but which others cannot. The important point is that

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Meaning

such a conclusion would not be in opposition to the thesis of
the potential inter subjectivity of every meaning. The fact that y x
and 7/2 do not stand in the relation of direct experience to each
other’s respective direct experience of Xi does not prevent them
both from directly experiencing Xi, or from indirectly desig-
nating (and so indirectly experiencing) by the use of signs the
experience relations in which the other stands — for under cer-
tain circumstances an object which cannot be directly expe-
rienced can, nevertheless, be denoted. Applying this result to
the case of a particular sign, y\ and y-i may differ in their direct
experience of the meaning situation and yet have the same
meaning in common and, in general, be able to decide what the
other means by a particular sign and the degree to which the
two meanings are the same or different. For the determination
of the meaning of Si (where Si is a sign vehicle) to y\ it is not
necessary that an investigator become 2/1 or have his experiences
of Si: it is sufficient to determine how Si is related to other signs
used by 2/1, under what situations 2/1 uses Si for purposes of
designation, and what expectations 2/1 has when he responds to
Si. To the degree that the same relations hold for y 2 as for y x ,
then Si has the same meaning to y x and yp, to the degree that
the relations in question differ for y x and 2/2, then Si has a
different meaning.

In short, since the meaning of a sign is exhaustively specified
by the ascertainment of its rules of usage, the meaning of any
sign is in principle exhaustively determinable by objective in-
vestigation. Since it is then possible, if it seems wise, to stand-
ardize this usage, the result is that the meaning of every sign is
potentially intersubjective. Even where the sign vehicle is in-
trinsically subjective there can be indirect confirmation that
there is such a sign vehicle with such and such meaning. It is
true that in practice the determination of meaning is difficult
and that the differences in sign usages among persons of even
the same social group may be rather great. But it is theoretical-
ly important to realize that the subjectivity of certain experi-
ences, and even experiences of semiosis, is compatible with the

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Foundations of the Theory of Signs

possibility of an objective and exhaustive determination of any
meaning whatsoever.

Having introduced the term ‘meaning’ only provisionally in
order to bring out the implications of the position here taken,
the use of the term will now be discontinued— it adds nothing
to the set of semiotical terms. It may be pointed out that the
preceding argument shows the agreement of what will be called
sign analysis with the requirements of scientific investigation.
Sign analysis is the determination of the syntactical, semantical,
and pragmatical dimensions of specific processes of semiosis; it
is the determination of the rules of usage of given sign vehicles.
Logical analysis is, in the widest sense of the term ‘logic,’ iden-
tical with sign analysis; in narrower usages, logical analysis is
some part of sign analysis, such as the study of the syntactical
relations of the sign vehicle in question. Sign analysis (i.e., de-
scriptive semiotic) can be carried on in accordance with all
recognized principles of scientific procedure.

13. Universal and Universality

Certain aspects of the “universality” (or generality) of signs
have long attracted attention, and their explanation has
been a source of many philosophical disputes. By viewing the
phenomena vaguely referred to under the overworked terms
‘universal and ‘universality’ through the prism of semiotical
analysis, the various components of the problems may be
separated and their relations seen.

The subject may be approached in terms of Peirce’s distinc-
tion between a sinsign and a legisign: a sinsign is a particular
something functioning as a sign, while a legisign is a “law”
functioning as a sign. A particular series of marks at a specific
place, such as house,’ is a sinsign; such a specific set of marks
is not, however, the Lnglish word house, for this word is “one,”
while its instances or replicas are as numerous as the various
employments of the word. It is a law or habit of usage, a“uni-
versal as over against its particular instances. Peirce was very
much impressed by this situation and made the difference basic
in his classification of signs; it gave an instance in the domain

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Universals and Universality

of signs of the phenomena of law (habit, Thirdness, mediation)
upon whose objectivity Peirce was so insistent.

The account which has here been given is compatible with
this general emphasis; the preceding section should have made
clear that semiosis, as a functional process, is just as real and
objective as are the component factors which function in the
process. It must also be admitted that in a given instance of
semiosis in which, say, ‘house’ functions as a sign vehicle, this
sinsign or this particular instance of semiosis is not identical
with the legisign house. What, then, is a legisign and where in
semiosis are “universals” and “universality” to be found? In
general, the answer must be that there is an element of uni-
versality or generality in all the dimensions and that confusion
results here as elsewhere when these are not distinguished and
when statements in the metalanguage are confounded with
statements in the thing-language.

It is experimentally confirmable that in a given process of
semiosis various sign vehicles may be substituted for the original
sign vehicle without the occurrence of any relevant change in
the remainder of the process. The metronome beat to which an
animal is conditioned may move faster or slower within certain
limits without the response of the animal undergoing change;
the spoken word ‘house’ may be uttered at different times by
the same or different persons, with various tonal changes, and
yet will awaken the same response and be used to designate the
same objects. If the word is written, the sizes may vary greatly,
the letters may differ in style, the media used may be of various
colors. The question of the limits of such variation and what
remains constant within this range is in a given case very diffi-
cult to determine even by the use of the most careful experi-
mental techniques, but of the fact of variability there is no
doubt possible. Strictly speaking, the sign vehicle is only that
aspect of the apparent sign vehicle in virtue of which semiosis
takes place; the rest is semiotically irrelevant. To say that a
given sign vehicle is “universal” (or general) is merely to say
that it is one of a class of objects which have the property or
properties necessary to arouse certain expectations, to combine

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Foundations of the Theory of Signs

in specified ways with other sign vehicles, and to denote certain
objects, i.e., that it is one member of a class of objects all of
which are subject to the same rules of sign usage. Thus ‘house’
and ‘house’ may be the same sign vehicles, but ‘house’ and
‘Haus’ are not; the fact that ‘the house is red’ conforms to the
rules of English while ‘the Haus is red’ does not, shows that the
sign vehicles are not the same, since the rules of usage are (in
part) different. None of the disciplines concerned with signs is
interested in the complete physical description of the sign ve-
hicle but is concerned with the sign vehicle only in so far as it
conforms to rules of usage.

In any specific case of semiosis the sign vehicle is, of course, a
definite particular, a sinsign; its “universality,” its being a
legisign, consists only in the fact, statable in the metalanguage,
that it is one member of a class of objects capable of performing
the same sign function.

Another component of the problem enters in connection with
the semantical dimension. The designatum of a sign is the class
of objects which a sign can denote in virtue of its semantical
rule. The rule may allow the sign to be applied to only one
object, or to many but not to all, or to everything. Here “uni-
versality” is simply the potentiality of denoting more than one
object or situation. Since such a statement is semantical, a
statement can be made in terms of the converse of the relation
of denotation: it can then be said that objects have the property
of universality when denotable by the same sign. In so far
as a number of objects or situations permit of a certain sign
being applied, they conform to the conditions laid down in the
semantical rule; hence there is something equally true of all of
them, and in this respect or to this degree they are the same —
whatever differences they may have are irrelevant to the par-
ticular case of semiosis. ‘Universality (or generality) of objects’
is a semantical term, and to talk as if ‘universality’ were a term
in the thing-language, designating entities (“universals”) in the
world, is to utter pseudo tiling-sentences of the quasi-semantical
type. This fact was recognized in the Middle Ages in the doc-
trine that ‘universality’ was a term of second intention rather

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Universals and Universality

than of first; in contemporary terms, it is a term within semiotic
and not a term in the thing-language. In the thing-language
there are simply terms whose rules of usage make them applica-
ble to a plurality of situations; expressed in terms of objects it
can only be said that the world is such that often a number of
objects or situations can be denoted by a given sign.

A similar situation appears in syntactics, where the relations
of sign vehicles are studied in so far as these relations are deter-
mined by formation or transformation rules. A combination of
sign-vehicles is a particular, but it may share its form with other
combinations of sign vehicles, i.e., a number of combinations of
different sign vehicles may be instances of the same formation or
transformation rule. In this case the particular sign combina-
tion has a formal or syntactical universality.

From the standpoint of pragmatics two considerations are
relevant to the problem in hand. One is the correlative of the
semantical situation which has already been described. The fact
that certain sign vehicles may denote many objects corresponds
to the fact that expectations vary in determinateness, so that a
number of objects may satisfy an expectation. One expects a
nice day tomorrow — and a number of weather conditions will
meet the expectation. Hence, while a response in a particular
situation is specific, it is a true statement within pragmatics that
similar responses are often called out by a variety of sign ve-
hicles and are satisfied by a variety of objects. From this point
of view the interpretant (in common with any habit) has a
character of “universality” which contrasts to its particularity
in a specific situation. There is a second aspect of sign univer-
sality distinguishable in pragmatics, namely, the social univer-
sality which lies in the fact that a sign may be held in common
by a number of interpreters.

It is accordingly necessary to distinguish in the universality
appropriate to semiosis five types of universality. Since the
term ‘universality’ has such a variety of usages, and is clearly
inappropriate in some of the five cases, the term ‘generality’
will be used instead. There are, then, five types of sign general-
ity: generality of sign vehicle, generality of form, generality of

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Foundations of the T heory of Signs

denotation, generality of the interpretant, and social generality. The
central point is that each of these kinds of generality can be
stated only within semiotic; generality is accordingly a rela-
tional concept, since all the branches of semiotic investigate
only relations. To speak of something as a “general” or a “uni-
versal” is merely to use a pseudo thing-sentence instead of the
unambiguous semiotical expression; such terms can only signify
that the something in question stands to something or other in
one of the relations embodied in the five kinds of sign generality
which have been distinguished. In this way there is kept what
is significant in the historical emphases of nominalism, realism,
and conceptualism, while yet avoiding the last traces of the
substantive or entitive conception of generality by recognizing
the level of discourse appropriate to discussions of generality
and the relational character of the terms employed at this level.

14. Interrelation of the Semiotical Sciences

Since the current tendency is in the direction of specialized
research in syntactics, semantics, or pragmatics, it is well to
stress emphatically the interrelations of these disciplines within
semiotic. Indeed, semiotic, in so far as it is more than these
disciplines, is mainly concerned with their interrelations, and
so with the unitary character of semiosis which these disciplines
individually ignore.

One aspect of the interrelation is indicated in the fact that
while each of the component disciplines deals in one way or
another with signs, none of them can define the term ‘sign’ and,
hence, cannot define themselves. ‘Syntactics’ is not a term
within syntactics but is a strictly semiotical term — and the same
is true of ‘semantics’ and ‘pragmatics.’ Syntactics speaks of for-
mation and transformation rules, but rules are possible modes
of behavior and involve the notion of interpreter; ‘rule’ is, there-
fore, a pragmatical term. Semantics refers explicitly only to
signs as designating objects or situations, but there is no such
relation without semantical rules of usage, and so again the
notion of interpreter is implicitly involved. Pragmatics deals
directly only with signs as interpreted, but ‘interpreter’ and

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Interrelation of the Semiotical Sciences

‘interpretant’ cannot be defined without the use of sign ve-
hicle’ and ‘designatum’— so that all of these terms are strictly
semiotical terms. Such considerations— themselves only a few
among many possible ones — show that, while the component
semiotical disciplines do not as sciences refer to one another, yet
they can be characterized and distinguished only in terms of the
wider science of which they are components.

It is also true that a person who studies some dimension of
semiosis uses terms which have all three dimensions and em-
ploys the results of the study of the other dimensions. The
rules which govern the sign vehicles of the language being
studied must be understood, and ‘understanding is a prag-
matical term. The rules for combining and transforming pos-
sible sign vehicles cannot be composed merely of possible sign
vehicles but must actually function as signs. In descriptive syn-
tactics there must be signs to denote the sign vehicles being
studied, and the aim must be to make true statements about
these sign vehicles— but ‘denote’ and ‘true’ are not syntactical
terms. Semantics will study the relation of a sign combination
to what it denotes or can denote, but this involves the knowl-
edge of the structure of the sign combination and the semantical
rules in virtue of which the relation of denotation may obtain.
Pragmatics cannot go far without taking account of the formal
structures for which it should seek the pragmatical correlate,
and of the relation of signs to objects which it seeks to explain
through the notion of habit of usage. Finally, the languages of
syntactics, semantics, and pragmatics have all three dimen-
sions: they designate some aspect of semiosis, they have a for-
mal structure, and they have a pragmatical aspect in so far as
they are used or understood.

The intimate relation of the semiotical sciences makes
semiotic as a science possible but does not blur the fact that
the subsciences represent three irreducible and equally legiti-
mate points of view corresponding to the three objective dimen-
sions of semiosis. Any sign whatsoever may be studied from
any of the three standpoints, though no one standpoint is ade-
quate to the full nature of semiosis. Thus in one sense there is

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Foundations of the Theory of Signs

no limit to either point of view, i.e., no place at which an inves-
tigator must desert one standpoint for another. This is simply
because they are studies of semiosis from different points of
view; in fastening attention upon one dimension, each deliber-
ately neglects the aspects of the process discernible in terms of
the other standpoints. Syntactics, semantics, and pragmatics
are components of the single science of semiotic but mutually
irreducible components.

Vli. Problems and Applications

15. Unification of the Semiotical Sciences

There remains the task of briefly showing the problems which
remain open within semiotic and the possible fields of applica-
tion. These may be roughly grouped under three headings : uni-
fication of the semiotical sciences, semiotic as organon of the
sciences, and humanistic implications of semiotic. The remarks
which follow aim merely to be suggestive— to indicate directions
rather than solutions.

The account which has been given has been adapted to the
purposes of an introduction. Large areas of the field were ig-
nored, exactitude in statement was often sacrificed to avoid
lengthy preliminary analysis, and the consideration of the ex-
amples which were introduced was carried only so far as to
illuminate the point at issue. Even though the larger outlines of
semiotic be correct, it is still far from the condition of an ad-
vanced science. Progress will require collaboration by many in-
vestigators. There is need both for fact-finders and for system-
atizes. The former must make clear the conditions under which
semiosis occurs and what precisely takes place in the process.;
the latter must in the light of available facts develop a precise
systematized theoretical structure which future fact-finders can
in turn use. One theoretical problem of importance lies in the
relation of the various kinds of rules. The theory of signs which
has been given opens up many points of contact with the con-
crete work of biologists, psychologists, psychopathologists, lin-
guists, and social scientists. Systematization can profitably
make use of symbolic logic; for, since semiotic deals throughout

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Unification of the Semiotical Sciences

with relations, it is peculiarly amenable to treatment in terms of
the logic of relations. The work of fact-finders and systematiz-
es is equally important and must go hand in hand, each pro-
vides material for the other. _

Semioticians should find the history of semiotic useful both
as a stimulus and as a field of application. Such hoary doctrines
as the categories, the transcendentals, and the predicables are
early sallies into semiotical domains and should be clarified by
the later developments. Hellenistic controversies over the ad-
monitive and the indicative sign, and the medieval doctrines of
intention, imposition, and supposition are worth reviving and
interpreting. The history of linguistics, rhetoric, logic, empiri-
cism, and experimental science offers rich supplementary ma-
terial. Semiotic has a long tradition, and in common with all
sciences it should keep alive its history.

In the development of semiotic the disciplines which now are
current under the names of logic, mathematics, and linguistics
can be reinterpreted in semiotical terms. The logical paradoxes,
the theory of types, the laws of logic, the theory of probability,
the distinction of deduction, induction, and hypothesis, the logic
of modality— all such topics permit of discussion within the
theory of signs. In so far as mathematics is knowledge of lin-
guistic structures, and not simply identified with some (or all)
of such structures, it too may be considered as part of semiotic.
Linguistics clearly falls within semiotic, dealing at present with
certain aspects of the complex sign structures which constitute
languages in the full semiotical sense of that term. It is possible
that the admittedly unsatisfactory situation with respect to
such terms as ‘word,’ ‘sentence,’ and ‘part of speech’ can be
clarified in terms of the sign functions which various linguistic
devices serve. Ancient projects of a universal grammar take on
a new and defensible form when translated into the stud} of the
way all languages perform similar sign functions by the use of
different devices.

Logic, mathematics, and linguistics can be absorbed in their
entirety within semiotic. In the case of certain other disciplines
this may occur only in part. Problems which are often classed

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Foundations of the Theory of Signs

as epistemological or methodological fall in large part under
semiotic: thus empiricism and rationalism are at heart theories
as to when the relation of denotation obtains or may be said to
obtain; discussions of truth and knowledge are inseparably
linked with semantics and pragmatics; a discussion of the pro-
cedures of scientists, when more than a chapter in logic, psy-
chology, or sociology, must relate these procedures to the cogni-
tive status of the statements which result from their application.
In so far as aesthetics studies a certain functioning of signs (such
as iconic signs whose designata are values), it is a semiotical
discipline with syntactical, semantical, and pragmatical com-
ponents, and the distinction of these components offers a base
for aesthetic criticism. The sociology of knowledge is clearly
part of pragmatics, and so is rhetoric; semiotic is the framework
m which to fit the modern equivalents of the ancient trivium of
logic, grammar, and rhetoric. It has already been suggested
that psychology and the human social sciences may find part
(if not the entire) basis of their distinction from other biological
and social sciences in the fact that they deal with responses
mediated by signs. The development of semiotic is itself a stage
in the unification of sciences dealing in whole or in part with
signs; it may also play an important role in bridging the gap
between the biological sciences, on the one hand, and the psy-
chological and human social sciences, on the other, and in
throwing a new light upon the relation of the so-called “formal”
and “empirical” sciences.

16 . Semiotic as Organon of the Sciences

Semiotic holds a unique place among the sciences. It may be
possible to say that every empirical science is engaged in finding
data which can serve as reliable signs; it is certainly true that
every science must embody its results in linguistic signs. Since
this is so, the scientist must be as careful with his linguistic
tools as he is in the designing of apparatus or in the making of
observations. The sciences must look to semiotic for the con-
cepts and general principles relevant to their own problems of
sign analysis. Semiotic is not merely a science among sciences
but an organon or instrument of all the sciences.

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Semiotic as Organon of the Sciences

This function can be performed in two ways. One is by mak-
ing training in semiotic a regular part of the equipment of the
scientist. In this way a scientist would become critically con-
scious of his linguistic apparatus and develop careful habits in
its use. The second way is by specific investigations of the lan-
guages of the special sciences. The linguistically expressed re-
sults of all the sciences is part of the subject matter of descrip-
tive semiotic. Specific analyses of certain basic terms and prob-
lems in the various sciences will show the working scientist
whatever relevance semiotic has in these fields more effectively
than any amount of abstract argument. Other essays in the
Encyclopedia may be regarded as contributing such studies.
Current scientific formulations embody many pseudo problems
which arise from the confusion of statements in the language of
semiotic and the thing-language — recent discussions of inde-
terminism and complementarity in the physical sciences abound
in illustrations. Empirical problems of a nonlinguistic sort are
not solved by linguistic considerations, but it is important that
the two kinds of problems not be confused and that nonlinguistic
problems be expressed in such a form as aids their empirical
solution. The classical logic thought of itself as the organon of
the sciences but was, in fact, unable to play the role it set itself ;
contemporary semiotic, embodying in itself the newer logical
developments and a wide variety of approaches to sign phe-
nomena, may again attempt to assume the same role.

17. Humanistic Implications of Semiotic

Signs serve other purposes than the acquisition of knowledge,
and descriptive semiotic is wider than the study of the language
of science. Corresponding to the various purposes which signs
serve, there have developed more or less specialized lan-
guages which follow to some extent the various dimensions of
semiosis. Thus the mathematical form of expression is well
adapted to stress the interrelation of terms in a language,
letting the relation to objects and interpreters recede into the
background; the language of empirical science is especially
suitable for the description of nature; the languages of morality,
the fine arts, and the applied arts are especially adapted to the

135

Foundations of the Theory of Signs

control of behavior, the presentations of things or situations as
objects of interest, and the manipulation of things to effect
desired eventuations. In none of these cases are any of the di-
mensions of semiosis absent; certain of them are simply sub-
ordinated and partially transformed by the emphasis upon one
of the dimensions. Mathematical propositions may have an em-
pirical aspect (many indeed were discovered empirically), and
mathematical problems may be set by problems in other fields,
but the language of mathematics subordinates these factors in
order to better accomplish the task it is developed to fulfil.
Empirical science is not really concerned with simply getting all
true statements possible (such as the statement of the area of
each mark on this page) but in getting important true state-
ments (i.e., statements that, on the one hand, furnish a secure
base for prediction and, on the other hand, that aid in the crea-
tion of a systematic science) — but the language of empirical
science is adapted to expressing the truth and not the impor-
tance of its statements. Lyric poetry has a syntax and uses
terms which designate things, but the syntax and the terms are
so used that what stand out for the reader are values and
evaluations. The maxims of the applied arts rest on true propo-
sitions relevant to the accomplishment of certain purposes (“to
accomplish a;, do so and so”); moral judgments may similarly
have an empirical component but, in addition, assume the de-
sirability of reaching a certain end and aim to control conduct
(“You ought to have your child vaccinated,” i.e., “Taking the
end of health for granted, vaccination is in the present situation
the surest way of realizing that end, so have it done”).

Semiotic provides a basis for understanding the main forms
of human activity and their interrelationship, since all these
activities and relations are reflected in the signs which mediate
the activities. Such an understanding is an effective aid in
avoiding confusion of the various functions performed by
signs. As Goethe said, “One cannot really quarrel with any form
of representation” — provided, of course, that the form of rep-
resentation does not masquerade as what it is not. In giving
such understanding, semiotic promises to fulfil one of the tasks

136

Selected Bibliography

which traditionally has been called philosophical. Philosophy
has often sinned in confusing in its own language the various
functions which signs perform. But it is an old tradition that
philosophy should aim to give insight into the characteristic
forms of human activity and to strive for the most general and
the most systematic knowledge possible. This tradition appears
in a modern form in the identification of philosophy with the
theory of signs and the unification of science, that is, with the
more general and systematic aspects of pure and descriptive
semiotic.

Selected Bibliography

Ajduxiewicz, K. “Sprache und Sinn,” Erkenntnis, Vol. IV (1934).
Benjamin, A. C. The Logical Structure of Science, chaps, vii, viii, and ix.
London, 1936.

Carnap, R. Philosophy and Logical Syntax. London, 1935.

. Logical Syntax of Language. Vienna, 1934; London, 1937.

. “Testability and Meaning,” Philosophy of Science. Vol. Ill (1936);

ibid., Vol. IV (1937).

Cassirer, E. Die Philosophic der symbolischen Formen. 3 vols. Berlin,
1923 ff.

Eaton, R. M. Symbolism and Truth. Cambridge, Mass., 1925.
Gatschenberger, R. Zeichen. Stuttgart, 1932.

Husserl, E. Logische LJntersuchungen, Vol. II, Part I. 4th ed. Halle, 1928.
Kokoszynska, M. “tjber den absoluten Wahrheitsbegriff und einige andere
semantische Begriffe,” Erkenntnis, Vol. VI (1936).

Mead, G. H. Mind, Self, and Society. Chicago, 1934.

. The Philosophy of the Act. Chicago, 1938.

Morris, C. W. Logical Positivism, Pragmatism, and Scientific Empiricism.
Paris, 1937.

Ogden, C. K., and Richards, I. A. The Meaning of ‘Meaning.’ London, 1923.
Peirce, C. S. Collected Papers, esp. Vol. II. Cambridge, Mass., 1931 ff.
Reichenbach, H. Experience and Prediction, chaps, i and ii. Chicago, 1938.
Schlick, M. Gesammelte Aufsatze, 1926-1936. Vienna, 1938.

Tarski, A. “Grundlegung der wissenschaftlichen Semantik,” Actes du
congres international de philosophe scientifique. Paris, 1936.

. “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studio.

philosophica, Vol. I (1935).

Wittgenstein, L. Tractatus logico-philosophicus . London, 1922.

137

Foundations of Logic
and Mathematics

Rudolf Carnap

Foundations of Logic and Mathematics

Contents:

I. Logical Analysis of Language: Semantics and
Syntax

1. Theoretical Procedures in Science ....

2. Analysis of Language

3. Pragmatics of Language B

4. Semantical Systems

5. Rules of the Semantical System B-S

6. Some Terms of Semantics

7. L-Semantical Terms

8. Logical Syntax

9. The Calculus B-C

143

145

147

148
150

153

154
158
160

II. Calculus and Interpretation

10. Calculus and Semantical System 163

11. On the Construction of a Language System 166

12. Is Logic a Matter of Convention? .... 168

III. Calculi and Their Application in Empirical
Science

13. Elementary Logical Calculi 171

14. Higher Logical Calculi 175

15. Application of Logical Calculi 177

140

Contents.

PAGE

16. General Remarks about Nonlogical Calculi

(Axiom Systems) 179

17. An Elementary Mathematical Calculus . 180

18. Higher Mathematical Calculi 184

19. Application of Mathematical Calculi 186

20. The Controversies over “Foundations” of

Mathematics 190

21. Geometrical Calculi and Their Interpretations 193

22. The Distinction between Mathematical and

Physical Geometry 195

23. Physical Calculi and Their Interpretations 198

24. Elementary and Abstract Terms 203

25. “Understanding” in Physics 209

Selected Bibliography 211

Index of Terms 213

141

Foundations of Logic and Mathematics

Rudolf Carnap

I. Logical Analysis of Language: Semantics and Syntax

1. Theoretical Procedures in Science

The activities of a scientist are in part practical : he arranges
experiments and makes observations. Another part of his work
is theoretical: he formulates the results of his observations in
sentences, compares the results with those of other observers,
tries to explain them by a theory, endeavors to confirm a theory
proposed by himself or somebody else, makes predictions with
the help of a theory, etc. In these theoretical activities, deduc-
tion plays an important part; this includes calculation, which
is a special form of deduction applied to numerical expressions.
Let us consider, as an example, some theoretical activities of
an astronomer. He describes his observations concerning a cer-
tain planet in a report, 0 X . Further, he takes into consideration
a theory T concerning the movements of planets. (Strictly
speaking, T would have to include, for the application to be
discussed, laws of some other branches of physics, e.g., concern-
ing the astronomical instruments used, refraction of light in the
atmosphere, etc.) From 0 x and T, the astronomer deduces a
prediction, P; he calculates the apparent position of the planet
for the next night. At that time he will make a new observation
and formulate it in a report 0 2 . Then he will compare the pre-
diction P with Oi and thereby find it either confirmed or not.
If T was a new theory and the purpose of the procedure de-
scribed was to test T, then the astronomer will take the con-
firmation of P by 0 2 as a partial confirmation for T; he will
apply the same procedure again and again and thereby obtain
either an increasing degree of confirmation for T or else a dis-
confirmation. The same deduction of P from 0 j and T is made
in the case where T is already scientifically acknowledged on the

143

Foundations of Logic and Mathematics

basis of previous evidence, and the present purpose is to obtain
a prediction of what will happen tomorrow. There is a third
situation in which a deduction of this kind may be made. Sup-
pose we have made both the observations described in 0 , and
in 0 2 ; we are surprised by the results of the observation de-
scribed in 0 2 and therefore want an explanation for it. This
explanation is given by the theory T; more precisely, by deduc-
ing P from 0 , and T and then showing that 0 2 is in accordance
with P (“What we have observed is exactly what we had to
expect”).

These simple examples show that the chief theoretical pro-
cedures in science namely, testing a theory, giving an explana-
tion for a known fact, and predicting an unknown fact — involve
as an essential component deduction and calculation; in other
words, the application of logic and mathematics. (These pro-
cedures will later be discussed more in detail, especially in
§§ 15, 19, and 23.) It is one of the chief tasks of this essay to
make clear the role of logic and mathematics as applied in em-
pirical science. We shall see that they furnish instruments for
deduction, that is, for the transformation of formulations of
factual, contingent knowledge. However, logic and mathe-
matics not only supply rules for transformation of factual sen-
tences but they themselves contain sentences of a different, non-
f actual kind. Therefore, we shall have to deal with the ques-
tion of the nature of logical and mathematical theorems. It
will become clear that they do not possess any factual content.
If we call them true, then another kind of truth is meant, one
not dependent upon facts. A theorem of mathematics is not
tested like a theorem of physics, by deriving more and more
predictions with its help and then comparing them with the
results of observations. But what else is the basis of their va-
lidity? We shall try to answer these questions by examining
how theorems of logic and mathematics are used in the con-
text of empirical science.

The material on which the scientist works in his theoretical
activities consists of reports of observations, scientific laws and
theories, and predictions; that is, formulations in language

1 44

Analysis of Language

which describe certain features of facts. Therefore, an analysis
of theoretical procedures in science must concern itself with
language and its applications. In the present section, in pre-
paring for the later task, we shall outline an analysis of lan-
guage and explain the chief factors involved. Three points of
view will be distinguished, and accordingly three disciplines
applying them, called pragmatics, semantics, and syntax. These
will be illustrated by the analysis of a simple, fictitious language.
In the later sections the results of these discussions will be
applied in an analysis of the theoretical procedure of science,
especially from the point of view of calculi, their interpretation,
and their application in empirical science.

2. Analysis of Language

A language, as, e.g., English, is a system of activities or,
rather, of habits, i.e., dispositions to certain activities, serving
mainly for the purposes of communication and of co-ordination
of activities among the members of a group. The elements of
the language are signs, e.g., sounds or written marks, produced
by members of the group in order to be perceived by other mem-
bers and to influence their behavior. Since our final interest in
this essay concerns the language of science, we shall restrict
ourselves to the theoretical side of language, i.e., to the use of
language for making assertions. Thus, among the different
kinds of sentences, e.g., commands, questions, exclamations,
declarations, etc., we shall deal with declarative sentences only.
For the sake of brevity we shall call them here simply sentences.

This restriction to declarative sentences does not involve, in
the investigation of processes accompanying the use of lan-
guage, a restriction to theoretical thinking. Declarative sen-
tences, e.g., ‘This apple is sour’, are connected not only with
the theoretical side of behavior but also with emotional, voli-
tional, and other factors. If we wish to investigate a language
as a human activity, we must take into consideration all these
factors connected with speaking activities. But the sentences,
and the signs (e.g., words) occurring in them, are sometimes
involved in still another relation. A sign or expression may con-

145

Foundations of Logic and Mathematics

cern or designate or describe something, or, rather, he who uses
the expression may intend to refer to something by it, e.g.,
to an object or a property or a state of affairs; this we call the
designatum of the expression. (For the moment, no exact defini-
tion for ‘designatum’ is intended; this word is merely to serve
as a convenient, common term for different cases — objects,
properties, etc. — whose fundamental differences in other re-
spects are not hereby denied.) Thus, three components have
to be distinguished in a situation where language is used. We
see these in the following example: (1) the action, state, and
environment of a man who speaks or hears, say, the German
word ‘blau’; (2) the word ‘blau’ as an element of the German
language (meant here as a specified acoustic [or visual] design
which is the common property of the many sounds produced at
different times, which may be called the tokens of that design) ;
(3) a certain property of things, viz., the color blue, to which
this man — and German-speaking people in general — intends to
refer (one usually says, “The man means the color by the
word”, or “The word means the color for these people”, or
“. . . . within this language”).

The complete theory of language has to study all these three
components. We shall call pragmatics the field of all those in-
vestigations which take into consideration the first component,
whether it be alone or in combination with the other com-
ponents. Other inquiries are made in abstraction from the
speaker and deal only with the expressions of the language and
their relation to their designata. The field of these studies is
called semantics. Finally, one may abstract even from the desig-
nata and restrict the investigation to formal properties — in a
sense soon to be explained — of the expressions and relations
among them. This field is called logical syntax. The distinction
between the three fields will become more clear in our subse-
quent discussions.

That an investigation of language has to take into consideration all the
three factors mentioned was in recent times made clear and emphasized
especially by C. S. Peirce, by Ogden and Richards, and by Morris (see Vol. I,
No. 2). Morris made it the basis for the three fields into which he divides

146

Pragmatics of Language B

semiotic (i.e., the general theory of signs), namely, pragmatics, semantics,
and syntactics. Our division is in agreement with his in its chief features.
For general questions concerning language and its use compare also Bloom-
field, Volume I, No. 4.

3. Pragmatics of Language B

In order to make clear the nature of the three fields and the
differences between them, we shall analyze an example of a
language. We choose a fictitious language B, very poor and
very simple in its structure, in order to get simple systems of
semantical and syntactical rules.

Whenever an investigation is made about a language, we call
this language the object-language of the investigation, and the
language in which the results of the investigation are formu-
lated the metalanguage. Sometimes object-language and meta-
language are the same, e.g., when we speak in English about
English. The theory concerning the object-language which is
formulated in the metalanguage is sometimes called metatheory.
Its three branches are the pragmatics, the semantics, and the
syntax of the language in question. In what follows, B is our
object-language, English our metalanguage.

Suppose we find a group of people speaking a language B
which we do not understand; nor do they understand ours.
After some observation, we discover which words the people
use, in which forms of sentences they use them, what these
words and sentences are about, on what occasions they are
used, what activities are connected with them, etc. Thus we
may have obtained the following results, numbered here for
later reference.

Pragm. 1 . — Whenever the people utter a sentence of the form
‘. . . ist kalt’, where ‘. . .’ is the name of a thing, they intend
to assert that the thing in question is cold.

Pragm. 2a . — A certain lake in that country, which has no
name in English, is usually called ‘titisee’. When using this
name, the people often think of plenty of fish and good meals.

Pragm. 2b . — On certain holidays the lake is called ‘rumber’;

147

Foundations of Logic and Mathematics

when using this name, the people often think— even during
good weather — of the dangers of storm on the lake.

Pragm.S.— The word ‘nicht’ is used in sentences of the form
nicht . . . , where ... is a sentence. If the sentence ‘. . .’ serves
to express the assertion that such and such is the case, the
whole sentence ‘nicht . . .’ is acknowledged as a correct asser-
tion if such and such is not the case.

In this way we slowly learn the designata and mode of use
of all the words and expressions, especially the sentences; we
find out both the cause and the effect of their utterance. We
may study the preferences of different social groups, age groups,
or geographical groups in the choice of expressions. We investi-
gate the role of the language in various social relations, etc.

lhe pragmatics of language B consists of all these and simi-
lar investigations. Pragmatical observations are the basis of
all linguistic research. We see that pragmatics is an empirical
discipline dealing with a special kind of human behavior and
making use of the results of different branches of science (prin-
cipally social science, but also physics, biology, and psychology).

4. Semantical Systems

We now proceed to restrict our attention to a special aspect
of the facts concerning the language B which we have found
by observations of the speaking activities within the group who
speak that language. We study the relations between the ex-
pressions of B and their designata. On the basis of those facts
we are going to lay down a system of rules establishing those
relations. We call them semantical rides. These rules are not
unambiguously determined by the facts. Suppose we have
found that the word ‘mond’ of B was used in 98 per cent of the
cases for the moon and in 2 per cent for a certain lantern. Now
it is a matter of our decision whether we construct the rules in
such a way that both the moon and the lantern are designata
of ‘mond’ or only the moon. If we choose the first, the use of
‘mond’ in those 2 per cent of cases was right— with respect to
our rules; if we choose the second, it was wrong. The facts do
not determine whether the use of a certain expression is right

148

Semantical Systems

or wrong but only how often it occurs and how often it leads
to the effect intended, and the like. A question of right or
wrong must always refer to a system of rules. Strictly speaking,
the rules which we shall iay down are not rules of the factually
given language B; they rather constitute a language system
corresponding to B which we will call the semantical system
B-S . The language B belongs to the world of facts ; it has many
properties, some of which we have found, while others are un-
known to us. The language system B-S, on the other hand, is
something constructed by us; it has all and only those properties
which we establish by the rules. Nevertheless, we construct B-S
not arbitrarily but with regard to the facts about B. Then we
may make the empirical statement that the language B is to
a certain degree in accordance with the system B-S. The previ-
ously mentioned pragmatical facts are the basis — in the sense
explained — of some of the rules to be given later (Pragm. 1
for SD 2 a and SL 1, Pragm. 2a , b for SD la, Pragm. 3 for SL 2).

We call the elements of a semantical system signs; they may
be words or special symbols like ‘O’, ‘ + \ etc. A sequence con-
sisting of one or several signs is called an expression. As signs
of the system B-S we take the words which we have found by
our observations to be words of B or, rather, only those words
which we decide to accept as “correct.” We divide the signs
of B-S — and, in an analogous way, those of any other semanti-
cal system — into two classes: descriptive and logical signs. As
descriptive signs we take those which designate things or prop-
erties of things (in a more comprehensive system we should
classify here also the relations among things, functions of
things, etc.). The other signs are taken as logical signs: they
serve chiefly for connecting descriptive signs in the construction
of sentences but do not themselves designate things, properties
of things, etc. Logical signs are, e.g., those corresponding to
English words like ‘is’, ‘are’, ‘not’, ‘and’, ‘or’, ‘if’, ‘any’, ‘some’,
‘every’, ‘all’. These unprecise explanations will suffice here.
Our later discussions will show some of the differentiae of the
two classes of signs.

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Foundations of Logic and Mathematics

Semantics as an exact discipline is quite new; we owe it to the very fertile
school of contemporary Polish logicians. After some of this group, especially
Lesniewski and Ajdukiewicz, had discussed semantical questions, Tarski, in
his treatise on truth, made the first comprehensive systematic investigation
in this field, giving rise to very important results.

5. Rules of fhe Semantical System B-S

In order to show how semantical rules are to be formulated
and how they serve to determine truth conditions and thereby
give an interpretation of the sentences, we are going to con-
struct the semantical rules for the system B-S. As preliminary
steps for this construction we make a classification of the signs
and lay down rules of formation. Each class is defined by an
enumeration of the signs belonging to it. The signs of B-S are
divided into descriptive and logical signs. The descriptive signs
of B-S are divided into names and predicates. Names are the
words ‘titisee’, ‘rumber’, ‘mond’, etc. (here a complete list of
the names has to be given). Predicates are the words ‘kalt’,
‘blau’, ‘rot’, etc. The logical signs are divided into logical
constants (‘ist’, ‘nicht’, ‘wenn’, ‘so’, ‘fuer’, ‘jedes’) and variables
(‘x’, ‘y’, etc.). For the general description of forms of expres-
sions we shall use blanks like ‘. . .’, ‘ ’, etc. They are not

themselves signs of B-S but have to be replaced by expressions
of B-S. If nothing else is said, a blank stands for any expression
of B-S. A blank with a subscript V, ‘p’, ‘s’, or V (e.g., ‘. . .„’)
stands for a name, a predicate, a sentence, or a variable, re-
spectively. If the same blank occurs several times within a rule
or a statement, it stands at all places for the same expression.

The rules of formation determine how sentences may be con-
structed out of the various kinds of signs.

Rules of formation. — An expression of B-S is called a sentence
(in the , semantical sense) or a proposition of B-S, if and only if
it ha£ 6ne of the following forms, F 1-4. F 1 : ‘. . .„ ist

p (e-g-. ‘mond ist blau’); F 2: ‘nicht. . .,’ (e.g., ‘nicht mond

ist blau’); F 3: ‘wenn . . .„ so ,’ (e.g., ‘wenn titisee ist rot, so

mond ist kalt’) ; F 4 : ‘fuer jedes where stands

for an expression which is formed out of a sentence not contain-
ing a variable by replacing one or several names by the variable

150

Rules of the Semantical System B-S

.„’ (e.g., ‘fuer jedes x, x ist blau’; ‘fuer jedes y, wenn y ist blau,
so y ist kalt’). The partial sentence in a sentence of the form
F 2 and the two partial sentences in a sentence of the form F 3
(indicated above by blanks) are called components of the whole
sentence. In order to indicate the components of a sentence in
case they are themselves compound, commas and square brack-
ets are used when necessary.

Rules B-SD. Designata of descriptive signs:

SD 1. The names designate things, and especially

a) each of the thing-names ‘titisee’ and ‘rumber’ desig-
nates the lake at such and such a longitude and lati-
tude.

h) ‘mond’ designates the moon.

Etc. [Here is to be given a complete list of rules for all
the names of B-S.)

SD 2. The predicates designate properties of things, and es-
pecially

a) ‘kalt’ designates the property of being cold.

b ) ‘blau’ designates the property of being blue.

c) ‘rot’ designates the property of being red.

Etc. [for all predicates].

Rules B-SL. Truth conditions for the sentences of B-S. These
rules involve the logical signs. We call them the L-semantical
rules of B-S.

SL 1. ‘ist’, form F 1. A sentence of the form ‘. . .„ ist f is

true if and only if the thing designated by ‘. . has the
property designated by ‘ p ’.

SL 2. ‘nicht’, form F 2. A sentence of the form ‘nicht . . ..’ is
true if and only if the sentence ‘. . .,’ is not true.

SL 3. ‘wenn’ and ‘so’, form F 3. A sentence of the form ‘wenn
. . .„ so - - V is true if and only if ‘. . is not true or
‘ .’ is true.

SL I)., ‘fuer jedes’, form F 4. A sentence of the form ‘fuer jedes
where is an expression formed out of a

sentence by replacing one or several names by the vari-
able *. is true if and only if all sentences of the follow-

151

Foundations of Logic and Mathematics

ing kind are true: namely, those sentences constructed
out of the expression by replacing the variable

\ at all places where it occurs within that expression
by a name, the same for all places; here names of any
things may be taken, even of those for which there is no
name in the list of names in B-S. (Example: The
sentence ‘fuer jedes x, x ist blau’ is true if and only if
every sentence of the form \ . ist blau’ is true; hence,
according to SL 1, if and only if everything is blue.)

The rule SL 1, in combination with SD, provides direct
truth conditions for the sentences of the simplest form; direct,
since the rule does not refer to the truth of other sentences.
SL 2-4 provide indirect truth conditions for the compound sen-
tences by referring to other sentences and finally back to sen-
tences of the simplest form. Hence the rules B-SD and SL to-
gether give a general definition of ‘true in B-S’ though not in
explicit form. (It would be possible, although in a rather com-
plicated form, to formulate an explicit definition of ‘true in
B-S’ on the basis of the rules given.) A sentence of B-S which
is not true in B-S is called false in B-S.

If a sentence of B-S is given, one can easily construct, with
the help of the given rules, a direct truth-criterion for it, i.e.,
a necessary and sufficient condition for its truth, in such a way
that in the formulation of this condition no reference is made
to the truth of other sentences. Since to know the truth condi-
tions of a sentence is to know what is asserted by it, the given
semantical rules determine for every sentence of B-S what it
asserts — in usual terms, its “meaning” — or, in other words, how
it is to be translated into English.

Examples: (1) The sentence ‘mond ist blau’ is true if and only if the moon
is blue. (2) The sentence ‘fuer jedes x, wenn x ist blau, so x ist kalt’ is true if
and only if every thing — not only those having a name in B-S — either is not
blue or is cold; in other words, if all blue things are cold. Hence, this sentence
asserts that all blue things are cold; it is to be translated into the English
sentence ‘all blue things are cold’.

Therefore, we shall say that we understand a language system,
or a sign, or an expression, or a sentence in a language system,

152

Some Terms of Semantics

if we know the semantical rules of the system. We shall also
say that the semantical rules give an interpretation of the lan-
guage system.

We have formulated the semantical rules of the descriptive signs by stat-
ing their designata, for the logical signs by stating truth conditions for the
sentences constructed with their help. We may mention here two other ways
of formulating them which are often used in the practice of linguistics and
logic. The first consists in giving translations for the signs and, if necessary,
for the complex expressions and sentences, as it is done in a dictionary. The
second way consists in stating designata throughout, not only for the descrip-
tive signs as in SD, but also for expressions containing the logical signs, corre-
sponding to SL. Example (corresponding to SL 1): A sentence of the form

• •!* ist - - -p designates (the state of affairs) that the thing designated by
‘. . -n has the property designated by ‘ p '.

6. Some Terms of Semantics

We shall define some more terms which belong to the meta-
language and, moreover, to the semantical part of the metalan-
guage (as is seen from the fact that the definitions refer to the
semantical rules). Any semantical term is relative to a semanti-
cal system and must, in strict formulation, be accompanied by
a reference to that system. In practice the reference may often
be omitted without ambiguity (thus we say, e.g., simply ‘syn-
onymous’ instead of ‘synonymous in B-S’).

Two expressions are said to be semantically synonymous, or
briefly, synonymous, with each other in a semantical system
S if they have the same designatum by virtue of the rules of
S. Hence, according to SD la, the signs ‘titisee’ and ‘rumber’
are semantically synonymous with one another in B-S. They
are, however, not what we might call pragmatically synony-
mous in B, as is shown by Pragm. 2a, b. Since the transition
from pragmatics to semantics is an abstraction, some proper-
ties drop out of consideration and hence some distinctions dis-
appear. Because of the semantical synonymity of the names
mentioned, the sentences ‘titisee ist kalt’ and ‘rumber ist kalt’
are also semantically synonymous. These two sentences have
the same truth conditions, although different pragmatical con-
ditions of application. Suppose that the lake is cold and hence
the sentence ‘titisee ist kalt’ is true. Then the sentence ‘rumber

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is kalt is also true, even if sinfully spoken on a working day.
If this happened by mistake, people would tell the speaker that
he is right in his belief but that he ought to formulate it — i.e.,
the same belief— in another way.

We shall apply the semantical terms to be defined not only
to sentences but also to classes of sentences. In what follows
we shall use ‘Si, ‘S 2 \ etc., for sentences; ‘C*, ‘C 2 , etc., for
classes of sentences; Ty , T 2 , etc., stand both for sentences and
for classes of sentences. (These ‘S' and ‘C’ with subscripts have
nothing to do with the same letters without subscripts, which
we use for semantical systems and calculi, e.g., ‘B-S’ and
‘B-C’.) We understand the assertion of a class of sentences C\
as a simultaneous assertion of all the sentences belonging to C\;
therefore, we make the following definition: a class of sentences
Ci is called true if all sentences of C x are true; false, if at least
one of them is false. T x and l\ (i.e., two sentences, or two
classes of sentences, or one sentence and one class) are called
equivalent with each other, if either both are true or both are
false. T 2 is called an implicate of T u if Ti is false or T 2 is true.
Ti is said to exclude T 2 if not both are true.

7. L-Semantical Terms

Let us compare the following two sentences: ‘Australia is
large (Si) and ‘Australia is large or Australia is not large’ (S 2 ).
We see that they have a quite different character; let us try
to give an exact account of their difference. We learn S x in
geography but S 2 in logic. In order to find out for each of these
sentences whether it is true or false, we must, of course, first
understand the language to which it belongs. Then, for Si
we have to know, in addition, some facts about the thing whose
name occurs in it, i.e., Australia. Such is not the case for S 2 .
Whether Australia is large or small does not matter here; just
by understanding S 2 we become aware that it must be right.
If we agree to use the same term ‘true’ in both cases, we may ex-
press their difference by saying that S x is factually (or empirical-
ly) true while S 2 is logically true. These unprecise explanations
can easily be transformed into precise definitions by replacing

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L-Semantical Terms

the former reference to understanding by a reference to seman-
tical rules. We call a sentence of a semantical system S (logi-
cally true or) L-true if it is true in such a way that the semantical
rules of S suffice for establishing its truth. We call a sentence
(logically false or) L-false if it is false in such a way that the
semantical rules suffice for finding that it is false. The two terms
just defined and all other terms defined on their basis we call
L-semantical terms. If a sentence is either L-true or L-false, it
is called L-deter minute, otherwise (L-indeterminate or) factual.
(The terms ‘L-true’, ‘L-false’, and ‘factual’ correspond to the
terms ‘analytic’, ‘contradictory’, and ‘synthetic’, as they are
used in traditional terminology, usually without exact defini-
tions.) If a factual sentence is true, it is called (factually true
or) F-true; if it is false, (factually false or) F-false. Every sen-
tence which contains only logical signs is L-determinate. This
is one of the chief characteristics distinguishing logical from de-
scriptive signs. (Example: ‘For every object x and every prop-
erty F, if x is an F then x is an F’ is L-true. There are no sen-
tences of this kind in the system B-S.)

Classification of sentences of a semantical system:

true false

L-true F-true F-false L-false

factual

Examples of sentences in B-S: (1) We found earlier (§ 5) that the sentence
‘mond ist blau’ (Si) is true in B-S if and only if the moon is blue. Hence, in
order to find out whether Si is true or false, not only must we know the
rules of B-S but we have to make observations of the moon. Hence Si is not
L-determinate but factual. (2) Let us analyze the sentence ‘wenn mond ist
blau, so mond is blau’ (Si). According to rule SL 3, a ‘wenn-so’ sentence is
true if its first component is not true or its second component is true. Now, if
Si is true, the second component of Si is true, and hence Si is true; and if Si
is not true, then the first component of Si is not true, and hence Si is again
true. Thus Si is true in any case, independently of the facts concerning the
moon; it is true merely in virtue of rule SL 3. Therefore S 2 is L-true. (3) The
sentence ‘nicht, wenn mond ist blau, so mond ist blau’ (S3) has Si as its com-

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ponent; and we found S 2 to be true on the basis of SL 3. Therefore, according
to SL 2, S 3 is not true but false. And, moreover, it is false not because some
fact happens to be the case but merely by virtue of the rules SL 3 and 2.
Hence, S 3 is L-false.

Terminological remark. The use of the word * true ' in everyday language
and in philosophy is restricted by some to factual sentences, while some
others use it in a wider sense, including analytic sentences. We adopted here
the wider use; it is more customary in modern logic (e.g., ‘truth function’,
truth-value-table’), and it turns out to be much more convenient. Otherwise,
we should always have to say in the semantical rules and in most of the se-
mantical theorems ‘true or analytic’ instead of ‘true’. Semantical rules stating
truth-conditions in the sense of ‘F-true’ would become very complicated and
indeed indefinite.

The definitions given can easily be transferred to classes of
sentences. Ci is called L-true if it is possible to find out that
Ci is true with the help of the semantical rules alone, hence if all
sentences of C\ are L-true. C\ is called L-false if it is possible to
find out with the help of the semantical rules that Ci is false,
i.e., that at least one sentence of C\ is false (in this case, how-
ever, all sentences of C x may be factual). If C\ is either L-true or
L-false, it is called L-determinate, otherwise factual.

If the semantical rules suffice to show that T, is an implicate
of T u we call T 2 an L-implicate of TV This relation of L-implica-
tion is one of the fundamental concepts in logical analysis of
language. The criterion for it can also be formulated in this
way: the semantical rules exclude the possibility of T x being
true and T 2 false; or in this way: according to the semantical
rules, if T x is true, l\ must be true. This last formulation of
the criterion shows that L-implication, as defined here, is
essentially the same as what is usually called logical conse-
quence or deducibility or strict implication or entailment, al-
though the form of the definitions of these terms may be differ-
ent. Our definition is a semantical one as it refers to the se-
mantical rules. Later we shall discuss the possibility of defining
a corresponding syntactical term.

Examples: (1) ‘mond ist rot’ (Si); ‘wenn mond ist rot, so titisee ist kalt’
(S 2 ), titisee ist kalt (S 3 ). We shall see that Sj is an L-implicate of the class
Ci consisting of Si and S 2 . According to the definition of ‘implicate’ (§ 6), if
S 3 is true, S 3 is an implicate of Ci. The same holds if Si is false because C\ is

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L-Semantical Terms

then also false. The only remaining case is that Si is true and S3 is false. In this
case, according to rule SL 3 (§ 5 ), S3 is false and, hence, C 1 is false too, and S3
is an implicate of C 1. Thus we have found, without examining the facts de-
scribed by the sentences, and merely by referring to the semantical rules,
that S 3 is an implicate of Ci. Therefore, S 3 is an L-implicate of C\. (2) ‘fuer
jedes x, x ist blau’ (S<); ‘mond ist blau’ (S5). We shall see that S5 is an L-
implicate of S 4 . If S 6 is true, S 6 is an implicate of S t . And if S 6 is not true, then
according to SL 4 (§ 5 ), S< is not true, and, hence, Sj is again an implicate of
S t . We found this result by merely referring to a semantical rule. Therefore,
Si is an L-implicate of S 4 .

Ti and T 2 are said to be L-equivalent if the semantical rules
suffice to establish their equivalence, in other words, if Z\ and
T 2 are L-implicates of each other. L-equivalent sentences have
the same truth conditions; therefore, they say the same thing,
although the formulations may be quite different.

Example: ‘mond ist kalt’ (Si); ‘nicht, mond ist kalt’ (S 2 ); ‘nicht, nicht,
mond ist kalt’ (S3). These sentences are factual; the semantical rules do not
suffice for finding out their truth or falsity. But they suffice for showing that
Si and S3 are equivalent. If Si is true, S 2 is, according to SL 2 (§ 5 ), false,
and hence S3 true. Therefore, in this case, Si and S3 are equivalent. And, if

51 is false, then S 2 is true and S3 is false; hence, Si and St are again equivalent.
Thus, on the basis of the semantical rules. Si and S3 cannot be other than
equivalent. Therefore they are L-equivalent.

If Si is an L-true sentence, then the truth of Si can be estab-
lished without any regard to the facts, e.g., to the properties of
the things whose names occur in Si. Therefore, Si does not con-
vey any information about facts; this is sometimes formulated
by saying that an L-true sentence has no factual content. Sup-
pose S 2 to be an L-implicate of the class of sentences C\. Then

5 2 is an implicate of C 1, and hence, if the sentences of C\ are true,
S 2 is also true; and, moreover, this relation between C\ and S 2
can be found to hold without taking into account any facts.
Therefore, S 2 does not furnish any new information concerning
facts that were not already given by C\. This is sometimes ex-
pressed by saying that logical deduction does not increase the
factual content of the premisses. The two characteristics just
explained of L-truth and L-implication (which have been espe-
cially emphasized by Wittgenstein) are very important for a
clear understanding of the relation between logic and empirical

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knowledge. We shall see later that they hold also for mathe-
matical theorems and mathematical deductions even if applied
in empirical science (§ 19 ).

8. Logical Syntax

We distinguished three factors in the functioning of language:
the activities of the speaking and listening persons, the desig-
nata, and the expressions of the language. We abstracted from
the first factor and thereby came from pragmatics to semantics.
Now we shall abstract from the second factor also and thus
proceed from semantics to syntax. We shall take into consid-
eration only the expressions, leaving aside the objects, proper-
ties, states of affairs, or whatever may be designated by the ex-
pressions. The relation of designation will be disregarded en-
tirely. As this relation is the basis of the whole semantical sys-
tem, it might seem as if nothing would be left. But we shall soon
see that this is not the case.

A definition of a term in the metalanguage is called formal if
it refers only to the expressions of the object-language (or, more
exactly , to the kinds of signs and the order in which they occur
in the expressions) but not to any extralinguistic objects and
especially not to the designata of the descriptive signs of the
object-language. A term defined by a formal definition is also
called formal, as are questions, proofs, investigations, etc., in
which only formal terms occur. We call the formal theory of an
object-language, formulated in the metalanguage, the syntax of
the object-language (or the logical syntax, whenever it seems
necessary to distinguish this theory from that part of linguistics
which is known as syntax but which usually is not restricted to
formal terms). A formal definition, term, analysis, etc., is then
also called syntactical.

The definitions of all semantical terms refer directly or indi-
rectly to designata. But some of these terms — e.g., ‘true’, ‘L-
true , L-implicate’ are attributed not to designata but only
to expressions; they designate properties of, or relations be-
tween, expressions. Now our question is whether it is possible

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Logical Syntax

to define within syntax, i.e., in a formal way, terms which cor-
respond more or less to those semantical terms, i.e., whose exten-
sions coincide partly or completely with theirs. The develop-
ment of syntax — chiefly in modern symbolic logic — has led to an
affirmative answer to that question. Especially is the possibility
of defining in a formal way terms which completely correspond
to ‘L-true’ and ‘L-implicate’ of fundamental importance. This
shows that logical deduction can be completely formalized.

A syntactical system or calculus (sometimes also called a for-
mal deductive system or a formal system) is a system of formal
rules which determine certain formal properties and relations of
sentences, especially for the purpose of formal deduction. The
simplest procedure for the construction of a calculus consists
in laying down some sentences as primitive sentences (some-
times called postulates or axioms) and some rules of inference.
The primitive sentences and rules of inference are used for two
purposes, for the construction of proofs and of derivations. We
shall call the sentences to which the proofs lead C-true sentences
(they are often called provable or proved sentences or theorems
of the calculus). A derivation leads from any not necessarily
C-true sentences, called the premisses, to a sentence, called the
conclusion. We shall call the conclusion a C-implicate of the
class of premisses (it is sometimes called derivable or derived or
[formally] deducible or deduced from the premisses or a [formal]
consequence of the premisses). A calculus may (but usually
does not) also contain rules which determine certain sentences
as C-false. If the rules of a calculus determine some sentence as
both C-true and C-false, the calculus is called inconsistent;
otherwise consistent. (If, as is usually done, no rules for ‘C-false’
are given, the calculus cannot be inconsistent.) In order to ex-
plain this procedure, we shall construct the calculus B-C as an
example.

Logical syntax has chiefly grown out of two roots, one being formal logic,
founded by Aristotle, the other the axiomatic method, initiated by Euclid.
The general idea of operations with calculi goes back to Leibniz; since the
middle of the last century it has been developed in the systems of symbolic
logic into a comprehensive discipline. Among the founders of symbolic logic,
or logistic, Boole (1854) is especially to be mentioned. More comprehensive

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systems (including the higher functional calculus [see § 14]) were created by
Schroeder (1890), Frege (1893), Peano (1895), and Whitehead and Russell
(1910). Frege was the first to formulate explicitly and to fulfil strictly the re-
quirement of formality, i.e., of a formulation of rules of logic without any ref-
erence to designata. Hilbert considerably developed the axiomatic method,
in its application both to geometry (see § 21) and to classical mathematics
(see §§18 and 20).

9. The Calculus B-C

While the sentences of a semantical system are interpreted,
assert something, and therefore are either true or false, within a
calculus the sentences are looked at from a purely formal point
of view. In order to emphasize this distinction, we sometimes
call sentences as elements of a semantical system propositions
and as elements of a calculus formulas.

We constructed earlier a semantical system B-S on the basis
of the language B, but not, as we have seen, uniquely deter-
mined by B. Analogously, we shall now construct a calculus
B-C on the basis of B. As preliminary steps for the construc-
tion of the syntactical rules proper, which we shall then call
rules of transformation, we have to make a classification of the
signs of B-C and to lay down syntactical rules of formation F c
1-4. But they correspond exactly to the classification and the
rules of formation F 1-4 of B-S (§5); these rules were already
formal. Therefore we shall not write them down again.

Calculus B-C. Rules of Transformation:

PS. A sentence of B-C is called a primitive sentence of B-C, if
it has one of the following forms, PS 1-4:

PS 1. ‘wenn . . . , so [wenn nicht .... so ]’.

PS 2. ‘wenn [wenn nicht . . . , so . . .], so . .

PS 3. wenn [wenn . . . , so ], so [wenn [wenn , so . - .

so [wenn . . . , so . - .

PS l ‘wenn [fuer jedes . so - . - . here \ is a

variable, - . - . - is a sentence which does not contain
fuer jedes but contains a name one or several

times, and is an expression constructed out of

■ ’ • by replacing . - . at one or several (not necessari-
ly all) places by the variable ‘. (Examples: [1] ‘wenn

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The Calculus B-C

[fuer jedes x, x ist rot], so mond ist rot’; [2] see sentence
(3) in the first example of a derivation, at the end of this
section.)

R. Rules of Inference: The relation of direct derivability holds
if and only if one of the following conditions is fulfilled.

R 1. Rule of Implication: From ‘wenn . . . , so ’ and

‘. . .’, ‘ ’ is directly derivable in B-C.

R 2. Rule of Synonymity: The words ‘titisee’ and ‘rumber’ may
be exchanged at any place (i.e., if S 2 is constructed out of
Si by replacing one of those words at one place by the
other one, then S 2 is directly derivable from Si in B-C).

A proof in B-C is a sequence of sentences of B-C such that each
of them is either a primitive sentence or directly derivable from
one or two sentences preceding it in the sequence. A sentence
Si of B-C is called provable in B-C if it is the last sentence of a
proof in B-C. A sentence of B-C is called C-true in B-C if and
only if it is provable in B-C; a sentence ‘. . .’ is called C -false
in B-C if and only if ‘nicht . . .’ is provable in B-C. (For B-C,
provability and C-truth coincide, and likewise derivability and
C-implication; for other calculi, this is in general not the case, as
we shall see.)

A derivation in B-C with a class C i of premisses is a sequence
of sentences of B-C such that each of them is either a sentence of
Cj or a primitive sentence or directly derivable from one or two
sentences preceding it in the sequence. The last sentence of a
derivation is called its conclusion. S 2 is called derivable from Ci
and also a C-implicate of Ci if it is the conclusion of a derivation
with the class of premisses Ci.

Both the rules of formation and the rules of transformation of
B-C do not in any way refer to designata; they are strictly for-
mal. Nevertheless, they have been chosen with regard to B-S
in such a way that the extension of the terms ‘C-true’, ‘C-false’,
and ‘C-implicate’ in B-C coincides with that of ‘L-true’,
‘L-false’, and ‘L-implicate’, respectively, in B-S. There are an
infinite number of other possible choices of primitive sentences
and rules of inference which would lead to the same result. This

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result gives the practical justification for our choice of the rules
of B-C. A calculus in itself needs no justification; this point will
be discussed later.

The calculus B-C corresponds to a restricted form of the so-called lower
functional calculus, as constructed by Hilbert and Bernays. PS 1-3 and R 1
correspond to the so-called sentential calculus. That the lower functional
calculus is complete, i.e., that it exhausts the extension of L-truth and L-im-
plication, has been shown by Godel.

Example of a proof in B-C. If in the following sequence the blank . .’ is
always replaced by the same sentence, e.g., ‘titisee ist blau’, the sequence fulfils
the conditions— as shown by the remarks on the left side— and therefore is a
proof. Hence any sentence of the form ‘wenn . . . , so . . .’ is provable and
C-true in B-C, e.g., Venn titisee ist blau, so titisee ist blau’.

PS 1 wenn . . . , so [wenn nicht . . . , so

PS 2 wenn [wenn nicht . . . , so . . .], so

PS 3 wenn [wenn .... so [wenn nicht .

so [wenn [wenn [wenn nicht . . . , so . . .
so [wenn . . . , so . . .]J

(here, Venn nicht . . . , so . . .’ has been taken for ‘ ’, and

‘. . .’ for ‘. - . -’)

(1) (3) R 1 wenn [wenn [wenn nicht . . . , so

so [wenn . . . , so . . .]

(2) (4) R 1 wenn . . . , SO . . .

, so .

*•••].

.], so . . .],

( 1 )

( 2 )

( 3 )

( 4 )

( 5 )

First example of a derivation in B-C:

Premisses

PS 4

titisee ist blau

fuer jedes x, [wenn x ist blau, so x ist kalt]
wenn [fuer jedes x, [wenn x ist blau, so x
ist kalt]], so [wenn titisee ist blau, so
titisee ist kalt]

(2) (3) R 1 wenn titisee ist blau, so titisee ist kalt
(1)(4) R 1 Conclusion: titisee ist kalt

0 )

( 2 )

( 3 )

( 4 )

( 5 )

If we interpret these sentences as in B-S, (1) says that a certain object is
blue, (2) says that all blue things are cold (see example [2] at the end of § 5),
(5) says that that object is cold. Here, however, the conclusion is derived from
the premisses in a formal way, i.e., without making use of an interpretation.

Second example of a derivation in B-C:

Premisses / Wenn mon< * ist b ^ au - so m °nd ist kalt
[nicht mond ist kalt

Provable: wenn [wenn mond ist blau, so mond ist kalt], so

[wenn nicht mond ist kalt, so nicht mond ist blau]

( 1 )

(2)

( 3 )

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Calculus and Semantical System

(1) (3) R 1 wenn nieht mend ist k. !t, so nicht mond ist blau (4)

(2) (4) R 1 Conclusion: nicht mono ist blau (5)

(3) is a provable sentence. To save space, we do not give its proof here.
Suppose that the proof of (3) has been constructed earlier, then the example
shows how its result can be used in a derive on. According to the definitions
previously given for ‘proof’ and ‘derivation any proof may also occur as a
part of a derivation. If this happens, we can abbreviate the derivation; we
write in the derivation not all the sentences of the proof, whose last sentence
we intend to use, but only this one sentence, as we have done in the example
given with sentence (3). In this way a sentence which has been proved once
can be used in derivations again and again. Later, in the discussion of the ap-
plication of calculi in empirical science we shall come back to this application
of proved sentences in derivations (§ 19).

II. Calculus and Interpretation

10. Calculus and Semantical System

We shall investigate the relations which may hold between
a calculus and a semantical system. Sometimes we shall use
as examples the calculus B-C and the semantical system B-S as
discussed before. Suppose a calculus is given — it may be desig-
nated by ‘Z-C’ or briefly ‘C’ — and a semantical system — desig-
nated by ‘Z-S’ or ‘S’. We call S an interpretation of C if the
rules of S determine truth criteria for all sentences of C; in
other words, if to every formula of C there is a corresponding
proposition of S; the converse is not required.

Suppose S fulfils the following condition: for any T u T 2 , T 3 ,
and T a , if J\ is a C-implicate of 7\ in C, T 2 is an implicate of
Ti in S; if T 3 is C-true in C, it is true in S; if T 4 is C-false in C,
it is false in S. If an interpretation S of C fulfils the condition
stated, we call it a true interpretation of C; otherwise sl false in-
terpretation. If the semantical rules suffice to show that S is a
true interpretation of C, then we call S an L-true interpretation
of C. In this case C-implication becomes L-implication; every
C-true sentence becomes L-true, and every C-false sentence be-
comes L-false. If, on the other hand, these semantical rules
suffice to show that S is a false interpretation, we call S an
L-false interpretation. If S is an interpretation but neither an

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Foundations of Logic and Mathematics

L-true nor an L-false interpretation of C, we call S a factual
interpretation of C. In this case, in order to find out whether
the interpretation is true, we have to find out whether some fac-
tual sentences are true; for this task we have to carry out em-
pirical investigations about facts. An interpretation S of C is
called a logical interpretation if all sentences of C become logical
sentences of S (i.e., sentences containing logical signs only),
otherwise a descriptive interpretation. A logical interpretation is
always L-determinate. Applying these definitions to the system
of our former example: B-S is a true and, moreover, L-true,
and descriptive interpretation of B-C.

The class of the sentences which are C-true in C is, interpreted
by S, a class of assertions; we call it the theory correlated to C
by S. If the interpretation is true, L-true or logical, respective-
ly, the correlated theory is likewise true, L-true or logical, re-
spectively ; the converse does not hold generally.

Previously we had a semantical system B-S and then con-
structed a calculus B-C “in accordance with” B-S. What was
meant by this can now be formulated : we intended to construct
B-C in such a way that B-S is a true interpretation of B-C. It is
easy to see that for any given semantical system S it is possible
to construct a calculus C of that kind. All we have to do is to se-
lect partial domains, as small as we wish, of the extensions of
‘implicate in S’, ‘true in S’, and ‘false in S’ (usually the null class) ,
and then lay down formal definitions of ‘C-implicate’, ‘C-true’,
and possibly ‘C-false’, in such a way that their extensions
correspond to these partial domains. On the other hand, it is
an important problem whether it is possible to construct for a
given system S a calculus C such that C is not only in accord-
ance with S, in the sense explained, but that the extensions of
C-implicate’, ‘C-true’, and (if defined at all) ‘C-false’ coincide
with those of ‘L-implicate’, ‘L-true’, and possibly ‘L-false,’ re-
spectively. If this is the case, we call C an L-exhaustive calculus
with respect to S. Thus B-C is L-exhaustive with respect to
B-S. (We do not define a term for the case that the extensions
of C-implicate , C-true’, and ‘C-false’ coincide with those of
implicate , true , and ‘false’ because that would be impossible

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Calculus and Semantical System

for any somewhat richer language system, e.g., for any language
system of a branch of science.)

In order to answer the question of the possibility of an L-exhaustive cal-
culus, we have to distinguish two fundamentally different kinds of rules of
transformation, which we call finite and transfinite rules. By finite rules we
understand those of the customary kind: primitive sentences and rules of in-
ference each of which refers to a finite number of premisses (in most cases one
or two). Almost all rules used by logicians up to the present time are finite.
Finite rules are applied in the construction of proofs and derivations, of the
usual kind, which are finite sequences of sentences, as we have seen in the
examples in B-C. A rule of transformation is called transfinite if it refers to an
infinite number of premisses. Because of this number being infinite, a trans-
finite rule cannot he used within a proof or derivation; a procedure of deduc-
tion of an entirely new kind is necessary. We call a calculus finite if all its
rules of transformation are finite, otherwise transfinite. It may be remarked
that some logicians reject transfinite rules.

We shall make the following terminological distinction: the terms ‘C-im-
plicate’ and ‘C-true’ are applied generally with respect both to finite and to
transfinite calculi. On the other hand, we shall restrict the corresponding
terms ‘derivable’ and ‘provable’ to finite calculi. Thus we call Ti a C-implicate
of T\ in C, if it is possible to obtain Ti from the premisses T 1 by a procedure
of deduction of any kind in C; and we call Ti C-true if it is possible to obtain
Ti by a procedure of deduction without premisses. If C is a finite calculus —
as, e.g., B-C — the deduction takes the form of a finite sequence of sentences,
either a derivation or a proof. In this case T; is called, moreover, derivable
from Ti , and T 3 is called, moreover, provable.

Now we come back to the problem whether it is possible to construct for a
given semantical system S an L-exhaustive calculus C. The answer can now
be formulated (but not proved here). The answer depends upon the degree of
complexity of S; more precisely, it depends upon whether there are in S a
sentence S 2 and an infinite class of sentences C\ such that S 2 is an L-implicate
of Ci but not an L-implicate of any finite subclass of C 1. (Example. S con-
tains a name for every object of an infinite domain: ‘ai’, ‘ai, ‘a 3 ’, etc. ‘P is
a descriptive predicate. C 1 is the [infinite] class of all sentences of the form
‘. . . is a P' where ‘. . .’ is one of the object names. Si is the sentence ‘for every
x, x is a P’.) If this is not the case, then there is a finite L-exhaustive calculus
C. If, however, it is the case, an L-exhaustive calculus C can be constructed
if and only if transfinite rules are admitted. For, because Ci is infinite, Si can-
not be derivable from Ci. If we decide in a given case to admit transfinite
rules, we have to accept the complications and methodological difficulties
connected with them. It was first shown by Godel that a calculus of the ordi-
nary kind (in our terminology, a finite calculus) cannot be constructed for the
whole of arithmetic.

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11. On the Construction of a Language System

We found earlier that the pragmatical description of a lan-
guage gives some suggestions for the construction of a corre-
sponding semantical system without, however, determining it.
Therefore, there is a certain amount of freedom for the selection
and formulation of the semantical rules. Again, if a semantical
s;y stem S is given and a calculus C is to be constructed in ac-
cordance with S, we are bound in some respects and free in
others. The rules of formation of C are given by S. And in the
construction of the rules of transformation we are restricted by
the condition that C must be such that S is a true interpretation
of C, as discussed before. But this still leaves some range of
choice. We may, for instance, decide that the class of C-true
sentences is to be only a proper subclass of the class of L-true
sentences, or that it is to coincide with that class (as we did in
constructing B-C), or that it is to go beyond that class and com-
prehend some factual sentences, e.g., some physical laws. When
the extensions of C-true and ‘C-implicate’ are decided, there is
still some possibility of choice in the construction of the rule::,
primitive sentences and rules of inference, leading to those
extensions. This choice, however, is not of essential impor-
tance, as it concerns more the form of presentation than the re-
sult.

If we are concerned with a historically given language, the
pragmatical description comes first, and then we may go by
abstraction to semantics and (either from semantics or immedi-
ately from pragmatics) to syntax. The situation is quite differ-
ent if we wish to construct a language (or rather a language sys-
tem, because we lay down rules), perhaps with the intention of
practical application, as for making communications or formu-
lating a scientific theory. Here we are not bound by a previous
use of language, but are free to construct in accordance with our
wishes and purposes. The construction of a language system Z
may consist in laying down two kinds of rules, the semantical
rules (Z-S or briefly S) and the syntactical rules (calculus Z-C
or C). As a common basis for both, according to our former dis-
cussion, we have to make a classification of the signs which we

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On the Construction of a Language System

intend to use and lay down rules of formation Z-F . Z-S consists
of two parts, rules for the descriptive signs (Z-SD or SD) and
rules for the logical signs (Z-SL or SL).

In constructing the system Z, we can proceed in two different
ways — different as to the order of b and C. Here the order is
not unessential, for, if we have chosen some rules arbitrarily, we
are no longer free in the choice of others.

The first method consists in first constructing S and then con-
structing C. We start with a classification of the kinds of signs
which we want, and rules F determining the forms of sentences
which we intend to use. Then we lay down the rules SD; we
choose objects, properties, etc., for which we wish to have direct
designations, and then signs to designate these objects, prop-
erties, etc. Next we construct the rules SL; we choose signs to
be used as logical signs and state for each of them the conditions of
the truth of the sentences constructed with its help. (As men-
tioned before, we may also proceed by indicating the translations
of the sentences containing logical signs, or giving their desig-
nata.) After this we proceed to syntax and construct the calculus
C, e.g., by stating primitive sentences and rules of inference. It
has been explained already that, if S is given or constructed, we
are limited in constructing C in some essential respects, because
C must be such that S is a true interpretation of C; but we are
free in other respects.

The second method for constructing Z is first to construct C
and then S. We begin again with a classification of signs and a
system F of syntactical rules of formation, defining ‘sentence in
C’ in a formal way. Then we set up the system C of syntactical
rules of transformation, in other words, a formal definition of
‘C-true’ and ‘C-implicate’. Since so far nothing has been de-
termined concerning the single signs, we may choose these defi-
nitions, i.e., the rules of formation and of transformation, in any
way we wish. With respect to a calculus to be constructed there
is only a question of expedience or fitness to purposes chosen,
but not of correctness. This will be discussed later.

Then we add to the uninterpreted calculus C an interpreta-
tion S. Its function is to determine truth conditions for the sen-

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Foundations of Logic and Mathematics

tences of C and thereby to change them from formulas to propo-
sitions. We proceed in the following way. It is already deter-
mined by the rules F which expressions are formulas in C. Now
we have to stipulate that each of them is also a proposition in
S. By the syntactical classification of the signs it is not yet com-
pletely settled which signs are logical and which descriptive.
In many cases there is still a considerable amount of freedom of
choice in this respect, as we shall see later in some examples.
After having stated which signs are to be logical and which de-
scriptive, we construct the rules SL for the logical signs. Here
our choice is restricted to some extent by the requirement that
the interpretation must be true.

Finally we establish the rules SD for the descriptive signs.
Here we have to take into account the classification of signs.
We choose the designata for each kind of signs and then for
each sign of that kind. We may begin with individual names.
First we choose a field of objects with which we wish to deal
m the language to be constructed, e.g., the persons of a certain
group, the towns of a certain country, the colors, geometrical
structures, or whatever else. Then we determine for each in-
dividual name, as its designatum, one object of the class chosen.
Then, for each predicate, we choose a possible property of those
objects, etc. In this way, a designatum for every descriptive
sign is chosen. If we decide to make S an L-true interpretation
of C, we have a great amount of freedom for the choice of the
rules SD. Otherwise, we find some essential restrictions. If
some of the C-true formulas are to become factual propositions,
they must be factually true. Therefore, in this case, on the basis
of our factual knowledge about the objects which we have
chosen as subject matter of Z, we have to take care that the in-
terpretations for the descriptive names, predicates, etc., i.e.,
their designata, are chosen in such a way that those factual C-
true sentences are actually true.

12. Is Logic a Matter of Convention?

There has been much controversial discussion recently on the
question whether or not logic is conventional. Are the rules on

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Is Logic a Matter of Convention?

which logical deduction is based to be chosen at will and, hence,
to be judged only with respect to convenience but not to cor-
rectness? Or is there a distinction between objectively right
and objectively wrong systems so that in constructing a system
of rules we are free only in relatively minor respects (as, e.g., the
way of formulation) but bound in all essential respects? Obvi-
ously, the question discussed refers to the rules of an interpreted
language, applicable for purposes of communication; nobody
doubts that the rules of a pure calculus, without regard to any
interpretation, can be chosen arbitrarily. On the basis of our
former discussions we are in a position to answer the question.
We found the possibility — which we called the second method — -
of constructing a language system in such a way that first a
calculus C is established and then an interpretation is given by
adding a semantical system S. Here we are free in choosing the
rules of C. To be sure, the choice is not irrelevant; it depends
upon C whether the interpretation can yield a rich language or
only a poor one.

We may find that a calculus we have chosen yields a lan-
guage which is too poor or which in some other respect seems
unsuitable for the purpose we have in mind. But there is no
question of a calculus being right or wrong, true or false. A true
interpretation is possible for any given consistent calculus (and
hence for any calculus of the usual kind, not containing rules
for ‘C-false’), however the rules may be chosen.

On the other hand, those who deny the conventional charac-
ter of logic, i.e., the possibility of a free choice of the logical
rules of deduction, are equally right in what they mean if not in
what they say. They are right under a certain condition, which
presumably is tacitly assumed. The condition is that the
“meanings” of the logical signs are given before the rules of de-
duction are formulated. They would, for instance, insist that

the rule R 1 of B-C (‘from ‘wenn . . . , so ’ and ‘. . .’, ‘ ’ is

directly derivable’ [§ 9]) is necessary; that it would be wrong
to change it arbitrarily, e.g., into R 1*: ‘from ‘wenn . . ., so
’ and ‘nicht . . ‘ ’ is directly derivable’. What they pre-
sumably mean is that the rule R 1 * is incorrect on the basis of

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Foundations of Logic and Mathematics

the presupposed “meaning” of the signs wenn’, ‘so’, and ‘nicht’.
1 hus they have in mind the procedure which we called the first
method (§11): we begin by establishing the semantical rules
SL or assume them as given obviously this is meant by saying
that the meaning is given — and then we ask what rules of
deduction, i.e., syntactical rules of transformation, would be in
accordance with the presupposed semantical rules. In this order
of procedure, we are, as we have seen, indeed bound in the
choice of the rules in all essential respects. Thus we come to a
reconciliation of the opposing views. And it seems to me that
an agreement should easily be attainable in the other direction
as well. The anti-conventionalists would certainly not deny
that the rule R 1* can also be chosen and can lead to correct re-
sults, provided we interpret the logical signs in a different way

(in the example given, we could interpret ‘wenn . . . , so ’,

e.g., as ‘. . . or ’).

The result of our discussion is the following: logic or the
rules of deduction (in our terminology, the syntactical rules of
transformation) can be chosen arbitrarily and hence are conven-
tional if they are taken as the basis of the construction of the
language system and if the interpretation of the system is later
superimposed. On the other hand, a system of logic is not a mat-
ter of choice, but either right or wrong, if an interpretation of the
logical signs is given in advance. But even here, conventions
are of fundamental importance; for the basis on which logic is
constructed, namely, the interpretation of the logical signs (e.g.,
by a determination of truth conditions) can be freely chosen.’

It is important to be aware of the conventional components
in the construction of a language system. This view leads to an
unprejudiced investigation of the various forms of new logical
systems which differ more or less from the customary form (e.g.,
the intuitionist logic constructed by Brouwer and Ileyting, the
systems of logic of modalities as constructed by Lewis and
others, the systems of plurivalued logic as constructed by Luka-
siewicz and Tarski, etc.), and it encourages the construction of
further new forms. The task is not to decide which of the dif-
ferent systems is the right logic” but to examine their formal

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Elementary Logical Calculi

properties and the possibilities for their interpretation and ap-
plication in science. It might be that a system deviating from
the ordinary form will turn out to be useful as a basis for the
language of science.

III. Calculi and Their Application in Empirical Science

13. Elementary Logical Calculi

For any given calculus there are, in general, many different
possibilities of a true interpretation. The practical situation,
however, is such that for almost every calculus which is actually
interpreted and applied in science, there is a certain interpreta-
tion or a certain kind of interpretation used in the great major-
ity of cases of its practical application. This we will call the
customary interpretation (or kind of interpretation) for the cal-
culus. In what follows we shall discuss some calculi and their
application. We classify them according to their customary in-
terpretation in this way: logical calculi (in the narrower sense),
mathematical, geometrical, and (other) physical calculi. The
customary interpretation of the logical and mathematical calcu-
li is a logical, L-determinate interpretation; that of the geo-
metrical and physical calculi is descriptive and factual. The
mathematical calculi are a special kind of logical calculi, dis-
tinguished merely by their greater complexity. The geometrical
calculi are a special kind of physical calculi. This classification
is rather rough and is only meant to serve a temporary, prac-
tical purpose.

To the logical calculi (in the narrower sense) belong most of
the calculi of elementary structure used in symbolic logic, above
all, the so-called sentential calculus and the so-called lower
functional calculus. The sentential calculus has approximately
the structure of B-C with F 4 and PS 4 omitted. The customary
interpretation corresponds to the rules B-SL 2, 3. The form
mostly used contains, however, only those signs which are logi-
cal in the customary interpretation, corresponding to the Eng-
lish words ‘not’, ‘if’, ‘or’, ‘and’, and the like, and sentential
variables. The lower functional calculus (or predicate calculus)

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Foundations of Logic and Mathematics

contains the sentential calculus and, in addition, general sen-
tences with individual variables, namely, universal sentences
(interpretation: ‘for every *,...’) and existential sentences (in-
terpretation: ‘there is an x such that Within symbolic

ogic, this calculus too is mostly used without descriptive signs
but with three kinds of variables: sentential variables, indi-
vidual variables (as in B-C), and predicate variables. The cus-
tomary interpretation is a logical one, as given by B-SL. In the
case of the logical calculi here explained the customary inter-
pretation is the only one which is ever used practically. (If the
calculi are supplemented in a certain way, it is even the only
possible true interpretation.) Therefore, we shall call it the
normal interpretation of the logical calculus.

If a calculus C is constructed with the intention of using it
mostly or exclusively with a certain interpretation S, it may
often seem convenient to use as signs of C not artificial sym-
bols but those words of the word-language whose ordinary use
is approximately in acccord with the interpretation intended
(a word with exact accordance will usually not be available).
Then we have in C the same sentences as in the interpreted lan-
guage S, which is perhaps to be applied in science; “the same
sentences” as to the wording, but in C they are formulas, while
they are propositions in S. This procedure is mostly chosen in
geometrical and other physical calculi (for examples see end
of § 17, beginning of § 22).

In what follows we shall do the same for the logical calculus
(where, for good reasons, it is usually not done). Thus, instead
of symbols, we shall use the words ‘not’, ‘if’, etc. It has been
shown (by H. M. Sheffer) that two primitive signs are sufficient
namely, ‘excludes’ (to be interpreted later) and ‘for every’. It
is not necessary to take as many primitive signs as we did in
B-C, corresponding to ‘not’, ‘if— then’, ‘for every’. The other
logical signs of the logical calculus can be introduced by defini-
tions. The primitive signs mentioned and all signs defined with
their help are called logical constants. We shall use three kinds
of variables: sentential variables (>’, ‘q’, etc.), individual vari-
ables ( x , V, etc., as in B-C), and predicate variables (‘ F ’,

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Elementary Logical Calculi

‘G’, etc.). For a sentential variable a sentence may be substi-
tuted, for an individual variable an individual name, for a predi-
cate variable a predicate, and for ‘Fx’ an expression of sentential
form containing the variable ‘x’.

A definition is a rule of a calculus which serves for introducing
a new sign. In simpler cases the rule states that the new sign
is to be taken as an abbreviation for a certain expression con-
sisting only of old signs (i.e., primitive signs or signs defined
earlier) . In other cases the rule states that sentences containing
the new sign and old signs are to be taken as abbreviations for
certain sentences containing old signs only. Rules of the first
kind are called explicit definitions (e.g., Defs. 11, 12, and 13 in
§ 14) ; those of the second kind are called definitions in use
(e.g., Defs. 1-7, below); we shall use still another kind of defini-
tion, the so-called recursive definitions frequently found in
arithmetic (e.g., Defs. 14 and 15 in § 14). The definitions in a
calculus are, so to speak, additional rules of transformation,
either primitive sentences or rules of inference, according to
their formulation; they are added in order to provide shorter
expressions. If a calculus C contains definitions and the inter-
pretation S contains semantical rules for the primitive signs of
C, the interpretation of the defined signs need not be given ex-
plicitly. The definitions, together with those rtiles of S, deter-
mine the truth conditions of the sentences containing the de-
fined signs and thereby the interpretation of these signs.

We shall formulate the definitions here in this form: ‘

for ‘ ’ This means that ‘. . is to serve as an abbreviation

for ‘ ’, i.e., that . .’ and ‘ ’, and likewise two expressions

constructed out of ‘. . and ‘ ’ by the same substitutions,

may always be replaced by each other. In this calculus, we
take as simplest form of sentences in the beginning ‘ Fx ’ (e.g.,
‘city Chicago’ instead of ‘Chicago is a city’); the usual form
with ‘is a’ is introduced later by Definition 7.

The expressions included in parentheses serve merely to facili-
tate understanding; in the exact formulation they have to be
omitted. The brackets and commas, however, are essential;
they indicate the structure of the sentence (cf. § 5).

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Foundations of Logic and Mathematics

Def. 1. ‘not p’ for ‘p excludes p' .

Def. 2. ‘p or q for ‘not p, excludes, not q .

Def. 3. ‘p and q’ for ‘not [p excludes g]’.

Def. 4. ‘if p then q’ for ‘not p, or q’.

Def. 5. ‘p if and only if q’ for ‘[if p then g] and [if q then p\’ .

Def. 6. ‘for some x, Fx for ‘not [for every x, not Fx]’.

Def. 7. 'x is an F’ for ‘Fx’.

The rules of transformation of the sentential calculus and the
functional calculus will not be given here. They are not essen-
tially different from those of B-C. It has been shown (by J.
Nicod) that, if ‘excludes’ is taken as primitive sign, one primi-
tive sentence is sufficient for the sentential calculus. For the
lower functional calculus we have to add one more primitive
sentence for ‘for every’, analogous to PS 4 in B-C.

The normal interpretation for the logical calculus is a logical
one. Therefore, if interpreted, it is, so to speak, a skeleton of a
language rather than a language proper, i.e., one capable of
describing facts. It becomes a factual language only if sup-
plemented by descriptive signs. These are then interpreted by
SD-rules, and the logical constants by SL-rules. As SL-rules
for the lower functional calculus we can state the following two
rules for the two primitive signs. For the sentential calculus the
first rule suffices.

1. A sentence of the form . . . excludes ’is true if and only

if not both ‘. . .’ and ‘ ’ are true.

2. A sentence of the form ‘for every . . . , ’is true if and

only if all individuals have the property designated by
‘- - with respect to the variable ‘. . (The individuals are
the objects of the domain described, which is to be deter-
mined by an SD-rule.)

The interpretation of the defined signs ‘not’, etc., is deter-
mined by rule (1) and Definition 1, etc. The interpretation of
‘not’ and ‘if— then’ is easily seen to be the same as that of
nicht , and wenn so in B-SL. (The truth conditions here
given by rule [1] and Definitions 1-5 are the same as those which
in symbolic logic usually are stated with the help of truth-value
tables for the corresponding symbols, the so-called connectives.)

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Higher Logical Calculi

14. Higher Logical Calculi

The lower functional calculus can be enlarged to the higher
functional calculus by the addition of predicates of higher levels.
The predicates occurring in the lower functional calculus are
now called predicates of first level ; they designate properties of
first level, i.e., properties of individuals. Now we introduce
predicates of second level, which designate properties of second
level, i.e., properties of properties of first level; predicates of
third level designating properties of third level, etc. Further,
new kinds of variables for these predicates of higher levels are
introduced. (In the subsequent definitions we shall use as vari-
ables for predicates of second level ‘m’ and V, for predicates of
third level ‘K’.) Expressions of the form ‘for every . . .’, and
analogously ‘for some . . .’ (Def. 6), are now admitted not only
for individual variables but also for predicate variables of any
level. Some new rules of transformation for these new kinds of
variables have to be added. We shall not give them here. Some
of them are still controversial.

The normal interpretation of the higher functional calculus
can again be given by two semantical rules. Rule (1) is kept, as
the sentential calculus remains the basis for the higher function-
al calculus. Rule (2) must be replaced by the subsequent rule
(2*), because of the extended use of ‘for every’. For individual
variables, (2*) is in accordance with (2). (It may be remarked
that there are some controversies and unsolved problems con-
cerning the properties of higher levels.)

2*. A sentence of the form ‘for every . . . , ’is true if and

only if all entities belonging to the range of the variable

*. . .’ have the property designated by ‘ ’ with respect to

‘. . .’. (To the range of an individual variable belong all
individuals, to the range of a predicate variable of level r
belong all properties of level r.)

To the definitions which we stated in the lower functional
calculus, new ones can now be added which make use of predi-
cates and variables of higher levels. We shall first give some

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Foundations of Logic and Mathematics

rough explanations of the new expressions and later the defini-
tions. First, identity can be defined; ‘x = y’ is to say that * is the
same object as y; this is defined by ‘x and y have all properties in
common (Def. 8). Then we shall define the concept of a car-
dinal number of a property, restricting ourselves, for the sake
of simplicity, to finite cardinal numbers. ‘F is an m’ is to say
that the property F has the cardinal number m; i.e., that there
are m objects with the property F. This concept is defined by a
recursive definition (for finite cardinals only). ‘F is a 0’ is de-
fined as saying that no object has the property F (Def. 9a).
Then ‘F is an m+’, where ‘m+’ designates the next cardinal num-
ber greater than m, i.e., m+1, is defined in the following way
in terms of in : there is a property G with the cardinal number
m such that all objects which have the property G, and, in addi-
tion, some object x, but no other objects, have the property F
(Def. 96). A property K of numbers is called hereditary if,
whenever a number m is a K, m+1 is also a K. Then ‘m is a
finite cardinal number can be defined (as Frege has shown) in
this way: m has all hereditary properties of 0 (Def. 10). The
numerals T, ‘2’, etc., can easily be defined by ‘0+’, T+\ etc.
(Def. 11, etc.). The sum (‘m+n’) and the product (‘mXn’) can
be defined by recursive definitions, as is customary in arith-
metic (Defs. 14 and 15).

Def. 8.

‘x-y’ for ‘for every (property) F, if x is an F then y is an F.
Analogously for any higher level.

9a. ‘F is a 0’ for ‘not [for some x, x is an F]’.

b. F is an to+ for for some G, for some x, tor every y \[y is an F if
and only if [y is a G or y = x}\ and G is an to and, not x is a G].
to is a finite cardinal number’ for ‘for every (property of numbers)
K, if [0 is a K and, for every n [if n is a K then n+ is a /fll then m
is a K\

Def. 11. ‘1’ for‘0+’.

Def. 12. ‘2’ for ‘1+’.

Def. 13. ‘3’ for ‘2+’.

Def.

Def. 10

Analogously for any further numeral.
Def. 14a. ‘to+ 0’ for ‘m’.

b. ‘m-\-n+’ for '[m-\-n] Jr ’ .

Def. 15a. ‘toXO’ for ‘O’.

b. ‘mXn+’ for ‘[toXw]+to\

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Application, of Logical Calculi

For the reasons mentioned before we have used, instead of
arbitrary symbols, words whose ordinary use agrees approxi-
mately with the interpretation intended. It is, however, to be
noticed that their exact interpretation in our language system
is not to be derived from their ordinary use but from their defi-
nition in connection with the semantical rules (1) and (2*).

We see that it is possible to define within the logical calculus
signs for numbers and arithmetical operations. It can further
be shown that all theorems of ordinary arithmetic are provable
in this calculus, if suitable rules of transformation are estab-
lished.

The method of constructing a calculus of arithmetic within a logical cal-
culus was first found by Frege (1884) and was then developed by Russell
(1903) and Whitehead (1910). (Defs. 9-15 are, in their essential features, in ac-
cordance with Frege and Russell, but make use of some simplifications due to
the recent development of symbolic logic.) We shall later outline another
form of an arithmetical calculus (§ 17) and discuss the problem of mathe-
matics more in detail (§ 20).

15. Application of Logical Calculi

The chief function of a logical calculus in its application to
science is not to furnish logical theorems, i.e., L-true sentences,
but to guide the deduction of factual conclusions from factual
premisses. (In most presentations of logical systems the first
point, the proofs, is overemphasized; the second, the derivations,
neglected.)

For the following discussions we may make a rough distinc-
tion between singular and universal sentences among factual
sentences. By a singular sentence of the language of science or
of an interpreted calculus we mean a sentence concerning one or
several things (or events or space-time-points), describing, e.g.,
a property of a thing or a relation between several things. By
a universal sentence we mean a sentence concerning all objects
of the field in question, e.g., all things or all space-time-points.
A report about a certain event or a description of a certain land-
scape consists of singular sentences; on the other hand, the so-
called laws of nature in any field (physics, biology, psychology,
etc.) are universal. The simplest kind of an application of the

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Foundations of Logic and Mathematics

logical calculus to factual sentences is the derivation of a singu-
lar sentence from other singular sentences (see, e.g., the second
example of a derivation in B-C, end of § 9). Of greater prac-
tical importance is the deduction of a singular sentence from
premisses which include both singular and universal sentences
We are involved in this kind of a deduction if we explain a
known fact or if we predict an unknown fact. The form of the
deduction is the same for these two cases. We have had this
form in the first example of a derivation in B-C (§ 9); we find
it again in the following example, which contains, besides signs
of the logical calculus, some descriptive signs. In an application
of the logical calculus, some descriptive signs have to be intro-
duced as primitive; others may then be defined on their basis.
SD-rules must then be laid down in order to establish the inter-
pretation intended by the scientist. Premiss (3) is the law of
thermic expansion in qualitative formulation. In later examples
we shall apply the same law in quantitative formulation (£>,
in § 19;£> 2 in §23). V

f 1. c is an iron rod.
Premisses: ( 2. c is now heated.

I 3. for every x, if x is an iron rod and x is heated, x expands.
Conclusion: 4. c now expands.

A deduction of this form can occur in two practically quite
different kinds of situations. In the first case we may have
found (4) by observation and ask the physicist to explain the
fact observed. He gives the explanation by referring to other
facts (1) and (2) and a law (3). In the second case we may have
found by observation the facts (1) and (2) but not (4). Here the
deduction with the help of the law (3) supplies the prediction
(4), which may then be tested by further observations.

The example given shows only a very short deduction, still
more abbreviated by the omission of the intermediate steps be-
tween premisses and conclusion. But a less trivial deduction
consisting of many steps of inference has fundamentally the
same nature. In practice a deduction in science is usually made
by a few jumps instead of many steps. It would, of course, be

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Nonlogical Calculi

practically impossible to give each deduction which occurs the
form of a complete derivation in the logical calculus, i.e., to dis-
solve it into single steps of such a kind that each step is the ap-
plication of one of the rules of transformation of the calculus,
including the definitions. An ordinary reasoning of a few sec-
onds would then take days. But it is essential that this dissolu-
tion is theoretically possible and practically possible for any
small part of the process. Any critical point can thus be put
under the logical microscope and enlarged to the degree de-
sired. In consequence of this, a scientific controversy can be
split up into two fundamentally different components, a factual
and a logical (including here the mathematical). With respect
to the logical component the opponents can come to an agree-
ment only by first agreeing upon the rules of the logical calculus
to be applied and the L-semantical rules for its interpretation,
and by then applying these rules, disregarding the interpreta-
tion of the descriptive signs. The discussion, of course, need
not concern the whole calculus; it will be sufficient to expand
the critical part of the controversial deduction to the degree re-
quired by the situation. The critical point will usually not be
within the elementary part of the logical calculus (to which all
examples of derivations discussed above belong) , but to a more
complex calculus, e.g., the higher, mathematical part of the
logical calculus, or a specific mathematical calculus, or a physi-
cal calculus. This will be discussed later; then the advantage of
the formal procedure will become more manifest.

16. General Remarks about Nonlogical Calculi (Axiom Systems)

In later sections we shall discuss certain other calculi which
are applied in science. The logical calculus explained previous-
ly is distinguished from them by the fact that it serves as their
basis. Each of the nonlogical calculi to be explained later con-
sists, strictly speaking, of two parts : a logical basic calculus and
a specific calculus added to it. The basic calculus could be ap-
proximately the same for all those calculi ; it could consist of the
sentential calculus and a smaller or greater part of the functional
calculus as previously outlined. The specific partial calculus

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does not usually contain additional rules of inference but only
additional primitive sentences, called axioms. As the basic
calculus is essentially the same for all the different specific cal-
culi, it is customary not to mention it at all but to describe only
the specific part of the calculus. What usually is called an
axiom system, is thus the second part of a calculus whose charac-
ter as a part is usually not noticed. For any of the mathematical
and physical axiom systems in their ordinary form it is necessary
to add a logical basic calculus. Without its help it would not be
possible to prove any theorem of the system or to carry out any
deduction by use of the system. Not only is a basic logical cal-
culus tacitly presupposed in the customary formulation of an
axiom system but so also is a special interpretation of the logical
calculus, namely, that which we called the normal interpreta-
tion. An axiom system contains, besides the logical constants,
other constants which we may call its specific or axiomatic con-
stants. Some of them are taken as primitive; others may be de-
fined. The definitions lead back to the primitive specific signs
and logical signs. An interpretation of an axiom system is given
by semantical rules for some of the specific signs, since for the
logical signs the normal interpretation is presupposed. If se-
mantical rules for the primitive specific signs are given, the
interpretation of the defined specific signs is indirectly deter-
mined by these rules together with the definitions. But it is also
possible— and sometimes convenient, as we shall see — to give
the interpretation by laying down semantical rules for another
suitable selection of specific signs, not including the primitive
signs. If all specific signs are interpreted as logical signs, the
interpretation is a logical and L-determinate one; otherwise it
is a descriptive one. (Every logical interpretation is L-deter-
minate; the converse does not always hold.)

17. An Elemenfary Mathematical Calculus

We take here as mathematical calculi those whose customary
interpretation is mathematical, i.e., in terms of numbers and
functions of numbers. As an example, we shall give the classical
axiom system of Peano for (elementary) arithmetic. It is usual-

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ly called an axiom system of arithmetic because in its customary
interpretation it is interpreted as a theory of natural numbers,
as we shall see. This interpretation is, however, by no means the
only important one. The logical basic calculus presupposed has
to include the lower functional calculus and some part of the
higher, up to expressions ‘for every F’ for predicate variables
of first level and Definition 8 for ‘ = ’ (§ 14). The specific primi-
tive signs are ‘b’, ‘N\ ‘ r . (The following axioms, of course, are,
within the calculus, independent of any interpretation. Never-
theless, the reader who is not familiar with them will find it
easier to conceive their form and function by looking at their in-
terpretation given below.)

Axiom, System of Peano:

Pi. b is an N.

P 2. For every x, if x is an N, then x' is an N.

P 3. For every x, y, if [x is an N and y is an .V and x' — y’\ then x = y.

P 4-. For every x, if a; is an N, then, not b = x’.

P 5. For every F, if [6 is an F and, for every x [if x is an F then x' is an F]\ then
[for every y, if y is an N then y is an F].

(Briefly: if F is any property of b which is hereditary [from x to x'\
then all N are F.)

It is easy to see that any number of true interpretations of this
calculus can be constructed. We have only to choose any in-
finite class, to select one of its elements as the beginning mem-
ber of a sequence and to state a rule determining for any given
member of the sequence its immediate successor. (An order of
elements of this kind is called a progression.) Then we inter-
pret in this way: ‘b’ designates the beginning member of the
sequence; if designates a member of the sequence then
‘. . ’ designates its immediate successor; ‘N’ designates the

class of all members of the sequence that can be reached from
the beginning member in a finite number of steps. It can easily
be shown that in any interpretation of this kind the five axioms
become true.

Example: ‘6’ designates August 14, 1938; if ‘. . .’ designates a day, ‘. . . ’’
designates the following day; ‘N' designates the class (supposed to be infinite)
of all days from August 14, 1938, on. This interpretation of the Peano system
is descriptive, while the customary one is logical.

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The customary interpretation of the Peano system may first
be formulated in this way: ‘b’ designates the cardinal number 0;
if designates a cardinal number n, then designates

the next one, i.e., n + 1 ; ‘N’ designates the class of finite cardinal
numbers. Hence in this interpretation the system concerns the
progression of finite cardinal numbers, ordered according to
magnitude. Against the given semantical rule ‘ ‘b’ designates
the cardinal number 0’ perhaps the objection will be raised that
the cardinal number 0 is not an object to which we could point,
as to my desk. This remark is right; but it does not follow that
the rule is incorrect. We shall give the interpretation in another
way, with the help of a translation.

In the investigation of calculi the procedure of translation of
one calculus into another is of great importance. A system of
rules of translation of the calculus K 2 into the calculus K x de-
termines for each primitive sign of K 2 an expression of
called its correlated expression, and for each kind of variable in
K 2 its correlated kind of variable in K y . The rules must be such
that the result of translating any sentence in K 2 is always a sen-
tence in K y . The translation is called C-true if the following
three conditions are fulfilled: (1) every C-true sentence in K 2
becomes, if translated, C-true in K y ; (2) every C-false sentence
in K 2 becomes C-false in K y ; (3) if the relation of C-implication
in K 2 holds among some sentences, then the relation of C-
implication in K y holds among those into which they are
translated. If we have an interpretation I y for the calculus K u
then the translation of K 2 into K y determines in connection
with I y an interpretation I 2 for K 2 . I 2 may be called a secondary
interpretation. If the translation is C-true and the (primary)
interpretation I y is true, I 2 is also true.

We shall now state rules of translation for the Peano system
into the higher functional calculus and thereby give a secondary
interpretation for that system. The logical basic calculus is
translated into itself ; thus w T e have to state the correlation only
for the specific primitive signs. As correlates for i b\ "’, ‘N\ we
take ‘O’, “+’, ‘finite cardinal number’ ; for any variable a variable

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two levels higher. Accordingly, the five axioms are translated
into the following sentences of the logical calculus.

P' 1. 0 is a finite cardinal number.

P' 2. For every m, if m is a finite cardinal number, then m+ is a finite cardinal
number.

P' 3. For every m, n, if [m is a finite cardinal number and n is a finite cardinal
number and m+ = n+] then m = n.

P' 4- For every m, if m is a finite cardinal number, then, not 0 = m+.

P' 5. For every K, if [0 is a K and, for every m [if m is a K then m+ is a A]]
then [for every n, if n is a finite cardinal number then n is a K\.

The customary interpretation of the Peano system can now be
formulated in another way. This interpretation consists of the
given translation together with the normal interpretation of the
higher functional calculus up to the third level. (P' 5 contains a
variable of this level.)

The whole interpretation is thus built up in the following way.
We have two L-semantical rules for the primitive signs ‘ex-
cludes’ and ‘for every’ of the logical calculus, indicating truth
conditions (rules [1] and [2*] in § 14). Then we have a chain of
definitions leading to Definitions 9a and b and 11 for ‘O’, ‘ + ’, and
‘finite cardinal number’ (§ 14). Finally we have rules of trans-
lation which correlate these defined signs of the logical calculus
to the primitive signs ‘b’, and ‘N’ of the Peano system.

If we assume that the normal interpretation of the logical
calculus is true, the given secondary interpretation for the Pe-
ano system is shown to be true by showing that the correlates
of the axioms are C-true. And it can indeed be shown that the
sentences P' 1-5 are provable in the higher functional calculus,
provided suitable rules of transformation are established. As
the normal interpretation of the logical calculus is logical and
L-true, the given interpretation of the Peano system is also
logical and L-true.

We can now define signs within the Peano axiom system
which correspond to the signs ‘O’, T’, etc., “ + ’, etc., of the logi-
cal calculus. For greater clarity we distinguish them by the sub-
script ‘P’. (In an arithmetical calculus, however — whether in
the form of Peano’s or some other — one ordinarily does not use

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arbitrary symbols like ‘b’ or ‘0 P \ ‘b r or ‘l P \ ‘ + P \ etc., but,
because of the customary interpretation, the corresponding
signs of the ordinary language ‘O’, ‘1’, “ + etc.)

Def.Pl. ‘Op for ‘b’.

Def . P Z. ‘lp’forV’.

Def. P 3. ‘2p’ for ‘b"\

Etc.

Def. P fa. ‘x + p Op for V.

b. 'x + P y r for ‘[a: + p y]’\

Def. P 5a. ‘x Xp Op’ for ‘Op’.

b. ‘x Xp l for ‘[rXp y] +p x.

Thus the natural numbers and functions of them can be de-
fined both in the logical calculus and in a specific arithmetical
calculus, e.g., that of Peano. And the theorems of ordinary
arithmetic are provable in both calculi. (Strictly speaking, they
are not the same theorems in the different calculi, but corre-
sponding theorems; if, however, the same signs are used — and,
as mentioned before, this is convenient and usual — then corre-
sponding theorems consist even of the same signs.)

18. Higher Mathematical Calculi

On the basis of a calculus of the arithmetic of natural num-
bers the whole edifice of classical mathematics can be erected
without the use of new primitive signs. Whether a specific calcu-
lus of arithmetic or the logical calculus is taken as a basis does not
make an essential difference, once the translation of the first into
the second is established. It is not possible to outline here the
construction of mathematics; we can make only a few remarks.
There are many different possibilities for the introduction of
further kinds of numbers. A simple method is the following one.
The integers (positive and negative) are defined as pairs of
natural numbers, the fractions as pairs of integers, the real
numbers as classes either of integers or of fractions, the complex
numbers as pairs of real numbers. Another way of introducing
any one of these kinds of numbers consists in constructing a new
specific calculus in which the numbers of that kind are taken as
individuals, like the natural numbers in the Peano calculus.

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This has been done especially for the real numbers. A specific
calculus of this kind can be translated in one way or another
into a more elementary specific calculus or into the logical calcu-
lus. (Example : The individual expressions of a specific calculus of
real numbers may be translated into expressions for classes of
integers or of fractions either in the Peano calculus or in the logi-
cal calculus.) For each of the kinds of numbers, functions (sum-
mation, multiplication, etc.) can be defined. Further, the con-
cept of limit can be defined, and with its help the fundamental
concepts of the infinitesimal calculus, the differential coefficient,
and the integral.

If a mathematical calculus is based on the Peano calculus by
the use of definitions, then its customary interpretation is de-
termined by that of the latter. If, on the other hand, a mathe-
matical calculus is constructed as an independent specific cal-
culus, we can give an interpretation for it by translating it either
into an enlarged Peano system or into an enlarged logical cal-
culus (as indicated above for a calculus of real numbers.) Here
we can scarcely speak of “the” customary interpretation, but
only of the set of customary interpretations. Their forms may
differ widely from one another; but they have in common the
character of logical interpretations. If the interpretation is
given by a translation either into the Peano system with refer-
ence to its customary interpretation or by a translation into the
logical calculus with reference to its normal interpretation, this
character is obvious. In a customary interpretation of a mathe-
matical calculus every sign in it is interpreted as a logical sign,
and hence every sentence consists only of logical signs and is
therefore L-determinate (see §7).

If we choose the form of the construction of mathematics
within the logical calculus, we do not even need a translation;
the interpretation is simply the normal interpretation of the
logical calculus. In this case every mathematical sign is defined
on the basis of the two primitive signs of the logical calculus, and
hence every mathematical sentence is an abbreviation for a sen-
tence containing, besides variables, only those two signs. In
most cases, though, this sentence would be so long that it would

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not be possible to write it down within a lifetime. Therefore,
the abbreviations introduced in the construction of mathe-
matics are not only convenient but practically indispensable.

19. Application of Mathematical Calculi

The application of mathematical calculi in empirical science
is not essentially different from that of logical calculi. Since
mathematical sentences are, in the customary interpretation,
L-determinate, they cannot have factual content; they do not
convey information about facts which would have to be taken
into consideration besides those described in empirical science.
.The function of mathematics for empirical science consists in
Providing, first, forms of expression shorter and more efficient
than non-mathematical linguistic forms and, second, modes of
logical deduction shorter and more efficient than those of ele-
mentary logic.

Mathematical calculi with their customary interpretation are
distinguished from elementary logical calculi chiefly by the oc-
currence of numerical expressions. There are two procedures in
empirical science which lead to the application of numerical ex-
pressions: counting and measurement (cf. Lenzen, Vol. I, No. 5,
§§ 4 and 5). Counting is ascertaining the cardinal number of a
class of single, separate things or of events. Measuring is ascer-
taining the value of a magnitude for a certain thing or place at a
certain time. For each physical magnitude, e.g., length, weight,
temperature, electric field, etc., there are one or several methods
of measurement. The result of a measurement is a fraction or a
real number. (Irrational real numbers can also occur, but only
if, besides direct measurement, calculation is applied.) If a de-
duction has to do with results of counting, we may apply, be-
sides an elementary logical calculus, a calculus of elementary
arithmetic. If it has to do with results of measurements, we
may apply a calculus of analysis, i.e., of real numbers.

Let us look at a very simple example of a logico-mathematical
deduction. We apply a certain part of the higher functional cal-
culus and an arithmetical calculus. We presuppose for the fol-
lowing derivation that in this arithmetical calculus the sentence

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Application of Mathematical Calculi

‘3+6 = 9’ (7) has been proved earlier. Whether we take the
arithmetical calculus in the form of a part of the higher func-
tional calculus (as in § 14) or in the form of a specific calculus
(as in § 17) does not make any essential difference ; in both cases
sentence (7) is provable. In order to keep in closer contact with
ordinary language, we use the following definition: ‘there are
to F’ s’ for ‘F is an to’; further, we write ‘n.i.t.r.’ for ‘now in this

1. There are 3 students n.i.t.r.

2. There are 6 girls n.i.t.r.

3. For every x [x is a person n.i.t.r. if and only if [x is a
student n.i.t.r. or x is a girl n.i.t.r.]].

4. For every x [if x is a girl n.i.t.r., then, not x is a stu-
dent n.i.t.r.].

5. For every F, G, H, m, n £if [m and n are finite cardi-
nal numbers and G is an m and H is an n and for
every x [x is an F if and only if, x is a G or x is an H]
and for every y [if y is a G then, not y is an H\] then
F is an m +nj.

[This says that, if a class F is divided into two
parts, G and H, the cardinal number of F is the sum
of the cardinal numbers of G and H .]

6. There are 3 + 6 persons n.i.t.r.

Arithmet. theorem: 7. 3+6 = 9.

(6) (7) Conclusion: 8. There are 9 persons n.i.t.r.

The premisses of this derivation describe some facts empiri-
cally established by observation (including counting). The con-
clusion is also a factual sentence; but its content, the amount of
factual information it conveys, does not go beyond that of the
premisses. We have discussed earlier (at the end of § 9) the
application of proved theorems in a derivation; here (5) and

(7) are examples of this method. These sentences do not con-
tribute to the factual content of the conclusion; they merely
help in transforming the premisses into the conclusion. To say
that the result (8) is “calculated” from the data (l)-(4), means
just this : it is obtained by a formal procedure involving a math-
ematical calculus. The effect of the application of a mathemati-

room

Premisses

Defs. 1-9, 14

(1)(2)(3)(4)(5)

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cal calculus is always, as in this example, the possibility of pre-
senting in a shorter and more easily apprehensible way facts al-
ready known.

Here an objection will perhaps be raised. That the applica-
tion of mathematics consists merely in a transformation of the
premisses without adding anything to what they say about the
facts, may be true in trivial cases like the example given. If,
however, we predict, with the help of mathematics, a future
event, do we not come to a new factual content? Let us discuss
an example of a derivation of this kind. The derivation — called
Di — leads to the prediction of a thermic expansion as in a former
example (§15), but now with quantitative determinations. The
premisses of A relate the results of measurements of the tem-
perature of an iron rod at two time-points and its length at the
first; further, the law of thermic expansion is one of the prem-
isses, but now in quantitative formulation; and, finally, there is
included a statement of the coefficient of thermic expansion.
1 he conclusion states the amount of the expansion of the rod.
We shall not represent A here in detail because a similar deriva-
tion A will be discussed later (§ 23) ; the premisses of A are not
only the sentences (l)-(5) of A but also (6) and (10); the con-
clusion in A is the same as in A- In A a calculus of real num-
bers (or at least of fractions) is applied. The conclusion de-
scribes a fact which has not yet been observed but could be test-
ed by observations. Now, the question is whether the deriva-
tion Di does not lead, with the help of a mathematical calculus,
to a factual content beyond that of the premisses. This might
seem so if we look only at the singular sentences among the
premisses. But two laws also belong to the premisses of A (the
sentences [6] and [10] of A). They are universal; they say that
certain regularities hold not only in the cases so far observed
but at any place at any time. Thus, these sentences are very
comprehensive. The conclusion merely restates what is already
stated by the universal premisses for all cases and hence also for
the present case, but now explicitly for this case. Thus, the
logico-mathematical derivation merely evolves what is implicit-
ly involved in the premisses. To be sure, if we state a new law

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Application of Mathematical Calculi

on the basis of certain observations, the law says much more
than the observation sentences known ; but this is not a deduc-
tion. If, on the other hand, a law is used within a derivation
with the help of a logico-mathematical calculus, then the law
must be among the premisses, and hence the conclusion does
not say more than the premisses. The situation is different in
the application of a physical calculus, as we shall see later
(§23).

On the basis of the presupposed interpretation, the premisses
and the conclusion of the derivation D\ are factual. But A also
contains sentences which are proved in a logico-mathematical
calculus and hence, when interpreted, are L-true, e.g., the sen-
tences which in A occur as (7) and (13) (§ 23). As explained
before, derivations are immensely simplified by the method of
laying down for any future use certain partial sequences occur-
ring in many derivations and containing only provable sentences.
Each sequence of this kind is a proof of its last sentence; wher-
ever it occurs in other proofs or derivations it may be represent-
ed by its last element, i.e., the theorem proved. Thus a logical
or mathematical theorem is, regarded from the point of view of
its application in empirical science, a device or tool enabling us to
make a very complex and long chain of applications of the rules
of the calculus by one stroke, so to speak. The theorem is itself,
even when interpreted, not a factual statement but an instru-
ment facilitating operations with factual statements, namely,
the deduction of a factual conclusion from factual premisses.
The service which mathematics renders to empirical science
consists in furnishing these instruments; the mathematician not
only produces them for any particular case of application but
keeps them in store, so to speak, ready for any need that may
arise.

It is important to notice the distinction between ‘primitive sentence’ and
‘premiss’. A primitive sentence of a calculus C (no matter whether it belongs
to the basic calculus or is one of the specific axioms, and no matter whether,
in an interpretation, it becomes L-true or factual) is stated as C-true by the
rules of the calculus C. Therefore, it has to become a true proposition in any
adequate (i.e., true) interpretation of C. The premisses of a derivation D in
C, on the other hand, need not be C-true in C or true in a true interpretation

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Foundations of Logic and Mathematics

of C. It is merely shown by D that a certain other sentence (the conclusion of
D) is derivable from the premisses of D and must therefore, in a true inter-
pretation, be true if the premisses happen to be true; but whether this is the
case is not determined by D.

20. The Controversies over "Foundations" of Mathematics

There have been many discussions in modern times about the
nature of mathematics in general and of the various kinds of
numbers, and, further, about the distinction and relationship be-
tween knowledge in mathematics and knowledge in empirical
science. In the course of the last century, mathematicians
found that all mathematical signs can be defined on the basis of
the signs of the theory of natural numbers.

The fundamental concepts of the infinitesimal calculus (differential co-
efficient and integral) were defined by Cauchy and Weierstrass in terms of the
calculus of real numbers, with the help of the concept ‘limit (of a sequence of
real numbers)’. Thereby they succeeded in entirely eliminating the dubious
concept of “infinitely small magnitudes” and thus giving the infinitesimal
calculus a rigorous basis in the theory of real numbers. The next step was
made by Frege and Russell, who defined real numbers as classes of natural
numbers or of fractions. (Fractions can easily be defined as pairs of natural
numbers.)

The reduction mentioned was entirely inside of mathematics.
Therefore, it left the more general and fundamental problems
unanswered. These have been discussed especially during the
last fifty years, usually under the heading “foundations of
mathematics”. Among the different doctrines developed in this
field, three are outstanding and most often discussed; they are
known as logicism, formalism, and intuitionism. We will indi-
cate briefly some characteristic features of the three movements.
Logicism was founded by Frege and developed by Russell and
Whitehead. Its chief thesis is that mathematics is a branch of
logic. This thesis was demonstrated by constructing a system
for the whole of classical mathematics within a logical calculus
(see § 14 and some remarks in § 18). Truth conditions for the
primitive signs of the logical calculus were given; thereby an
interpretation for the whole mathematical system was deter-
mined. In this interpretation all mathematical signs became logi-
cal signs, all mathematical theorems L-true propositions.

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Foundations" of Mathematics

Formalism, founded by Hilbert and Bernays, proposed, in
contradistinction to logicism, to construct the system of classical
mathematics as a mere calculus without regard to interpreta-
tion. The theory developed is called metamathematics; it is,
in our terminology, a syntax of the language system of mathe-
matics, involving no semantics. Hilbert’s system is a com-
bination of a logical basic calculus with a specific mathematical
calculus using as specific primitive signs ‘0’ and as did
Peano’s system (§ 17). The controversy between the two doc-
trines concerning the question whether first to construct logic
and then mathematics within logic without new primitive signs,
or both simultaneously, has at present lost much of its former
appearance of importance. We see today that the logico-mathe-
matical calculus can be constructed in either way and that it does
not make much difference which one we choose. If the method of
logicism is chosen, constructing the system of mathematics as
a part of the logical calculus, then by the normal interpretation
of the latter we get an interpretation, and moreover the cus-
tomary one, of the former. The formalists have not concerned
themselves much with the question how the mathematical cal-
culus, if constructed according to their method, is to be inter-
preted and applied in empirical science. As already explained
(§ 17), the interpretation can be given by rules of translation
for the specific primitive signs into the logical calculus. Another
way would be to lay down L-semantical rules for these signs,
stating the truth conditions for the descriptive sentences in
which they occur. Formalists do not give an interpretation for
the mathematical calculus and even seem to regard it as im-
possible for the nonelementary parts of the calculus, but they
emphasize very much the need for a proof of the consistency
for the mathematical calculus and even regard it as the chief
task of metamathematics. There is some relation between the
two questions; if a proof of consistency for a calculus can be
given, then a true interpretation and application of the calculus
is logically possible. So far, a proof of consistency has been
given only for a certain part of arithmetic; the most comprehen-
sive one has been constructed by Gentzen (1936).

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Godel has shown (1931) that it is not possible to construct a proof for the
consistency of a calculus C containing arithmetic, within a metalanguage pos-
sessing no other logical means (forms of expression and modes of deduction)
than C. Hilbert s aim was to construct the proof of consistency in a “finitist”
metalanguage (similar to an intuitionist system, see below). At the present,
it is not yet known whether this aim can be reached in spite of Godel’s result.
In any case, the concept of “finitist logic” is in need of further clarification.

The doctrine of intuitionism was originated by Brouwer
(1912) and Weyl (1918) on the basis of earlier ideas of Kroneck-
er and Poincare. This doctrine rejects both the purely formal
construction of mathematics as a calculus and the interpreta-
tion of mathematics as consisting of L-true sentences without
factual content. Mathematics is rather regarded as a field of
mental activities based upon “pure intuition”. A definition, a
sentence, or a deduction is only admitted if it is formulated in
“constructive” terms; that is to say, a reference to a mere pos-
sibility is not allowed unless we know a method of actualizing it.
Thus, for instance, the concept of provability (in the mathe-
matical system) is rejected because there is no method which
would lead, for any given sentence S, either to a proof for S or
to a proof for the negation of S. It is only allowed to call a
sentence proved after a proof has been constructed. For similar
reasons, the principle of the excluded middle, the indirect proof
of purely existential sentences, and other methods are rejected.
In consequence, both elementary logic and classical mathe-
matics are considerably curtailed and complicated. However,
the boundary between the admissible and the nonadmissible is
not stated clearly and varies with the different authors.

Concerning mathematics as a pure calculus there are no sharp
controversies. These arise as soon as mathematics is dealt with
as a system of “knowledge”; in our terminology, as an inter-
preted system. Now, if we regard interpreted mathematics as
an instrument of deduction within the field of empirical knowl-
edge rather than as a system of information, then many of the
controversial problems are recognized as being questions not of
truth but of technical expedience. The question is: Which form
of the mathematical system is technically most suitable for the
purpose mentioned? W r hich one provides the greatest safety?

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Geometrical Calculi

If we compare, e.g., the systems of classical mathematics and of
intuitionistic mathematics, we find that the first is much simpler
and technically more efficient, while the second is more safe
from surprising occurrences, e.g., contradictions. At the pres-
ent time, any estimation of the degree of safety of the system of
classical mathematics, in other words, the degree of plausibility
of its principles, is rather subjective. The majority of mathe-
maticians seem to regard this degree as sufficiently high for all
practical purposes and therefore prefer the application of classi-
cal mathematics to that of intuitionistic mathematics. The lat-
ter has not, so far as I know, been seriously applied in physics by
anybody.

The problems mentioned cannot here be discussed more in detail. Such
discussion is planned for a later volume of this Encyclopedia. A more detailed
discussion can be found in those of the books which deal with mathematics
mentioned in the “Selected Bibliography” at the end of this monograph.

21. Geometrical Calculi and Their Interpretations

When we referred to mathematics in the previous sections, we
did not mean to include geometry but only the mathematics of
numbers and numerical functions. Geometry must be dealt
with separately. To be sure, the geometrical calculi, aside from
interpretation, are not fundamentally different in their char-
acter from the other calculi and, moreover, are closely related to
the mathematical calculi. That is the reason why they too have
been developed by mathematicians. But the customary inter-
pretations of geometrical calculi are descriptive, while those of
the mathematical calculi are logical.

A geometrical calculus is usually constructed as an axiom
system, i.e., a specific calculus presupposing a logical calculus
(with normal interpretation). Such a calculus describes a struc-
ture whose elements are left undetermined as long as we do not
make an interpretation. The geometrical calculi describe many
different structures. And for each structure, e.g., the Euclidean,
there are many different possible forms of calculi describing it.
As an example let us consider an axiom system of Euclidean
geometry. We choose a form having six primitive signs; three

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Foundations of Logic and Mathematics

for classes of individuals, ( P X \ ‘P 2 \ ‘P 3 \ and three for relations,
I , B , R. . We write I(x,y) for the relation I holds between
x and y\ and ‘B(x,y,z)’ for ‘the (triadic) relation B holds for
x,y,z\ We will give only a few examples out of the long series of
axioms :

G 1. For every x, y [if [x is a P x and y is a P,] then, for some z [2 is a P 2 and
I(x,z) and I{y,z)]\.

G 2. For every x [if x is a P 3 then, for some y [y is a P x and, not Z(y,x)]].

G 3. For every x, y, z [if B(x,y,z) then, not B(y,x,z)].

G 4 . For every x, y, z [if [x is a P x and y is a P 2 and z is a P 3 and I(x,z ) and
I(y,z) and, not I{x,y)} then there is (exactly) 1 u such that [n is a P 2 and
I(x,u) and I ( u,z ) and, for every t [if I(t,u) then, not 7(f,y)]]] . (Euclidean
parallel axiom.)

For a geometrical calculus there are many interpretations,
and even many quite different and interesting interpretations,
some of them logical, some descriptive. The customary inter-
pretation is descriptive. It consists of a translation into the
physical calculus (to be dealt with in the next section) together
with the customary interpretation of the physical calculus.
Rules of translation: (1) TV is translated into ‘point’, (2) ‘P 2 ’
into ‘straight line’, (3) ‘P*’ into ‘plane’, (4) ‘I(x,y)’ into ‘x is
lying on y’ (incidence), (5) 'B(x,y,z)’ into ‘the point x is be-
tween the points y and zona straight line’, (6) ‘K(x,y,u,v)’ into
‘the segment x,y is congruent with the segment u,v (i.e., the
distance between x and y is equal to the distance between u and
v) . It is to be noticed that the words point’, etc., are here signs
of the physical calculus in its customary interpretation. Hence
we may think of a point as a place in the space of nature;
straight lines may be characterized by reference to light rays in
a vacuum or to stretched threads; congruence may be char-
acterized by referring to a method of measuring length, etc.
Thus the specific signs of a geometrical calculus are interpreted
as descriptive signs. (On the other hand, the specific signs of a
mathematical calculus are interpreted as logical signs, even if
they occur in descriptive factual sentences stating the results
of counting or measuring; see, e.g., the logical sign ‘3’, defined
by Def. 13, § 14, occurring in premiss [1], § 19.) The axioms and

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Mathematical and Physical Geometry

theorems of a geometrical calculus are translated into descrip-
tive, factual propositions of interpreted physics; they form a
theory which we may call physical geometry, because it is a
branch of physics, in contradistinction to mathematical geom-
etry i.e., the geometrical calculus. As an example, the four
axioms stated above are translated into the following sentences
of the physical calculus (formulated here, for simplicity, in the
forms of ordinary language).

PG 1. For any two points there is a straight line on which they lie.

PG 2. For any plane there is a point not lying on it.

PG 3. If the points x, y, and z lie on a straight line and x is between y and z,
then y is not between x and z.

PG If. If the point x and the line y lie in the plane z, but i not on y, then
there is one and only one line u in the plane z such that x lies on u and
no point is both on u and y (hence u is the parallel to y through x).

22. The Distinction between Mathematical and Physical Geometry

The distinction between mathematical geometry, i.e., the
calculus, and physical geometry is often overlooked because
both are usually called geometry and both usually employ the
same terminology. Instead of artificial symbols like ‘Pi’, etc.,
the words ‘point’, ‘line’, etc., are used in mathematical geometry
as well. The axioms are then not formulated like G 1-4 but like
PG 1-4, and hence there is no longer any difference in formula-
tion between mathematical and physical geometry. This pro-
cedure is very convenient in practice — like the analagous pro-
cedure in the mathematical calculus, mentioned previously—
because it saves the trouble of translating, and facilitates the
understanding and manipulating of the calculus. But it is essen-
tial to keep in mind the fundamental difference between mathe-
matical and physical geometry in spite of the identity of for-
mulation. The difference becomes clear when we take into con-
sideration other interpretations of the geometrical calculus.

Of especial importance for the development of geometry in
the past few centuries has been a certain translation of the
geometrical calculus into the mathematical calculus. This leads,
in combination with the customary interpretation of the mathe-
matical calculus, to a logical interpretation of the geometrical

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Foundations of Logic and Mathematics

calculus. The translation was found by Descartes and is known
as analytic geometry or geometry of coordinates. TV (or in
ordinary formulation, ‘point’) is translated into ‘ordered triple
of real numbers’; ‘P 3 ’ (‘plane’) into ‘class of ordered triples of
real numbers fulfilling a linear equation’, etc. The axioms,
translated in this way, become C-true sentences of the mathe-
matical calculus; hence the translation is C-true. On the basis
of the customary interpretation of the mathematical calculus
the axioms and theorems of geometry become L-true proposi-

1 he difference between mathematical and physical geometry
became clear in the historical development by the discovery of
non-Euc lulean geometry, i.e., of axiom systems deviating from
the Euclidean form by replacing the parallel axiom (G 4) by
some other axiom incompatible with it. It has been shown that
each of these systems, although they are incompatible with one
another, does not contain a contradiction, provided the Euclid-
ean system is free from contradictions. This was shown by giv-
ing a translation for each of the non-Euclidean systems into the
Euclidean system. Mathematicians regarded all these systems
on a par, investigating any one indifferently. Physicists, on the
other hand could not accept this plurality of geometries; they
asked . \\ hich one is true? Has the space of nature the Euclid-

ean or one of the non-Euclidean structures?” It became clear by
an analysis of the discussions that the mathematician and the
p ysicist were talking about different things, although they
themselves were not aware of this in the beginning. Mathe-
maticians have to do with the geometrical calculus, and with
respect to a calculus there is no question of truth and falsity.
Physicists, however, are concerned with a theory of space, i.e
of the system of possible configurations and movements of
odies, hence with the interpretation of a geometrical calculus.

* hen an interpretation of the specific signs is established— and
to a certain extent, this is a matter of choice-then each of the
calculi yields a physical geometry as a theory with factual con-
tent bince they are incompatible, at most one can be true
(truth of a class of sentences [see § 6]). The theories are factual

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Mathematical and Physical Geometry

The truth conditions, determined by the interpretation, refer to
facts. Therefore, it is the task of the physicist, and not of the
mathematician, to find out whether a certain one among the
theories is true, i.e., w'hether a certain geometrical structure is
that of the space of nature. (Of course, the truth of a system of
physical geometry, like that of any other universal factual
sentence or theory, can never be known with absolute certainty
but at best with a high degree of confirmation.) For this pur-
pose, the physicist has to carry out experiments and to see
whether the predictions made with the help of the theory under
investigation, in connection with other theories confirmed and
accepted previously, are confirmed by the observed results of
the experiments. The accuracy of the answer found by the
physicist is, of course, dependent upon the accuracy of the in-
struments available. The answer given by classical physics was
that the Euclidean system of geometry is in accordance with the
results of measurements, within the limits of the accuracy of
observations. Modern physics has modified this answer in the
general theory of relativity by stating that the Euclidean
geometry describes the structure of space, though not exactly,
yet with a degree of approximation sufficient for almost all prac-
tical purposes; a more exact description is given by a certain
non-Euclidean system of geometry. Physical geometry is in its
methods not fundamentally different from the other parts of
physics. This will become still more obvious when we shall see
how other parts of physics can also take the form of cal-
culi (§ 23).

The doctrine concerning geometry acknowledged by most
philosophers in the past century was that of Kant, saying that
geometry consists of “synthetic judgments a priori”, i.e., of
sentences which have factual content but which, nevertheless,
are independent of experience and necessarily true. Kant at-
tributed the same character also to the sentences of arithmetic.
Modern logical analysis of language, however, does not find any
sentences at all of this character. We may assume that the
doctrine is not to be understood as applying to the formulas of
a calculus; there is no question of truth with respect to them

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Foundations of Logic and Mathematics

because they are not assertions; in any case they are not syn-
thetic (i.e., factual). The doctrine was obviously meant to ap-
ply to arithmetic and geometry as theories, i.e., interpreted
systems, with their customary interpretations. Then, however,
the propositions of arithemetic are, to be sure, independent of
experience, but only because they do not concern experience or
facts at all; they are L-true (analytic), not factual (synthetic).
For geometry there is also, as mentioned before, the possibility
of a logico-mathematical interpretation; by it the sentences of
geometry get the same character as those of mathematics. On
the basis of the customary interpretation, however, the sen-
tences of geometry, as propositions of physical geometry, are
indeed factual (synthetic), but dependent upon experience, em-
pirical. The Kantian doctrine is based on a failure to distinguish
between mathematical and physical geometry. It is to this dis-
tinction that Einstein refers in his well-known dictum : “So far
as the theorems of mathematics are about reality they are not
certain; and so far as they are certain they are not about
reality.”

The question is frequently discussed whether arithmetic and
geometry , looked at from the logical and methodological point
of view, have the same nature or not. Now we see that the an-
swer depends upon whether the calculi or the interpreted sys-
tems are meant. There is no fundamental difference between
arithmetic and geometry as calculi, nor with respect to their
possible interpretations; for either calculus there are both logical
and descriptive interpretations. If, however, we take the sys-
tems with their customary interpretation— arithmetic as the
theory of numbers and geometry as the theory of physical
space— then we find an important difference: the propositions
of arithmetic are logical, L-true, and without factual content;
those of geometry are descriptive, factual, and empirical.

23. Physical Calculi and Their Interpretations

The method described with respect to geometry can be ap-
plied likewise to any other part of physics: we can first con-
struct a calculus and then lay down the interpretation intended

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Physical Calculi arid Their Interpretations

in the form of semantical rules, yielding a physical theory as an
interpreted system with factual content. The customary for-
mulation of a physical calculus is such that it presupposes a
logico-mathematical calculus as its basis, e.g., a calculus of real
numbers in any of the forms discussed above (§18). To this
basic calculus are added the specific primitive signs and the
axioms, i.e., specific primitive sentences, of the physical calculus
in question.

Thus, for instance, a calculus of mechanics of mass points can
be constructed. Some predicates and functors (i.e., signs for
functions) are taken as specific primitive signs, and the funda-
mental laws of mechanics as axioms. Then semantical rules are
laid down stating that the primitive signs designate, say, the
class of material particles, the three spatial coordinates of a
particle x at the time t, the mass of a particle x, the class of
forces acting on a particle x or at a space point s at the time t.
(As we shall see later [§ 24], the interpretation can also be given
indirectly, i.e., by semantical rules, not for the primitive signs,
but for certain defined signs of the calculus. This procedure
must be chosen if the semantical rules are to refer only to ob-
servable properties.) By the interpretation, the theorems of the
calculus of mechanics become physical laws, i.e., universal
statements describing certain features of events; they constitute
physical mechanics as a theory with factual content which can
be tested by observations. The relation of this theory to the
calculus of mechanics is entirely analogous to the relation of
physical to mathematical geometry. The customary division
into theoretical and experimental physics corresponds roughly
to the distinction between calculus and interpreted system.
The work in theoretical physics consists mainly in constructing
calculi and carrying out deductions within them; this is essen-
tially mathematical work. In experimental physics interpreta-
tions are made and theories are tested by experiments.

In order to show by an example how a deduction is carried
out with the help of a physical calculus, we will discuss a
calculus which can be interpreted as a theory of thermic expan-
sion. To the primitive signs may belong the predicates ‘Sol’ and

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Foundations of Logic and Mathematics

‘Fe\ and the functors ‘lg’, ‘te\ and ‘th’. Among the axioms may
be A 1 and A 2. (Here, V, ‘/3’ and the letters with subscripts
are real number variables; the parentheses do not contain ex-
planations as in former examples, but are used as in algebra and
for the arguments of functors.)

A 1. For every x, t u t 2 , h, Z 2 , T u T 2 , /3 [if \x is a Sol and lg(x,<i) = l ,

andlg (x,t 2 ) = l 2 and te (x,t l ) = Ti and t e{x,ti)=Ti and th(x) = /8]
thenf 2 = Z 1 X(H-/3X(7 , 2 -7’ 1 ))].

A 2. For every x, if [x is a Sol and x is a Fe] then th(r) = 0.000012.

The customary interpretation, i.e., that for whose sake the
calculus is constructed, is given by the following semantical
rules, ‘lg (x,ty designates the length in centimeters of the body
x at the time t (defined by the statement of a method of meas-
urement); ‘t e(x,t)’ designates the absolute temperature in centi-
grades of x at the time t (likewise defined by a method of
measurement); ‘th(x)’ designates the coefficient of thermic ex-
pansion for the body x; ‘Sol’ designates the class of solid bodies;
‘Fe’ the class of iron bodies. By this interpretation, A 1 and A 2
become physical laws. A 1 is the law of thermic expansion in
quantitative form, A 2 the statement of the coefficient of ther-
mic expansion for iron. As A 2 shows, a statement of a physical
constant for a certain substance is also a universal sentence. Fur-
ther, we add semantical rules for two signs occuring in the sub-
sequent example: the name ‘c’ designates the thing at such and
such a place in our laboratory; the numerical variable V as
time coordinate designates the time-point t seconds after August
17, 1938, 10:00 a.m.

Now we will analyze an example of a derivation within the
calculus indicated. This derivation D 2 is, when interpreted by
the rules mentioned, the deduction of a prediction from prem-
isses giving the results of observations. The construction of the
derivation D 2 , however, is entirely independent of any inter-
pretation. It makes use only of the rules of the calculus, name-
ly, the physical calculus indicated together with a calculus of
real numbers as basic calculus. We have discussed, but not
written down, a similar derivation D y (§ 19), which, however,
made use only of the mathematical calculus. Therefore the

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Physical Calculi and T heir Interpretations

physical laws used had to be taken in D x as premisses. But here
in D 2 they belong to the axioms of the calculus (A 1 and A 2,
occurring as [6] and [10]). Any axiom or theorem proved in a
physical calculus may be used within any derivation in that

calculus without belonging to the premisses of the derivation, in
exactly the same way in which a proved theorem is used within a
derivation in a logical or mathematical calculus, e.g., in the first
example of a derivation in § 19 sentence (7), and in D x (§ 19)
the sentences which in D 2 are called (7) and (13). Therefore
only singular sentences (not containing variables) occur as
premisses in D 2 . (For the distinction between premisses and
axioms see the remark at the end of § 19.)

Derivation Dr.

Premisses

1. c is a Sol.

2. c is a Fe.

3. te(c,0) = 300.

4. te(c,600) =350.

5. lg(c,0) = 1,000.

Axiom A 1

6. For every x, t u t 2 , l u l 2 , T u T 2 , /3 [if [x is a Sol
and lg(x,<!)=Zi and lg(x,Z 2 ) = k and te(x,t l ) = T i
and te(x,< 2 ) = T 2 and th(x) = /3]

then l 2 =kX(l+/3X(T 2 -T 1 ))].

Proved mathem.
theorem :

7. For every l u l 2 , T u T 2 , (3 [/ 2 -Z, = ZiX/ 3X(7’ 2 - 7\)
if and only if Z 2 = ZiX (l +/3X (7j— Pi))].

(6) (7)

8. For every x, t u ... (as in [6]) ... [if [ ]

then l 2 — li ! =l l XdX(T 2 — ft)].

(1)(S)(4)(8)

9. For every l u l 2 , @ [if [th(c) = /3 and lg(c, 0 )=h and
lg(c,600) = Z 2 ] then Z 2 -Zi = Z!X/3X(350-300)].

Axiom A 2

10. For every x, if [x is a Sol and x is a Fe] then
th(x) = 0.000012.

(1) (2) (10)

11. th(c) = 0.000012.

(9)(11)(5)

Proved mathem.

12. For every l u Z 2 [if [lg(c,0) = Zi and lg(c,600) = Z 2 ]
then Z 2 — Zi= 1,000 X0. 000012 X (350— 300)].

theorem: 13. l,000X0.000012X(350-300) = 0.6.

(12) (13) Conclusion: 14. lg(c,600)— lg(c,0) = 0.6.

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Foundations of Logic and Mathematics

On the basis of the interpretation given before, the premisses
are singular sentences concerning the body c. They say that
c is a solid body made of iron, that the temperature of c
was at 10:00 a.m. 300° abs., and at 10:10 a.m. 350° abs., and
that the length of c at 10 :00 a.m. was 1,000 cm. The conclusion
says that the increase in the length of c from 10:00 to 10:10
a.m. is 0.6 cm. Let us suppose that our measurements have con-
firmed the premisses. Then the derivation yields the conclusion
as a prediction which may be tested by another measurement.

An y physical theory, and likewise the whole of phvsics, can
in this way be presented in the form of an interpreted system,
consisting of a specific calculus (axiom system) and a system of
semantical rules for its interpretation; the axiom system is, tac-
itly or explicitly, based upon a logico-mathematical calculus
with customary interpretation. It is, of course, logically pos-
sible to apply the same method to any other branch of science as
well. But practically the situation is such that most of them
seem at the present time to be not yet developed to a degree
which would suggest this strict form of presentation. There is
an interesting and successful attempt of an axiomatization of
certain parts of biology, especially genetics, by Woodger (Vol. I,
No. 10). Other scientific fields which we may expect to be soon
accessible to this method are perhaps chemistry, economics, and
some elementary parts of psychology and social science.

Within a physical calculus the mathematical and the physical
theorems, i.e., C-true formulas, are treated on a par. But there
is a fundamental difference between the corresponding mathe-
matical and the physical propositions of the physical theory, i.e.,
the system with customary interpretation. This difference is
often overlooked, lhat physical theorems are sometimes mis-
taken to be of the same nature as mathematical theorems is per-
haps due to several factors, among them the fact that they con-
tain mathematical symbols and numerical expressions and that
they are often formulated incompletely in the form of a mathe-
matical equation (e.g., A 1 simply in the form of the last equa-
tion occurring in it). A mathematical proposition may contain
only logical signs, e.g., ‘for every m, n, m + n = n + m, or

202

Elementary and Abstract Terms

descriptive signs also, if the mathematical calculus is applied in
a descriptive system. In the latter case the proposition, al-
though it contains signs not belonging to the mathematical
calculus, may still be provable in this calculus, e.g., ‘lg(c) +
lg (d) = lg(d) + lg(c)’ (‘lg’ designates length as before). A phys-
ical proposition always contains descriptive signs, because
otherwise it could not have factual content; in addition, it usual-
ly contains also logical signs. Thus the difference between math-
ematical theorems and physical theorems in the interpreted
system does not depend upon the kinds of signs occurring but
rather on the kind of truth of the theorems. The truth of a
mathematical theorem, even if it contains descriptive signs, is
not dependent upon any facts concerning the designata of these
signs. We can determine its truth if we know only the semanti-
cal rules; hence it is L-true. (In the example of the theorem just
mentioned, we need not know the length of the body c.) The
truth of a physical theorem, on the other hand, depends upon
the properties of the designata of the descriptive signs oc-
curing. In order to determine its truth, we have to make ob-
servations concerning these designata; the knowledge of the
semantical rules is not sufficient. (In the case of A 2, e.g., we
have to carry out experiments with solid iron bodies.) There-
fore, a physical theorem, in contradistinction to a mathematical
theorem, has factual content.

24. Elementary and Abstract Terms

We find among the concepts of physics— and likewise among
those of the whole of empirical science — differences of abstract-
ness. Some are more elementary than others, in the sense that
we can apply them in concrete cases on the basis of observations
in a more direct way than others. The others are more abstract;
in order to find out whether they hold in a certain case, we have
to carry out a more complex procedure, which, however, also
finally rests on observations. Between quite elementary con-
cepts and those of high abstraction there are many intermediate
levels. We shall not try to give an exact definition for ‘degree of
abstractness’; what is meant will become sufficiently clear by

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Foundations of Logic and Mathematics

the following series of sets of concepts, proceeding from ele-
mentary to abstract concepts: bright, dark, red, blue, warm,
cold, sour, sweet, hard, soft (all concepts of this first set are
meant as properties of things, not as sense-data); coincidence;
length; length of time; mass, velocity, acceleration, density,
pressure; temperature, quantity of heat; electric charge, electric
current, electric field; electric potential, electric resistance, co-
efficient of induction, frequency of oscillation; wave function.

Suppose that we intend to construct an interpreted system of
physics — or of the whole of science. We shall first lay down a
calculus. Then we have to state semantical rules of the kind
SD for the specific signs, i.e., for the physical terms. (The SL-
rules are presupposed as giving the customary interpretation of
the logico-mathematical basic calculus.) Since the physical
terms form a system, i.e., are connected with one another,
obviously we need not state a semantical rule for each of them.
For which terms, then, must we give rules, for the elementary or
for the abstract ones? We can, of course, state a rule for any
term, no matter what its degree of abstractness, in a form like
this: ‘the term ‘te’ designates temperature’, provided the meta-
language used contains a corresponding expression (here the
word ‘temperature’) to specify the designgtum of the term in
question. But suppose we have in mind the following purpose
for our syntactical and semantical description of the system of
physics : the description of the system shall teach a layman to
understand it, i.e., to enable him to apply it to his observations
in order to arrive at explanations and predictions. A layman is
meant as one who does not know physics but has normal senses
and understands a language in which observable properties of
things can be described (e.g., a suitable part of everyday non-
scientific English). A rule like ‘the sign ‘P’ designates the prop-
erty of being blue’ will do for the purpose indicated ; but a rule
like ‘the sign ‘Q’ designates the property of being electrically
charged’ will not do. In order to fulfil the purpose, we have to
give semantical rules for elementary terms only, connecting
them with observable properties of things. For our further dis-

204

Elementary and Abstract Terms

cussion we suppose the system to consist of rules of this kind, as
indicated in the following diagram.

first second

method method

abstract terms:

elementary terms:

interpretation of
elementary terms:

Now let us go back to the construction of the calculus. We
have first to decide at which end of the series of terms to start
the construction. Should we take elementary terms as primitive
signs, or abstract terms? Our decision to lay down the semanti-
cal rules for the elementary terms does not decide this question.
Either procedure is still possible and seems to have some reasons
in its favor, depending on the point of view taken. The first
method consists in taking elementary terms as primitive and
then introducing on their basis further terms step by step, up to
those of highest abstraction. In carrying out this procedure, we
find that the introduction of further terms cannot always take
the form of explicit definitions; conditional definitions must also
be used (so-called reduction sentences [see Vol. I, No. 1, p. 50]).
They describe a method of testing for a more abstract term, i.e.,
a procedure for finding out whether the term is applicable in
particular cases, by referring to less abstract terms. The first
method has the advantage of exhibiting clearly the connection
between the system and observation and of making it easier to
examine whether and how a given term is empirically founded.
However, when we shift our attention from the terms of the

‘electric field’,

‘temperature’,

‘length’,

‘yellow’, ‘hard’, . . .

observable properties
of things

primitive

terms

primitive

terms

fi-

ts

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Foundations of Logic and Mathematics

system and the methods of empirical confirmation to the laws,
i.e., the universal theorems, of the system, we get a different
perspective. Would it be possible to formulate all laws of phys-
ics in elementary terms, admitting more abstract terms only as
abbreviations? If so, we would have that ideal of a science in
sensationalistic form which Goethe in his polemic against New-
ton, as well as some positivists, seems to have had in mind. But
it turns out — this is an empirical fact, not a logical necessity —
that it is not possible to arrive in this way at a powerful and
efficacious system of laws. To be sure, historically, science
started with laws formulated in terms of a low level of abstract-
ness. But for any law of this kind, one nearly always later found
some exceptions and thus had to confine it to a narrower realm
of validity. The higher the physicists went in the scale of terms,
the better did they succeed in formulating laws applying to a
wide range of phenomena. Hence we understand that they are
inclined to choose the second method. This method begins at the
top of the system, so to speak, and then goes down to lower and
lower levels. It consists in taking a few abstract terms as primi-
tive signs and a few fundamental laws of great generality as
axioms. Then further terms, less and less abstract, and finally
elementary ones, are to be introduced by definitions; and here,
so it seems at present, explicit definitions will do. More special
laws, containing less abstract terms, are to be proved on the
basis of the axioms. At least, this is the direction in which phys-
icists have been striving with remarkable success, especially in
the past few decades. But at the present time, the method can-
not yet be carried through in the pure form indicated. For many
less abstract terms no definition on the basis of abstract terms
alone is as yet known; hence those terms must also be taken as
primitive. And many more special laws, especially in biological
fields, cannot yet be proved on the basis of laws in abstract
terms only; hence those laws must also be taken as axioms.

Now let us examine the result of the interpretation if the first
or the second method for the construction of the calculus is
chosen. In both cases the semantical rules concern the elemen-
tary signs. In the first method these signs are taken as primi-

206

Elementary and Abstract Terms

tive. Hence, the semantical rules give a complete interpretation
for these signs and those explicitly defined on their basis. There
are, however, many signs, especially on the higher levels of
abstraction, which can be introduced not by an explicit defini-
tion but only by a conditional one. The interpretation which the
rules give for these signs is in a certain sense incomplete. This is
due not to a defect in the semantical rules but to the method by
which these signs are introduced; and this method is not arbi-
trary but corresponds to the way in which we really obtain
knowledge about physical states by our observations.

If, on the other hand, abstract terms are taken as primitive —
according to the second method, the one used in scientific
physics — then the semantical rules have no direct relation to the
primitive terms of the system but refer to terms introduced by
long chains of definitions. The calculus is first constructed float-
ing in the air, so to speak ; the construction begins at the top and
then adds lower and lower levels. Finally, by the semantical
rules, the lowest level is anchored at the solid ground of the
observable facts. The laws, whether general or special, are not
directly interpreted, but only the singular sentences. For the
more abstract terms, the rules determine only an indirect inter-
pretation, which is — here as well as in the first method — incom-
plete in a certain sense. Suppose ‘ B ’ is defined on the basis of
‘A’; then, if ‘A’ is directly interpreted, ‘B’ is, although indirect-
ly, also interpreted completely; if, however, ‘B’ is directly inter-
preted, ‘A’ is not necessarily also interpreted completely (but
only if ‘A’ is also definable by ‘B’).

To give an example, let us imagine a calculus of physics constructed, ac-
cording to the second method, on the basis of primitive specific signs like
‘electromagnetic field’, ‘gravitational field’, ‘electron’, ‘proton’, etc. The sys-
tem of definitions will then lead to elementary terms, e.g., to ‘Fe’, defined as a
class of regions in which the configuration of particles fulfils certain conditions,
and ‘Na-yellow’ as a class of space-time regions in which the temporal dis-
tribution of the electromagnetic field fulfils certain conditions. Then seman-
tical rules are laid down stating that ‘Fe’ designates iron and ‘Na-yellow’
designates a specified, yellow color. (If ‘iron’ is not accepted as sufficiently
elementary, the rules can be stated for more elementary terms.) In this way

207

Foundations of Logic and Mathematics

the connection between the calculus and the realm of nature, to which it is to
be applied, is made for terms of the calculus which are far remote from the
primitive terms.

Let us examine, on the basis of these discussions, the example
of a derivation D 2 (§ 23). The premisses and the conclusion of
A are singular sentences, but most of the other sentences are
not. Hence the premisses and the conclusion of this as of all
other derivations of the same type can be directly interpreted,
understood, and confronted with the results of observations.
More of an interpretation is not necessary for a practical appli-
cation of a derivation. If, in confronting the interpreted prem-
isses with our observations, we find them confirmed as true,
then we accept the conclusion as a prediction and we may base
a decision upon it. The sentences occurring in the derivation be-
tween premisses and conclusion are also interpreted, at least
indirectly. But we need not make their interpretation explicit
in order to be able to construct the derivation and to apply it.
All that is necessary for its construction are the formal rules of
the calculus. This is the advantage of the method of formaliza-
tion, i.e., of the separation of the calculus as a formal system
from the interpretation. If some persons want to come to an
agreement about the formal correctness of a given derivation,
they may leave aside all differences of opinion on material ques-
tions or questions of interpretation. They simply have to ex-
amine whether or not the given series of formulas fulfils the
formal rules of the calculus. Here again, the function of calculi
in empirical science becomes clear as instruments for transform-
ing the expression of what we know or assume.

Against the view that for the application of a physical cal-
culus we need an interpretation only for singular sentences, the
following objection will perhaps be raised. Before we accept a
derivation and believe its conclusion we must have accepted the
physical calculus which furnishes the derivation; and how can
we decide whether or not to accept a physical calculus for appli-
cation without interpreting and understanding its axioms? To
be sure, in order to pass judgment about the applicability of a
given physical calculus we have to confront it in some way or

208

"Understanding" in Physics

other with observation, and for this purpose an interpretation is
necessary. But we need no explicit interpretation of the axioms,
nor even of any theorems. The empirical examination of a phys-
ical theory given in the form of a calculus with rules of inter-
pretation is not made by interpreting and understanding the
axioms and then considering whether they are true on the basis
of our factual knowledge. Rather, the examination is carried
out by the same procedure as that explained before for obtaining
a prediction. We construct derivations in the calculus with
premisses which are singular sentences describing the results of
our observations, and with singular sentences which we can test
by observations as conclusions. The physical theory is indirect-
ly confirmed to a higher and higher degree if more and more of
these predictions are confirmed and none of them is discon-
firmed by observations. Only singular sentences with elemen-
tary terms can be directly tested; therefore, we need an explicit
interpretation only for these sentences.

25. "Understanding" in Physics

The development of physics in recent centuries, and especially
in the past few decades, has more and more led to that method
in the construction, testing, and application of physical theories
which we call formalization, i.e., the construction of a calculus
supplemented by an interpretation. It was the progress of
knowledge and the particular structure of the subject matter
that suggested and made practically possible this increasing
formalization. In consequence it became more and more pos-
sible to forego an “intuitive understanding” of the abstract
terms and axioms and theorems formulated with their help.
The possibility and even necessity of abandoning the search for
an understanding of that kind was not realized for a long time.
W T hen abstract, nonintuitive formulas, as, e.g., Maxwell’s equa-
tions of electromagnetism, were proposed as new axioms, phys-
icists endeavored to make them “intuitive” by constructing a
“model”, i.e., a way of representing electromagnetic micro-
processes by an analogy to known macro-processes, e.g., move-
ments of visible things. Many attempts have been made in this

209

Foundations of Logic and Mathematics

direction, but without satisfactory results. It is important to
realize that the discovery of a model has no more than an
aesthetic or didactic or at best a heuristic value, but is not at all
essential for a successful application of the physical theory.
The demand for an intuitive understanding of the axioms was
less and less fulfilled when the development led to the general
theory of relativity and then to quantum mechanics, involving
the wave function. Many people, including physicists, have a
feeling of regret and disappointment about this. Some, especial-
ly philosophers, go so far as even to contend that these modern
theories, since they are not intuitively understandable, are not
at all theories about nature but “mere formalistic construc-
tions”, “mere calculi”. But this is a fundamental misunder-
standing of the function of a physical theory. It is true a theory
must not be a “mere calculus” but possess an interpretation, on
the basis of which it can be applied to facts of nature. But it is
sufficient, as we have seen, to make this interpretation explicit
for elementary terms; the interpretation of the other terms is
then indirectly determined by the formulas of the calculus,
either definitions or laws, connecting them with the elementary
terms. If we demand from the modern physicist an answer to
the question what he means by the symbol V’ of his calculus,
and are astonished that he cannot give an answer, we ought to
realize that the situation was already the same in classical
physics. There the physicist could not tell us what he meant by
the symbol ‘E’ in Maxwell’s equations. Perhaps, in order not to
refuse an answer, he would tell us that ‘E’ designates the electric
field vector. To be sure, this statement has the form of a seman-
tical rule, but it would not help us a bit to understand the
theory. It simply refers from a symbol in a symbolic calculus
to a corresponding word expression in a calculus of words. We
are right in demanding an interpretation for ‘E’, but that will be
given indirectly by semantical rules referring to elementary
signs together with the formulas connecting them with ‘E\
This interpretation enables us to use the laws containing ‘E’ for
the derivation of predictions. Thus we understand ‘E’, if “un-
derstanding” of an expression, a sentence, or a theory means

210

Selected Bibliography

capability of its use for the description of known facts or the
prediction of new facts. An “intuitive understanding” or a di-
rect translation of ‘ E ’ into terms referring to observable prop-
erties is neither necessary nor possible. The situation of the
modern physicist is not essentially different. He knows how to
use the symbol ‘\p’ in the calculus in order to derive predictions
which we can test by observations. (If they have the form of
probability statements, they are tested by statistical results of
observations.) Thus the physicist, although he cannot give us a
translation into everyday language, understands the symbol
\p and the laws of quantum mechanics. He possesses that kind
of understanding which alone is essential in the field of knowl-
edge and science.

Selected Bibliography

Carnap, R. Abriss der Logislik. Wien, 1929.

. “Die Mathematik als Zweig der Logik,” Blatter fur deutsche Philoso-
phic, Vol. IV (1930).

. Logical Syntax of Language. (Orig., Wien, 1934.) London and New

York, 1937.

. “Formalwissenschaft und Realwissenschaft,” Erkenntnis, Vol. V

(1935).

. “Testability and Meaning,” Philosophy of Science, Vols. Ill (1936)

and IV (1937).

Dedekind, R. Was sind und was sollen die Zahlen? Braunschweig, 1888.
Einstein, A. Geometrie and Erfahrung. Berlin, 1921.

Fraenkel, A. Einleitung in die Mengenlehre. 3d ed. Berlin, 1928.

Frank, P. Interpretations and Misinterpretations of Modern Physics. Paris,
1938.

Frege, G. Die Grundlagen der Arithmetik. Breslau, 1884.

. Grundgesetze der Arithmetik, Vols. I and II. Jena, 1893 and 1903.

Hahn, H. Logik, Mathematik und Naturerkennen. Wien, 1933.

Heyting, A. Mathematische Grundlagenforschung, Intuitionismus, Beweis-
theorie. Berlin, 1934.

Hilbert, D. “Axiomatisches Denken,” Math. Annalen, Vol. LXXVIII
(1918).

Hilbert, D., and Ackermann, W. Grundzuge der theoretischen Logik. Berlin,
1928. 2d ed., 1938.

Hilbert, D., and Bernays, P. Grundlagen der Mathematik, Vol. I. Berlin,
1934.

21 1

Foundations of Logic and Mathematics

Lewis, C. I., and Langford, C. H. Symbolic Logic. New York and London,
1932.

Menger, K. “The New Logic,” Philosophy of Science, Vol. IV (1937).
Morris, C. W. Logical Positivism, Pragmatism, and Scientific Empiricism.
Paris, 1937.

Peirce, C. S. Collected Papers (esp. Vol II). Cambridge, Mass., 1931 ff.
Quine, W. V. “Truth by Convention,” in Philosophical Essays for A. N.

Whitehead. London and New York, 1936.

Reichenbach, H. Philosophic der Raum-Zeil-Lehre. Berlin, 1928.

Russell, B. The Principles of Mathematics. Cambridge, 1903. 2d ed., 1938.

. Introduction to Mathematical Philosophy . London, 1919. 2d ed., 1920.

Scholz, H. Geschichte der Logik. Berlin, 1931.

Tarski, A. “Der Wahrheitsbegriff in den formalisierten Sprachen,” Studia
philosophica, Vol. I (1935).

. “Grundlegung der wissenschaftlichen Semantik,” in Actes du congres

international de philosophic scientifique. Paris, 1936.

. Einfiihrung in die mathematische Logik. Wien, 1937.

Waismann, F. Einfiihrung in das mathematische Denken. Wien, 1936.
Whitehead, A. N. and Russell, B. Principia mathematica, Vols. I, II, and
III. Cambridge, (1910) 1925, (1912) 1927, and (1913) 1927.
Wittgenstein, L. Tractatus logico-philosophicus. London, 1922.

212

Index of Terms

[The numbers refer to the sections of this monograph.]

Abstract term, 166
Axiom system, 158

Basie calculus, 158

C- . . . , 150, 151
Calculus, 150
Classification of signs, 147
Conclusion, 151
Consistent, 150
Convention, 154
Customary interpretation, 155

Definition, 155
Derivable, derivation, 151
Descriptive sign, 146
Designatum, 144

Elementary term, 166
Equivalent, 148
Exclude, 148
Explanation, 143, 157
Expression, 146

F- . . . , factual, 149
False, 147

Finite rule, calculus, 152
Formal, 150
Formalism, 162
Formalization, 167
Formula, 151

Geometry, 163

Implicate, 148
Interpretation, 152
Intuitionism, 162

L- .... 149
L-exhaustive, 152
Logical calculi, 155, 156

Logical sign, 146
Logicism, 162

Mathematical calculi, 159
Metalanguage, 145

Name, 147

Normal interpretation, 155, 156
Number, 156, 159, 160

Object-language, 145

Physical calculi, 165
Pragmatics, 144, 145
Predicate, 147
Prediction, 143, 157
Premiss, 151, (161)

Primitive sentence, 151, (161)
Proof, provable, 151
Proposition, 147, 151

Rules of formation, 147, 151
Rules of inference, 151
Rules of transformation, 151

Semantics, 144, 146
Sentence, 144
Sentential calculus, 155
Sign, 146

Singular sentence, 157
Specific calculus, sign, 158
Synonymous, 148
Syntax, 144, 150

Transfinite rule, calculus, 152
Translation, 159
True, 147

Understanding, 147, 167
Universal sentence, 157

213

Linguistic Aspects of
Science

Leonard Bloomfield

Linguistic Aspects of Science

Contents:

I. Introduction , a0! .

1. Language and Science 218

2. The Present State of Linguistics 218

3. Scope of This Essay 220

4. Terminology 222

5. Delimitation 223

6. Writing and Language 224

7. Language and Handling Activity .... 226

8. Apriorism 229

9. Mentalism 230

10. Solipsism 231

II. The Function of Language

11. The Speech Community 233

12. Relayed Speech 234

13. Verbal Self-stimulation 234

14. Meaning 235

15. Acquisition of Meaning 237

III. The Structure of Language

16. Phonemic Structure 239

17. Grammatical Structure 242

216

Contents:

PAGE

18. The Sentence 245

19. Constructional Level and Scope 246

20. Varieties of Reference 248

21. Substitution and Determination 251

IV. Precision in Natural Language

22. Reporting Statements 251

23. Negation 253

24. Abstraction and Approximation 254

25. Counting 256

26. Infinite Classes of Speech-Forms 257

V. Scientific Language

27. Development of Scientific Language .... 260

28. General Character of Scientific Language 263

29. Publicity and Translatability 264

30. Postulational Form 265

31. Sentence-Forms of Scientific Language . 267

32. Syntactic Features 269

33. Special Features 271

VI. Summary

34. The Place of Linguistics in the Scheme of Sci-
ence 272

35. Relation of Linguistics to Logic and Mathemat-
ics 273

Selected Bibliography 275

Index of Technical Terms 277

217

Linguistic Aspects of Science

Leonard Bloomfield

I. Introduction

1. Language and Science

Language plays a very important part in science. A typical
act of science might consist of the following steps: observation,
report of observations, statement of hypotheses, calculation,
prediction, testing of predictions by further observation. All
but the first and last of these are acts of speech. Moreover, the
accumulation of scientific results (the “body” of science) con-
sists of records of speech utterance, such as tables of observed
data, a repertoire of predictions, and formulas for convenient
calculation.

The use of language in science is specialized and peculiar. In
a brief speech the scientist manages to say things which in ordi-
nary language would require a vast amount of talk. His hearers
respond with great accuracy and uniformity. The range and
exactitude of scientific prediction exceed any cleverness of
everyday life: the scientist’s use of language is strangely effec-
tive and powerful. Along with systematic observation, it is
this peculiar use of language which distinguishes science from
non-scientific behavior.

The present essay attempts to state briefly such general con-
siderations of linguistics as may throw light upon the procedure
of science.

2. The Present State of Linguistics

The ancient Greeks succeeded in describing the main syn-
tactic and inflectional forms of their language. The description,
to be sure, was incomplete and contained some serious miscon-
ceptions; especially the method of definition was unscientific.

219

Linguistic Aspects of Science

With slight modifications, the ancient Greek doctrine has come
down into our schools and constitutes today the equipment, as
to linguistic knowledge, of the educated man. The ancient
doctrine was able to persist because the principal modern lan-
guages of Europe are related to ancient Greek and have much
the same structure.

Around the beginning of the nineteenth century the Sanskrit
grammar of the ancient Hindus became known to European
scholars. Hindu grammar described the Sanskrit language com-
pletely and in scientific terms, without prepossessions or philo-
sophical intrusions. It was from this model that Western schol-
ars learned, in the course of a few decades, to describe a lan-
guage in terms of its own structure. The greatest progress in
this respect occurred perhaps in connection with the native lan-
guages of America. The study of languages of diverse structure
broke down parochial misconceptions according to which spe-
cial features of Indo-European grammar were universal in hu-
man speech; it destroyed the pseudo-philosophical dogmas
which were built up on these misconceptions. The Greek and
medieval notions of linguistics, which still hold sway, it would
seem ineradicably, in our schools, have thus been long out of date.

Acquaintance with the Sanskrit grammar and, at the same
time, a new mastery of historical perspective brought about, at
the beginning of the nineteenth century, the development of
comparative and historical linguistics. The method of this
study may fairly be called one of the triumphs of nineteenth-
century science. In a survey of scientific method it should serve
as a model of one type of investigation, since no other historical
discipline has equaled it. Historical linguistics, like descriptive,
has failed to influence our schools or to enter higher education,
except for a haphazard and usually incorrect statement here and
there in our schoolbooks. A confusion, strangely enough, of
linguistic relationship with the breeding, in races, of domestic
animals has found its way into popular sociology.

In the last half-century the study of variations within a lan-
guage, especially of local dialects, has given us a new insight
into the minute processes which make up linguistic change.

220

Scope of T his Essay

Unlike the science of chemistry, which is only a decade or so
older, modern linguistics, perhaps for lack of immediate eco-
nomic use, has failed to put its results into public possession.
Not only the man in the street but also the educated public and
even the practitioners of closely related branches of science are
generally untouched by linguistic knowledge. Under the spell
of the traditional dogma which is carried along in the schools
the non-linguist responds often enough with incredulous sur-
prise to well-established results that have long ago become com-
monplace among linguists.

3. Scope of This Essay

In this essay we are concerned only with such phases of lin-
guistics as may throw light upon scientific procedure. Histori-
cal-comparative linguistics will not here concern us, since the
use of language in science presupposes complete stability in the
habits of speech. The linguistic changes which occur within a
generation or two are relatively minute; such as they are, they
disappear entirely under the agreements, explicit and other,
which underlie the scientist’s use of language.

As to descriptive linguistics, the vastly greater part of its data
also can here be left aside, for we need not enter upon any ex-
amination of the widely differing types of structure that appear
in different languages. It is enough for us to know that nearly
all the structural features of our language which we are inclined
to accept as universal — features such as the actor-action sen-
tence, the elaborate part-of-speech system, or the special in-
flections of our nouns and verbs — are peculiarities of the Indo-
European family of languages and are by no means universal in
human speech. In part, moreover, the uniformity with which
these features seem to appear in the languages that figure in our
schooling is an illusion produced by the inaccuracy of the con-
ventional description.

For our purpose we need scarcely touch upon the structural
features of any one language; we shall be concerned only with
the most general aspects of linguistic structure and with the
function and effect of language. Beyond this, we shall need to

221

Linguistic Aspects of Science

consider the peculiarities of only one type of specialized dialect,
for the language activity of scientists, in so far as it differs from
other uses of language, constitutes the jargon or dialect of a
craft, comparable to the dialects, say, of fishermen, miners, or
carpenters.

This great restriction of our task does not remove the diffi-
culties which are bound to arise. Even simple and well-estab-
lished statements about language are likely to disturb or baffle
the non-linguist, not because of any intrinsic difficulty but be-
cause of his prepossession with ancient doctrines that in the
course of time have taken on the tenacity of popular supersti-
tions. For the rest, our task demands that we go beyond such
simple and well-established facts to matters which are subject to
controversy. The writer can only do his utmost to speak plain-
ly, in the hope that the reader will read attentively and, so far
as may be, without injecting the prepossessions of our common-
sense views about language.

4. Terminology

Linguists naturally have no respect for words. Engaged in
describing structural systems of the widest diversity, they are
accustomed, at need or at convenience, to invent and to discard
all manner of technical terms. The varying terminologies of dif-
ferent linguists will concern only the reader who chooses to go
farther afield: within this essay only one set of terms will ap-
pear.

On the other hand, the technical terms which are established
among linguists often conflict or overlap with the meaning of
these same terms as they occur in general scholastic usage and
in philosophical discussion. It would be impractical in this essay
to avoid them and to introduce the reader, instead, to fanciful
innovations. At the time of definition technical terms will be
given in italics, and an index will facilitate later reference to
these definitions.

Since this essay, moreover, tries to make a specialized and
hitherto neglected application of linguistics, we shall be com-

222

Delimitation

pelled to use some temporary and unaccepted terms, and these,
too, are likely to conflict with popular or scholastic usage.

Talking about the forms of speech is not easy. Perhaps only
linguists are inured to this kind of discourse; others are likely to
find it confusing. This difficulty is superficial and should be
overcome without much trouble. Linguists are accustomed in
printing to italicize the speech-forms which are under discussion
or else to inclose them in square brackets; in this essay, how-
ever, conforming to more general usage, we shall place them be-
tween single quotation marks (for example: the word ‘word’).

5. Delimitation

Deep-seated inhibitions block the observation of language.
To the unaccustomed observer language seems shapeless and
fleeting. One does not talk about language. Among us an ave-
nue of rationalization is provided in the traditional lore of the
schoolroom. Where this does not suffice, we are accustomed to
several lines of evasion : instead of observing language, one ob-
serves or, at any rate, discourses upon such matters as correct-
ness of speech, literary values, or writing.

The notion that there is such a thing as “correct” and “incor-
rect” speech, that a speaker may “make mistakes” if he speaks
“carelessly” in his native language, and that some kind of au-
thoritative ruling may be sought, arose in the eighteenth cen-
tury as an outgrowth of a peculiar social development. It has
a realistic background (from which, however, it departs almost
beyond recognition) in the existence, in large modern speech
communities, of a standard dialect, spoken by the privileged and
educated classes and employed in public speech (school, church,
stage, court of law) and in nearly all writing; the standard dia-
lect, common to all of a speech area (English) or to a large part
of it (German, Dutch-Flemish), contrasts with local or regional
dialects which enjoy a less favored position.

The discussion of literary values — that is, of the artistic use
of language by specially gifted individuals — enjoys general favor
as a substitute for the observation of language. This exercise, in

223

Linguistic Aspects of Science

spite of its prevalence in journalistic and academic spheres,
need not here concern us.

6. Writing and Language

When the non-linguist sets out to talk about language, he
very often lapses into discourse about writing. In order to avoid
this confusion, we shall here use the term ‘language only of the
conventional use of vocal sound (“spoken” language), distin-
guishing this from substitutes, such as writing or drum signals,
and from other actions, such as facial mimicry, which may serve
in communication. In the present essay the term ‘language’ will
therefore have a more restricted meaning than in other sections
of this Encyclopedia. Similarly, phrases like ‘a language’ will
here be used only of established (“natural”) languages that pre-
vail in communities, and not of restricted systems of symbolism,
such as appear in mathematics and logic. For all linguistic
study it is of primary importance to know that writing is not
the same thing as language. Failure to make this distinction
was one of the chief factors that prevented a beginning of lin-
guistic science in the seventeenth and eighteenth centuries. The
popular-scholastic contrast of “spoken language” and “written
language” is entirely misleading.

In point of time language appears as a characteristic of the
human species; men spoke as far back as we know anything
about them; we must reckon here with biological reaches of
time. To conjecture about the origin of language is to conjecture
about the origin of the human species. Writing appears, by con-
trast, as a modern invention. At the earliest points of use (Meso-
potamia, Egypt) it is only a few thousand years old.

Writing is a device for recording language by means of visual
marks. By “recording” we mean that the beholder, if he knows
the language of the writer and the system of writing, can repeat
the speech which the writer uttered, audibly or internally, when
he set down the marks.

Writing has been practiced in only few communities, and in
these, until recently, almost always as a special skill of a few
people. Anything like widespread literacy is a recent develop-

224

Writing and Language

ment covering only a small part of the world. Languages are
quite the same whether their speakers practice writing or not;
where there is widespread literacy, writing produces certain sur-
face irregularities in the historical development of a language;
these are not extensive, but their exact scope has not been de-
termined. The best-known effects of this sort are the so-called
spelling pronunciations, as when some people pronounce a d
in ‘Wednesday’ or a t in ‘often.’

A literate person finds it easy to talk about writing, because
he acquired this skill under explicit verbal instruction, and be-
cause the permanent written markings record the movements of
the writing hand. His acquisition of language, on the other
hand, was unaccompanied, in the nature of the case, by any
verbal accounting, and the sounds of language are fleeting and
intangible. Hence the normal speaker finds it difficult to talk
about language and cannot even describe the movements with
which he produces speech. Some of these movements, in fact,
though every child learns to make them, have so far baffled the
physiologist’s attempt at determination. The non-linguist, when
he discusses linguistic matters, is likely to mistake writing for
language.

A mathematical writer, for example, appreciating the advan-
tages of a duodecimal system of numbers, invents two written
symbols, say t for ‘ten’ and e for ‘eleven’ (so that 10 now repre-
sents ‘twelve,’ 84 . represents ‘one hundred,’ lOe ‘one hundred
and fifty-five,’ etc.), and he is disappointed when people find
this system too difficult to adopt. He has confused writing with
language. Since the number-words of our language are decimal
in structure, the proposed written forms are totally out of gear
with our language and can be interpreted only by a cumbersome
process of calculation.

All this concerns us here because the language of science tends
to appear in written notation; in fact, its most characteristic and
powerful form, mathematical discourse, can be transmitted only
by means of a written record. The branching-out of written no-
tation into forms which cannot be matched in actual speech
represents a peculiar and highly specialized development of

225

Linguistic Aspects of Science

writing. This will concern us in the sequel; here we need say
only that, in order to understand the specialized development,
we must have a clear and correct picture of the underlying
habit.

What applies to writing applies also to less elaborate com-
municative systems, such as gestures or signals. A period when
such systems may have been independent of language, or co-
ordinate with it, would, in any case, lie very far behind us. The
conditions which might have prevailed in such a period are
quite unknown. In times that are accessible to us and under
human conditions as we know them, any system of gestures or
signals that goes beyond vague beckoning appears simply as an
outgrowth and substitutive reproduction of language.

The confusion, extremely common among literate but lin-
guistically untrained persons, between language and its deriva-
tive substitutes often takes the form of inverting this relation or
of ascribing some kind of “independence” to the substitutes.
Alterations, no matter how fantastic, in a system of writing do
not affect the language which is represented. If a new system of
writing came into wide use in a community, it might in time
produce some very modest linguistic modifications. This must
not lead us to suppose that a change, say, from the Arabic to the
Latin alphabet, such as has recently been effected for Turkish,
has any immediate bearing on the language.

7. Language and Handling Activity

The difference between utterance of speech and all other
events is naturally paramount in the study of language. The
human acts which are observable under ordinary conditions are
thus divided into language, the vocal utterance of the conven-
tional type, and handling activities, a somewhat narrow name
to cover all other normally observable acts, including not only
manipulation but also mimicry, gesture, locomotion, acts of ob-
servation, etc. The linguist studies sequences in which language
mediates between non-linguistic events.

Language creates and exemplifies a twofold value of some hu-
man actions. In its biophysical aspect language consists of

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Language and Handling Activity

sound-producing movements and of the resultant sound waves
and of the vibration of the hearer’s eardrums. The biosocial
aspect of language consists in the fact that the persons in a
community have been trained to produce these sounds in cer-
tain situations and to respond to them by appropriate actions.
The biosocial function of language arises from a uniform, tradi-
tional, and arbitrary training of the persons in a certain group.
They have been trained to utter conventional sounds as a sec-
ondary response to situations and to respond to these slight
sounds, in a kind of trigger effect, with all sorts of actions.

In their turn, on a tertiary level of response, handling actions
may be subjected biosocially to language: specific handling ac-
tions are conventionalized as a response to specific forms of
speech, and these handling actions or their products serve as
stimuli to call forth these same forms of speech. These sub-
stitutes for speech occur scantily in simple communities but
play a great part in our civilization. Apart from speculations
about a prehuman or semihuman antiquity, in man as we see
him, all handling actions which bear elaborate communicative
value owe this value to their biosocial subjection to language.

The most familiar example of this is the art of writing: per-
manent visible marks are produced conventionally as responses
to speech and serve as stimuli for the production of speech. For
this it is necessary, of course, that the marks be conventionally
associated with features of language. These features may be of
various kinds : Chinese characters and our numerical digits rep-
resent words ; the letters of the alphabet represent single typical
speech-sounds. The biophysical aspect of writing consists in the
materials used for producing the marks, in the shape of the
marks, and in the movements by which one makes them. Once
we have established the communal habit or convention which
attaches them biosocially to speech-forms, the biophysical fea-
tures may vary within the convention. Thus, the variety of
paper, pencil, or ink; the shape, always within our convention,
of the letters — script, printed, Roman or italic, small or capital,
etc. — and the movements of the writer’s hand or the kind of

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Linguistic Aspects of Science

printing press are all indifferent so far as the verbal message is
concerned.

The biophysical features of substitutes for language are im-
portant enough on their own level. Systems of writing may dif-
fer as to the time and labor required for making the marks and
as to their legibility. The permanence of the materials is some-
times important. Mechanical devices like the typewriter and
the printing press may yield great advantages.

More important for our discussion are the conventions which
associate the written marks with forms of language. The alpha-
bet, which associates, in principle, each mark with a typical
sound of speech, makes reading and writing relatively easy,
since it requires only a few dozen different characters; for the
same reason it lends itself easily to mechanical transmission
and reproduction. The system of writing, best exemplified in
Chinese, where a character is assigned to a word of the language,
is much harder to use and to mechanize. On the other hand, a
piece of writing can be read in various languages, provided that
these agree in structure and that minor divergences are bridged
by means of supplementary conventions. Moreover, word-
writing has the value of compactness and easy survey. These
advantages appear very clearly in the case of our numeral
digits. Here, by confining a system of word-writing to a very
limited domain of speech-forms, we make it readable, with the
aid of supplementary conventions, in any language. Thus, a
graph like 71 is read in English as ‘seventy-one,’ in German as
‘one and seventy,’ in French as ‘sixty-eleven,’ in Danish as
‘one and half-four-times’ (sc., ‘twenty’), etc.

Our numeral digits illustrate the advantage of a well-arranged
written symbolism. Simple discourses which move entirely in
this domain can be carried out by means of a calculating ma-
chine. Short of this extreme case, we may cite cable codes,
where the written forms are revised in the direction of brevity;
or, again, systems of notation such as are used in symbolic logic,
where a stock of symbols is so devised as to yield extremely sim-
ple rules of discourse. The construction of a calculating machine
or the rules of arrangement and substitution in a system of logis-

228

A prior ism

tics require great care and present great interest : it is decidedly
advantageous to study them for their own sake and without
reference to the linguistic end points of the sequences in which
they operate. Nevertheless, if they are to operate, they must be
so planned that, starting with speech-forms, they yield speech-
forms in the end. No serious use would be made of a calculating
machine, or of a code, or of a system of logistics, which failed
to deliver a linguistically significant end result. By way of con-
trast, the reader may find it not without value to consider such
non-linguistic systems as musical notation, say for the piano, or,
again, the moves of a game of chess. Persons unaccustomed to
the consideration of language are prone to the error of overlook-
ing the linguistic character of notational or mechanical subsidi-
aries of language and viewing them as “independent” systems;
at the same time they may resort to the metaphor of calling
these systems “languages.” This metaphor is dangerous, since it
may lead to the notion that such systems can liberate us from un-
certainties or difficulties which inhere in the working of language.

Some students of language, among them the present writer,
believe that widespread and deep-seated errors in supposedly
scientific views of human behavior rest upon misconceptions of
this sort — upon failure to distinguish between linguistic and
non-linguistic events, upon confusion of the biophysical and the
biosocial aspects of language and its subsidiaries, and, above all,
upon a habit of ignoring the linguistic parts of a sequence and
then calling upon metaphysical entities to bridge the gap. Sev-
eral phases of this error will here demand our attention.

8. Apriorism

In the prescientific view of these matters, a term such as
“reasoning” covers, on the one hand, observations which cost
no great labor and, on the other hand, utterances of speech
which are not recognized for what they are. Thus there arises
the notion that knowledge may be obtained by a process of
“reasoning a priori.” Everyday observations, generally human
or systematized by tribal tradition, are viewed as innate data of
reason, and the ensuing deductions are clothed in a mystic valid-

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Linguistic Aspects of Science

ity. If the deductions are correctly made, the “a priori” proce-
dure differs from an ordinary act of science only in that the basic
observations are unsystematic and remain untested.

In scientific procedure we mean by deduction the purely verbal
part of an act of science which leads from the report of observa-
tion and the hypotheses to a prediction. If we replace the re-
port of observation by arbitrarily invented postulates, the dis-
course makes no pretense to validity in the sphere of handling
actions. Deductive discourse of this kind is produced in logic,
mathematics, or the methodology of science. It is made to fit
some type of observational data, or else it exists for its own sake,
in readiness for the emergence of observational data to which it
may be applied. Until modern times. Euclidean geometry was
viewed as an “a priori” system: the underlying everyday ob-
servations about the spatial character of objects were viewed as
inborn and unquestionable truths. Today the same system,
apart from the correction of flaws, is treated as a purely verbal
discourse of deduction from postulates. It is especially useful
because these postulates, by virtue of their historic origin, are
such as to make the discourse applicable to the placing of ob-
jects, as this placing is observed in the first approximation that
is customary in everyday handling. We have learned, however,
that astronomical magnitudes make sensible the error in these
postulates, and they accordingly demand a different discourse,
based upon other postulates which, in turn, will be chosen so as
to fit the new observations. One employs postulates which fit
the observed data within the margin of error. The postulates
are chosen so as to yield a simple calculation ; the discrepancies
are set aside to await more accurate measurements, which, in
their turn, will make possible a more closely approximative dis-
course.

9. Mentalism

In the common sense of many peoples, perhaps of all, lan-
guage is largely ignored, and its effects are explained as owing to
non-physical factors, the action of a “mind,” “will,” or the like.
These terms, as well as the many others connected with them.

230

Solipsism

yield service in daily life, in art, and in religion ; that they have
no place in science is the contention of many scientists. Men-
talistic terms do not figure in the procedure of physics, biology,
or linguistics, but many students of these subjects employ them
in the theoretical parts of their discourse.

An individual may base himself upon a purely practical, an
artistic, a religious, or a scientific acceptance of the universe,
and that aspect which he takes as basic will transcend and in-
clude the others. The choice, at the present state of our knowl-
edge, can be made only by an act of faith, and with this the issue
of mentalism should not be confounded.

It is the belief of the present writer that the scientific descrip-
tion of the universe, whatever this description may be worth, re-
quires none of the mentalistic terms, because the gaps which
these terms are intended to bridge exist only so long as language
is left out of account.' If language is taken into account, then
we can distinguish science from other phases of human activity
by agreeing that science shall deal only with events that are ac-
cessible in their time and place to any and all observers (strict
behaviorism) or only with events that are placed in co-ordinates
of time and space {mechanism ),- or that science shall employ only
such initial statements and predictions as lead to definite han-
dling operations {operationalism ) , or only terms such as are de-
rivable by rigid definition from a set of everyday terms concern-
ing physical happenings {physicalism) . These several formula-
tions, independently reached by different scientists, all lead to
the same delimitation, and this delimitation does not restrict
the subject matter of science but rather characterizes its meth-
od. It is clear even now, with science still in a very elementary
stage, that, under the method thus characterized, science can
account in its own way for human behavior — provided, always,
that language be considered as a factor and not replaced by the
extra-scientific terms of mentalism.

10. Solipsism

Much of the procedure of science and particularly the end
result of each act of science consists of speech in a somewhat

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Linguistic Aspects of Science

specialized form. In order to give an account of science, we
must know something of language. Linguistics, however, in
any feasible arrangement, occupies a rather late place among the
divisions of science, since it comes after biology and all studies
of the animal (and not specifically human) circumstances of
man.

The fact that we are thus inevitably caught in a circle should
not be exploited in favor of a naive solipsism which is surprising-
ly prevalent even among otherwise competent men of science.
It is true, of course, that science can deal with only such events
as have called forth a verbal response on the part of some per-
son — at any rate, of the scientist who undertakes to deal with a
given event. More generally, the solipsist says that the uni-
verse consists, of necessity, only of his “experiences” or “per-
ceptions.” The truth of this axiom cannot be affected by the
study of language or of “experiences” or “perceptions,” since
these studies occupy a late place in science and deal with only a
small part of the universe.

If it were strictly applied, the solipsist’s axiom would remain
irrelevant and harmless. Any predicate that is applied to every-
thing in the world will play no part in science, for science deals
only with correlations of events within the universe. The
solipsist, having tagged everything, indifferently and without
exception, as his “percept” or “experience,” goes on to study
things exactly as other scientists study them. He does not allow
his solipsism to interfere with work in his own field of science.
In more general discussion, however, the solipsist commits the
fallacy of using his meaningless axiom as a bludgeon to knock
over special points of discourse. He is most likely to do this in
the face of any attempt at scientific study of human behavior,
and especially in the face of any linguistic considerations. It is
as though in practical life the solipsist should suddenly recall
that an oncoming railway train was only a creature of his per-
ceptions.

232

The Speech Community

II. The Function of Language

11. The Speech Community

Language makes human behavior extremely different from
that of animals because it establishes a minute and accurate
interaction between individuals. Among animals such interac-
tion, where it exists, concerns only a few gross situations or else
is rigidly set in an unvarying pattern. Apart from this, each
individual’s response to stimuli is limited by the powers of this
individual s body. In man these ordinary responses, to which
we give the name of ‘handling actions,’ are paralleled by a sec-
ond set of responses — the utterance of speech. The speech
sounds act as a stimulus upon other persons who may then per-
form a handling response such as to give a biologically favorable
outcome to the speaker’s situation. Thus a person who is in
need of food but unable by his own bodily power to get it
may, by speech, prompt others to get it for him.

Thus, in addition to the normal biological series S — > R, man
has also the series

S — » r s — » R .

Here r s denotes the act of language; the biologically effec-
tive stimulus S and response R are no longer confined to occur-
rence within one body. Language bridges the gap between the
individual nervous systems. It makes possible a minute division
of labor and high specialization of individual abilities.

Much as single cells are combined in a many-celled animal,
separate persons are combined in a speech community — a high-
er and more effective type of organization. If the wrnrd ‘organ-
ism’ be not confined to denote an individual animal, we may
speak here, without metaphor, of a social organism. Primarily,
the social organism is the speech community — the community
of persons speaking one language — but bilingual or multilingual
persons mediate everywhere between these communities: cul-
ture areas, such as Europe with her daughter-nations, approach
the coherence of a single-speech community; some degree of
communication now subsists between all persons on earth.

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Linguistic Aspects of Science
12. Relayed Speech

The simplest case of speech utterance is that in which the
hearer performs a handling action in response to a non-linguistic
stimulus which impinges on the speaker. Very often, however,
the hearer responds by addressing speech to the first speaker or
to other persons. Many relays of speech may intervene between
a non-linguistic stimulus and a handling response. The stimuli
of various individuals may contribute to the sequence of speech;
the sequence itself may be multiple, carried on simultaneously
by many chains or a network of speaker-hearer connections; and
the handling response may be performed by many persons, co-
operatively or in independent actions. The intermediate utter-
ances in such a chain or net exemplify the case where speech is
prompted by no immediate non-verbal stimulus and calls forth
no immediate handling response but figures merely as a part of
a sequence of linguistic and other events in a community. The
word ‘sumach,’ for instance, in its primary and simple use will
be uttered in the presence (under the visual stimulus) of a cer-
tain kind of tree, but a speaker hitherto ignorant of the word,
once this is pointed out to him, will be able, without further ex-
planation, to take part in sequences of action and discourse in
which the word ‘sumach’ occurs in absence of the primary stim-
ulus. Other speech-forms, such as most of those used in this
monograph, adhere to no simple stimulus of that sort but occur
entirely in complex situations of discourse.

Speech utterance itself, on a different level, may serve as a
relatively final response, as in instruction in a foreign language
or mathematics, or in a discussion leading to agreement as to
the use of some technical term. If we follow a sequence far
enough, we expect, of course, to find some modification of non-
verbal activity: even poetry or fiction will in the end lead to a
more than verbal result.

13. Verbal Self-stimulation

The normal human being, totally uncritical as to language,
sees himself not only as a separate body but also as a source of
effect upon other persons. However pervasive and at times pow-

234

Meaning

erful this effect may be, it is nearly always uncertain. In con-
trast with this, he is able, with rare exceptions, to co-ordinate
his earlier speech (“intention”) with later handling actions of
his own. This contributes toward the view of the “self” or
“ego” as a more than bodily entity.

What here interests us is not merely the relative sureness but
also the usefulness of verbal self-stimulation. The utterance
of a speech-form is not only a response but serves also as a
stimulus to the speaker himself. The utterance can be easily
repeated and replaces conveniently an evanescent or remote
stimulus. The varied situations of counting (as, say, a flock of
sheep before and after a storm) can be adduced to illustrate
this.

A child talks to himself at first out loud; then he learns to
mumble or whisper; finally, he suppresses all audible and even
all visible movements of speech. In the popular view no move-
ment at all is supposed to remain. This is a view which we
could adopt only by abandoning the basic assumptions of
physics. We must suppose rather that the movements of speech
— which, as we shall see, consist of a small number of contrast-
ing units — are replaced by internal movements, at first pre-
sumably as mere reductions of the normal movements of speech,
but capable, in the course of time, of any degree of substitution.
This inner speech accounts for the main body of the vaguely
bounded system of actions that in everyday parlance goes by
the name of “thinking.”

14. Meaning

In the sequence S — » r s — > R and its complex deriva-
tives, the speech or sequence of speeches has no immediate
gross biological effect and serves merely as a “sign” mediating
between the more practically important stimuli and responses,
which are here represented by the large letters S and R. These,
to be sure, may themselves consist of speech, but, in the inten-
tion of the formula, this speech will then be on a lower level than

the speech represented by the small r s; an example of this

would be discourse about the validity of an earlier speech, or,

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Linguistic Aspects of Science

with another difference of level, discourse about the forms of
some language. These special cases need not here concern us:
we may speak of the end points S and R as speaker’s stimulus
and hearer’s response, without regard to their verbal or non-
verbal character. Ihe two together constitute, in linguistic

terminology, the meaning of the speech utterance r s. This

holds good even under a mentalistic view: in this view it is
merely supposed that the speaker’s stimulus and the hearer’s
response are “ideas,” “concepts,” or the like, which may be
postulated in more or less exact accommodation to the uttered
speech-forms and serve to link these to the actually observable
stimulus and response.

The term ‘meaning,’ which is used by all linguists, is neces-
sarily inclusive, since it must embrace all aspects of semiosis
that may be distinguished by a philosophical or logical analysis:
relation, on various levels, of speech-forms to other speech-
forms, relation of speech-forms to non-verbal situations (ob-
jects, events, etc.), and relations, again on various levels, to the
persons who are participating in the act of communication.

When the correlations of speech-forms with meanings are
known, some utterances turn out to be conditioned more im-
mediately by the speaker’s stimulus, and others by the hearer’s
response. Contrast, for example, a report, such as ‘It’s raining,’
with a command, such as ‘Come here!’ In earlier stages of an
infant’s language-learning, speaker’s meaning and hearer’s mean-
ing may be imperfectly correlated, but in the normal use of
language both these aspects of meaning are so firmly knit to the
speech-form that no distinction is possible. In the description
of a language we need not define each form twice over, determin-
ing first the situations in which it is uttered and then the re-
sponses to which it leads. In general, we define in terms of stim-
ulus because the earlier step in the sequence exhibits less varia-
tion. The speaker has been trained, as part of his acquisition
of language, to play indifferently either part in the interchange
of speech. To be sure, each speaker understands a wider range
of speech-forms than he utters, but this does not bring about
any discrepancy between the two aspects of meaning: in the

236

Acquisition of Meaning

adult speaker they are entirely merged in a firm correlation
which is guaranteed by the evident working of language.

15. Acquisition of Meaning

Hearing a speech in a language of which we are entirely ig-
norant, we must include in our first estimate of the meaning the
total situation of the speaker and all the ensuing actions of the
hearer. To consider meaning from any less inclusive position
would lay us open to prepossessions which lurk close by. The
more striking features of gesture, situation, and immediate re-
sponse will often lead us to a hypothesis, to be removed or con-
firmed by further experience and by the correlations that will be
gradually built up. After we have thus acquired the use of a
small supply of speech-forms, becoming, to this extent, members
of the community, we may induce full-fledged speakers to talk
about their language: we ask them the names of things and
lead them to define speech-forms in terms of other speech-forms.
In the history of travel and adventure all this has many times
occurred; the student of language naturally prefers to seek the
mediation of a bilingual speaker. For the infant’s acquisition of
language, however — though little is known of it — we must pos-
tulate exactly this process. The elders, of course, facilitate the
process by uttering simple speech-forms with strikingly appar-
ent and much repeated practical emphasis upon the relevant
features of situation and response. This elementary process has
been called demonstration; this use of the word is distinct, of
course, from its use in logic and mathematics as a synonym of
‘proof.’ A speech-form has been acquired by demonstration
when the learner utters it in the conventionally appropriate
situations and responds to it by the conventionally appropriate
actions.

Teacher and learner may develop an agreement under which
the demonstration of meaning, for certain types of speech-
forms, is shortened to a single act, such as holding up an object
or unmistakably pointing at it while speaking its name.

Another factor of learning is the occurrence of a speech-form
in contexts whose other components are familiar. Certain types

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Linguistic Aspects of Science

of forms, such as the words ‘and,’ ‘or,’ ‘because,’ ‘the,’ are pre-
sumably acquired in this way.

Finally, there is the process of definition. For the most part,
definition is informal: a speech-form is equated with roughly
synonymous forms. Thus we may tell a child that ‘rapid’ means
‘quick’ or ‘fast,’ and we expect the statistical effect of future
demonstration and context to train him to use the new word in
its more exact conventional setting. Bilingual statements of
meaning, such as ‘cheval : horse’ are in general of this rough
type; even apart from differences of structure and function, the
meanings in two languages almost never coincide.

We call a definition formal when, for some sphere of discourse,
the new form and the old one by which it is defined are freely
and completely interchangeable. Formal definition occurs in
ordinary life only in the case of a very few of the simplest and
most abstract speech-forms. The notable instance is furnished
by the number words : ‘eight’ means ‘seven and one.’ Other in-
stances doubtless occur: ‘down’ could be defined as ‘in the
direction in which heavy things fall when we drop them.’ The
chief place of formal definition, however, is in explicit agree-
ment. This is familiar to us from scientific practice: for a cer-
tain discourse, and perhaps for whole types of discourse, par-
ticipants in some technical procedure agree to use a new term
as the exact equivalent of some older speech-form. Thereafter
they initiate learners into this agreement. Formal definitions
may be bilingual (as ‘Kraft: force’), provided abstraction is
made from structural and functional differences (for instance,
gender of a noun in French or German, as opposed to English).

It will be seen that, even in the case of formal definition, the
degree of uniformity among the speakers and even the consisten-
cy of any one speaker will be subject to any uncertainty as to
the meaning of the defining speech-forms. These may in turn
have been formally defined, but in the end we must come to
forms which have been only roughly defined and, for each speak-
er, to forms which he got in infancy from context or, at the very
beginning, from demonstration. The meaning of a speech-form
in the habit of any one speaker contains factors and fringes of

238

Phonemic Structure

variation; these may be greatly reduced, but probably never
entirely eliminated, for the observer by consideration of many
speakers, and for the speakers themselves by copious and con-
sistent interchange of speech upon some topic that demands ac-
curate agreement.

III. The Structure of Language

16. Phonemic Structure

In ordinary discussion human situations and responses appear
perhaps less flowing and vague than we have here described
them in the guise of meanings, for there we are free (as we are
not here, in our linguistic discussion) to endow them with some
of the permanence and neat outline which they obtain by virtue
of their correlation with the forms of language. In language we
order and classify the flowing phenomena of our universe; our
habit of doing this is so pervasive that we cannot describe
things as they may appear to an infant or a speechless animal.
The price we pay is a sensible inadequacy of our speech, offset
by the privilege of any degree of approximation such as may be
seen in some microscopic investigation of science or in the work
of the poet. It is the entire task of the linguist to study the
ordering and formalization which is language; he thus obtains
the privilege of imagining it removed and catching a distant
glimpse of the kind of universe which then remains.

The ordering and formalizing effect of language appears,
first of all, in the fact that its meaningful forms are all composed
of a small number of meaningless elements. We should obtain,
in this respect, a parallel to language if, with a dozen or so of
different flags, we devised a code in which the exhibition of
several flags (in the limiting case, of one flag) in a fixed position
and arrangement would constitute a meaningful sign.

The forms of every language are made up out of a small num-
ber — ranging perhaps between fifteen and seventy -five — of typi-
cal unit sounds which have no meaning but, in certain fixed
arrangements, make up the meaningful forms that are uttered.
These signals are the phonemes of the language. The speakers

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Linguistic Aspects of Science

have the habit of responding to the characteristic features of
sound which in their language mark off the various phonemes
and of ignoring all other acoustic features of a speech. Thus, a
German who has not been specially trained will hear no differ-
ence between such English forms as ‘bag’ and ‘back,’ because
the difference in his language is not phonemic; it is one of the
acoustic differences which he has been trained to ignore. In the
same way, a speaker of English will hear no difference, until he is
trained to do so, between two Chinese words which sound to him,
say , like man, and differ as to their scheme of pitch; we fail to
hear the difference because in our language such a difference is
not connected with a difference of meaning and is consistently
ignored whenever it chances to occur. The acoustic features
which set off a phoneme from all other phonemes in its lan-
guage, and from inarticulate sound, exhibit some range of
variation. It is not required that this range be continuous:
acoustically diverse features may be united, by the habit of the
speakers, in one phoneme.

The number of phonemes which will be stated as existing in
any one language depends in part upon the method of counting.
For instance, we shall recognize an English phoneme [j] which
appears initially in forms like ‘yes,’ ‘year,’ ‘young,’ and another
phoneme [e] in the vowel sound of words like ‘egg,’ ‘ebb,’ ‘bet.’
The longer vowel sound in words like ‘aim,’ ‘say,’ and ‘bait’ may
then be counted as another phoneme, or else one may describe
it as a combination of the phonemes [e] and [j]. This option would
not exist if our language contained a succession of [e] plus [j]
which differed in sound, and as to significant forms in which it
occurred, from the vowel sound of ‘aim,’ ‘say,’ ‘bait.’ Thus, the
English sound [c], which appears in words like ‘chin,’ ‘rich,’
church,’ must be counted as a single phoneme and not as a
combination of [t] as in ten’ and [§] as in ‘she,’ because in forms
like ‘it shall’ or ‘courtship’ we have a combination of [t] plus [si
which differs in sound and as to significant forms from [c] in
‘itch Al’ or ‘core-chip.’ The fact that these last two forms are
unusual or nonsensical does not affect the distinction. The

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Phonemic Structure

count of phonemes in Standard English will vary, according to
economy, from forty-odd to around sixty.

For the most part, the phonemes appear in utterance in a
linear order. Where this is not the case, the arrangement is so
simple that we can easily put our description into linear order.
For instance, the noun ‘convict’ has a phoneme of stress (loud-
ness) which starts with the beginning of the word and covers
the first vowel phoneme; the verb ‘convict’ differs in that the
same stress phoneme is similarly placed upon the beginning of
the partial form ‘-vict.’ If we wish to put our description of
these forms into linear order, we need only agree upon a con-
vention of aligning a symbol for the stress phonemes, e.g.,
'convict and con’ vict.

Thus, every speech in a given dialect can be represented by a
linear arrangement of a few dozen symbols. The traditional
system of writing English, with its twenty-six letters and half-
dozen marks of punctuation, does this very imperfectly but
sufficiently well for most practical needs.

This rigid simplicity of language contrasts with the con-
tinuous variability of non-linguistic stimulation and response.
For this reason linguists employ the word ‘ form ’ for any mean-
ingful segment of speech, in contrast with their use of ‘meaning’
for stimulus and response.

The sound produced in a speech is to all ordinary purposes a
continuum. To determine which features are phonemic, we
must have some indication of meaning. A German observer,
say, who, studying English as a totally unknown language, no-
ticed in a few utterances the acoustic difference between ‘bag’
and ‘back,’ could decide that this is a phonemic difference only
when he learned that it goes steadily hand in hand with a
difference of meaning.

Two utterances, say of the form ‘Give me an apple,’ no matter
how much they may differ in non-phonemic features of sound,
are said to consist of the same speech-form; utterances which
are not same are different. The decision of the speakers is prac-
tically always absolute and unanimous. This fact is of primary
concern to us, since by virtue of it the speakers are able to ad-

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Linguistic Aspects of Science

here to strict agreements about speech-forms and to establish
all manner of correspondences, orderings, and operations in this
realm. To take an everyday instance: anyone can look up a
word in a dictionary or a name in a directory.

It would not do to overlook the fact that the phonemes of a
language are identifiable only by differences of meaning. For
this, however, a relatively small number of gross differences will
suffice: once the phonemes are established, any form of the
language is completely and rigidly definable (apart from its
meaning) as a linear or quasi-linear sequence of phonemes. We
do not possess a workable classification of everything in the
universe, and, apart from language, we cannot even envisage
anything of the sort; the forms of language, on the other hand,
thanks to their phonemic structure, can be classified and or-
dered in all manner of ways and can be subjected to strict agree-
ments of correspondence and operation. For this reason, lin-
guistics classifies speech-forms by form and not by meaning.
When a speech-form has been identified, we state, as well as
may be, its meaning: our success depends upon the perfection
of sciences other than linguistics. The reverse of this would be
impossible. For instance, we shall usually seek a given word in a
thesaurus of synonyms by looking it up in the alphabetical in-
dex. We could not use a telephone directory which arranged the
names of the subscribers not in their alphabetical order, but ac-
cording to some non-verbal characteristic, such as weight,
height, or generosity.

17. Grammatical Structure

Some utterances are partly alike in form and meaning; for
instance :

Poor John ran away.

Our horses ran away.

Poor John got tired.

Our horses got tired.

This forces us to recognize meaningful constituent parts, such
as ‘poor John,’ ‘our horses,’ ‘ran away,’ ‘got tired.’

A form which can be uttered alone with meaning is a free

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Grammatical Structure

form; all our examples so far are free forms. A form (‘form’ al-
ways means ‘meaningful form’) which cannot be uttered alone
with meaning is a bound form. Examples of bound forms are the
suffix ‘-ish’ in ‘boyish,’ ‘girlish,’ ‘childish,’ or the suffix ‘-s’ in
‘hats,’ ‘caps,’ ‘books.’

A free form which does not consist entirely of lesser free for ms
is a word. Thus, ‘boy,’ which admits of no further analysis into
meaningful parts, is a word; ‘boyish,’ although capable of such
analysis, is a word, because one of the constituents, the suffix
‘-ish,’ is a bound form; other words, such as ‘receive,’ ‘perceive,’
‘remit,’ ‘permit,’ consist entirely of bound forms.

A free form which consists entirely of lesser free forms is a
'phrase; examples are ‘poor John’ or ‘poor John ran away.’

Sets of words, such as ‘perceive: receive: remit’ or ‘per-
ceive: permit: remit,’ establish a parallelism between the ex-
tremes, ‘perceive’ and ‘remit.’ The habit which is thus revealed
is a morphologic construction. In the same way, sets of phrases,
such as ‘John ran: John fell: Bill fell’ or ‘John ran: Bill ran:
Bill fell,’ establish a parallelism between the extremes ‘John
ran’ and ‘Bill fell,’ and illustrate a syntactic construction. The
parts of a form which exhibits a construction are the constituents
of the form : the form itself is a resultant form.

In the study of an unknown language we proceed as above:
partial similarities between forms reveal their complexity, and
we progressively recognize constituents and determine, often
with some difficulty, whether they are free or bound. In pre-
senting the description of a language, however, we begin with
the constituents and describe the constructions in which they
appear.

A construction, morphologic or syntactic, consists in the ar-
rangement of the constituents. In addition to the meaning of
the constituents, the resultant form bears a constructional
meaning, which is common to all forms that exhibit the same
construction. Even more than other elements of meaning, con-
structional meanings are likely to present difficulties of defini-
tion, for they are often remote from simple non-linguistic events.

The features of arrangement differ in different languages.

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Linguistic Aspects of Science

Modulation is the use of certain special phonemes, secondary
phonemes, which mark certain forms in construction. In Eng-
lish, features of stress play a large part as secondary phonemes.
We have seen this in the contrast between the verb ‘convict’
and the noun ‘convict.’ In syntax it appears in the absence of
word-stress on certain forms. Thus, in a phrase like ‘the house,’
the word ‘the’ is unstressed; on the other hand, it may receive a
sentence stress when it is an important feature of the utterance.

Phonetic modification is the substitution of phonemes in a
constituent. For instance, ‘duke,’ when combined with the
suffix ‘-ess’ or ‘-y’ is replaced by ‘duch-’; in syntax the words
‘do not’ are optionally replaced, with a slight difference of
meaning, by ‘don’t.’ Neither modulation nor phonetic modifica-
tion plays any part in the specialized scientific uses of language;
it is otherwise with the features of arrangement which we now
have to consider.

1 he selection of the constituent forms plays a part apparently
in all languages. If we combine the word ‘milk’ with words like
‘fresh,’ ‘cold,’ ‘good,’ we get designations of special kinds of
milk: fresh milk,’ ‘cold milk,’ ‘good milk’; if we combine it with
words like ‘drink,’ ‘fetch,’ ‘use,’ we get designations of acts:
drink milk, fetch milk,’ ‘use milk.’ The difference in construc-
tional meaning goes hand in hand with the selection of the
forms. We describe these habits by saying that the construc-
tion has two (or more) positions which are filled by the constitu-
ents. A function of a form is its privilege of appearing in a cer-
tain position of a certain construction. The function, collec-
tively, of a form is the sum total of its functions. Forms which
have a function in common constitute a form-class. Thus, the
forms ‘milk,’ ‘fresh milk,’ ‘cold water,’ ‘some fine sand,’ etc.,
are in a common form-class, since all of them combine with
forms of the form-class ‘drink,’ ‘don’t drink,’ ‘carefully sift,’
etc., in the construction of action-on-object. In syntax, as these
examples indicate, words and phrases appear in common form-
classes. If words alone are considered, their largest inclusive
form-classes are known as parts of speech. In many languages,
and very strikingly in English, the form-classes of syntax over-

244

T he Sentence

lap in so complex a fashion that various part-of-speech classifi-
cations are possible, according to the functions which one
chooses primarily to take into account.

The forms of a form-class contain a common feature of mean-
ing, the class meaning. The traditional grammar of our schools
gets into hopeless difficulties because it tries to define form-
classes by their class meaning and not by the formal features
which constitute their function.

The use of order as a feature of arrangement is by no means as
widespread as the use of selection, but, on account of its sim-
plicity and economy, it plays a great part in the scientific
specializations of language. In English the order of the constitu-
ents is a feature of nearly all constructions; thus, in ‘fresh milk’
or ‘drink milk’ the constituents appear only in this order. In
some instances, features of order alone distinguish the positions:
contrast, for example, ‘John hit Bill’ with ‘Bill hit John.’

18. The Sentence

In any one utterance a form which, in this utterance, is not a
constituent of any larger form is a sentence. By definition, any
free form and no bound form can occur as a sentence. Various
supplementary features are used in different languages to mark
the sentence, especially its end. In English, secondary pho-
nemes of pitch are used in this way. In much the same manner
as constructions, sentence types are distinguished by features of
arrangement. The meanings of these types have to do largely
with the relation of speaker and hearer (“pragmatic” features of
meaning). Thus, pitch and, in part, selection and order deter-
mine in English such types as statement (‘at four o’clock’),
yes-or-no question (‘at four o’clock?’), and supplement question
(‘at what time?’).

In many languages, perhaps in all, certain free forms are
marked off as especially suited to sentence use. A sentence
which consists of such a form is a full sentence. In English the
favorite sentence forms are phrases which exhibit certain con-
structions. The most important is the actor-action construc-
tion in which a nominative substantive expression is joined

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Linguistic Aspects of Science

with a finite-verb expression: ‘Poor John ran away.’ ‘John ran.’
‘I’m coming at four o’clock.’ ‘Can you hear me?’ A sentence
which does not consist of a favorite sentence form is a minor
sentence: ‘Yes.’ ‘Fire!’ ‘At four o’clock.’ ‘If you can hear me.’

English and many other languages distinguish clearly a type
of sentence whose type-meaning can perhaps be described by
the term ‘report.’ In English the report sentences are full-
sentence statements exhibiting the actor-action construction or
a co-ordination of several actor-action phrases. A great deal of
labor has been spent upon attempts at giving a precise definition
of this type-meaning, in disregard of the likelihood of its differ-
ing in different languages and in oblivion of the danger that our
sociology may not be far enough advanced to yield such a
definition. For our purpose, at least a rough outline of this
meaning will be needed. In the normal response to a report the
hearer behaves henceforth as if his sense organs had been
stimulated by the impingement of the reported situation upon
the sense organs of the speaker. Since the meaningful speech-
forms of the report, however, constitute at bottom a discrete
arrangement, the hearer’s responses can correspond to the
speaker’s situation to the extent only that is made possible by
the approximative character of the report. Thus, when a
speaker has said, ‘There are some apples in the pantry,’ the
hearer behaves as though his sense organs had been stimulated
by the impingement of the apples upon the speaker’s sense
organs — as though the speaker’s adventure with the apples, to
the extent that it is represented by the meanings of the speech-
forms, had been witnessed by the hearer, not visually, but
through some sense organ capable of a certain discontinuous
range of stimulation.

Irony, jest, mendacity, and the like represent derived types
of speech and response; they need not here concern us.

19. Constructional Level and Scope

Constructions are classified, first of all, by the form-class of
the resultant form.

If the resultant form differs, as to the big distinctions of form-

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Constructional Level and Scope

class, from the constituents, the construction is said to be
exocentric. For instance, actor-action phrases like ‘John ran
away’ or ‘He ran away’ differ in form-class from nominative
substantive expressions like ‘John’ or ‘he’ and from finite-verb
expressions like ‘ran away.’ Similarly, the functions of prep-
ositional phrases, such as ‘in the house’ or ‘with him,’ differ
from those of a preposition (‘in,’ ‘with’) and from those of an
objective substantive expression (‘the house,’ ‘him’).

If the resultant form agrees as to the major distinctions of
form-class with one or more of the constituents, then the con-
struction is said to be endocentric. For instance, the phrase
‘bread and butter’ has much the same function as the words
‘bread,’ ‘butter.’

If, as in this example, two or more of the constituents have
the same function as the resultant form, the construction is co-
ordinative and these constituents are the members of the co-
ordination. If only one constituent agrees in form-class with the
resultant form, the construction is subor dinative; this con-
stituent is the head of the subordination, and any other con-
stituent is an attribute of this head. Thus, in ‘fresh milk,’ the
head is ‘milk’ and the attribute is ‘fresh’; in ‘this fresh milk,’ the
head is ‘fresh milk’ and the attribute ‘this’; in ‘very fresh,’
the head is ‘fresh’ and the attribute ‘very’; in ‘very fresh
milk,’ the head is ‘milk’ and the attribute is ‘very fresh.’

The difference of analysis in these two cases is worth ob-
serving :

this / fresh milk

very fresh / milk

Although we are unable to give precise definitions of meaning,
especially of such ethnically created ranges as constructional
meanings and class meanings, yet the mere subsistence of like
and unlike sets determines schemes of construction. Only in
rare cases does the structure of a language leave us a choice be-
tween different orders of description. At each step of analysis
we must discover the immediate constituents of the form; if we
fail in this, our scheme will be contradicted by the construc-
tional meanings of the language.

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Linguistic Aspects of Science

If a form contains repeated levels of endocentric construction,
there will be a word or co-ordinated set of words which serves
as the center of the entire phrase. Thus, in the phrase ‘this very
fresh milk,’ the word ‘milk’ is the center.

The formal features of construction — selection of constituent
forms, order, phonetic modification, and modulation by means
of secondary phonemes — differ greatly in various languages and
sometimes lead to very complex structures of word or phrase,
but they seem nowhere to permit of an unlimited box-within-
box cumulation. Even simple formations may lead to ambiguity
because the scope — that is, the accompanying constituents on
the proper level — of a form may not be marked. For instance,
‘an apple and a pear or a peach’ may mean exactly two pieces
of fruit: then the immediate constituents are ‘an apple / and /
a pear or a peach,’ and the phrase ‘a pear or a peach’ and the
phrase ‘an apple’ constitute the scope of the form ‘and.’ On the
other hand, the phrase may mean either two pieces of fruit or
one piece: then the immediate constituents are ‘an apple and
a pear / or / a peach,’ and the scope of ‘and’ now consists, on
its level, of the phrases ‘an apple,’ ‘a pear.’ Similarly, ‘three
times five less two times two’ may mean 26, 18, 11, or 3. These
uncertainties are not tolerable in the scientific use of language;
it is a striking peculiarity of this use that they are removed only
in written notation — as especially by the parentheses and
brackets that are used in algebra. The result is a system of
writing which cannot be paralleled in actual speech.

20. Varieties of Reference

A thoroughgoing comparison of speech-forms, say in some
one language, with features of the non-linguistic world is impos-
sible at the present state of our knowledge. Our system of re-
sponses, with its neat discrimination of objects, classes, positions,
qualities, movements, etc., results very largely from our use of
language. We cannot return to the animal’s or the infant’s
state of speechless response.

In order to find out how much of our world is independent of
any one language, we might try to compare the grammars and

248

Varieties of Reference

lexicons of different languages. At present we have reasonably
complete data for a few languages only; at some future time,
when this task can be undertaken, the results will be of great
interest. The forms of any one language could scarcely serve
as a frame of reference: we should need, instead, a non-lin-
guistic scale by which to measure.

It is the task of science to provide a system of responses which
are independent of the habits of any person or community.
These responses are twofold, in accordance with the universal
scheme of human behavior: science provides relevant handling
responses and clarified speech-forms. In the nature of the case,
however, the entire result is transmitted and preserved in a
verbal record. If science had completed its task, we could ac-
curately define the meanings of speech-forms.

Even the most favorable type of meanings will show the diffi-
culty of definition. Clusters of stimuli which produce roughly
the same elementary responses in all people and, in accordance
with this, are not necessarily tied up with communal habits,
have been successfully studied: this is the domain of so-called
external phenomena, the domain of physical and biological
science. Here some of the simpler lexical classifications of lan-
guage correspond in the main to the classifications of science, as,
for instance, in the names of familiar species of plants and ani-
mals. However, there is often some gross divergence, as when
several species are called by the same name, or one species by
several names, and there is a great deal of less manageable
vagueness at the borders — species which sometimes are and
sometimes are not included in a designation. Even in this sim-
plest sphere, the meaning of many speech-forms involves ethno-
logical features. Here, too, we encounter, on the simplest level,
speech-forms which have no extra-linguistic validity, unless it
be in the designation of secondarily created artifacts: dragons,
griffins, unicorns, etc.

Where science reveals a continuous scale of phenomena, such
as color, the segments included under linguistic terms vary greatly
in different languages ; they overlap and grow vague at the edges ;
and they are subject to extraneous limitations (‘bay,’ ‘roan’).

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Linguistic Aspects of Science

When we come to meanings which are involved in the habit
of communities and individuals, we fall even farther short of
accurate definition, since the branches of science which deal
with these things are quite undeveloped. In practice we resort
here to artistic, practical and ethical, or religious terminologies
of definition, and these, however valuable for our subsequent
conduct, fail to satisfy the peculiar requirements of science.

In all spheres the structure of languages reveals elements of
meaning which are quite remote from the shape of any one situa-
tion and are attached rather to constellations which include,
often enough, personal or ethnical features. Relatively simple
instances are words like ‘if,’ ‘concerning,’ ‘because,’ or the sub-
tle difference, so important in English, between the types ‘he
ran’ : ‘he was running’ : ‘he has run’ : ‘he has been running.’
The difficulty is even greater in the case of bound forms, which
cannot be isolated in their language; consider, for instance, the
deprecative feature in some of the uses of ‘-ish’ (‘mannish’).

Constructional meanings and class meanings pervade a lan-
guage, in part as universally present categories; they generally
defy our powers of definition. The singular and plural categories
of nouns in English are relatively manageable, but include some
troublesome features, such as ‘wheat’ versus ‘oats.’ Gender-
classes, as in French or German, are almost entirely ethnological
in character. The normal speaker, without special training, is
incapable of talking about these features; they are not reflected
in any habits beyond their mere presence in the structure of the
language, dhe major form-classes are remote from any extra-
linguistic phenomena. If we assign to the English class of sub-
stantives some such meaning as “object,” then words like ‘fire,’
wind,’ and ‘stream’ require an ethnologic commentary. The
mechanics of a language often require that otherwise similar
designations occur in more than one grammatical class. Thus,
in English, as a center for the actor in the actor-action construc-
tion, we require a noun : hence we have forms like ‘height’ be-
side ‘high’ or ‘movement’ beside ‘(it) moves.’ Duplications of
this kind are not symptoms of any special level of culture but
result merely from a rather common grammatical condition.

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Reporting Statements

21. Substitution and Determination

Apparently, all languages save labor by providing substitute
forms whose meaning rests wholly upon the situation of speaker
and hearer, especially upon earlier speech. Since these occur
more frequently than specific forms, they are easily uttered and
understood; moreover, they are nearly always short and, often
enough, bound forms. Thus T and ‘y° u ’ replace names, and
‘this’ and ‘that’ the naming of a thing which may be identified
by gesture. The most important type of substitution, for our
subject, is anaphora: the substitute replaces repetition of a
speech-form which has just been uttered or is soon to be uttered.
Thus, the set ‘he, she, it, they’ replaces noun expressions, and
the set ‘do, does, did’ replaces finite-verb expressions (‘I’ll go
by train if John does’). A form competent to fill one position
of a construction may suffice for anaphora of a phrase embody-
ing the whole construction: ‘Mary dances better than Jane’;
here ‘Jane’ serves as the anaphoric substitute for ‘Jane dances.’

Akin to the substitute forms, and very often identical with
them, are determiners, which indicate a range within the class
of phenomena that is designated by an ordinary speech-form:
‘this apple,’ ‘the apple’ (anaphora), ‘every apple,’ ‘all apples,’
etc. We shall be interested in determiners which leave the speci-
men entirely unrestricted: ‘an apple,’ ‘some apple,’ ‘any apple.’
If only one specimen is involved, anaphora is easily made (‘it,’
‘the apple,’ ‘this apple’); but, where several specimens are in-
volved, English, like other languages, provides very poor means
for distinguishing them. To provide for the identification of
more than one variable, we must look to other phases of lan-
guage which contain the germs of a more accurate system of
speech.

IV. Precision in Natural Language

22. Reporting Statements

To discuss the meaning of all the varieties of utterance would
be equivalent to outlining a complete sociology. We need deal

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Linguistic Aspects of Science

only with a single type, the report (§ 18 ), since it alone is re-
quired for science.

The welfare of a community depends, so far as the actions of
people are concerned, most directly upon simple handling activi-
ties whose occasion and performance are plainly observable —
activities such as the gathering of food, hunting, fishing, con-
struction of dwellings, boats, and containers, manufacture of
clothing and tools, etc. These are manipulations of non-human
objects, satisfactory in their biophysical aspect. Even where
human bodies figure as objects, as in surgery or conflict, these
actions suffice in themselves, with a minimum of biosocial sig-
nificance. The situation in which an act of this sort will succeed
does not always present itself in full to the performer; another
person may mediate by speech: ‘There are berries beyond the
cliff ; ‘The fish are biting today’ ; ‘My moccasins need patching’ ;
etc. Reports like these concern matters where behavior is uni-
form: in general, people will agree on the outcome of a test.
This is the sphere of ordinary life out of which science grows
forth. Natural science grows forth directly; the scientific study
of man is hampered by the difficulty of subjecting biosocially
conditioned behavior to such simple and testable reports.

If we try for a moment, and with full recognition of inade-
quacy, to ignore the forms of language, we may perhaps say
that a report of this kind conveys, in the first place, the verbal
substitute of a stimulus: ‘It’s raining.’ Here the ‘it’ is an empty
and merely formal indication of a point of reference, for which
there is in this case no practical need. In most instances, how-
ever, some other stimulus, which has already affected the hearer
or has been verbally represented to him, serves as a point of
reference for the placing of the new stimulus which the hearer
has not experienced: ‘There are some apples in the pantry’—
the last three words may represent nothing new, a familiar
complex of stimuli to which the apples are now added. This
adherence of a new stimulus to an old one is perhaps the practi-
cal background that is formalized in the actor-and-action or
subject-and-predicate constructions which appear in the favor-
ite sentence types of many languages: ‘The fish | are biting |

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Negation

today’; Russian ‘Ivan doma,’ literally ‘John | at home.’ How-
ever, there is no rigid agreement between the structure of the
practical stimuli and the formal structure of the utterance.

The behavior of the speakers distinguishes very well between
the features of the report which convey the relevant handling
stimulus and the features of purely social and personal signifi-
cance, such as especially the formal structure. Two reports of
different structure (‘There are berries beyond the cliff’ and ‘Be-
hind the steep cliff over there you can find some berries’) may
mediate the same simple handling sequence; it is only the ac-
companying personal and social adjustments which differ. It
is a well-tried hypothesis of linguistics that formally different
utterances always differ in meaning; they may be equivalent,
however, as to some one partial phase of meaning. Of this, the
best example is the practical phase of the simple report; out of
it there grows forth the equivalence of variously worded state-
ments in scientific discourse. Here, as in the simple reporting of
ordinary language, the equivalence covers the phase of meaning
which is observable indifferently by all persons.

23. Negation

Every language, apparently, contains inhibitory forms, such
as ‘Don’t!’ The report may be thrown into reverse by utterance
in a negative form: ‘There aren’t any apples in the pantry.’ As
this English example shows, the negative version may differ
formally in complicated ways from the positive. The human
voyage in space and time is linear: response cannot be made
simultaneously to a report and its negative. If both are re-
ceived as sentences, without comment as to biosocial values, the
hearer is no better able to act than before; he has heard a
contradiction.

It is very important, but not always easy, to distinguish, in
this matter, between formal grammatical features and features
of meaning. Contradiction is a feature of meanings, not of
grammatical forms. In the normal case, where the contradic-
tory phrases appear as separate sentences, there is, by defini-
tion, no grammatical nexus between them. In the more difficult

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Linguistic Aspects of Science

case, where they appear in the same sentence (‘It is green be-
cause it is not green’), there is no grammatical incongruity. It is
the meaning of the negating speech-forms which is here involved.

If we wished, using the full extent of a language (like English)
in its ordinary form, to guard ourselves against contradiction
in every discourse, no matter how long, we should have to mas-
ter the universe in what would amount to omniscience.

One is likely to be deceived about this because in simple cases
the rules for non-contradiction may take on a formal character.
For instance, in Old English and in most dialects of modern
English, a negated sentence often contains more than one
negative unit ( I ain t got none’) ; in Latin and in present-day
Standard English two negatives are separately superadded so
as to cancel each other (‘I did not have nothing’). What is here
involved is the meaning of the negative words. The contradic-
tion in a phrase like a round square’ rests upon the meaning of
the terms ‘round’ and ‘square’ or upon that of the basic terms
which underlie successive definitions.

In miniature linguistic systems, such as are produced in logic
and mathematics, the conventions for avoiding contradiction
are rules of meaning (semantic rules). Thus, in a system of
logistic, it may be agreed that a statement and the same state-
ment preceded by the symbol ‘~’ may not, upon pain of con-
tradiction, appear as sentences in any one discourse. This agree-
ment defines the meaning of

The dichotomy of contradictory sentences inheres in the na-
ture of human speech: a sentence in a discourse excludes its
contrary. Thus we obtain implication. The sentences ‘Socrates
is a man’ and ‘All men are mortal’ exclude the sentence ‘Socrates
is not mortal,’ and this is the same as saying that they imply the
sentence ‘Socrates is mortal.’ These effects are due to the mean-
ings of the words all, is,’ ‘a,’ and ‘not,’ and to the meanings of
the grammatical forms.

24. Abstraction and Approximation

A report concerning a simple handling activity covers, in a
rough way, only so much of the situation as is useful. If the

254

Abstraction and Approximation

speaker is prompted, say by an additional question (‘What color
are the apples?’), he is usually able to extend his report. More-
over, he can often subject himself, in the way of continued ob-
servation, to further stimuli of the complex in question (‘I’ll
look and see how many apples there are’). Every apple has a
color, and every set of apples a number, but these features are
not communicated in the report, ‘There are some apples in the
pantry.’ It is a trick of pseudo-philosophy to postulate a meta-
physical “concept” of an apple to account for the imperfect
reporting function of the word ‘apple.’ The obvious fact is
simply that a speech does not mention every feature of stimulus.
Since the ranges of stimulation and of predisposition are to all
practical purposes continuous, and language can provide only a
discrete set of forms, this abstract character of language is in-
evitable: not all the features of a situation appear in the report.
If we do not consider the extension of an object, we may speak
of it as a ‘point’ ; if we speak of one dimension only, we may call
it a ‘line’; if of two only, we call it a ‘surface’; terms like ‘straight
line,’ ‘plane,’ ‘triangle,’ etc., add further characteristics, but
still leave unmentioned certain simple features which are
present in every object. This does not create a world of “con-
cepts.”

By lengthening his report, the speaker may tell more: we
distinguish degrees of approximation. So far as the speaker has
observed, or else perhaps only under reservation of irrelevant
detail, the linear object if turned in a certain way would occupy
the same position as before; hence the speaker says that the
‘line’ is a ‘straight’ one. It may be true that no object has been
found to fulfill this condition to such an extent as to appear
‘straight’ under our best observation with mechanical aids.
This means merely that the speaker did not employ these aids
or else that his report was incomplete in this respect.

Although the meanings of language are discrete, there is no
limit to their cumulation. By extending his utterance, the
speaker may come closer and closer to a full picture of the situa-
tion. This is familiar, for one thing, from the art of fiction. In
the realm of handling operation one rarely approximates be-

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Linguistic Aspects of Science

yond the features that are useful. In the scientific expansion of
this domain, however, one often dwells upon features which
have no immediate use in practical life— such features, for in-
stance, as appear in the botanist’s systematic classification of
families, genera, and species. Accordingly, scientists and spe-
cialists in practical operation invent technical terms, either by re-
defining everyday expressions or by borrowing or creating new
words. A technical term, then, replaces long phrases, or even a
complicated discourse, and its meaning is fixed by an agreement
of definition, which, in science, receives explicit formulation and
strict adherence.

The useful approximation, in a simple society, will be a rough
one, as civilization progresses, usefulness is discovered in closer
approximations. Utterances and responses become more vari-
able. Ihis variability of response in individuals, and, thanks to
manifold specialization of individuals, in the community, may
yield a basis for a scientific definition of what we mean in every-
day life by such words as ‘welfare’ or ‘happiness.’

25. Counting

I he part of ordinary language which most immediately opens
into the language-forms of science is the number complex. The
equality and inequality of sets of discrete objects are defined
in terms of the result of placing the objects of the sets in one-to-
one correspondence. Actual placing together of the objects is
often inconvenient or impossible. The fingers (and, in suitable
climate and surroundings, the toes) serve as an intermediate
set. Speech-forms are more convenient. The quinary, decimal,
and vigesimal groupings of these forms reflect the habit of
counting on the fingers. Some languages provide only a few
number- words; many, however, have an indefinitely extensible
system. If any number-words are otherwise meaningful, this
does not affect their use in counting: all that is needed, and,
in most instances, all that is provided, is a set of otherwise
meaningless free forms which every child learns to recite in a
fixed order.

The set is more easily remembered, and can become indefi-

256

Infinite Classes of Speech-Forms

nitely extensible, if it consists of a short sequence repeated over
and over again, with an auxiliary counting of the repetitions.
Thus, in the decadic system we have:

one, two nine, ten;

ten one, ten two ten nine, two tens;

two tens one, two tens two, two tens nine, three tens;

ten tens one, ten tens two ten tens nine, ten tens ten;

ten tens ten one

The longer forms may be replaced by shorter substitutes, as
‘ten tens’ by ‘one hundred,’ ‘ten hundreds’ by ‘one thousand,’
etc. Later in the sequence such abbreviative forms may be
agreed upon by specialists, as ‘billion,’ which in England means
‘a million millions,’ but in the United States ‘a thousand mil-
lions.’

The system thus outlined appears plainly in Chinese; in
most languages it is encumbered, but not changed, by irregu-
larities, such as our ‘eleven’ for ‘ten one,’ ‘twenty’ for ‘two tens,’
etc.

Evidently there is no point at which the recitation has to end.
No matter what number expression be given, one can always
cap it by one which belongs later in the sequence. This is what
we mean, of course, when we say that the system is indefinitely
extensible.

Of all the features of ordinary language, the number expres-
sions enter most directly into the scientific use of language, and
they continue there to occupy the principal place. Putting dis-
crete objects in one-to-one correspondence is perhaps as uni-
form, from person to person, as any operation. Scientists try to
reduce their tasks to this shape. Our uniformity in the use of
number expressions becomes the basis of complex systems of
discourse which confine themselves largely to these expressions
and enable us to perform long calculations without leaving this
safe ground.

26. Infinite Classes of Speech-Forms

The indefinite extensibility of the system of number expres-
sions of ordinary language leads to the infinite classes of mathe-

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Linguistic Aspects of Science

rnatics. This linguistic background is worth examining, if only
to save mystical aberrations.

Given any set of number expressions, every speaker is able
to utter, without doubt or dispute, a number expression that is
not in the set. This is what we mean by an infinite class of
speech-forms.

In mathematics a less direct definition is useful: to say that a
class is infinite means that it can be put in one-to-one corre-
spondence with a genuine part of itself. Thus, we can put the
set of all (positive integer) numbers in one-to-one correspond-
ence with the set of all even (positive integer) numbers by
simply assigning to each number its double.

This mathematical definition demands something more than
the linguistic definition which we have given above, since we
can define a one-to-one correspondence only if there is some
order or system in the class. This condition is fulfilled, of
course, in the case of the number expressions of ordinary lan-
guage, and it will be fulfilled in any infinite class of speech-
forms that is at all useful in discourse. In practice mathemati-
cians sometimes employ the linguistic version, as, for instance,
in Euclid’s proof of the theorem that the class of prime numbers
is infinite.

The popular or religious use of the word ‘infinite’ has no
place in science, for in this use the word means something
which cannot be dealt with, even verbally — something which
cannot be grasped or understood— by the powers of a human
being. In the sense with which we are concerned, the only
infinite classes are classes of speech-forms. One might perhaps
devise an indefinitely extensible system of other actions, such
as gestures or graphic markings, but our saturation in language
would from the very outset force us to assign verbal substitutes
to the forms of such a system. Persons not accustomed to the
observation of language are likely, in this matter, to see ob-
jects where only speech-forms are present. To name or define
an infinite class of hypothetical objects or events is merely to
adduce a class of speech-forms. Archimedes showed that one

258

Infinite Classes of Speech-Forms

may name a number greater than the possible number of grains
of sand in the solar system.

Any infinite class of speech-forms can be put into linear order,
for each form consists of a linear (or quasi-linear, § 16) se-
quence of phonemes, and the forms may be alphabetized ac-
cording to an arbitrary rank assigned to the phonemes. Apart
from the imperfections of our traditional system of writing,
which do not affect the matter, this is what we do in our every-
day alphabetizing. No matter what new word may come into
English, or what strange name into a list of telephone sub-
scribers, there is no difficulty as to its place in the alphabetical
sequence.

If we make the additional demand that the order be like
that of the number expressions in counting, such that every
form has an immediate successor (with no intervening forms),
then we must call in the number expressions : the speech-forms
must be grouped first according to the number of phonemes (or,
in traditional writing, of letters) in each one and, then, within
each of these groups, alphabetically.

Mathematicians, to be sure, set themselves a much harder
task, since they deal with infinite classes each of whose members
is an infinite class of speech-forms, as, for instance, with classes
of unending decimal expressions. One begins with such forms as
‘one-third’ or ‘the square root of two or pi (this last defined
as the limit sum of a simply constructed infinite series). Then
one may demand some special form of expression (such as that
of a decimal) whose constituents (such as the digits of the deci-
mal) may require laborious calculation, one by one. 1 hen,
further, one may define new classes whose members depend
upon these singly calculated forms, as, for example, an unend-
ing decimal whose digits depend, according to some stated
formula, upon the corresponding digits of the decimal expres-
sion of pi. Various types of discourse can be agreed upon, ac-
cording to the kind of infinite classes that may be admissible.
The actually uttered speech-forms, no matter how elaborate,
which define or name any class, can always be alphabetized.

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Linguistic Aspects of Science

The utterance of all members of an infinite class can be re-
quired neither in mathematics nor anywhere else.

V. Scientific Language

27. Development of Scientific Language

Persons who carry on a specialized activity develop technical
terms and locutions; these shorten speech and make response
more accurate. Such are the special vocabularies of fishermen
carpenters, miners, and other craftsmen. These terminologies
contribute to the dialectal differentiation which exists in all fair-
sized speech communities. The special vocabularies and turns
of speech which are used in the various branches of science be-
long in this same general type; only, as scientific observation
reaches beyond the interests of ordinary life, the vocabulary of
science becomes very large. From timid neologisms it grows to
a state where some scheme of word creation stands at the service
of every member of the guild. In this way European and Ameri-
can scientists freely coin words by derivation and composition
of Latin and ancient Greek stems; such words, with adaptations
of grammar and phonetics, are accepted as loan-words in the
scientific dialects of the several languages.

The exact responses, and the careful and often complex cal-
culations of science, enforce an unusually meticulous style of
speech. The syntactic scope of forms and the domain of substi-
tutes have to be clearly indicated. This, with the elimination of
personal factors, produces a general scientific style of utterance.

he sentences may extend to great length and may evoke an
immediate response only in hearers or readers who are favorably
predisposed by training; on the other hand, the message, once
grasped, is unmistakable. In pseudo-science the difficulties but
not the advantages are imitated; the clinical symptoms are
locutions which do not lead to handling response, appeals to
personal or ethnical connotation, and, above all, an obscurity
which remains even under analysis by a trained recipient.

At an advanced stage the demands for exhibition of data and
for complex but unerring calculation lead to speech-forms and

260

Development of Scientific Language

especially to written discourses which move outside the sphere
of ordinary language. The ancient Greeks carried on mathe-
matical demonstrations largely in ordinary language; it was the
development, in the early modern period, of arithmetic and
algebra, with its box-within-box markings of scope, that di-
vorced scientific calculation not only from ordinary language
but, to all practical purposes, from vocal utterance. People
learned to calculate rapidly and accurately by visual reception
and graphic manipulation of a small stock of characters in
simple arrangements. Thus there arose the plan of conducting,
or at least outlining and testing, scientific discourse by means of
simple and rigidly manipulated graphic systems.

Without an entirely sharp boundary, we have, then, linguis-
tically, two types of scientific discourse which we shall here dis-
tinguish by the names informal and formal. (We are here using
the term ‘formal’ in its everyday sense and not with the tech-
nical meaning which it has in logic.) Informal scientific dis-
course uses ordinary language with the addition of technical
words and turns of phrase and with syntactic and stylistic re-
strictions in favor of uniform response. It is generally capable
of reception by a qualified listener. Formal scientific discourse
uses a rigidly limited vocabulary and syntax and moves from
sentence to sentence only within the range of conventional
rules. In general, it can be carried on only in writing, mainly
because no vocal equivalents have been devised or practiced for
the elaborate markings of scope.

Within formal scientific discourse it is customary to dis-
tinguish, again without a sharp boundary, between mathemati-
cal discourse in general and a special type, symbolic logic, which
is devised to establish and test the basic rules of scientific dis-
course.

Within mathematical discourse there is probably no linguistic
difference between applied mathematics that is, calculations
which form part of a scientific discourse — and pure mathe-
matics, where calculation is made for its own interest, with
arbitrary axioms replacing the observational data and hypoth-
eses of scientific procedure. Linguistic variety consists primarily

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Linguistic Aspects of Science

in the use or non-use of the number vocabulary. The term
mathematics’ is ordinarily applied only to formal discourse
whose basic vocabulary is numerical, and to geometry even
w en numbers are absent or play a subsidiary part. Other ex-
amples of non-numerical mathematical discourse appear, how-
ever, in such symbolisms and calculations as those of chemistry
and of linguistics. One tries, wherever it is possible, to employ
numerical discourse, because of the favorable character of the
number-words and because of the stock of ready-made devices
and calculations which has been accumulated in the pursuit of
pure mathematics.

In all this development we have not left the domain of lan-
guage. We reach its outskirts in the use of written discourses
which will fail of effect if given in vocal form (§ 19 ). In general,
to be sure, the separate written characters have been agreed
upon as substitutes for specific words or phrases. In many cases,
however, we manage best by ignoring these values and confining
ourselves to the manipulation of the written symbols; systems of
symbolic logic, especially, may be viewed, in a formal way, as
systems of marks and conventions for the arrangement of these
marks. The transition to non-linguistic activity could be made
if we devised such marks and conventions without any initial
or final linguistic interpretation, quite as in children’s games
that are played with paper and pencil. The marks could then
no longer be qualified as writing. Actually, our formal sys-
tems serve merely as written or mechanical mediations between
utterances of language. On the other hand, important linguistic
conditions may be obscured when we speak of a formal system,
like a system of symbolic logic, as a “language.” The inter-
pretation, initial and final, of the procedure is made in terms of
some natural language (such as English), and the system as a
whole is meaningless to a reader to whom it has not been inter-
preted^ these terms. It differs even from an “artificial lan-
guage,” such as Esperanto, since the latter supplies a complete
set of speech responses for ordinary use, while a mathematical
system or a system of symbolic logic supplies only a limited set
of responses that mediate between acts of speech.

262

General Character of Scientific Language

By way of parenthesis it may be well to add that no artificial
language, so far, has reached the function of a natural lan-
guage. The artificial language is devised by speakers of a nat-
ural language, inevitably in terms of this and with little seman-
tic deviation, and it is acquired, in similar terms, by persons
who already speak a natural language. It would attain the
status of such a language if a group of infants acquired it, by
the usual process of demonstration, as their native speech. The
only actual event even approaching this has been the implanting
of Hebrew upon infants in Palestine: preserved only in written
records, Hebrew had for some two thousand years been no
person’s native language.

Even such a thing as a tabulation of numbers is linguistic;
apart from the verbal character of the elements, the arrange-
ment leads to a linguistic interpretation. We leave the domain
of language only when we come to a drawing, a geometrical
diagram, or a map: here, indeed, we employ a non-verbal object
directly as a representation of another non-verbal object.

A formal dialect, such as a system of symbolic logic, may
very well be used to state the rules which govern its use. This
means simply that the formal dialect suffices for statements
about word-order, selection, and substitution. It neither, on the
one hand, removes the system from its linguistic status nor, on
the other hand, gives it the standing of an independent “lan-
guage.”

28. General Character of Scientific Language

An act of speech is a happening in the world and, as such, an
object of science; the branch of science which studies it is lin-
guistics. Scientists, however, are speakers and may agree to ut-
ter speech in certain ways; thanks to the simplicity of phonetic
structure, they are able quite accurately and uniformly to ad-
here to these agreements. Accordingly, they treat their own
utterances not as an object of science but as a part of scientific
procedure. In this sense, and to the extent that social agree-
ments about speech can be maintained, scientists may be said to

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Linguistic Aspects of Science

“control” the forms of their technical dialect, in contrast with
the world of meanings — that is, the world of events, including
the utterance of speech other than that which forms part of the
scientist’s own activity. This outside world is reportable and
predictable only to the extent that earlier acts of science have
mastered it — at best imperfectly.

The scientist may construct a discourse, as in pure mathe-
matics, in which the speech-forms have no meaning beyond that
which is created by the scientific agreements governing their
use — a type of discourse anticipated by the natural numbers of
ordinary speech and, in most instances, based upon them. Such
a discourse produces a calculation, made for its own interest, or
as a model, or with a view to eventual use; about the outside
world it tells nothing. We may be sure of its correctness because
it moves only within the verbal agreements upon which it is
based.

On the other hand, as soon as we admit meanings of the out-
side world, we risk error, and our certainty is then only such as
may result from earlier acts of science. In a priori reasoning we
see a sleight of hand which introduces observations or beliefs of
everyday life into a seemingly pure calculation.

It is our task to discover which of our terms are undefined or
partially defined or draggled with fringes of connotation, and to
catch our hypotheses and exhibit them by clear statements, in-
stead of letting them haunt us in the dark. The mentalist fails
to list his undefined terms or to state his hypotheses ; he wonders
at the obtuseness with which we refuse to accept certain “con-
cepts” which are necessary in the vocabulary of everyday life.

29. Publicity and Translatability

Science deals with the phases of response that are alike for all
normal persons. Its observations and predictions can be tested
by anyone. Where a temporal feature is present (as in the pas-
sage of a meteor), the question concerns the acceptability of a
report and is usually settled by agreement of qualified inde-
pendent observers. Unique personal or communal behavior fig-
ures in science as an object, which may be observed like any

264

Publicity and T ranslatability

other; but it does not figure as a part of scientific procedure.
This exclusion demands a redefinition of speech-forms; even
number words like ‘seven’ or ‘thirteen’ have to be stripped of
superstitious connotation. In the terminology of physics, the
most advanced branch of science, one can see how far this
stripping has been carried and surmise how much farther it still
must go. Linguistically, as well as in handling, science is a
public activity. In scientific colloquy recognition is granted nei-
ther to the private predispositions of the participants nor to the
private connotations which they attach to forms of language;
each participant burns his own smoke.

This does not mean that such predispositions, private ad-
ventures, or connotations are excluded from the object range of
science: as objects they will be studied in psychology, biogra-
phy, history, aesthetics, etc.

As, by convention and training, the participants in scientific
discourse learn consistently to ignore all private factors of
meaning, the lexical, grammatical, and stylistic features of their
informal discourse become indifferent: each scientist responds
to each discourse only with the relevant operations or their
linguistic substitutes. Thus, half-a-dozen differently worded
treatises, say on elementary mechanics, will produce the same
result, so far as science is concerned. In this uniformity the dif-
ferences between languages (as English, French, German), far-
reaching and deep-seated as they are, constitute merely a part
of the communicative dross. We say that scientific discourse is
translatable, and mean by this that not only the difference be-
tween languages but, within each language, the difference be-
tween operationally equivalent wordings has no scientific effect.

30. Postulational Form

This stripping-down of meanings and exclusion of silent hy-
potheses has cost mankind much labor and many heartaches,
and will cost more. Twists of meaning and of belief that prevail
in our community may deceive us all; whoever has detected one
or another finds himself misunderstood, since his fellows often
relapse into attaching the traditional meaning to his words or

265

Linguistic Aspects of Science

inserting the traditional hypothesis into their reading of his
calculations.

In order to unmask blind connotations and beliefs and to
bring disagreements into the light of operation, where they may
be subject to test, scientists resort — not often enough, to be
sure — to the method of postulates, an explicit statement of what
is taken for granted. One lists the undefined terms which one
takes from everyday language or, more often, from other
branches of science where they have been defined. Physicalism,
as we have seen (§ 9), is the hypothesis which supposes that, in
the unified vocabulary of science which is thus laid out, the
ultimately undefined words will be simple terms in the domain
of elementary physics — which, in turn, of course, rest upon
physical terms of everyday life. In the same way, alongside the
observed data, and on a par with them as starting-points for
calculation, one states the hypotheses — the suppositions, gen-
erally results of other branches of science, which one accepts as
prior to the work in hand. All new terms, after that, are rigidly
defined : the new term is to be fully and freely interchangeable
with the defining phrase. Only predictions, reached by accept-
able calculation, may be added, by way of testing, to the list of
hypotheses. If the terms are few enough and the structure of
hypothesis and observation is simple enough, a mathematical
system may be found or devised for the calculations. A dramat-
ic instance of this was the invention of co-ordinate geometry.
At best, of course, the uncertainties of all human response will
play into the choice and character of our observation and, in-
directly, through the failings of other sciences, into our structure
of hypothesis. In the sciences that deal with man there is little
enough that we have even learned to observe. It is all the more
desirable that we lay bare our situation and our doubts by the
frank survey of the postulational method.

It is the task of logic to examine the consistency of our cal-
culations, mathematical or informal. If calculation can be per-
formed in terms of numerical mathematics, its validity stands
or falls with the validity of this discipline, and this, so far, has
been the main concern of modern logic. If we cannot translate

266

Sentence-Forms of Scientific Language

our discourse into an accepted dialect of mathematics, we had
better travel with care. Formal rules of calculation would be
equivalent to the meaning of our terms and to the content of
our hypotheses, but these are often too vague or complex for
formal discourse. Most scientific investigations are born with
the makeshift help of informal methodology rather than with
the professional guidance of formal logic.

31. Sentence-Forms of Scientific Language

The translatability of scientific discourse allows informal dis-
course to maintain a wide range of variation in vocabulary,
grammar, and style; but, by the same token, this variation
plays no serious part in the work of science. In principle we
could agree upon any number of formal dialects, each capable of
a uniform version of the results of science. The character of
such a dialect is limited only by the fundamental linguistic type
of scientific discourse. However, we have seen that certain
speech-forms, most strikingly the number expressions, offer ad-
vantages great enough to make our discourse tend always in their
direction. Apart from the special vocabularies of the several
branches of science — or rather, apart from the great vocabulary
of science — our discourse tends always toward the shape that is
provided by mathematics and formal logic. Accordingly, we
need here outline only the features which seem necessary in a
formal dialect that is to serve for the use of science, and we may
ignore the possibilities of expansion and variation.

Of sentence types, only the full-sentence statement comes into
consideration. Every sentence will consist either of a state-
ment-phrase (§ 18) or of several such in co-ordination. State-
ments, of course, will figure also as subordinate parts of large
sentences.

1 his restriction would make bound forms of all parts of utter-
ance other than the favored type of phrase: the structure of our
statements would be a matter of morphology. In fact, it is pos-
sible to devise systems of discourse in which, for example, every
phoneme (or, graphically, every letter) will have a meaning.

267

Linguistic Aspects of Science

Such a system will be very compact but inconveniently remote
from ordinary language.

It is customary, instead, even in formal systems, to use words
and phrases of ordinary language as the smallest meaningful
units. 1 his is done because, on the one hand, the morphology
of most languages is whimsical and complicated, and, on the

other hand, we wish in explanatory discourse, in definition, and
in logical discussion to isolate every meaningful form. Hence
we speak of forms like ‘not,’ ‘and,’ ‘plus,’ etc., as “words,” even
though within the system of formal speech they dare not appear
as sentences.

Calculation may be viewed as a process of exclusion. A set of
initial sentences (report of observation and hypotheses) ex-
cludes certain other sentences, including the negatives of certain
other sentences; a sentence whose negative is excluded is there-
by included (implied) in our discourse. All other sentences are
irrelevant. Accordingly, a scientific dialect will contain such
forms as ‘not,’ ‘excludes,’ ‘implies’ (‘therefore,’ ‘if .... , then
. . . .’). One or several such must be undefined; the rest can be
defined. The dichotomy of including and excluding statements
inheres in the nature of language. A system which contained
intermediate values (as, say, of probability) would still have to
provide this dichotomy for statements in its discourse.

A report transmits either the mere existence of a stimulus or
the accompaniment of a known stimulus by a new one. In either
case, the report may give some analysis of the stimuli. Hence
we may expect statements of such types as ‘There exists . . . .’
and .... is ’ Formally, if we make the necessary agree-

ments, the mere presentation of a speech-form or of a sequence
of speech-forms in a fixed order might be made to suffice; we
approximate ordinary language by using words or phrases of
existence and of predication.

In most languages the constituents of a predication (two-part
statement) belong to different form-classes : ‘John [ ran’ ; ‘This
apple | is red.’ Senseless locutions like ‘Ran | ran’ or ‘Is red | is
red’ are grammatically impossible. However, the provision of
duplicate forms, such as ‘red’ and ‘redness’— lexically similar

268

Syntactic Features

expressions in more than one form-class — makes it grammatical-
ly possible to say ‘Redness | is red.’ If this is not apprehended
as nonsense, contradictions may arise. Thus, if a class of three
members is a “triad,” and in our discussion there appear exactly
three such classes, A, B, C, then we may be tempted to say that
the “class” of triads (T) “is a member of itself”; then, however,
there are four such classes — A, B, C, T — and the “class” of
triads is not “a member of itself”; but, remove T, and there are
again three — a hopeless contradiction. Accordingly, if one
wishes to use locutions of the type ‘Red is a color,’ one must
distinguish different levels among the forms which enter into
predication.

If the stimuli are broken into parts, more than one speech-
form may appear in statements of existence, and more than two
in predicative statements: ‘There exists (a pair of men in the
relation of) father-son’; and, George and Edward being known,
‘George (and) Edward are father-son.’ Here ‘father-son’ figures
as a unit combining with the two units ‘George’ and ‘Edward.’

A third type of sentence will be equational, of a form like

‘. . . . equals . . . .’ or ‘. . . . means ’ It is this type which

makes definition possible; accordingly, it must remain unde-
fined. We have seen that, in scientific speech, definition implies
complete interchangeability of the new term with the old: if
this is not agreed upon, we cannot develop the kind of dialect
that serves in scientific use.

Finally, we shall have a form for the exclusion of statements,

such as ‘not ’ From this most strikingly, but actually not

more than from our other fundamental expressions, there arise
the rules of calculation. These rules embody such meaning as is
granted to the undefined forms of a system. They govern the
sequence of sentences : they are not grammatical (morphologic
or syntactic) but lexical.

32. Syntactic Features

The type of statement which says that (the known) George is
the father of (the known) Edward will present a formal shape
something like ‘FS (George, Edward).’ Here the two names are

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Linguistic Aspects of Science

syntactically co-ordinate but not interchangeable, since the
order (father first) has been agreed upon. A commutative con-
struction would appear if we said that Edward and George were
brothers, as ‘BB (Edward, George).’ We require a construction
which co-ordinates any number of terms, to begin with, in a
linear sequence, with order fixed or free. Such sequences, famil-
iar from Cartesian geometry and from physics, are vectors. The
definitions of this term which appear in elementary treatises put
the cart before the horse, since by appealing to “direction” or
specifying conditions for a “change of co-ordinates,” they very
improperly use an application or a special eventuality of reckon-
ing to define a form of speech. It is only after we have defined a
numerical vector as a sequence of numbers in fixed order that it
devolves upon the geometer or physicist to show that the vector
can be used to state a direction or to discuss how its terms will
be altered by a new frame of reference. Otherwise we are ex-
posed to mysticism: the vector may become in the end a “crea-
tion of our thoughts” (“ein Gedankending”).

A more complex constellation may require a higher ordering:
matrices are syntactically of two dimensions, as when we name
someone’s grandparents, say paternal first and men first:

John Edna

Thomas Lilian

The written form allows of two syntactic dimensions; if we used
a model in space, say of wires and significant beads, we could
construct super-matrices of three syntactic dimensions. It
would be easy to formulate problems which required more than
three syntactic dimensions, but difficult to devise an appro-
priate symbolism.

Scientific speech inevitably follow’s ordinary language in
designating sets of similar phenomena by one term, but it then
requires a far more systematic determination (§ 21 ). If several
phenomena of the set play different parts in the discourse, the
term itself is a variable, and some unmistakable identification
distinguishes the separate specimens (as, in algebra, ‘x u ’ ‘x 2 ,’
‘x 3 ,’ etc.). This allows us to dispense with the ‘a,’ ‘any,’ ‘the’ of

270

Special Features

ordinary language. By means of existence statements one can
then define precise terms in the sphere of our ‘some,’ ‘all,’ ‘no.’

33. Special Features

We can bind ourselves to no limit upon the number of oc-
currences of any one type of phenomenon which will be studied
or upon the minuteness of subdivision to which a phenomenon
may be subjected. This is equivalent to a demand that our
dialect contain infinite classes of speech-forms. In this matter
philosophers and mathematicians have often failed to recognize
fundamental linguistic facts. So far as concerns anything acces-
sible to science, the only infinite classes are classes of speech-
forms; such situations as are referred to by terms like ‘limit,’
‘dense,’ and ‘continuous’ arise only from our agreements as to
the use of speech-forms and will be sought in vain in that outer
world which is studied by science.

In the class of all English sentences that may be uttered,
without limit upon length, we have an infinite class of the lowest
type, and, if we imagine ourselves incapable of phonetics and
ignorant of writing, this class lacks anything like order. By
selecting from it and agreeing upon types of order, we define the
infinite classes which serve us in scientific discourse. Our ordi-
nary speech furnishes the simplest of these in the shape of the
number-words, linearly and discretely ordered, with a first
member ‘one.’

Linguistic devices like these are entirely the creation of so-
ciety, their usefulness, but never their operation, depends upon
our handling activity. Viewed from the strictly linguistic level,
it is an accident that the divisibility of matter goes beyond the
power of our finest instruments, and an accident, no less, that
our physics postulates molecular structure. Regardless of the
latter circumstance, the former brings to us an engineering de-
mand for a densely and linearly ordered infinite class: in the
order there shall be a form between any two forms that may be
named. This demand is met by means of number-pairs (two-
place vectors) in the shape of the rational numbers, with the
familiar agreement that, if ad precedes be, then among the

271

Linguistic Aspects of Science

rational numbers (a, b ) shall precede (c, d ). However, one could
also agree upon an ordering of the natural numbers that would
make them dense. From this it follows, that, leaving the de-
cadic syntax of the natural numbers, we could devise a densely
ordered class out of repetitions of a single speech-form and, in
greater variety, out of phrases built, say, of two speech-forms—
witness the decimal expressions consisting of 0’s and l’s.

A “continuous” object, such as a piece of wire, presents itself
between any two of its points, no matter how we swing the
knife. A straight line, in geometry, is met by lines at points
which are not designated by rational co-ordinates. Given a lin-
early and densely ordered class of speech-forms, we require,
therefore, that any expression (within the vocabulary and syn-
tax of our dialect) which divides (“cuts”) the class in the famil-
iar manner of Dedekind s postulate shall be a member of a new
class. This class is then continuous ; it contains a member to
match every member of the old class, since these members make
the required division, and it contains, further, any other phrases
(within our dialect) which define a cut. Mathematicians some-
times ignore the obvious fact that the possibility of defining the
“cuts” depends entirely upon the vocabulary and syntax of our
discourse and by no means upon our “perception of time” or
upon any mystical realm of “ideas.” In the discourse of num-
bers the agreements of multiplication (‘square,’ etc.) make pos-
sible one kind of cut, and those of limits another. It is for the
speakers to decide what expressions they will admit into their
discourse.

VI. Summary

34. The Place of Linguistics in the Scheme of Science

The subject matter of linguistics, of course, is human speech.
Other activities, such as writing, which serve as substitutes for
speech, concern linguistics only in their semiotic aspect, as rep-
resentations of phonemes or speech-forms. Since the meanings
of speech cover everything (designata, including denotata; syn-
tactic relations; pragmatic slants), linguistics, even more than

272

Relation of Linguistics to Logic and Mathematics

other branches of science, depends for its range and accuracy
upon the success of science as a whole. For the most part, our
statements of meaning are makeshift. Even if this were not the
case, linguistics would still study forms first and then look into
their meanings, since language consists in the human response to
the flow and variety of the world by simple sequences of a very
few typical speech-sounds.

Linguistics is the chief contributor to semiotic. Among the
special branches of science, it intervenes between biology, on
the one hand, and ethnology, sociology, and psychology, on the
other: it stands between physical and cultural anthropology.

Language establishes, by means of sound waves and on the
basis of a communal habit, an ever ready connection between
the bodies of individuals — a connection between their nervous
systems which enables each person to respond to the stimuli
that act upon other persons. The division of labor, civilization,
and culture arise from this interaction. Popularly and even, to
a large extent, academically, we are not accustomed to observ-
ing language and its effects : these effects are generally explained
instead by the postulation of “mental” factors. In the cosmos,
language produces human society, a structure more complex
than the individual, related to him somewhat as the many-
celled organism is related to the single cell.

35. Relation of Linguistics to Logic and Mathematics

Specialized uses of language involve no great alterations of
structure; the specialization consists rather in the way that
language is applied. Thus, the study of literature requires that
we investigate the institutions and traditions of the community
and the psychology (physiology, social status, biography) of the
creative individual. In connection with science, language is spe-
cialized in the direction of forms which successfully communi-
cate handling responses and lend themselves to elaborate re-
shaping (calculation). To invent and to employ these forms is
to carry on mathematics. The critique and theory of scientific
speech is the task of logic. Logic is a branch of science closely
related to linguistics, since it observes how people conduct a

273

Linguistic Aspects of Science

certain type of discourse. In contrast with this, the invention
and skilful manipulation of speech-forms is not a science but a
skill, craft, or art; it is as such that we class mathematics.
Mathematics appears as a science only so long as we believe
that the mathematician is not creating speech-forms and dis-
courses but exploring an unknown realm of “concepts” or
“ideas.”

Since mathematics is a verbal activity and logic a study of
verbal activities, both of these disciplines presuppose linguistics.
However, the forms of language which enter into mathematics,
and are to be examined by logic, are simple and fairly normal:
in principle, neither pursuit requires any technical knowledge of
linguistics. In practice, since we labor under a load of tradi-
tional and popular misconception about language, a great deal
of doubt, error, and dispute will be avoided if mathematicians
and logicians acquire enough linguistics to remove these mis-
conceptions. The principal sources of difficulty are twofold.
Our popular belief distorts the relation of writing to language,
placing the two on a par (“written” and “spoken” language) or
even reversing the dependence, so as to suppose that a change in
writing equals or prompts a change in language. Our popular
belief replaces the function and effects of language by “mental”
factors, which, whatever their place in literature or religion, are
excluded, as non-physical, from the subject matter and proce-
dure of science.

274

Selected Bibliography

Selected Bibliography

§ 2. On the doctrine of the ancient Greeks: B. Delbrllck, Einleitung in das Studium
der indogermanischen Sprachen (“Bibliothek indogermanischer Grammatiken,” Vol. IV
[6th ed.; Leipzig, 1919]).

For the Sanskrit grammar: Language, V (1929), 267. The comparative method:
A. Meillet, La Methods comparative en linguistique historique (“Instituttet for sammen-
lignende kulturforskning,” Ser. A, No. 2 [Oslo, 1925]). History of modern linguistics:
H. Pedersen, Linguistic Science in the Nineteenth Century (Cambridge, Mass., 1931).
Elementary outline of the subject: L. Bloomfield, Language (New York, 1933; London,
1935). Dialect geography: E. C. Roedder in Germanic Review, I (1926), 281.

§ 4. Distinction between speech-forms of discourse and speech-forms under dis-
cussion: R. Carnap, Logical Syntax of Language (Vienna, 1934; London and New York,
1937), sec. 42.

§ 5. On “correctness” in language: C. C. Fries, The Teaching of the English Lan-
guage (New York, 1927); S. A. Leonard, The Doctrine of Correctness in English Usage,
1700-1800 (“University of Wisconsin Studies in Language and Literature,” Vol. XXV
[Madison, 1929]); see also American Speech, II (1927), 432.

§6. Writing: Pedersen, op. cit.; E. H. Sturtevant, Linguistic Change (Chicago,
1917). Duodecimal notation confused with speech-form: Atlantic Monthly, CLIV
(1934), 459. Gesture: W. Wundt, Volkerpsychologie, Vol. I: Die Sprache (3d ed.;
Leipzig, 1911), p. 143.

§ 7. Biophysical and biosocial: A. P. Weiss, A Theoretical Basis of Human Behavior
(2d ed.; Columbus, 1929), p. 84.

§ 9. Mentalistic doctrines of various types: A. P. Weiss in Psychological Review,
XXIV (1917), 301, 353; in Journal of Psychology, XVI (1919), 626. See also Psycho-
logical Review, XXVI (1919), 327; ibid., XXIX (1922), 329. Behaviorism: Weiss,
Theoretical Basis of Human Behavior; operationalism: P. W. Bridgman, The Logic of
Modern Physics (New York, 1927); physicalism: R. Carnap, The Unity of Science
(“Psyche Miniatures: General Series,” No. 63 [London, 1934]); Philosophy and Logical
Syntax (“Psyche Miniatures: General Series,” No. 70 [London, 1935]); O. Neurath in
The Monist, XLI (1931), 618; see also Language XII (1936), 89.

§ 10. A. P. Weiss in Psychological Review, XXXVIII (1931), 474.

§ 14. Meaning is discussed with bibliography, on a mentalistic basis, to be sure,
by G. Stern, Meaning and Change of Meaning (“Goteborg hogskolas Ursskrift,” Vol.
XXXVIII, No. 1 [Gothenburg, 1932]). See also C. W. Morris’ monograph. Founda-
tions of the Theory of Signs (“International Encyclopedia of Unified Science,” Vol. I,
No. 2 [Chicago, 1938]).

§ 16. The phoneme as a signal: E. Sapir, Language (New York, 1921).

§ 18. The sentence: in Language, VII (1931), 204. On “pragmatic” features of
meaning: Morris, op, cit.

§ 19. Levels and ranks of construction: O. Jespersen, The Philosophy of Grammar
(London and New York, 1924).

§ 20. Categories: F. Boas, “Introduction” in Handbook of American Indian lan-
guages (Smithsonian Institution, Bureau of American Ethnology, Bull. 40 [Washing-
ton, 1911]), Vol. I.

275

Linguistic Aspects of Science

§ 24. See in Philosophy of Science, II (1935), 499; Language, XII (1936), 94. On
variability: Weiss, Theoretical Basis of Human Behavior, p. 134.

§26. Euclid’s proof, conveniently given in L. E. Dickson, Introduction to the
Themy^of Numbers (Chicago, 1929), p. 4. Archimedes, ed. J. L. Heilig (Leipzig, 1913),

§31. Calculation: K. Menger in Philosophy of Science, IV (1937), 299.

§ 33. Elementary outline: F. Waismann, Einfuhrung in das matliematische Denken
(Vienna, 1936).

276

Index of Technical Terms

[Numbers refer to pages of this monograph.]

Abstraction, 255
Actor-action, 245
Anaphora, 251
Approximation, 255
Attribute, 247

Behaviorism, 231
Biophysical, 226
Biosocial, 227
Bound form, 243

Categories, 250
Center, 248
Class meaning, 245
Constituent, 243
Construction, 243
Contradiction, 253
Co-ordinative, 247

Deduction, 230
Definition, 238
Demonstration, 237
Determiner, 251
Different, 241

Endocentric, 247
Equation, 269
Equivalent, 253
Existence, 268
Exocentric, 247

Fill, 244
Form, 241
Formal, 261
Formal definition, 238
Form-class, 244
Free form, 242
Full sentence, 245
Function, 244

Handling, 226
Head, 247

Immediate, 247
Implication, 254
Infinite, 258

Informal, 261
Inner speech, 235

Language, 224

Matrix, 270
Meaning, 236
Mechanism, 231
Member, 247
Mentalism, 230
Minor sentence, 246
Modulation, 244
Morphology, 243

Negation, 253

Operationalism, 231
Order, 245

Parts of speech, 244
Phoneme, 239
Phonetic modification, 244
Phrase, 243
Physicalism, 231
Position, 244
Predication, 268

Resultant, 243

Same, 241
Scope, 248

Secondary phoneme, 244
Selection, 244
Sentence, 245
Sentence type, 245
Subordinative, 247
Substitute, 251
Syntax, 243

Technical term, 256

Variability, 256
Variable, 270
Vector, 270

Word, 243
Writing, 224

Procedures of Empirical
Science

Victor F. Lenzen

Procedures of Empirical Science

Contents:

I. Introduction pace

1. The Problem of Empirical Science .281

2. The Subject Matter of Empirical Science 283

II. Observation

3. Perception 284

4. Counting 286

5. Measurement of Length 289

6. Measurement of Time 295

7. Measurement of Weight 302

8. Observation through Causality 304

9. Observation of Microphysical Entities 306

10. Partition between Object and Observer . . 308

III. Systematization

11. Classification 311

12. The Correlation of Events 315

13. Successive Approximation 321

14. Successive Definition 324

15. Atomism 326

16. Statistics in Quantum Theory 331

17. Atomism in Biology 333

IV. Conclusion

18. Unity of Science 335

Selected Bibliography 338

280

Procedures of Empirical Science

Victor F. Lenzen
I. Introduction

1. The Problem of Empirical Science

The problem of empirical science is the acquisition and system-
atization of knowledge concerning the things and phenomena
experienced in observation.

The basic procedures of empirical science occur in daily life.
Prior to his cultivation of science an individual perceives rela-
tively stable things in space and time, describes them with the
aid of symbols which record and communicate the results of
observation, and explains perceptible phenomena in terms of
causes. Simple experimental techniques are also used in the
ordinary conduct of life. The child learns the operations of
counting, of measuring length and time, of weighing with a
balance. The builder uses tools, the housewife applies heat to
produce the chemical reactions of cookery, the farmer culti-
vates crops. Such procedures are based upon prehistoric dis-
coveries and inventions which have become the heritage of the
race. In our historic era empirical science criticizes, augments,
and systematizes practical experience. The science of one gen-
eration becomes incorporated in the technology of the succeed-
ing one. Science and practice co-operate in the adjustment of
man to his environment.

The preceding description of the interdependence of science
and practice may be illustrated by a sketch of the origin of sci-
ence in ancient Greece. It is traditional that the Greeks cre-
ated European science, but their achievement was founded on
knowledge and procedures which had been inherited from earlier
Babylonian and Egyptian civilizations. The observations of
correlations between the apparent positions of the heavenly

281

Procedures of Empirical Science

bodies by the Babylonians provided a basis for the measure-
ment of time and furnished data for a geometrical picture of
celestial motions. The procedures used by the Egyptians in
their surveys of land were formulated in propositions whose de-
ductive relations were set forth in Euclidean geometry, which
has been the classic model for systems of science. The Egyptians
practiced medicine and surgery, and the Greeks under the lead-
ership of Hippocrates developed this field by observation and
experiment. Out of elements derived from practice was created
the science of biology, which was especially developed by Aris-
totle, who made observations on animals and invented a sys-
tem of classification. Machines were employed prior to the cre-
ative Greek period; on Egyptian temples the god Osiris is de-
picted weighing the soul with a balance. The principles implied
in such machines are formulated in the science of statics which
was founded by Archimedes. Thus the Greeks built upon the
particular knowledge and techniques of the Babylonians and
Egyptians. They organized the observations and procedures of
their predecessors into theories founded upon principles.

The Greek interest in systematization also led to the creation
of constructive hypotheses which reduced the diversity of per-
ceptible things to a unitary basis. The theories concerning the
nature of things initiated by Thales and his successors eventual-
ly led to the creation of the atomic theory, which in the modern
era has provided a unified interpretation of physical and chemi-
cal phenomena. The Greeks emphasized the rational factor in
science at the expense of observation. They were fertile in the
creation of theories, but failed to develop adequately the tech-
nique of experiment. Greek science lost its creative power as
ancient civilization decayed, but with the revival of science dur-
ing the dawn of the modern era the method of controlled experi-
mentation came to be adequately appreciated. Vesalius’ in-
sistence upon dissection for anatomy, Galileo’s experiments on
falling bodies, and Bacon’s attempt to formulate the method of
induction recognized the need for observation and experiment in
empirical science. The experimental point of view was clearly
stated by Newton in his assertion that physical properties are

282

Subject Matter of Empirical Science

to be derived from experiments. The properties of things are
modes of reaction to conditions that are subject to experimental
control. Significant experimentation requires the guidance of
hypotheses which serve to predict the results of observation.
Accordingly, in the well-developed empirical sciences a theory
which applies to a selected universe of discourse assumes the
form of a hypothetico-deductive system, and scientific pro-
cedure consists in deriving predictions which are tested by the
results of experiments. This union of the Greek conception of
theory with experimental procedure has resulted in the modern
development of science, which through technology created the
industrial revolution and today is even more radically trans-
forming the practice of daily life. Refinements in technique are
providing the instruments for future developments in the the-
ories w T hich systematize the results of observation.

2. The Subject Matter of Empirical Science

In preparation for the analysis of procedures I shall sketch
the general character of the subject matter of empirical science.
The initial objects of science are the things experienced in per-
ception, and their most general characters are position in space
and time. The systematic and ultimately quantitative investi-
gation of space-time order may be called generalized physics.
Thus one arrives at the doctrine of physicalism, which asserts
that the concepts of empirical science are reducible to those
which express the properties of spatio-temporal things.

Carnap has shown that by a method of reduction, of which
definition is a special case, the terms of a science are introduced
by statements which involve terms designating perceptible
things and properties. An outline of physicalist analysis, with
especial reference to biology, is given in Volume I, Number 1.
I present an analysis, adapted from Carnap, 1 of some biological
terms. An organism is a special type of space-time structure,
and hence the designation of terms such as ‘species,’ ‘genus,’
‘events in organisms,’ is always determined on the basis of per-
ceptible criteria. Thus “ ‘fertilization’ is defined as the union
of spermatazoon and egg; ‘sperinatazoon’ and ‘egg’ are defined

283

Procedures of Empirical Science

as cells of specified origin and specified perceptible properties;
‘union’ as an event consisting of a specified spatial redistribu-
tion of parts.” Such biological terms as ‘metabolism,’ ‘growth,’
regeneration, cell division,’ etc., are also introduced in terms
of perceptible criteria.

In behavioristics the responses of organisms to their environ-
ment are investigated, and in sociology the relations of human
beings to one another and to the environment. As Neurath has
emphasized, sociology deals with space-time structures. The
essential difference between mechanics and sociology is that the
relative simplicity of mechanical phenomena renders possible
the repetition of experiments, whereas the complexity of social
processes makes controlled experimentation difficult.

The formation of precise and quantitative concepts for the
properties of spatio-temporal things is the first problem of phys-
ics, which plays a fundamental role in empirical science. The
analysis of physical procedure is therefore basic in this mono-
graph.

II. Observation

3. Perception

The basic procedure of empirical science is observation. Sci-
entific description starts with observation; confirmation of the
hypotheses of a theory is attained when phenomena predicted
on the theory are observed. The term ‘observation’ has a mean-
ing which is relative to a scientific situation. In daily life obser-
vation consists of perception with the unaided senses, in as-
tronomy observation of the heavenly bodies is mediated by
telescopes, in biology observation is through a microscope, in
atomic physics observation is theoretical interpretation of ex-
perimentally controlled phenomena. The observation of micro-
physical entities requires the explicit use of physical principles
as instruments of interpretation, but all observation involves
more or less explicitly the element of hypothesis. In the present
part of this monograph I shall describe the different types of
observation in the order of increasing explicitness of the hypo-
thetical factor. Let us begin with perception.

284

Perception

The defining characteristic of perception is the occurrence of
a sense datum and its interpretation as the aspect of an objec-
tive thing. Perception thus involves the hypothesis that there
exists an object to which given aspects are referred. The hy-
pothesis that various common things exist is continually being
confirmed by the reproduction of perceptions of them. I re-
peatedly have perceptions which are described as responses to a
desk with relatively stable properties. Different types of per-
ception are found to be correlated; thus, upon sight of my desk
I expect to touch it. The truth of a perception is confirmed or
disconfirmed by testing the predictions derivable from it. The
development of the concept of an object is completed by the
hypothesis of the identity of the perceptible objects of a society
of observers. Thus the concept of objective thing is social; sci-
ence is tested by social procedure . 2 The scientific criterion of
objectivity ultimately rests upon the possibility of occurrence
of predicted perceptions to a society of observers.

In daily life perception usually passes unreflectively into ac-
tion. As I walk along the street, I perceive someone coming to-
ward me and automatically step out of the way. I reach the
curb and start to cross the street, but I perceive an automobile
and hesitate. Nevertheless, even in the almost unreflective
practice of common experience there are flashes of discrimina-
tion of qualities and comparisons between them. I notice that
the approaching automobile is red; I judge that one man is
taller than his companion. A phase of perception, accordingly,
is discrimination and comparison of given qualities and rela-
tions. The results of such procedure can be recorded and com-
municated only if the experienced qualities and relations are
symbolized. When things are perceived to have aspects that
are similar in a specific respect, they are assigned a common qual-
ity or relation which is designated by a general name. The des-
ignation of the name of a simple quality such as redness can be
understood only if one has immediately experienced the quality.
General names are instruments of analysis. A qualitative anal-
ysis of the data of perception provides the basis for the descrip-
tion of a thing in terms of predicates. For example, an apple

285

Procedures of Empirical Science

may be described as red, round, smooth, sweet, etc. Thus per-
ception is completed by analysis of perceptible properties and
the expression of the results of observation in descriptions. This
procedure is a first step in the systematization of knowledge.
Description of things is a mode of classification which in turn
furnishes material for a more general classification. In its de-
scriptive stage, which is based upon perception, empirical sci-
ence introduces order into cognitions by classification.

On the perceptual level of knowledge the properties of a thing
are commonly discovered by manipulation. An object which
visually appears to be at hand may be denied reality and called
a hallucination because it cannot be grasped.

But things may also be investigated by sight; indeed, for the
study of microscopic objects the procedure is to observe them
with the aid of optical instruments. In virtue of the capacity of
glass to refract light, it is possible to make microscopes and tele-
scopes, so that on looking at an object through an optical instru-
ment its visual aspect is magnified. The phenomenon of mag-
nification renders it possible to infer the existence of objects
which do not appreciably affect the unaided senses. Observa-
tion with instruments is an extension of perception in which the
hypothetical element is more explicit than in the ordinary per-
ceptions of daily life. Yet the hypotheses have been so often
confirmed that practical certainty is achieved.

In the development of empirical science qualitative descrip-
tion of things and phenomena is supplemented by quantitative
representation. In addition to characterizing a group of objects
as many, one may assign to it a number; one may not rest with
the description of a rod as long or short, but record that it is so
many meters long. Quantitative representation requires pro-
cedures for assigning numbers to measurable properties. I shall
sketch the methods for the assignment of numbers to collections
and to continuous manifolds such as space and time.

4. Counting

The fundamental quantitative procedure of science is count-
ing a set of objects in order to characterize it by a number. A

286

Counting

collection may consist of similar things such as the sheep in a
flock; or the members of a collection may be as dissimilar as the
books, papers, and other things on my desk. For the purpose of
counting, differences in characteristics of the members of a col-
lection are ignored, but their distinguishability must be pre-
served.

The basic operation in counting is illustrated in the following
quotation from Conant’s The Number Concept . 3 “The savage
can form no mental concept of what civilized man means by
such a word as soul; nor would his idea of the abstract number 5
be much clearer. When he says five, he uses, in many cases at
least, the same word that serves him when he wishes to say
hand, .... and his only comprehension of the number is, ‘these
objects are as many as the fingers on my hand.’ ” This example
suggests that counting is an operation of determining similarity
between collections. Two collections are defined to be similar if
they can be put into one to one correspondence, so that to every
member of one collection is correlated a member of the other
collection. Similar collections, or classes, are assigned the same
number. The null class is characterized by the number zero,
unit classes by one, couples by two, trios by three, etc. In-
deed, Whitehead and Russell define a cardinal number as the
class of all similar classes. Thus we may define the natural num-
bers which serve to describe collections of objects.

The number of a collection is determined by counting. As
one counts, the members are put into one to one correspondence
with the natural numbers. If one has used the natural numbers
from 1 to n, it is established that the number of objects in the
collection is the same as the number of natural numbers from
1 to n, and, since this number is n, the collection counted has n
objects. The number of a collection is independent of the order
in which the counting occurs.

Given two collections we may combine them to form a new
collection. For example, the addition of a collection of three ob-
jects to a collection of two objects yields a collection of five ob-
jects, and the result is the same if the collection of two is added
to the collection of three. The formation of the new collection

287

Procedures of Empirical Science

by addition may be represented by the addition of the numbers
of the collections added, thus 3 + 2 = 2 + 3 = 5. A similar
relation holds for all instances of addition, and hence we may
infer a general law, a + b = b + a, the commutative law for
addition.

Laws for multiplication are also derived from observation.
If one combines two collections each containing three objects,
the resultant collection contains six objects. The combination
of three collections each containing two objects also yields six
objects. The formation of collections by multiplication may be
represented by the numerical expression ‘2 X 3 = 3 X 2 = 6.’
A similar relation holds for every multiplication of collections,
and, hence, one infers the general commutative law for multipli-
cation, a X b = b X a.

The preceding discussion of the commutative laws of addition
and multiplication demonstrates that the fundamental proposi-
tions of ordinary algebra are initially derived from observation.
Numbers are first discovered as characters of collections; but,
once principles expressing their relations are set up, the numbers
acquire properties defined in terms of their relations to one an-
other. They become the subject matter of an abstract theory
which defines their properties independently of any empirical
application. As the number system is extended from the natural
numbers to include negative, rational, irrational, and complex
numbers, the original connection between numbers and collec-
tions is lost. The real numbers find a geometrical representation
in the points of a line, and complex numbers in vectors.

The operation of counting is performed by pointing or other-
wise indicating the members of a collection to which one assigns
the numbers 1, 2, etc., respectively. The physicist, however, has
invented mechanical counters so that, for example, the number
of revolutions of a wheel is recorded. In contemporary physics
the number of microphysical events, such as the passages of
cosmic-ray particles through a Geiger counter, can be registered
by a mechanical device. In order to determine the number of
molecules in a specific volume of gas, however, it is necessary to
employ indirect methods based upon theories.

288

Measurement of Length

Statistical surveys are important for many problems, and
counting is their initial procedure. I have indicated the impor-
tance of statistics in physics, but the social sciences are even
more dependent upon counting for their materials. Counts are
taken of the number of inhabitants in defined areas, of the num-
ber of births per year, and similarly of deaths, marriages, homi-
cides, etc. From such statistical data there are calculated birth-
rates, death-rates, and other results which give useful informa-
tion about a given society.

5. Measurement of Length

Measurement is the general procedure of assigning numbers
to the properties of objects. A measurable property is usually
called a magnitude, but the term ‘quantity’ is also used. A basic
measurement is that of length or distance by a procedure which
was invented in remote antiquity. In order to measure the
length of a rod, one selects a standard of length and counts the
number of times that it can be laid off on the rod. I shall ex-
plain the general procedure of measurement by analyzing that of
length.

The basic principle of measurement is that the space-time
coincidence of objective events is perceptible to a community of
observers. The most direct exemplification of this principle is
offered by the measurement of length, which depends upon the
determination of contact of two things from the contact of their
aspects in perception. The empirical concept of contact is only
approximately precise, but for theory one sets up the postulate
that two coincident points are perceived to be coincident.
Brunswik has explained how coincidence is the basis of objec-
tivity. Two coincident points are perceived under exactly the
same conditions; hence identical spatial perceptions are stimu-
lated by the objects. If, for example, perception is mediated
by light which is scattered by the coincident points, light from
both points will travel over the same path to the same point on
the retina and produce the same perceptual reactions. The ob-
jectivity of measurement is based upon the fundamental func-
tion of the perception of coincidence.

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Procedures of Empirical Science

The physical concept of length is empirically derivable from
solid bodies, which in view of the properties to be described may
be called practically rigid bodies. Two points on a rigid body
determine a stretch which is defined to include its end points.
Let us place two bodies adjacent to each other so that two points
on the one coincide with two points on the other. Assuming
constancy in external conditions, the coincidence of the two sets
of end points of the stretches remains constant in time. It is
also independent of position in space, for, if the two bodies are
displaced together, coincidence is preserved. Accordingly, one
characterizes the two stretches as congruent. Congruence is ob-
served to be a symmetrical relation, that is, if A is congruent
to B, B is congruent to A; it is also transitive, that is, if A is
congruent to B, and B is congruent to C, then A is congruent to
C. To the congruent stretches one assigns the same number, the
length of each stretch or the distance between its end points.

Thus far I have described the test for congruence of adjacent
stretches. The test breaks down if the two stretches are sepa-
rated in space. Observation shows, however, that, if separated
stretches are again brought together, the coincidence of end
points is restored. Furthermore, stretches that may successive-
ly be exhibited as congruent to a standard stretch may be dem-
onstrated to be congruent when brought together at a distance
from the standard. In view of such experimental results, one is
led to postulate that stretches are congruent at a distance if
they prove themselves to be congruent when adjacent. In
effect, we postulate that the distance between two points on a
solid body is unchanged in a displacement. In the present con-
text it is meaningless to ask if this convention is really true.
Such a question would be significant only if there were a more
fundamental definition of length or distance. Accordingly, we
may adopt as a standard of length the distance between two
marks on some solid body. I he international legal standard of
length is the meter. It was originally defined as 1/10,000,000 of
the distance from either pole to the equator and was approxi-
mately exemplified by the distance between two scratches on a
bar, the Metre des Archives, at the temperature of melting ice

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Measurement of Length

under atmospheric pressure. Later the distance between the
two scratches was arbitrarily chosen as the standard. The wave-
length of a red spectral line emitted by cadmium has been meas-
ured in terms of the standard meter, thus preparing the way for
the adoption of this wave-length as the standard of length.

Thus far we have considered congruence of stretches and
identity of length of congruent stretches. I have not yet ex-
plained the procedure for measuring the length of any stretch
in terms of a standard. Let us suppose we are given a line. In
nature a line may be realized by a ray of light, by a stretched
cord, or by the edge of some solid. One may measure the length
of a line by the following procedure: Beginning at one end,
place stretches, congruent to the standard stretch, end to end
until the other end of the line is reached. The number of
stretches is the numerical measure of the length of the line rela-
tive to the adopted standard. In practice the congruent
stretches are constructed by placing a standard alongside the
line and making points on it which coincide with the end points
of the stretch. The accuracy of a measurement of length can
be indefinitely increased by decreasing the standard of length.

Our description of procedure in measuring length has presup-
posed constancy of the external environment and of the stand-
ard. In practice, however, corrections must be made for a
change in conditions. Suppose that we have measured the di-
mensions of various bodies with a standard rod. If the measure-
ments are repeated with a heated rod, it will be found that the
numerical measures are smaller. As Carnap has shown, we
have in principle a choice between two methods of explaining
the change. We may postulate that the standard rod retains
the same length but that the heating of the rod has produced,
by an action at a distance, changes in the dimensions of the
bodies in the environment. Or we may infer a change in the
standard rod. If the standard rod is heated by a flame in con-
tact with it, the second explanation is in accordance with a
principle of contiguous causality which states that a physical
change is to be attributed to an agency in the neighborhood
of the body undergoing the change. This universally accepted

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kind of explanation allows the standard of length to vary with
temperature. Accordingly, the definition of the standard of
length requires specification of its temperature. But the defi-
nition of temperature presupposes the concept of length; for
example, temperature may be defined in terms of the length of a
thread of mercury in a glass tube. Thus we seem to have be-
come involved in a vicious circle. In order to measure length,
we must correct for the temperature of the rod, but, in order to
measure temperature, we need to measure length.

I he circle is avoided by the method of successive approxima-
tion. Let the standard of length be the distance between two
scratches on a particular rod; this constitutes a definition to the
first approximation. With the aid of our standard we can con-
struct a thermometer with which we may discover the law for
the dependence of length on temperature. Thus the definition
of a standard to the first approximation provides a basis for a
definition of temperature in terms of which the standard can
then be defined to a second approximation. The possibility of
this procedure is based upon the fact that a change in tempera-
ture produces a relatively small change in length. An error in
the measurement of temperature resulting from a small error
in the standard of length used to construct the scale of the ther-
mometer gives rise only to errors of the second order of small
quantities in the standard of length that is specified in terms of
the temperature. The method of successive approximation is
also used in controlling the constancy of other physical proper-
ties of the rod. The procedure of measuring length thus involves
the performance of certain operations under controlled condi-
tions. Our description of the measurement of length illustrates
the experimental basis and operational nature of scientific con-
cepts.

The results of measurement need to be appropriately sym-
bolized in order to enter as elements in a mathematical develop-
ment. The result of a measurement of length is stated by a
proposition such as, ‘The length of this rod is 2 meters.’ Carnap
has shown that the foregoing proposition expresses a relation

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Measurement of Length

between the number 2 and the object to which it is assigned.
The proposition may then be written

2 (length in meters) this rod.

The last proposition is of the form xRy, where the relation R is
one-many; there are many rods to which the number has the
relation designated by ‘length in meters.’ Whitehead and Rus-
sell have shown that a one-many relation gives rise to a descrip-
tive function; thus from xRy one obtains

x = the R of y ,

or, in symbols,

x = R‘y .

Applying this analysis to the present problem one obtains

2 = the length in meters of this rod ,

2 = the length in meters ‘this rod .

Relations between measurable properties are represented by
functional relations between their numerical measures. This
may be illustrated by the law of addition which characterizes
length as an extensive quantity. Two stretches A and B on a
straight line may be adjoined to form a stretch C. If the lengths
of the stretches are measured in terms of a standard, L, the re-
sults of measurement may be recorded by ‘a = the length rela-
tive to L of A,’ etc. The adjunction of the stretches is then rep-
resented by the proposition which expresses a relation between
measures, ‘a + b = c.’ The goal of exact empirical science is
the expression of natural laws as functional relations between
numerical values.

The quantitative concept of length is the basis of metrical
geometry which expresses the spatial properties of figures. The
geometer initially studies the spatial relations of practically
rigid bodies, but he also makes figures by stretching cords and
by drawing with a ruler and compass. Since the paths of light
rays are usually straight, triangles and other figures may be

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formed of light rays. Indeed, the rectilinear propagation of
light furnishes a practical test of straightness; the carpenter
tests the straightness of an edge by sighting along it.

The properties of triangles may be used to measure distances
by calculation from observations. In the method of tnangula-
tion one side of a triangle is the straight line between two stakes,
and the other two sides are rays of light from a distant object to
the stakes. The surveyor measures the length of the straight
line between the stakes with a tape; from measures of the angles
that the other two sides make with the base line and the general
laws for triangles he calculates the distances of the object from
the stakes.

Geometry deals initially with fixed structures at rest in a
frame of reference such as the surface of the earth. If geometri-
cal figures are changing, one may specify their instantaneous
properties. The motion of figures relative to a given frame of
reference introduces new problems. In classical kinematics it
was assumed that geometrical figures are independent of their
state of motion. Analysis of the operation of measuring the
length of a body in motion reveals that the concept of simul-
taneity is presupposed. Suppose, for example, that a straight
rod is moving with uniform velocity with respect to a given
frame of reference. The procedure for measuring its length that
may be adopted by an observer at rest in that frame is to mark
on the frame the simultaneous positions of the end points of the
rod. The length of the moving rod relative to the given frame is
the distance between the two points marked on the frame. The
special theory of relativity determines simultaneity to be rela-
tive to the frame of reference, so that the length of a rod rela-
tive to a frame in which the rod is moving is different from its
length relative to a frame in which the rod is at rest.

The geometrical study of the spatial relations of bodies yields
a concept of physical space. In order to increase definiteness,
use is made of relatively small bodies or small portions of bodies
which are called points. Examples of points are dots, pinholes,
knots in cords. For maximum definiteness there is formed the
limiting concept of a point which has a definite position. Space

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Measurement of Time

may then be described as the system of relative positions of
points. The metrical structure of space is defined in terms of
the properties of configurations of rigid bodies. In view of rela-
tivity, space is defined for a selected frame of reference. The
surface of the earth was used by the Greeks as the frame for
physical space, but in the modern era the preferred frame has
been one with its origin at the center of mass of the solar sys-
tem, with the axes oriented with respect to the fixed stars. In
this frame physical space is Euclidean to at least the first ap-
proximation. This means, for example, that it is possible to
construct a cubical lattice out of equal rods, that is, a Cartesian
coordinate system. The general theory of relativity, however,
involves the view that matter determines physical space to be
non-Euclidean.

The propositions of geometry are to be viewed as initially gen-
eralizations from observations on the spatial properties of struc-
tures of practically rigid bodies. In Euclidean geometry the
science is based upon a small number of axioms from which the
propositions can be deduced as theorems. In abstract geometry
the original axioms have been transformed into postulates which
define implicitly the fundamental concepts of geometry. From
this point of view the postulates of Euclidean geometry are
descriptions of the formal spatial properties of rigid bodies.
That the concepts of these forms are applicable to the objects of
perception is a hypothesis to be confirmed and limited by ob-
servation.

6. Measurement of Time

The problem of the measurement of time is to invent a pro-
cedure for assigning to events numbers, called the times or dates
of events. An event is a process of relatively short duration, the
shortness depending on the precision with which one intends to
describe phenomena. In theory one forms the concept of in-
stantaneous event as a theoretical construct. The basic tem-
poral relations are simultaneity and succession of events. Two
events are objectively simultaneous if they are simultaneously
perceptible to a community of observers. A similar criterion

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may be given for succession. We must distinguish between
local time, the time system at a specific place, and extended
time, the time system throughout a space.

The procedure of measuring local time is based upon some
concrete temporal process, usually a periodic motion. The be-
havior of a standard clock defines the scale of time. Now, vari-
ous physical processes can be used for the definition of metrical
time: the rotation of the earth about its axis, the revolution of
the earth around the sun, the vibration of a pendulum, the revo-
lution of the moon around the earth, etc.

The procedure for measuring time may be illustrated by
adopting as the fundamental temporal process the vibration of
a pendulum. The fundamental principle is that successive vi-
brations of the pendulum take equal times. This principle has
an empirical basis. We have a qualitative estimate of the dura-
tion of processes and can judge, for example, that the time of
one vibration of a pendulum is less than the time between sun-
rise and sunset. Hence we may estimate that two successive
vibrations of a pendulum take the same time. Again, all pendu-
lums which are similar in structure vibrate in synchronism,
that is, if two pendulums of equal length are released simul-
taneously, they vibrate together, passing through correspond-
ing points at the same time. But the principle that successive
vibrations take the same time, although suggested by experi-
ence, is a definition of equal intervals of time. It is a conven-
tion concerning which it is meaningless to ask whether it is true
or false. The question of truth or falsity would presuppose an-
other standard, for which one would have to assume that each
performance of a periodic process requires the same time.

Let us then suppose that we have chosen the vibration of a
pendulum as the basis of a metrical structure for time. We may
select the beginning of a particular vibration as the origin of
time; to the ends of successive vibrations are assigned the num-
bers 1, 2, 3, etc. The time of any event is expressed by assign-
mg to !t the number which is correlated with the vibration, the
end of which is simultaneous with the event. If the end of the
vibration and the event are not simultaneous, we may imagine

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Measurement of Time

clocks with shorter and shorter periods and thus, by interpola-
tion, approximate as accurately as we please to a precise assign-
ment of time to the event. The duration of a process is, then,
expressed numerically by the difference of the times of its be-
ginning and end.

The physical process that serves for the definition of a time
scale must be subject to constant conditions. The period of a
pendulum, for example, is constant only if the conditions of the
environment are invariable. For example, the earth’s gravita-
tional field must remain constant, the action of electric and
magnetic fields must be negligible, the temperature must not
change, etc. The control of conditions is accomplished by the
method of successive approximation. We have initially a quali-
tative estimate of constancy of conditions, and therefore we
may apply the principle that a specific pendulum keeps the
same time on various occasions. For daily life such a definition
may be adequate, but for scientific observation the conditions
must be quantitatively defined. The definition of time to the
first approximation enables one to define acceleration, force,
‘electric field,’ etc., quantitatively. We may then specify quan-
titatively the constant conditions for the clock, and thus the
time scale can be defined to a second approximation.

In practice it may not be possible to control the conditions
to which clocks are subject. If one has a quantitative descrip-
tion of the actual conditions, however, it may be possible to
correct the times assigned in terms of the actual clock, and
thereby find the times which would be indicated by an undis-
turbed clock. The earth is the standard clock for astronomical
measurements; the definition of time is given by the convention
that the angular velocity of the earth about its axis is constant.
Now, certain anomalies in the moon s motion can be explained
on the hypothesis that the earth is slowing down on account of
tidal friction. This explanation presupposes the laws of mechan-
ics, which have been confirmed by experiments in which time
was measured by the rotation of the earth. I he explanation
implies that the earth-clock indicates only an approximate time.
One may say, however, that time is to be defined by the rota-

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tion of the earth— assuming that there are no frictional forces.
The empirically indicated time would then have to be corrected
in order to find the time that would be indicated by a friction-
less earth. The astronomer does not actually compute the cor-
rection from the friction and the laws of mechanics but obtains
the correction by comparison with the moon. This illustration,
however, shows that corrections may be made for disturbing
conditions.

In the historical development different processes have been
employed for the measurement of time. The ancients measured
time by the motions of the heavenly bodies, but they also meas-
ured time by the amount of sand that ran out of an hourglass.
The two methods agree approximately on account of the empiri-
cally observed correlation between the flow of sand out of the
hourglass and celestial motions. We choose the astronomical
method as the standard and ascribe the inaccuracy of hourglass
time to variable conditions such as differences in the size of
grains, smoothness of surfaces, etc. That is, the disagreement
between hourglass time and astronomical time is explained by
the hypothesis that differences in the grains affect their motions
and not the motion of the heavenly bodies. The use of accepted
physical principles in the definition of physical quantities is here
exemplified.

I supplement the description of procedure by a discussion of
the considerations which determine the choice of a standard
clock. In the first place, one seeks a process that is as perma-
nent as possible; the rotation of the earth and the motion of the
moon especially satisfy this requirement. Furthermore, the
clock is to be as free as possible from disturbing influences. The
clocks on the earth are subject to disturbances, hence the astro-
nomical clocks are preferable. A pendulum, however, is readily
reproducible and is useful as a secondary clock when it has been
standardized. Most desirable of all for a definition of time is in-
dependence of special properties of matter. A pendulum re-
quires the earth’s gravitational field; the mainspring of a watch
is dependent on the elastic properties of a particular substance;
the heavenly bodies are subject to destruction. The preferred

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Measurement of Time

definition is in terms of functional relations between physical
quantities. For example, we may define time as the independent
variable in the equations of dynamics. The equations which ex-
press physical laws then constitute an implicit definition of time.
In particular, time may be thought of as defined by the first
law of motion which states that a body acted upon by no forces
persists in a state of uniform rectilinear motion. By this defi-
nition, equal intervals of time are indicated by equal distances
passed over by a body under no forces. This definition of time
in terms of the first law achieves the dissociation of the time
scale from special processes or bodies. The change in defini-
tion exemplifies again the method of successive definition. Initi-
ally, physical quantities are defined in terms of special opera-
tions and provide the basis for the empirical discovery of gen-
eral laws, which may then be transformed into implicit defi-
nitions of the physical quantities involved in them. But it then
becomes an experimental fact that the measure of a quantity
which is obtained by special operations approximately satis-
fies the definition of the quantity in terms of the general laws.
For example, the measure of time by a pendulum approximately
satisfies the definition of time in terms of the first law of mo-
tion.

So far, the discussion has been restricted to local time; I now
turn to the problem of the time system throughout a space.
If P, and Pi are two separated points in space, how are we to
define a time scale at Pi which is the same as that at Pi? We
must postulate a law of connection between local times.

In this discussion the points of space will be referred to a defi-
nite frame of reference. It is assumed that similar clocks at rest
may be placed at any point in the given space. Similar clocks
have the property that, if at rest at the same place, they keep
the same time. The problem is: Given the time scale at Pi,
how shall we extend it to P 2 ? Suppose that at Pi there is a set
of similar clocks. If one of the clocks is moved slowly to P 2 , we
may transport its time scale by the postulate that the clock at
P 2 is synchronous with the clocks at Pi. Since no immediate
perception of the synchronism is possible when the clocks are

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Procedures of Empirical Science

separated, the postulate is a definition by which the tune svstem
at Pi is extended to the point P 2 . There is some empirical foun-
dation for this definition. If two clocks which are synchronous
at Pi are transported to P 2 , they will be synchronous at P 2 .
This experimental result is found even if the two clocks are
transported along different paths. Again, if a clock which is
synchronous with the clocks at P x is transported and then re-
turned to Pi, it will again be synchronous with the clocks at the
starting-point. All these experimental results support the con-
vention that a time system is extended throughout space by the
slow transport of a clock. This method of extending a scale of
time is employed in daily life. Thus, the locomotive engineer
extends along the track the time system of the place at which he
sets his watch. If a ship carries a chronometer, it extends a
time system along its course. In transporting a clock, we must
not unduly accelerate and disturb it; but the accelerations to
which ordinary clocks are subject do not affect them appreci-
ably.

Ihe preferred method of extending a time system employs
light signals. At time h as indicated by the clock at Pi a signal
is sent and arrives at P 2 at the time f 2 as indicated by the clock
there. The clocks at the separated points keep the same time
if U = ti + l/c, where l is the distance between the two points
and c is the speed of the signal. In effect, the principle of the
constancy of the speed of light in a vacuum is postulated as a
definition by which a time system is extended throughout space.
Recognition of this procedure occurs in the special theory of
relativity.

Ihus far I have described procedures developed by astrono-
mers and physicists for the exact measurement of time. Esti-
mates of time are also made by the geologist, who investigates
the history of the earth. A fundamental problem of geology is
the temporal order of strata. If it is possible to assume that
strata have not been appreciably displaced from the positions
in which they were created, the following principle is applied:
The order of superposition is the order in time, the oldest stra-
tum being at the bottom and the newest at the top. This applica-

300

Measurement of Time

tion of the principle is restricted by the circumstance that
strata may disappear underground or be continued by a differ-
ent series. William Smith, however, discovered a method of
correlating strata in different regions. From observations upon
exposed sections of strata in England, he found that each group
of strata contained characteristic fossils, so that the order of
superposition of the strata indicated the order of succession of
the fossils. This discovery furnished the empirical basis for the
procedure of correlating the strata in different regions by means
of their characteristic fossils. The hypothesis implied by this
procedure is that strata which have been laid down in widely
separated periods of time will contain quite different fossils,
while those which were formed approximately contemporane-
ously will contain similar fossils.

The preceding discussion explains the procedure of determin-
ing simultaneity and succession of strata. The geologist also
seeks to make quantitative estimates of time. In order to de-
termine the age of a sedimentary deposit, he may make an esti-
mate of the present rate of deposition and, from the thickness of
the deposit, calculate the time required to produce it. The same
procedure is used in estimating the age of the earth from the
mass of salt in the ocean and the rate at which it is entering from
the rivers. The most accurate method of estimating geologic
time is based upon radioactive phenomena. Radioactive ele-
ments like uranium and thorium spontaneously disintegrate at
a definite rate and give rise to elements of lower atomic weight.
The final products are metallic lead, of atomic weight 206 if de-
rived from uranium, and of atomic weight 208 if the final prod-
uct is of the thorium series. If a rock is rich in uranium, one
may measure the masses of uranium and lead which it contains
and, from the rate of transformation of uranium into lead, cal-
culate the age of the rock. Since helium is a product of radio-
active disintegration, it may also be used to estimate the age of
rocks. Chemical analyses of rocks from all parts of the world
show that radioactive elements are widely distributed; these
radioactive clocks in rocks indicate their ages.

Lecomte du Nolly has defined biological time by the depend-

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Procedures of Empirical Science

ence of the rate of reparation of cells upon the age of organisms.
He found that the rate of cicatrization of wounds decreases
with age as ordinarily determined; in a man of sixty the rate is
one-fifth that of a child of ten. Processes that are uniform when
measured in terms of astronomical time become nonuniform on
the biological time scale.

7. Measurement of Weight

d he measurements of length and time are observations in
which the results of operations are perceived and interpreted
with the aid of principles. In fundamental measurements of
length the basic operation consists of laying off the standard on
a line. Since a measuring rod is operated by the observer and
its state is experimentally controlled, the procedure of measuring
length is experimental. Now, it may be contended that there is
direct perception of length and time, and hence measurements
of them are frequently called direct. The term ‘direct’ con-
trasts length and time with properties that are known only as
they manifest themselves in experiments which are construc-
tively interpreted. A discussion of some of these experimentally
exhibited properties will reveal further the method of experi-
ment and will illustrate Newton’s dictum that the properties of
things are derived from experiments.

Ihe first example is the physical property, weight. In a sense
we perceive length and experience duration, but observation of
weight requires constructive interpretation. In order to ob-
serve that a material body has weight, I may support the body
with my hand. The muscles attached to the hand exert a force
which is experienced through kinesthetic sensations. Now, a
fundamental principle of statics states that, if a body is at rest,
the forces on it must be in equilibrium. This principle of equi-
librium accordingly requires that I conceive of the body as hav-
ing a weight, directed downward, which is balanced by the up-
ward force exerted through the hand. This simple experiment
therefore exhibits weight as a force that balances some other
force. The weight is assumed to continue in existence when the
upward force is removed, so that, if the body is released and

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Measurement of Weight

falls freely to the earth, the weight of the body is interpreted
to be the cause of its acceleration.

The measurement of weight may be based upon a lever such
as is used in a beam balance. The empirical foundation for the
procedure is the fact that two bodies may be found such that if
they are attached respectively to the ends of a level with equal
arms, the lever remains horizontal. It is also an empirical fact
that if two bodies maintain one lever horizontal, they will main-
tain all others horizontal, regardless of position, length, materi-
al, color, etc. The two bodies behave similarly in different con-
texts, and hence it is convenient to define them to be equal in
weight. If two bodies of equal weight are attached to one end
of the lever, a body at the other end which maintains the lever
horizontal may be assigned a weight twice that of the first. By
this procedure one may build up a set of bodies with weights
that are integral multiples of a standard weight. One may also
define fractional weights: If two standard weights are balanced
by three equal weights, each of the latter is two-thirds of the
standard. The scale of weights is constructed so that it satisfies
the law of addition for extensive quantities. The present discus-
sion interprets the principle that equal weights maintain an
equal arm lever horizontal, which was selected by Archimedes
to be a fundamental principle of statics, as an empirically found-
ed definition of equality of weight. Of course, one could use an-
other definition of equality of weight, and then the principle of
the lever would be an experimental law. But the alternative
definition would presuppose the selection of some other principle
of mechanics as a definition.

After weight has been ascribed to a body, it is possible to use
this force to discover other forces. If a body is attached to a
spring balance, the spring is stretched before equilibrium is at-
tained; it is assumed that the end of the spring is acted upon by
two forces in equilibrium, the weight of the body and the tension
in the spring which is equal in magnitude but upward in direc-
tion. If one attaches bodies of different weight to the spring and
measures its extension, one can discover Hooke’s law that the
stress in a spring is proportional to its extension. Hooke’s law

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Procedures of Empirical Science

may be used as an alternative definition of force in statics. To
conclude: Given a single type of force, weight, it is possible to
interpret statical experiments as exhibiting weight to be bal-
anced by other forces such as tensions in springs and electric
and magnetic forces. Observation of forces consists in construc-
tive interpretation of mechanical experiments in the light of
principles which play the role of definitions. The physical con-
cept of force expresses the assignment of numbers in order to
describe the conditions of motion. The procedure involves the
element of hypothesis. The physical quantities which are as-
signed by constructive interpretation of experiments must be
instrumental in predicting the results of other observations.

8. Observation through Causality

I have thus far analyzed observation of perceptible things and
properties. Nonperceptible entities are also inferred to exist as
the hypothetical causes of perceptible phenomena. Such infer-
ence through causality eventually becomes observation. I use
here a crude concept of efficient causality, but in the pages on sys-
tematization I analyze it as expressing functional relationship.

The transformation of explanation by constructive hy-
potheses expressing causation into observation may be illus-
trated by an analysis of vision. Let us suppose that one may
touch a thing but cannot see it. Under appropriate conditions
the thing becomes visible, that is, it becomes possible to experi-
ence visual aspects which are correlated with the data of tactual
perception. For example, if I introduce a lighted candle into a
dark room, things become visible; with my eyes open I may see
a table. Hence we must think of the visual aspect as produced,
at least in part, by the candle. If explanation is to be expressed
in terms of the perceptible, one must conceive of this production
as an action at a distance, since the candle and the thing per-
ceived are separated in space. The physicist, however, inter-
prets the process as an action by contiguous causality; he de-
fines radiation through the principle that the visual aspect of
a thing is functionally dependent on radiation which travels
from a source to the thing and is scattered toward the observer.

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Observation through Causality

Thus a principle of contiguous causality yields a definition of
radiation, which is assumed as a construct. The definition is
justified, since the assumed radiation is instrumental in predict-
ing new phenomena. The definition accordingly plays the role of
a hypothesis which is tested by its predictions.

Again, if we keep the illuminated thing constant, as estimated
by touch, and change the source of radiation, the visual aspect
changes — in color, for example. The color of an aspect is the
basis of assigning a specific property of radiation, the wave-
length as measured in Young s double-slit experiment. The
properties of radiation are thus defined in terms of the visual
aspects which it makes possible. Radiation not only produces
visual aspects but also affects a photographic plate. On allow-
ing radiation to pass through a prism or grating, it is spread into
a spectrum; radiation of a specific wave-length produces a line
having a specific position on the plate. To the spectroscopist the
perception of a spectral line constitutes an observation of the
radiation which produced the line. Experiments on the pressure
of light have shown that it exerts a pressure; radiation may
therefore be observed through the momentum which it com-
municates to some directly perceptible object.

A visual aspect is partly produced by radiation, but the aspect
also depends upon the thing perceived. If the source of radia-
tion is kept constant and the thing illuminated is varied, the
visual aspect changes; for example, the visual aspect of a chair
differs from that of a desk in the same light. Hence we must
think of the visual aspect as functionally dependent on the
radiation and thing conjointly. The visual aspect manifests the
reaction of the thing to radiation; in physical language, the
visual aspect depends upon the reflection of radiation by the
thing. The preceding statement expresses a law which may
be viewed as a generalization from observation based upon the
definition of a thing in terms of tactual aspects and the principle
of contiguous causality. In the development of physics, how-
ever, the principle that a thing reflects radiation becomes an
essential element in the definition of physical reality. This is
the basis of observation by photography. An object reflects

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radiation which is focused by a system of lenses so that an
image is produced on a plate. Perception of the image is the
basis of inference to the structure of the object.

9. Observation of Microphysical Entities

Observation through causality, which is exemplified by the
assumption of radiation, is the procedure for the study of micro-
physical entities such as electrons, protons, neutrons, etc. Ob-
servation of such entities consists in the perception of macro-
physical phenomena that may be interpreted as effects of micro-
physical objects. In this field, principles become constructive
instruments of interpretation, and so observation is more sub-
ject to the uncertainties of hypotheses than is perception of com-
mon things.

I present some examples of the observation of electrified par-
ticles through their effects. If a high-speed alpha particle
strikes an appropriate screen, it produces a scintillation which
consists of light assumed to be emitted by countless atoms and
molecules that have been excited by the particle. Thus, a single
particle can be detected by visual perception of the macrophysi-
cal scintillation that it produces. If an electrified particle with
sufficient energy passes through a cloud chamber, it ionizes the
molecules that it strikes, producing ions on which water vapor
condenses. Perception of the track of waterdrops completes the
observation of the ionizing particle. Again, a Geiger counter
is a tube in which a momentary current flows when a particle
of sufficient energy passes through and produces ionization.
The momentary current is amplified and actuates a mechanical
counter which registers the number of particles that pass
through the tube. Hence the production of scintillations and
condensation tracks, and the actuation of Geiger counters, en-
able the physicist to detect individual microphysical entities.
If many particles impinge on an appropriate plate, a perceptible
pattern may be produced. The ultimate elements of physical
theory are observed through perceptible macrophysical phe-
nomena.

The physicist determines the properties of elementary par-

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Observation of Microphysical Entities

tides by investigating their behavior under experimentally con-
trolled conditions. For example, an electron may be subjected
to magnetic or electric fields, and from the curvature of the
path, observed in a cloud chamber by the condensation track
and in a cathode-ray tube by the deflection of the effect on a
screen, it is possible to determine the ratio of electric charge to
mass as defined by assumed physical principles. In Millikan s
measurement of electronic charge, electrons were captured by
perceptible oil drops the behavior of which was observed under
specified physical conditions. The measurement was completed
by a calculation of the charge from the data of the experiment
and the assumed physical principles.

An observation in which the element of hypothesis is quite
explicit is that of the energy levels of the atom. According to
contemporary theory, an atom consists of a positively electrified
nucleus to which are bound electrons distributed among con-
centric shells. The electronic shells are defined by quantum
conditions and constitute the basis for a discrete set of station-
ary states of constant energy of the atom. As electrons jump
from one shell to another, the stationary state of the atom
changes discontinuously with the absorption or emission of
radiation, the frequency of which is determined by the differ-
ence between the energies in the initial and final states. Radia-
tion emitted or absorbed produces spectral lines upon a photo-
graphic plate, and from the serial order of the lines it is possible
to determine the energy levels of the atom. These levels, or
quantum states, are the object of a constructive hypothesis
which serves to correlate and predict spectral data, but the
theory has been so often confirmed that an observation of spec-
tral lines now seems practically a direct observation of the en-
ergy levels of the atom. In the development of physical theory
during the last two decades there has been a transition from the
active construction of hypotheses for atomic theory to the al-
most unreflective acknowledgment of its objects as real. Indi-
rect observation, consisting of measurements on spectral lines
plus the theory from which calculations are made, has achieved
the practical certainty of direct observation.

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10. Partition between Object and Observer

As one traces the development of observation from percep-
tion through measurements of length, time, and weight to the
detection of radiation and the electrified particles of microphys-
ics, it is evident that the function of apparatus becomes more
and more important. Bohr has emphasized the fact that the
observer and his instruments must be presupposed in any in-
vestigation, so that the instruments are not part of the phe-
nomenon described but are used. The problem accordingly
arises of defining the partition between object and the observer
and his apparatus.

I present some examples of the partition between object and
observer. Tactual perception is an interaction between a body
and end organs such as those in the tip of a finger. If one touches
a desk with a finger, the partition is between them. An observer,
however, may be extended by mechanical devices. Bohr has
cited the following example: If one firmly grasps a long stick in
one’s hand and touches it to a body, the body touched is the
object of observation, and the stick is an apparatus that may
be viewed as part of the observer. It is a psychological fact
that one locates the tactual aspect at the end of the stick, so
that the partition is between the body and the end of the stick.
If, however, the stick is held loosely in the hand, the stick be-
comes the perceived object, and the partition is between stick
and hand.

In vision perception is mediated by radiation which travels
from the object to the observer. On account of the finite speed
of propagation of radiation, the state of the object is prior to,
and distant from, the perception of that state. The position of
the object is of primary biological significance, and so the per-
ceiver habitually places the partition at the object and unre-
flectively accepts the radiation which the object emits or re-
flects as part of the observer. For example, the visual percep-
tion of a stick is dependent on the sunlight which is reflected
from the stick to the eye. In daily life the object of interest is
the stick, and hence the light is treated as an instrument of ob-

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servation which is an integral part of the observer. But, if the
stick is in water and appears to be bent, the explanation of this
illusion of vision in terms of the refraction of light by the water
presupposes that the partition be placed between the eye and
the light.

In perception the partition is determined by the biological
needs of the observer and is ordinarily difficult to displace.
Brunswik, however, has shown that a change in perceptual at-
titude is possible; for example, we can compare objects with re-
spect to the size of the pattern on the retina produced by the
light from them. In such a case we are to think of the partition
as at the retina. In view of the extension of the concept of ob-
servation to include interpretation in term of principles, it is
desirable to distinguish between cognitive and perceptual par-
titions. The object may be observed through causation of per-
ceptible effects, so that the two partitions are different. The
position of the cognitive partition depends upon the interest
of the observer. If a physicist is looking at a pointer on a scale,
its status depends on the purpose of the observation. If he is
using the instrument to measure an electric current, the pointer
is an extension of the observer; the object is the electric current.
If the physicist is calibrating his instrument, the pointer is part
of the object of observation; the light by which the pointer is
seen is then an instrument which belongs to the observer. If he
studies the properties of light by photographing spectral lines,
the light is the object of observation, and a line is registered on
the plate, which plays the role of observer. But if the physicist
examines the spectral line, it becomes the object, and the light
reflected by it in turn is an instrument.

In the analysis of observation the cognitive partition between
object and observer may be displaced in opposite directions.
The naive observer probably assumes that in visual perception
he is in direct contact with a distant object; indeed, the child
reaches for the moon. In perception the partition is habitually
at the object. But it is a hypothesis that our visual perception
of a distant object is dependent on the radiation that produces
its visual aspects. Physical investigation reveals that the radia-

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tion is affected by an intervening medium, and so the cognitive
partition is placed at the organism to which the observer is now
restricted. The problem of the physicist ends when he reaches
the boundary of the organism, but the biologist displaces the
partition into the body of the observer. The optometrist places
the partition at the retina of the eye, and the physiologist of the
nervous system displaces it still farther into the organism.
While the biologist displaces the partition farther and farther
into the organism, the physicist displaces it farther and farther
into the physical object. In macrophysics he observes the prop-
erties of bodies that are expressible in terms of perceptible phe-
nomena; in microphysics he penetrates to the molecule, the
atom, and now the nucleus. The perceptible effects of micro-
physical entities, which are not directly perceptible, serve as
instruments of observation which thereby become incorporated
into the observer. After much experience with the condensation
tracks produced by electrons and other elementary particles,
one may come to view the perception of the tracks as an obser-
vation of the particles. It is almost as if the perceptual parti-
tion had been displaced so that it coincides with the cognitive
one.

The position of the cognitive partition is of fundamental sig-
nificance in quantum theory. In an observation of a micro-
physical quantity there occurs an interaction between object
and instrument; the instrument reacts against the object and
may produce an unpredictable, finite change in the value of
a quantity that is canonically conjugate to the one being ob-
served. In such observations it is not possible to control the
action of the measuring instrument upon the object, for the
instruments cannot be investigated while serving as means of
observation. It is impossible to assign definite canonically con-
jugate quantities simultaneously, since the experimental ar-
rangements through which the effects of microphysical entities
are registered are mutually exclusive. The situation in micro-
physics is expressed by the concept of complementarity and may
be illustrated by the following example. The position of a par-
ticle may be observed by allowing it to impinge on a screen.

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Classification

which in order to define position must be rigidly attached to the
frame of reference. During the interaction between particle and
screen there is an unpredictable exchange of momentum be-
tween them. The momentum given to the screen is absorbed
by the frame of reference, and its value cannot be substituted
in the principle of conservation of momentum to calculate the
momentum given to the particle. The procedure for observing
position excludes the possibility of using the screen to determine
the momentum of the particle. If the screen is mobile in the
frame of reference, it may be used to measure momentum, but
then specification of position is lost. The observations of micro-
physics require interpretation in terms of classical concepts, but
the fundamentally unpredictable, finite effects of the disturb-
ances by the instruments of observation lead to a restriction
in the applicability of classical concepts to microphysical ob-
jects. The cognitive partition between object and apparatus
is the seat of an indeterminacy which limits theoretical physics
to the statistical prediction of the results of classically inter-
preted experiments. The state of the object on one side of the
partition is represented by a wave function that changes in ac-
cordance with a differential equation which exemplifies a prin-
ciple of causality. On the other side of the partition there occurs
a sequence of events which proceeds from the measuring ap-
paratus to the observer in accordance with classical laws. The
statistical interpretation of the wave function states that it is
an instrument for predicting the results of observations regis-
tered with classically controlled apparatus and perceived after
the manner of common experience. The quantum mechanical
theory of microphysics has resulted in a limitation of the possi-
bilities of observation.

III. Systematization

11. Classification

In empirical science particular cognitions are organized into
systems of knowledge. The particular cognitions of daily life
already involve order: the perception of a thing involves the
hypothesis that the actual perception is correlated with possible

Procedures of Empirical Science

perceptions; the concept of thing expresses a relatively invari-
able correlation of properties attributed to it. Science pre-
supposes and continues the systematizing procedures of com-
mon experience.

A basic procedure for introducing order into knowledge is
classification. Classification is founded on the similarities be-
tween things or events; it is based upon the fact that things are
similar in specific respects and dissimilar in others. The order-
ing of things into classes is especially characteristic of descrip-
tion in sciences like botany and zoology, but the procedure
originates in prescientific experience. The occurrence of general
names in language indicates that classification is a primitive
mental process. The word ‘animal,’ for example, shows that
there has been formed a class of living things that have the
power of self-movement. The employment of names to de-
note any one of a number of common things is evidence that
particular cognitions of things have been systematized by
classification.

The classifications of common experience are based upon
superficial or striking properties of things, and this dependence
on superficial resemblances also characterizes the early phases
of science. Scientific classification, however, eventually comes
to be based upon essential characters which may be discovered
only by careful observation and experimentation. The ancients
classified material things in terms of earth, air, fire, and water;
this set of substances has been superseded by a system of ele-
ments, initially classified in terms of atomic weight but now in
terms of atomic number. Physics is not ordinarily thought of as
a classificatory science, yet it offers classifications such as those
of spectra on the basis of quantum numbers characterizing the
energy levels in the atom or molecule.

Classification is a characteristic procedure of the descriptive
study of living things. The many plants and animals are classi-
fied according to the presence or absence of specific properties,
functions, etc. A distinction is frequently made between artifi-
cial and natural classification. An artificial classification is
based upon some superficial similarity in structure, color,

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Classification

habitat, etc. For example, animals may be classified as terres-
trial or aquatic depending upon whether they live on land or
in water. A natural classification is based upon the fundamental
characters of things. The basic criterion is similarity of struc-
ture, the study of which is called morphology or anatomy, but
similarity of function and behavior are also used. While classi-
fication occurs in daily life, science aims at systems of classifi-
cation.

The basic classification in biology is the division of the organic
world into the plant and animal kingdoms. A kingdom is
generally progressively subdivided into phyla, subphyla,
classes, orders, families, genera, and species. The scientific
name of an organism is compounded of the names designating
genus and species. Thus man is called Homo sapiens, tie be-
longs to the species sapiens, the genus Homo, the family
Hominidae, the order Primates, the class Mammalia, the sub-
phylum Vertebrata, and the phylum Chordata of the animal
kingdom. The natural systems of classifications of biology are
designed to exhibit genetic relationships.

The principles of procedure for research on problems of classi-
fication have been set forth by Jepson, who has systematically
described the flora of California. He stresses the importance of
field studies of habits, life-history, soil exposure, and associated
species. Field records must be made at the place and time of
observation and should be validated by specimens for a her-
barium. Observation of plants in their natural environment
should be supplemented by experimentation with garden cul-
tures. Jepson transplanted individuals of Eschscholtzia cali-
fornica (California poppy) from the Great Valley to the sea-
coast. Characters attributed to assumed other species de-
veloped, and hence there was justified a reduction in the num-
ber of species. For the work of classification he sought the
following data: (1) entire life-history of a species; (2) bio-
geographic status; (3) characters at the limits of the area in
which it has its greatest development; (4) structure, character,
and presence or absence of plant organs; (5) variation in the
organs of a species, especially from one individual or from a

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series of individuals where these have common parentage; (6)
results of multilation of an individual.

The concepts of species and genus are fundamental instru-
ments of classification in biology, and Jepson shows how the
definitions of these concepts depend upon the state of knowl-
edge. In his work on the flora of California he lists seven species
of the genus Eschscholtzia, while Fedde in Das Pflanzenbuch
reports one hundred species in California. In Jepson the genus
Ptelea is represented by one species, whereas Greene has six
species for California. In his discussion of criteria for genera
and species Jepson states that the species must consist of indi-
viduals bound together by intimate genetic connection as de-
termined by the morphology, detailed structure, life-history,
genetic evidence, geographic history, and ecologic status. The
species should represent a natural unit, especially from the geo-
graphic standpoint, and every effort should be made to give
it precise definition. Jepson states that a genus should include
all species of close genetic connection which have a marked
natural resemblance or are closely bound together by structural
peculiarities which indicate a close line of descent or form a com-
pact natural group. Genera so founded are sufficiently large to
establish relationships on a recognizable scale and to bring out
the intimate relationships which exist between floras of different
regions or countries as a result of past migrations. Genera hav-
ing marked characters should not be subject to a segregation
which reduces the generic character to the level of a species
character. It is, however, necessary that the limits of genera
should with increase of knowledge of their structure be subject
to revision and modification. That the characters for classifi-
cation must be adapted to the specific problem, Jepson illus-
trates by the genus Arctostaphylos. By morphologic characters,
or by biometric measurements, or by other methods determined
in advance, there would scarcely be more than five or six species
for California. He distinguishes twenty-five species on the basis
of differences in reaction to chaparral fires; the responses are
constant and fundamentally unlike and are further correlated
with geographic and ecologic segregation.

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Correlation of Events

A thing is characterized by a relatively invariable correlation
of simultaneously existing properties. Similarity between mem-
bers of a class signifies that such correlation is exemplified by
each member. A classification therefore expresses general laws
of correlation which, since they apply to simultaneously ex-
isting properties, express uniformities of coexistence. But, since
functions are considered in classification, laws expressing cor-
relations of properties involve those expressing correlations of
events.

12. The Correlation of Events

The formulation of correlations of events is an important
phase of systematization. Indeed, it has been argued that
events are constitutive of reality. Examples of events are a
flash of lightning, an eclipse of the sun, an earthquake, the
birth of a living being. In daily life and qualitative science an
event may extend through an appreciable duration, but for
precision an event is idealized as the occurrence of properties
at an instant. The records of events constitute the raw material
for the systematizing activity of science.

A stage in the investigations of correlations of events is the
determination of temporal sequences. The history of political
events, historical geology, and paleontology are arrangements of
events in temporal order. Science in the form of history system-
atizes observations of events by fitting them into schemes of
development of the cosmos, life, and society. Empirical science
also formulates laws which express regularities in the correla-
tion of the general characters of events. Such laws provide a
basis for prediction; in fact, their confirmation or disconfirma-
tion depends on the occurrence or nonoccurrence of events pre-
dicted from them. On the view that events are the basic con-
stituents of reality, laws of coexistence become special cases of
laws of correlation of events.

Correlations of events in nature are generally complex, so
that in science a phenomenon is analyzed into constituent ele-
ments. An example of resolution into concurrent factors is
Galileo’s analysis of the motion of a projectile. Neglecting the

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resistance of the air, the projectile describes a parabolic path.
Galileo analyzed the process into a superposition of two motions
which may be considered independently: a horizontal motion
with constant velocity and a vertical motion subject to the
acceleration of gravity, lhe analysis of a phenomenon into
successive elements is achieved by differential calculus. Kepler
formulated the laws of planetary motion in terms of concepts
that characterize the motion as a whole. One of his laws states
that the orbits of the planets are ellipses. Newton discovered
the differential equation which describes the instantaneous
character of the motion. He expressed the acceleration, which
is a derivative with respect to time, as determined by a force
which is a function of the distances and masses of surrounding
bodies. The differential analysis gives an exact formulation of
the popular concept of causality.

The concept of causality expresses a relation between phe-
nomena, such that one phenomenon is viewed as the cause of
some other phenomenon, the effect. The cause is the condition
of the effect; a description of the causes of a phenomenon con-
stitutes an explanation. One may exemplify causality by il-
lustrations drawn from all realms of experience. For example,
the action of a force on a body is the cause of its acceleration ;
the application of a stimulus to an organism is the cause of a
response. The concept of causality is best understood by an
analysis of the physical concept of causality. Indeed, the
hypothesis has been entertained that all natural phenomena are
analyzable into physical processes. The typical physical cause
is the action of force.

In the life of the individual it is probable that the concept of
force is derived from experiences of his own exertions and that
of his fellows. The individual exerts a force, for example, with
a hand, and observes that it produces effects. A father forces
a child to fill the wood box. Bodies are set in motion or brought
to rest, forces can act in opposite directions and annul one
another. The primitive concept of force is derived from experi-
ences of our own exertions and of our fellows; it expresses pro-
duction, creation, generation, efficacy.

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Correlation of Events

Since the concept of force was originally derived from the
forces exerted by an individual, it is understandable that primi-
tive attempts at explaining natural phenomena were in terms
of the' theory of animism. Preceding the origin of empirical
science men invented animistic explanation of natural phe-
nomena. Natural bodies were interpreted to be the abodes of
principles of life; for example, the lodestone, which attracts
iron, was interpreted to be the seat of a soul. It is to be recog-
nized that animism was an expression of the demand for causal
explanation. Scientific progress began when men directed their
attention to the perceptible properties of things, observed cor-
relations of properties and sequences of phenomena, and ex-
plained the behavior of things in terms of the causal efficacy of
other natural things. Animism did not, however, disappear
with the development of science. Thales, the first historical
figure of Greek science stated, “All things are full of gods.”
Empedocles conceived of the forces of attraction and repulsion
as instances of love and hate. In modern times Kepler thought
of the planets as guided in their orbits by angels, and it may be
contended that vitalism in biology is a relic of animism.

Comte has explained in his law of the three stages that every
science starts at a theological stage, passes through a meta-
physical one, and then achieves final form in a positivist stage.
The several sciences are at different stages of development. The
operational point of view in physics is recognition that physical
science has most completely realized the final form; this may be
exemplified by the content of the physical concept of causality.
Physical causality may be illustrated by the Newtonian theory
of the motion of a body in a gravitational field. In this theory
material bodies are conceived to have the physical property,
mass. Between two material bodies there is a force of attrac-
tion which varies directly as the product of the masses and in-
versely as the square of the distance between them. Fixing our
attention upon one of the bodies, we may say that it is acted
upon by a force which is exerted by the other body, or, more
precisely, that the first body has an acceleration which depends
upon the distances and masses of surrounding bodies. Thus one

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seeks the causes of a phenomenon pertaining to a given body in
the correlated states of other bodies. The cause of the accelera-
tion of the body is described not in terms of a vital principle
within the body but in terms of the observable properties of
other bodies. In the seventeenth century the exertion of a
force was viewed as a state of activity of one body that pro-
duced an acceleration in another body. Causality was thus
interpreted to be the expression of power, efficiency, produc-
tion, necessary connection. The concept of causality was then
in a metaphysical stage which retained traces of animism.

The concept of efficient causality was criticized by Hume. He
analyzed the causal process in the collision of two billiard balls.
If a moving ball strikes a ball of equal mass which is at rest,
the first ball is brought to rest, while it communicates its state
of motion to the second ball. We say that the motion of the
first ball is the cause of the motion of the second ball, which is
the effect. In observing this process, Hume found only a sequence
of phenomena; the state of motion of the one ball was succeeded
by the state of motion of the second ball. On close observation
one would observe that as the two balls come in contact there is a
deformation of the surfaces of contact and then a recovery from
the deformation, during which process the first ball is brought
to rest and the second ball moves away with the original state
of motion of the first ball. At no time while the balls are in
contact can one see why the process must occur as it does and
not in some other way. Prior to the observation of a collision
of two balls one could not predict the outcome of the collision.
According to Hume, the concept of efficient causality merely
expresses a uniformity of sequence of phenomena.

The concept of causality as correlation was definitely formu-
lated by John Stuart Mill in his canons of induction. Mill’s
canons are exemplified by the method of concomitant varia-
tions which states that whatever varies in any manner when-
ever another phenomenon varies in some particular manner is
either a cause or an effect of that phenomenon, or is connected
with it through some fact of causation. Mill’s canons offer cri-
teria by which one can determine whether or not there is a

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Correlation of Events

causal relation between specific phenomena. The mathematical
form of causality purges it of the last vestiges of efficiency.
Causal laws are stated as functional relations between numeri-
cal measures of variable quantities.

The mathematical expression of causal laws may be illus-
trated by the Newtonian theory of gravitation. If two bodies
having masses mi and m 2 are at a distance r, each of these bodies
will exert a gravitational force upon the other which is directly
proportional to the product of the masses and inversely pro-
portional to the square of the distance between them. Let us
fix our attention upon mi. The physical description of the
phenomenon is that m x is accelerated by a force which is
exerted by m^. The mathematical expression of the law of
motion is given by the differential equation

mi

dv i
dt

= G

mimi

One should also add that, while ?n 2 is exerting a force upon mi,
the latter is exerting an equal and opposite force upon the
former. The causal process in the system consisting of the two
bodies is a mutual action.

A special case is that in which m 2 is the mass of the earth and
mi is the mass of a freely falling body. For the region in the
neighborhood of the surface of the earth we may assume that
the factor Grm/r 2 is a constant g. Let s be the distance measured
downward from a point above the surface of the earth. Since
dvi/dt = d 2 s/df, the differential equation becomes

d 2 s d' 2 s

m 'd? =mg df =g -

One can integrate this equation and express the coordinate s
and the speed as functions of the time and two arbitrary con-
stants. Thus

* = so + v 0 (t — 1„) + \g it - t 0 ) 2 .

The constants s 0 and v 0 are the values of the distance and the
speed at an initial time t = t Q . Hence, if one knows the position

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and speed at an initial time, one can calculate the position and
speed at some past or future time.

Hume characterized cause and effect as contiguous in time.
If in our example the force exerted by m? is the cause and the
acceleration of mi is the effect, it appears that cause and effect
are simultaneous, for the acceleration is simultaneous with the
force. We could, however, say that the cause precedes the
effect if, after a body were introduced into the vicinity of m u
it would take a finite time for the gravitational force to begin
to act on mi. But, according to the Newtonian theory, gravi-
tational attraction is propagated with an infinite speed. Hence
we can find no meaning for the statement that the cause pre-
cedes the effect. Again, contiguity of cause and effect in space
means that the body exerting the force is in contact with the
one accelerated. On the Newtonian theory of action at a dis-
tance spatial contiguity is also lost. In the present example
discontinuity in causal action gives rise to an ordinary differen-
tial equation for the motion. The contiguity of a causal process
in space and time is exemplified by wave motion in an elastic
medium. If a portion of the medium is set in vibration, the
state of motion is communicated by contiguous action .to
neighboring parts and travels with a finite speed. The space-
time contiguity of the process is represented by a partial dif-
ferential equation for the displacement as a function of the
space coordinates and the time. The causal factor is represented
in the differential equation by the dependence of the velocity
upon the density and elasticity of the medium.

The search for causes is guided by a principle of causality.
In qualitative terms the principle is expressed by the proposi-
tion, ‘the same cause produces the same effect.’ For example, if
for a system subject to a definite law of force the same initial
conditions are realized at some other time and in some other
place, the motion will be the same. Suppose that I drop a body
from a certain height above the surface of the earth; a specific
motion will occur. The principle of causality predicts that if at
some other time and place a body is dropped from the same
height, the motion will be similar to the first one.

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Successive Approximation

The principle of causality thus appears to assert the existence
of causal laws that are independent of time and place. This
would imply that causal laws are significant because the order
of nature is one in which certain patterns in phenomena recur.
In view of the universal interrelatedness of things, however, it
would appear that the state of the whole universe is the condi-
tion of every phenomenon. But since the state of the universe
apparently never recurs, we could not observe the repetition
of a specific process in some other place at another time. At
first glance it would appear that the principle of causality is
empty.

The answer to the foregoing objection is that it is possible
progressively to isolate systems and processes. In a mechanical
experiment on the surface of the earth it is possible in practice
to ignore the influence of the heavenly bodies. In general, the
universe is a set of loosely coupled systems. Physical forces
usually vary inversely as the second power of the distance and
therefore decrease rapidly with distance. For ordinary accuracy
the physicist may view his laboratory as an isolated system.
Against a background which may be considered constant, the
same relative initial conditions give rise to similar processes.
Causal processes can be isolated by controlling the state of the
environment.

13. Successive Approximation

The control of the conditions under which a causal law is
exemplified proceeds by successive approximation. In pre-
liminary experiments one must presuppose that the conditions
are constant and thereby obtain an approximate law. With the
aid of approximate laws one can then define the conditions of
an experiment more precisely, or correct for the disturbing in-
fluences, and thus determine a law to a higher order of ap-
proximation. Galileo’s first experiments on the acceleration of
a ball rolling down an inclined plane were performed with crude
apparatus. After the laws of dynamics were discovered from
the action of gravity, it was possible to define the conditions of

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the experiment more accurately and indeed to explain how
Galileo’s experiments were disturbed by friction.

Examples of the method of successive approximation are
offered by contemporary physics. It is incorrect to think that
the theory of relativity and the quantum theory have destroyed
classical physics. Classical theory has to be assumed to a first
approximation in order to define the experimental conditions
under which relativity holds to a higher approximation. Thus
in his first paper on the special theory Einstein begins the sys-
tematic discussion with the statement, “Let us have given a
system of coordinates, in which the equations of Newtonian
mechanics hold to the first approximation.” The formulation of
the postulates of the theory which corrects classical dynamics
requires a system of coordinates which is defined by the con-
dition that the classical theory holds in it to the first approxi-
mation. In order that the foregoing procedure may apply, the
more precise theory must contain the first approximation as a
limiting case. In the special theory of relativity the length and
mass of a body are functions of its speed; for speeds that are
small in comparison with the speed of light one obtains the
classical assumption that length and mass are independent of
speed. The classical dynamics is a limiting case of relativistic
dynamics.

According to the general theory of relativity, the path of a
ray of light is curved in a gravitational field; as is well known,
this prediction has been verified by the measurement of the de-
flection by the sun of light from a star. The observations and
calculation’s whereby this result is verified presuppose, how-
ever, that the law of the rectilinear propagation of light holds
in a region near the surface of the earth. The experimental
procedure is justified because the earth’s gravitational field is
relatively small, and in the limiting case of zero gravitational
fields relativistic theory reduces to the limiting assumption
that light travels in a homogeneous medium in Euclidean
straight lines.

Classical physics is a necessary basis for quantum physics.
Quantum phenomena occur under macrophysical conditions

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Successive Approximation.

and are measured by apparatus which registers results that are
interpreted in terms of classical concepts. The employment of
classical physics in the control of microphysical phenomena is
justified by the fact that the classical laws are limiting cases of
the quantum laws. Since observations of microphysical phe-
nomena are subject to statistical laws, the causal laws of
classical physics render possible the definition of experimental
conditions in which causality fails.

On account of the complexity of organisms, it is difficult to
control the conditions of a process in a single living system. The
biologist therefore uses a procedure, called comparative experi-
mentation by Claude Bernard, in which another system serves
as a control. Suppose, for example, that it is desired to deter-
mine the effect of modification or removal of a deep-seated
organ of an animal. Such an experiment requires an operation
which disturbs neighboring organs. In order to distinguish be-
tween the effect of the operative procedure and that of dis-
turbing the specific organ, the biologist performs a similar
operation on a similar animal, but without disturbing the organ
under investigation. A controlled experiment requires com-
parison of the results upon at least two organisms, under con-
ditions which are the same in each experiment except in one
respect. Comparative experimentation may use two animals
of the same species, or the same animal at different times or
different parts of one animal at the same time.

Controlled experimentation in biology is facilitated by the
use of cultures. The problem of plant nutrition is the determina-
tion of the substances necessary for the structural composition
and metabolism of the higher plants. Experimental procedure
consists in placing the roots of the plant in a water culture,
which provides a controllable external medium from which the
plant can absorb the nutrient substances. The nutritional re-
quirements of all sorts of plants have been discovered by vary-
ing the kinds and amounts of salts in the medium. An im-
portant procedure for physiology is the cultivation of fragments
of tissues and organs of animals outside of the organism. Dr.
Carrel deposited small fragments of living tissues in fluid

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plasma or in an artificial medium. The maintenance of life in
such culture mediums renders possible controlled experimenta-
tion on living processes.

14. Successive Definition

With the development of science there has been discussed the
problem of the genesis of scientific laws. At first sight it ap-
pears to be evident that the laws of empirical science are gen-
eralizations from observation. But laws serve to define con-
cepts, and observation employs laws as principles of interpreta-
tion. In the spirit of Kant, who taught that the category of
causality is an a priori condition of the possibility of experience,
Dingier holds that the principles of physics are postulates which
ought to be chosen in accordance with a principle of simplicity.
In the present work a solution of the problem is offered by the
theory of successive definition. According to this theory, the
status of a natural law may change in the development of
science. A law which originates as a generalization from ex-
perience may be transformed into a convention that expresses
an implicit definition of the concepts it involves. The theory of
successive definition has been anticipated in the discussions of
arithmetic, geometry, time, and weight. I shall now use classi-
cal dynamics for a more detailed explanation. Some concepts
must be initially assumed as understood; I shall assume that,
in addition to the concepts of geometry and time, we have a
statical concept of force. Examples of forces are weight and the
force exerted by a stretched spring. If a body is supported at
rest by a stretched spring, the interpretation of the phenome-
non in statics is that the weight acting downward is balanced by
the equal and opposite upward force exerted by the spring.

If the body is released, it falls under the action of its weight,
which is assumed to be the same as when the body is at rest!
Similarly, if a spring is stretched a given distance, it is assumed
to exert the same force whether it is accelerating a body or
is balanced by some other force.

Let us now apply forces to various bodies. It is found that
the velocity increases in the direction of the force. In order to

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Successive Definition

describe the action of force, we may assign to a body in motion
the physical quantity momentum, I, which may be defined by
the postulates that a body at rest has zero momentum and that
the momentum communicated to a body by a constant force
is proportional to the product of force and time. Experiment
shows that for velocities small in comparison with the velocity
of light the momentum is directly proportional to the velocity.
We may introduce a factor of proportionality m and write
/ = mV. The usefulness of our definition of momentum depends
on the fact that regardless of the kind of force employed to
generate momentum in a given body the factor m is the same.
Hence the physical quantity m, the mass, is viewed as an in-
trinsic property of the body, which for classical dynamics is
independent of velocity, but for relativistic dynamics depends
on velocity. The outcome thus far is that we have defined
momentum in terms of an impulse equation and have trans-
formed the empirical law of the proportionality between
momentum and velocity into a definition of mass.

The foregoing dynamical example illustrates the fact that
generalizations from experience become definitions of new con-
cepts. I cite some further examples. The law expressing the
dependence of the stress in an elastic body upon the strain be-
comes a definition of elastic constant. The law w'hich states the
functional relation between the length of a wire and its tem-
perature becomes the definition of coefficient of linear expan-
sion. Electrical resistance and the constant in the general gas
law are other examples of constants that are defined in terms
of empirical laws which have been transformed into definitions.
A related example of the transformation of empirical laws into
definitions is given by the principle of the conservation of
energy. In dynamics it is possible to define energy so that in
isolated ideal systems it is constant. On the basis of experi-
ments which showed that a definite amount of heat can be pro-
duced by the performance of a definite amount of work, the
mechanical principle was extended to a general principle for all
natural processes. As Poincare noted, this principle, which was
initially an empirical hypothesis confirmed by experiments

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demonstrating the mechanical equivalence of heat, has ac-
quired the status of a definition. If in a physical process the
total change of known forms of energy is not zero, a new kind
of energy is assumed in order to preserve the principle.

Further study of dynamics reveals an alternating procedure
! n successive definition. We initially adopted force from statics,
but, having determined how to assign masses to bodies, we may
calculate momentum from I = mV and define force in the more
general sense by Newton’s equation of motion, which states
that force is equal to the rate of change of momentum. The
usefulness of this definition depends on the fact that simple laws
of force may be found in important applications, for example
the inverse square law for gravitation and electrostatics, and
Hooke s law which states that in elastic bodies stress is pro-
portional to strain. Thus a limited concept may provide the
basis for a law which is used to define a new concept. Then the
law may be employed with the new concept to define a general
concept which replaces the original limited one. In the fore-
going example the equation of motion was used alternately to
define momentum and force. The fundamental laws of dy-
namics have become conventions which implicitly define the
concepts that they involve. With C. I. Lewis we view the
principles of fundamental sciences like geometry and dynamics
as definitions of concepts which serve to interpret the data of
observation. It must be recognized, however, that the suita-
bility and applicability of a conceptual scheme are the subject
matter of hypotheses which must be confirmed or disconfirmed
by observation.

15. Atomism

Thus far we have studied procedures for the systematization
of cognitions of directly observable things and phenomena I
shall now consider systematization from a more detailed and
constructive point of view. Perceptible phenomena can be
analyzed into nonperceptible processes; empirical laws can be
derived from more ultimate laws. In physics the transition is
from a macrophysical level to a deeper microphysical one

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Atomism

The more ultimate point of view may be called atomism in a
general sense. Atomic theory vividly illustrates the method
of hypothesis.

The fundamental ideas of an atomic theory -were created by
the early Greek philosophers. As Meyerson has so clearly
shown, the mind seeks identities in the natural world. This
disposition expressed itself in early Greek science in the en-
deavor to interpret natural things as modes of a permanent
substance. Attempts to provide for change in a theory of sub-
stance led to the atomic theory of Leucippus and Democritus.
According to atomism, substance consists of countless atoms
in the void, and these constitute natural bodies. Natural
phenomena consist of changes in the groupings of atoms. Thus
the theoretical demand for persistent substance is reconciled
with the acknowledgment of the reality of change. It is unfair
to state that the ancient atomic theory was only speculative.
As is exemplified in the exposition of Lucretius, the Greeks ex-
plained many phenomena qualitatively in terms of atomism.

Quantitative development of the atomic hypothesis began
when it was used by Dalton to explain the laws of chemical
combination in the opening decade of the nineteenth century .
Chemical processes are exemplified by the rusting of iron upon
exposure to the atmosphere and the decomposition of w T ater
into hydrogen and oxygen by electrolysis. Such processes are
characterized as composition and decomposition of substances.
The criteria of substances are initially qualitative properties
like color, odor, hardness, but are ultimately physical proper-
ties such as density, melting-point, boiling-point, specific heat,
electrical conductivity, spectrum, etc. Thus the procedures for
observing physical properties are basic to chemistry. The
chemist, furthermore, has characteristic techniques for separat-
ing substances in physical mixtures by using differences in
boiling-point and solubility, by filtering, etc. He employs
methods such for producing chemical changes as heating, mixing
solutions, electrolysis. In chemical phenomena the physical
properties of the final substances are usually quite different
from those of initial ones. Chemistry originated in antiquity

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with procedures of cooking, the working of metals, the tanning
of leather; and the study of chemical phenomena was furthered
by the alchemists’ attempt to transform baser metals into gold.
But the quantitative development of chemistry only began in
earnest during the closing decades of the eighteenth century,
when Lavoisier demonstrated the conservation of mass in
chemical reactions. The use of the chemical balance to weigh
materials which are mixed or produced in a reaction renders
possible the quantitative formulation of an atomic theory.

The atomic theory of chemistry is based upon three funda-
mental laws: the law of conservation of mass, the law of definite
proportions, and the law of multiple proportions. The fore-
going laws, which are derived from chemical experiments, are
readily explicable by the atomic theory in which the masses of
individual atoms are assumed constant and the same for a given
element. The constancy of atomic masses provides a basis for
the conservation of mass. In recent years it has been discovered
that there exist isotopes, that is, atoms of the same element
possessing different atomic weights. It may also be noted that
in order to extend the law of conservation to nuclear reactions
the relativistic equivalence between mass and energy must be
invoked. The hypothesis that the molecules of a substance are
always composed of the same kinds of atoms explains the law
of definite proportions. The hypothesis that molecules may be
formed of different numbers of the same kinds of atoms ex-
plains the law of multiple proportions.

Atomism in the theory of matter furnished a basis for an
atomic theory of electricity. Faraday discovered the laws of
conduction of electricity through electrolytes. An explanation
of these laws is immediately given by the hypothesis that in an
electrolyte molecules are dissociated into positively and nega-
tively charged ions which carry integral multiples of a unit of
electric charge.

Further progress in atomism was made by the kinetic-
molecular theory of heat. This theory was based upon the dis-
covery of the mechanical equivalence of heat, which provided
the empirical foundation for the hypothesis that the heat con-

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Atomism

tent of a body consists of molecular mechanical energy. The
kinetic theory of matter has been especially developed for
gases. In this theory a homogeneous gas consists of countless
molecules, similar in mass, and moving with high speeds. In a
collision between molecules there is conservation of momentum
and energy. In order to make calculations from the hypotheses,
it is necessary to use statistical assumptions and methods. It
is the statistically defined quantities, however, that are signifi-
cant for observable phenomena; an important part of the theory
consists of assumptions correlating the statistical quantities
with experimentally measurable quantities such as pressure and
temperature. For example, the pressure of a gas is assumed to
be equal to the time average of the total transport of momentum
to unit area per unit time. That is, the pressure indicated by a
manometer is the resultant force per unit area produced by the
reflection of the molecules by the boundary of the gauge. The
temperature of a gas is assumed to be proportional to the time
average of the kinetic energy per degree of freedom. On sim-
plifying assumptions it is possible to deduce the general gas
law and other laws for gases. Thus the observable, macro-
physical properties of gases are explained in terms of the action
of microphysical bodies.

Another advance in atomic theory was the molecular explana-
tion of the Brownian movement. The success of the atomic
theory in chemistry and in the theory of gases failed to con-
vince notable doubters. The kinetic picture was viewed as a
fiction which served only as an instrument for economical
thought. Atomic weights were merely combining ratios and not
significant of realities. The doubters were practically all si-
lenced, however, by the molecular explanation of the Brownian
movement. If colloidal particles suspended in a liquid are
viewed through an ultramicroscope, the particles are observed
to move in an irregular and random manner. The kinetic ex-
planation is that the particles are sufficiently light so that they
are appreciably affected on impact with a molecule. The ir-
regular, zigzag motion of the particles is caused by irregular
bombardment by the molecules of the liquid. In observations

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Procedures of Empirical Science

of the Brownian movement, one therefore observes the effects
of a single molecule.

hhe present illustration shows that a principle of contiguous
causality is a factor in the observation of physical objects. The
properties of nonperceptible entities are inferred from the be-
havior of perceptible bodies in collisions which are assumed to
satisfy principles of the conservation of momentum and energy.
The molecule must be assumed in order to interpret perceptible
phenomena in accordance with accepted physical principles.
Thus the physical existence of molecules, atoms, electrons,
radiation, etc., is like the existence of the physical properties
of perceptible bodies. If a colloidal particle exists, then so does
the molecule which interacts with it in conformity to physical
principles. If a zinc sulphide screen exists, then so does the
alpha particle which excites it to scintillate. The functional
relations which are expressed by the laws of nature relate the
physical properties of perceptible bodies to the physical proper-
ties of microphysical entities. The physical world is the object
of a hypothetico-deductive system which is assumed in order
to interpret the results of observation — initially to interpret a
datum of perception as an aspect of a body to which one at-
tributes mass, temperature, electric charge, etc., on the basis
of its behavior; further to interpret phenomena such as Brown-
ian movements, scintillations, and condensation tracks as the
effects of microphysical entities; and then to express macro-
physical processes as statistical resultants of microphvsical
processes.

The development of atomism is one of increasing detail in its
representation of physical objects. Dalton failed to distinguish
between atoms and molecules. In simplified kinetic theory the
volumes of molecules are neglected and also the forces between
them, except in impact. I he atoms were initially assumed equal
in mass, but isotopes were discovered. Electric phenomena in
gases led to the discovery of the electron, and Rutherford’s dis-
covery of the nucleus provided a basis for the picture of the
atom as a positive nucleus surrounded by shells of electrons.
A first clue to the structure of the nucleus was given by radio-

330

Statistics in Quantum Theory

activity, which was interpreted as the disintegration of unstable
nuclei and the correlated emission of alpha particles, beta parti-
cles, and gamma rays. In recent years instruments have been
invented that will communicate to electrified particles energies
which are high enough to shatter the nucleus. At present the
nucleus is assumed to be constituted of protons and neutrons.

The preceding discussion shows that the concept of atom is
relative to experimental procedure. The generic meaning of
the term is something that cannot be divided, but the concept
of indivisibility is relative to the instruments employed. The
chemical atom of the nineteenth century was not divisible by
ordinary chemical methods, but its outer structure w r as shat-
tered by electric discharges, and now its inner structure is being
smashed by the projectiles from powerful atomic guns.

In the history of physics theories of continuity have com-
peted with atomism. The Cartesian vortices of the seventeenth
century, the nineteenth-century ether models of the atom, and
recent electromagnetic field theories of the electron testify to
the scientific impulse to reduce apparent discontinuity to
continuity. In a field theory of matter the fundamental physical
quantities are those that characterize the electromagnetic field
in space-time. Regions of high value of the field are interpreted
to be electrified particles, the inertial mass of which consists
of the energy of the surrounding field. Since the field is con-
tinuous, the separateness of the electrified particles is only an
appearance. Despite attempts to build theories of matter by
modifying Maxwell’s equations and by generalizing the theory
of relativity, theories of continuity are not playing a vital role
in contemporary physics.

16. Statistics in Quantum Theory

The discussion of atomism in physics may be supplemented
by an exposition of the basic significance of statistical laws in
quantum theory. The wave function that represents the state
of a microphysical system satisfies a differential equation which
expresses determinism. But the results of observations on
quantities characterizing a system are subject to statistical

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laws. Statistical laws, which in classical kinetic theory are
founded on determinism, are fundamental in quantum theory.

fhe statistical laws of quantum theory are exemplified in
observations on the state of polarization of light. If ordinary
light enters a Nicol prism, the emergent beam consists oi plane-
polarized light. Since light exhibits corpuscular properties, the
beam is conceived to consist of photons which are in a state
that is defined by the plane of polarization. An observation on
the state of polarization of light is performed by allowing the
beam to enter a crystal of tourmaline. If all the light passes
through, it has been observed to be plane-polarized in a plane
perpendicular to the optic axis of the crystal. If no light passes
through, its plane of polarization has been observed to be
parallel to the axis. If the fraction of the original beam that
passes through is expressed as sin ' 2 a, the plane of polarization
has been observed to be inclined at an angle a to the axis of
the crystal. For a constant experimental arrangement the re-
sults of an experiment are reproducible. The effects registered
in observation are produced by the mass effect of many photons
in accordance with the statistical law that a fraction sin 2 a of
the total number passes through and a fraction cos 2 a is ab-
sorbed. The result of an experiment with a single photon, how-
ever, is not predictable with certainty.

In order to describe the individual experiment, I shall, in con-
formity with Dirac, represent the state of polarization perpen-
dicular to the optic axis by ft and the state parallel to it by ft.
The state inclined at an angle a, and represented by ft, may
then be expressed as formed by the superposition of states
ft and ft. If Cl and c 2 designate the weights of the component
states, we may write ft = c,ft + c 2 ft. The action of the
tourmaline in an observation on the photon in state ft is to
force it to jump unpredictably into state ft or ft, and the
probabilities are respectively proportional to the squares of the
weights. The probability of jumping into state ft and passing
through is sin 2 a, and the probability of jumping into state ft
and being absorbed is cos 2 a. If the photon is initially in state
ft, the result of an observation may be predicted with certainty.

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Atomism in Biology

The general formulation is as follows. Of any atomic system
one may say that it can be prepared so that it is in a determi-
nate state. The preparation is carried out with the aid of de-
terministic macrophysics. The quantum theory assumes that
whenever a system is in a determinate state it can be regarded
as formed by the superposition of two or more states. In con-
sequence of this superposition the results of observation are in
general not predictable with certainty. There is a finite proba-
bility that a result will be observed which is characteristic of
one component state, and a finite probability that the result
will be characteristic of another component state. If the system
is initially in one of the component states, the result of an ob-
servation is predictable with certainty.

17. Atomism in Biology

The systematization and explanation introduced by atomism
are exemplified in biology. The phenomena of heredity are
explained by a theory of genes which is analogous to atomic
theory.

The concept of heredity expresses the empirical generaliza-
tion that organisms are similar to their parents, dhe scientific
study of heredity presupposes an analysis which distinguishes
different characters of an organism, so that a complex process
may be analyzed into its constituents. In organisms created by
sexual union traits arise that are combinations of those of their
progenitors.

The fundamental law of heredity was formulated by Mendel
as a result of his experiments on plant hybridization. He
studied in varieties of garden peas the inheritance of selected
characters such as length of stem, whether tall or short; form
of ripe seeds, whether round or wrinkled; color of food material
within the seeds; etc. Mendel crossed two forms having distinct
characters and counted the number of descendants in successive
generations possessing one or the other of these characteristics.
He first determined that the selected characters were constant
for certain varieties, a procedure analogous to the isolation of
pure substances in chemistry. He then crossed varieties having

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Procedures of Empirical Science

different characters and observed the offspring. When a tall
variety was crossed with a short one, the offspring in the first
filial generation were all tall. If these individuals were bred
among themselves, the result was a ratio of three tall to one
short. This ratio was a consequence of the fact that in the
second generation the proportions of pure tall, impure tall,
and pure short were 1:2:1. Correns crossed a white-flowered
variety of Mirabilis, the four o’clock, with a red-flowered one
and obtained blended, pink hybrids. The offspring of the pink
flowers were white, pink, and red in the proportions 1:2:1.- The
Mendelian law of heredity is analogous to the law of definite
proportions in chemistry.

The explanation of Mendelian inheritance is based upon the
constitution of the germ cell. The occurrence of pure dominants
and pure recessives from hybrid parents is explained by the
hypothesis that the germ cells carry only specific unit charac-
ters, for example, the factors for red or white flowers. Since all
germ cells are pure with respect to a type of character, the
hybrid offspring of parents having contrasting characters will
produce in equal numbers germ cells bearing the dominant
character and those bearing the recessive character. The chance
combination of these two classes of male and female cells will
yield the typical Mendelian proportion: 1DD, 2D(R), 1RR.
The law of heredity therefore has an atomistic basis similar to
that for chemistry.

The explanatory stage of systematization in the science of
heredity is devoted to the factors in the germ cells which
determine traits or characters. Within the nucleus of the cell
occurs a substance called chromatin, which takes the form of
threads, or chromosomes, preparatory to cell division. The
chromosomal theory of heredity is that the determinants of
particular traits, called genes, are located in linear series in the
chromosomes. By the use of special fixing and staining meth-
ods and observation under a microscope it may be observed
that the chromosomes contain distinguishable structures called
chromomeres. John Belling set forth the parallelism between
the serial arrangement, structure, attraction between homol-

334

Unity of Science

ogous entities, rate of division, and number of chromomeres,
and the corresponding properties of the genes. Belling con-
cluded that correct scientific procedure demands the adoption,
as a working hypothesis, of the assumption that chromioles and
chromomeres are genes, doubtless with more or less of an
envelope.

Confirmation of the preceding hypothesis has been obtained
by a study of the large chromosomes in the salivary gland of
Drosophila melanog aster. Painter and his co-workers separated
the elongated threads within the nuclear wall and observed on
the chromosomes a great variety of bands, some broad and
deeply staining, others narrow or made up of a series of dots.
The patterns of bands and lines were characteristic of a given
element, and it was possible to recognize the same element in
the nuclei of different individuals. Irradiation with X-rays pro-
duced changes in the structure 01 a chromosome; there occurred
translocations of elements, deletions or breaks, and these were
observed under the microscope. The correlation of a short de-
letion with the absence of a trait fixed the location of the cor-
responding gene. By a genetic study of the characters trans-
mitted, and a cytological study of the changes in the structure
of the chromosomes, it was possible to correlate traits with
position and thereby to locate the genes.

IV. Conclusion

18. Unify of Science

The present monograph on the procedures of empirical
science may properly be concluded with an appraisal of the
prospects for a unified science.

A basis for unity is the circumstance that the initial subject
matter of the several sciences is furnished by common experi-
ence. The first objects of empirical science are things and
phenomena given in perception, space-time structures whose
properties are described in terms of perceptible qualities and
relations. The motion of bodies, the behavior of organisms,
and the relations between societies are initially described by

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Procedures of Empirical Science

concepts abstracted from perception. A body whose law of fall
is investigated in physics may be characterized as extended,
heavy, smooth, and gray. Gold is described as heavy and
yellow. Living organisms are characterized by growth, repro-
duction, and death. Carnap has shown how the symbols for
such perceptible things and properties may be used to construct
a language for science.

Unity is introduced into science by the fundamental position
of physics. In generalized physics there are fashioned precise
concepts of space and time which serve as instruments for the
description of the general properties of matter and energy in
specialized physics. General and specialized physical concepts
are instruments for the description of chemical* biological, be-
havioristic, and social phenomena. The chemist identifies sub-
stances by their physical properties, such as density, specific
heat, boilmg-pomt, characteristic spectrum. The biologist
studies the chemical constitution and reactions of matter con-
stituting a living organism, investigates the exchanges of energy
between organism and environment, and interprets phenomena
of the nervous system to be electrical. In order to study the re-
sponses of an organism to its environment, the behaviorist sub-
jects it to experiments with physically controlled apparatus.
It therefore appears that the methods of observation and experi-
mentation of physics constitute a unified procedure for science.
Science starts with the perceptions, analyses, and operations of
daily life, and progressively applies the measuring rods, clocks,
balances, thermometers, and ammeters of the physicist in pur-
suit of precise data. Claude Bernard especially emphasized the
significance of physico-chemical techniques for biology. The
procedures of specialized physics, however, must be applied
with discrimination in investigations of organisms. New
methods must continually be devised to solve new problems in
physics. It is therefore to be expected that the special tech-
niques of physics will have to be appropriately modified on ap-
plication to living processes. It is an open question whether or
not all natural laws are reducible to the laws of specialized
physics. However, since the phenomena of science are spatio-

336

Unity of Science

temporal processes, one must recognize with the doctrine of
physicialism that the procedures of generalized physics are the
basis of all empirical science. In this unity of procedure resides
the unity of science.

One may further ask whether or not it is possible to construct
a single deductive theory for all science. Rationalists in the
past have sought to unify all knowledge by a single principle,
but the logical study of deductive systems has shown that a set
of postulates is required for a fruitful theory. It may therefore
be doubted that a limited set of principles will systematize both
physical and biological laws. In agreement with Bohr, Heisen-
berg has stated 4 that new phenomena require new concepts and
laws for their description and has stressed the relatively closed
character of systems for different realms of phenomena.

In considering the possibility of a single system of science, it
is instructive to review the progress toward systematic unity in
physics. In the seventeenth century Descartes sought to re-
duce all the phenomena of the material world to matter in mo-
tion. A great impetus to a mechanical theory of nature was
given when Newton systematized the knowledge of gravita-
tional phenomena, expressed in Galileo’s law of falling bodies
and Kepler’s laws of planetary motion, by the application of a
law of gravitation to the laws of dynamics. Huyghens pro-
vided a basis for the phenomenon of light by a material ether,
while Newton proposed a corpuscular theory. During the nine-
teenth century elastic solid theories of the ether were developed,
and heat was reduced to the mechanical energy of the systems
of molecules into which perceptible bodies were resolved. Max-
well created the electromagnetic theory of light with the aid of
Faraday’s theory of a medium and attempted to explain it in
terms of the mechanical properties of an ether. Thus unity in
physical theory was sought on the basis of a mechanical theory
of nature. But the mechanical theory no longer provides a basis
for systematic unity in physics. Electrodynamic theories have
been proposed which make electric charges and their electro-
magnetic fields fundamental. Since the advent of the general
theory of relativity, in which the force of gravitation is re-

337

Procedures of Empirical Science

placed by space-time curvature, theories have been developed
which reduce physical quantities to characteristics of curved
space-time. Quantum mechanics had to be developed in order
to systematize observations on microphysical processes.

In the face of apparent disunity, developments in contem-
porary physics inspire the hope that quantum mechanics and
the theory of relativity may be united in a single theory. And
because of the basic function of generalized physics and the
ever increasing development and adaptation of the techniques
of specialized physics, the progress of physics toward unity
augurs well for the unity of all empirical science.

NOTES

1. R. Carnap, The Unity of Science (London, 1934).

2. Cf. C. W. Morris, Logical Positivism, Pragmatism, and Scientific Empiricism
(Paris, 1937), p. 35.

3. Quoted by Harold Chapman Brown, “Intelligence and Mathematics, ” in Crea-
tive Intelligence , ed. John Dewey (New York, 1917).

4. \\ . Heisenberg, 11 andlungen in den Grundlagen der Naturicissenschaft (Leipzig,

Selected Bibliography

Belling, John. The Ultimate Chromosomes of Lilium and Aloe with Regard to
the Numbers of Genes. University of California Publications in Botany,”
Vol. XIV, No. 11. Berkeley, 1928.

Bernard, Claude. Experimental Medicine. Trans. Henry Copley Greene.
New York, 1927.

Bohr, N. Atomtheorie und Naturbeschreibung. Berlin, 1931 .

Brown, Harold Chapman. “Intelligence and Mathematics” in Creative
Intelligence. Ed. John Dewey. New York, 1917.

Bruns wik, Egon. Wahrnehmung und Gegenstandswelt. Leipzig and Wien
1934.

Campbell, N. What Is Science? London, 1921 .

Carnap, R. Physikalische Begrijfsbildung. Karlsruhe, 1926.

. The Unity of Science. Trans. M. Black. London: Kegan Paul, 1934.

. “Testability and Meaning,” Philosophy of Science, October, 1936,
and January, 1937.

Dirac, P. A. M. Principles of Quantum Mechanics. Oxford, 1935.
Heisenberg, W. W andlungen in den Grundlagen der Naturwissenschaft. Leip-
zig, 1935.

Jepson, W. L. A Manual of the Flowering Plants of California. Berkeley, 1925.

338

Selected Bibliography

Lenzen, V. F. The Nature of Physical Theory. New York, 1931.

. Physical Causality. “University of California Publications in

Philosophy,” Vol. XV. Berkeley, 1932.

. The Schema of Time. “University of California Publications in

Philosophy,” Vol. XVIII. Berkeley, 1936.

. “The Partition between Physical Object and Observer,” American

Physics Teacher , Vol. V (1937).

Mackensen, Otto. “Locating Genes on Salivary Chromosomes,” Journal
of Heredity, Vol. XXVI (1935).

More, L. T. Isaac Newton. New York, 1934.

Morris, C. W. Logical Positivism, Pragmatism, and Scientific Empiricism.
Paris, 1937.

Neurath, Otto. Empirische Soziologie. Wien, 1931.

Nouy, Lecomte du. Biological Time. London, 1936.

Painter, Theophilus S. “Salivary Chromosomes and the Attack on the
Gene,” Journal of Heredity, Vol. XXV (1934).

Reichenbach, H. Philosophic der Raum-Zeit Lehre. Berlin and Leipzig,
1928.

Russell, B. Introduction to Mathematical Philosophy. London, 1920.

Whitehead, A. N. The Principle of Relativity. Cambridge, 1922.

Wolf, A. Essentials of Scientific Method. London, 1928.

339

I

!

f

i

j

( Continued from front flap)

main subdivisions of semiotic (the theo-
ry of signs) — syntactics, semantics, and
pragmatics.

FOUNDATIONS OF LOGIC
AND MATHEMATICS

RUDOLF CARNAP begins his study by
examining the basic categories of logic,
illustrating them by a consideration of
simple artificial linguistic systems. He
then considers the nature and interpreta-
tion of a calculus and calculi and their
application in empirical science.

LINGUISTIC ASPECTS
OF SCIENCE

LEONARD BLOOMFIELD’S monograph
is concerned with the nature of linguis-
tics as a science, the devices of natural
languages to attain precision, the special
characteristics of the language of sci-
ence, the place of linguistics in the
scheme of science, and the relation of
linguistics to logic and mathematics.

PROCEDURES OF EMPIR-
ICAL SCIENCE

VICTOR F. LENZEN’s treatment of this
topic centers around two main problems:
the techniques of observation and the
procedures of systematization in empiri-
cal science. In his analysis of observation
the author discusses perception, count-
ing, measurement, observation through
causality, and observation of microphys-
ical entities. Systematization is analyzed
by considering classification, correlation,
the methods of successive approximation
and successive definition, the use of sta-
tistical methods in quantum theory, and
atomism. This monograph concludes
with a brief discussion of the place of
physics in the unity of science.

International

Encyclopedia

of

Unified Science

• -r-- a.',

• - -«*,• M

VOLUME I PART 1

PART 1

Encyclopedia and Unified Science — Otto Neurath, Niels Bohr, John
Dewey, Bertrand Russell, Rudolf Carnap, Charles W. Morris.
Foundations of the Theory of Signs — Charles W. Morris. Foundations
of Logic and Mathematics — Rudolf Carnap. Linguistic Aspects of
Science — Leonard Bloomfield. Procedures of Empirical Science —

Victor F. Lenzen.

PART 2

Principles of the Theory of Probability — Ernest Nagel. Foundations
of Physics — Philipp Frank. Cosmology — E. Finlay-Freundlich
Foundations of Biology — Felix Mainx. The Conceptual Framework
of Psychology — Egon Brunswik.

Critical Comment on Part 1

"The excellence of these separate monographic studies more than
justifies the project. . . . Their excellence is a tribute to the vision
and judgment of the editors, who are to be complimented for
organizing the project.” — I. BERNARD COHEN, Isis.

"There is no better introduction to the study of signs than this work
of Morris.”— PAUL WEISS, Ethics.

"[Carnap] outlines expertly the objectives and many of the
achievements of the modern logical analysis of language. ... A
valuable guide to the field.” — Journal of Philosophy.

"[Bloomfield’s study is] excellent. ... I know of nothing in anywhere
near the same amount of print which would go farther toward
resolving a great many profitless controversies in the social sciences.” —
GEORGE A. LUNDBERG, American Sociological Review.

"[Lenzen gives] a clear and concise account of the genesis of many
physical concepts and of the methods employed by scientists to
introduce greater accuracy and precision into their observations. . . .
Presented very lucidly and supported by many fresh illustrations.” —
MASON GROSS, Philosophic Abstracts.

j i, ^ 7 , i • » v

THE UNIVERSITY OF CHICAGO PRESS