# RUDOLF CARNAP ON SEMANTICAL SYSTEMS AND

W.V.O. QUINE'S PRAGMATIST CRITIQUE

## BOOK III - Page 3

**Semantical Systems: Physics and
the Reduction of Theories**

Even before Carnap had published his
*Introduction to Semantics*,
he had formulated his concept of science as a semantical system,
and this concept did not change fundamentally for the duration
of his contributing career.
The early statements of this concept are set forth in his
“Logical Foundations of the Unity of Science” and “Foundations
of Logic and Mathematics” in the
*International Encyclopedia
of Unified Science* (1938).
In these works he asserts that philosophy of science is
not the study of the activities of scientists,
*i.e.*, the pragmatics
of science, but rather is the study of the results of the
activity, namely the resulting linguistic expressions, which
constitute semantical systems.
More specifically the philosopher treats the language of
science as an object language, and develops a metatheory about
the semantics and syntax of this object language. The metatheory is
expressed in a metalanguage.

A physical theory is an interpreted semantical system. Procedurally a calculus
is firstly constructed, and then semantical rules are laid down
to give the calculus factual content. The resulting physical
calculus will usually presuppose a logical mathematical calculus
as its basis, to which there are added the primitive signs which
are descriptive terms, and the axioms which are the specific
primitive sentences of the physical calculus in question. For example a calculus of
mechanics of mass points can be constructed with the fundamental
laws of mechanics taken as axioms. Semantical rules are laid
down stating that the primitive signs designate the class of
material particles, the three spatial coordinates of a particle
**
x** at time

*t**,*the mass of a particle

*x**,*and the class of forces acting on a particle

*x**or on a space*

*at time*

**s**

*t**.*Thus by semantical interpretation the theorems of the calculus of mechanics become physical laws, that constitute physical mechanics as a theory with factual content that can be tested by observations. Carnap views the customary division of physics into theoretical and experimental physics as corresponding to the distinction between calculus and interpreted system. The work in theoretical physics consists mainly in the essentially mathematical work of constructing calculi and carrying out deductions with the calculi. In experimental physics interpretations are made and theories are tested by experiments.

Carnap maintains that any physical theory and even the whole of physics can be presented in the form of an interpreted system consisting of a specific calculus, an axiom system, and a system of semantical rules for interpretation. The axiom system is based on a logicomathematical calculus with customary interpretation for the nondescriptive terms. The construction of a calculus supplemented by an interpretation is called “formalization”. Formalization has made it possible to forgo a so-called intuitive understanding of the theory. Carnap says that when abstract, nonintuitive formulas such as Maxwell’s equations of electromagnetism were first proposed as new axioms, some physicists endeavored to make them intuitive by constructing a “model”, which is an analogy to observable macroprocesses. But he maintains that the creation of a model has no more than aesthetic, didactic, or heuristic value, because the model offers nothing to the application of the physical theory. With the advent of relativity theory and quantum theory this demand for intuitive understanding has waned.

A more adequate and mature treatment of physics as a
semantical system, and especially of the problem of abstract or
theoretical terms in the semantical system, can be found in
Carnap’s “The Methodological Character of Theoretical Concepts”
(1956) and in his *
Philosophical Foundations of Physics: An Introduction to the
Philosophy of Science* (1966).
Firstly Carnap makes some preliminary comments about
terms and laws: All the descriptive terms in the object
languages used in science may be classified as either
prescientific or scientific terms. The prescientific terms
are those that occur in what Carnap calls the physicalist or
thing-language.
This language is not the same as the phenomenalist language
advocated by Mach.
Carnap had earlier in his career attempted to apply
constructionalist procedures to the construction of a
phenomenalist language in his
*Logical Structure of the
World *(1928).
But later he decided to accept a language, in which the idea of
a physical thing is not linguistically constructed out of
elementary phenomena, because he came to believe that all
science could be reduced to the thing-language. This thing-language
refers to things and to the properties of things.

In Russell’s predicate calculus things and properties are symbolized as two distinct types of signs: instantiation signs and predicate signs. But the thing language is also expressible in a natural language such as English. The predicates or other descriptive signs referring to properties are of two types: observation terms and disposition terms. Observation terms are simply names for observable properties such as “hot” and “red”. These words are called “observable thing-predicates.” Disposition terms express the disposition of a thing to a certain behavior under certain conditions. They are called “disposition predicates” and are exemplified by such words as “elastic”, “soluble”, and “flexible”. These terms are not observable thing-language properties, but by use of conditional reduction sentences they are reducible to observation predicates. Opposed to prescientific terms are scientific terms. Carnap classified all scientific terms as “theoretical terms” in a broad sense, even though physicists, as he notes, customarily refer to such terms as “length” and “temperature” as observation terms, because their measurement procedures are relatively simple. More abstract theoretical terms are exemplified by “electron” or “electrical field.”

A discussion of theoretical terms requires some further discussion of semantical rules in physical theory. There are two types of semantical rules: definitions and conditional reduction sentences. A reduction sentence for a descriptive sign is a conditional statement that gives for the sign the conditions for its application by reference to other signs. The reduction sentence does not give the complete meaning for the descriptive sign, but it gives part of its meaning. It is a “method of determination” enabling the user to apply the term in concrete cases. A definition is a special case of a reduction sentence that gives all of the meaning of a descriptive term, because it is an equivalence or biconditional sentence. There is never more than one definition for a univocal term, but there may be many reduction sentences for a univocal term, each of which contributes to the term a part of its meaning. Unfortunately Carnap seems never to have elaborated on how the meanings of terms can have parts. Both types of semantical rules – definitions and reduction sentences – introduce new terms into an object language. If one language is such that every descriptive term in it is expressible by reduction sentences in terms of another language, then the second language is called a “sufficient reduction basis” for the first language. For all scientific terms the scientist always knows at least one method of determination, and all such methods always either are reduction sentences or are introduced into an axiomatic system of physics by explicit definition in the axiomatic system.

Carnap states that he disagrees with the philosophy of
the physicist Paul W. Bridgman, who stated in his
*Logic of Modern Physics*
(1927) that, any concept is nothing more than a set of
operations; it is synonymous with the corresponding set of
operations. This principle is called “operationalism”, and it
implies for example that there are as many different concepts of
temperature or length as there are different ways of measuring
temperature or length.
Carnap maintains that these different operational rules
for measurement should not be considered definitions giving the
complete meaning of the quantitative concept. He prefers his idea of
reduction sentences in which statements of operational
procedures are semantical rules giving only part of the meaning
of the theoretical term.
In Carnap’s philosophy what distinguishes theoretical
terms from observation terms is precisely the fact that the
meanings of theoretical terms are always partial and
incomplete. This
view distinguishes Carnap from Heisenberg and from other
positivists such as Nagel, who prefer equivocation to partial
meanings. In
Carnap’s view statements of operational rules understood as
reduction sentences together with all the postulates of
theoretical physics function to give partial interpretations to
quantitative concepts.
These partial interpretations are never final, but rather
are continually increased or “strengthened” by new laws and new
operational or measurement rules that develop with the advance
of physics. Such in
brief is Carnap’s taxonomy of terms.

Consider next Carnap’s taxonomy of scientific laws: Carnap classifies scientific laws as empirical laws and theoretical laws. This division does not correlate exactly to the division between observation terms and theoretical terms in the broader and less abstract sense of his meaning of “theoretical term.” The distinction is based on how the laws are developed. Empirical laws are also called empirical generalizations, because they are developed by inductive generalization, which to Carnap means recognition of regularities by observation of repeated instances. The empirical laws contain observation predicates or magnitudes that are measured by relatively simple procedures that can be expressed in reduction sentences or definitions. Empirical laws therefore may contain theoretical terms in the broad sense, such as “temperature”, “volume”, and “pressure”, as occur in the gas laws, as well as observation terms as may occur in such universal generalizations as “every raven is black.” The scientist makes direct observations or repeated measurements, finds certain regularities, and then expresses the regularities in an empirical law. Theoretical laws on the other hand cannot be made by inductive generalization, because they contain theoretical terms in the narrower or more abstract sense; these theoretical terms are too abstract for making laws by generalization. Examples of these terms are “electron”, “atom”, “molecule”, and “electromagnetic field.” These are the descriptive terms that the physicists also call theoretical and unobservable, and measurements associated with these theoretical terms cannot be acquired in simple or direct ways. The development of theoretical laws proceeds by the physicists’ imaginative construction of theories in the object language of their science.

Having examined Carnap’s classification of the types of
terms and of scientific laws, it is now possible to discuss the
construction of physical theories. Logically there is
firstly a calculus.
Conceivably the calculus might be completely uninterpreted, but
most often the calculus is supplied by what Carnap calls the
logicomathematical calculus with its semantical rules for its
logical terms with their “customary” interpretations. In other words the
physicist seldom develops his own logic or mathematics and may
use some pre-existing mathematics that may never have previously
been used in physics, *e.g.*,
a non-Euclidian geometry.
The physicist then postulates certain axioms, and the
descriptive terms in the axiomatic system will either be
primitive terms or will be completely defined by reference to
primitive terms given in the axioms. In the axiom system the
primitive terms may be classified either as elementary terms or
as theoretical terms in either the narrow or more the abstract
sense. Elementary
terms are either observation terms, or are simple magnitudes
which are theoretical terms in the less abstract sense. The elementary terms are
given their semantical interpretation by semantical rules that
either define them or give methods of determination by
conditional reduction sentences.

The aim of the early positivists was to make all the primitive terms elementary terms. In this way the semantics of the primitive terms would be given by semantical rules that would either designate them as observation predicates, or designate them by reference to experimental measurement procedures. And since none of the abstract theoretical terms are primitive in the axiomatic system, any such terms would have to be defined by reference to the primitive terms. This method would completely satisfy the early positivist requirement that all the semantics in the physical theory be supplied by semantical rules that constitute an effective reduction of the theory to observations or to experimentally based measurements. This would insure that there would be no contamination of science by metaphysical “nonsense”.

However, there is a problem with this approach, even though it would satisfy the requirements of the early positivists. The theories actually constructed by physicists contain abstract theoretical terms that cannot be defined by reference to elementary descriptive terms having semantical rules directly giving them their empirical meanings. As Carnap states, what physicists actually do is not to make all the primitive terms elementary terms, but rather to make the abstract theoretical terms primitive in the axiomatic system and to make the axioms of the systems very general theoretical laws. In this constructional procedure the semantical rules initially have no direct relation to the primitive theoretical terms. Carnap borrows Carl G. Hempel’s metaphorical language describing the axioms with their primitive terms as “floating in the air”, meaning that the theoretical hypotheses are firstly developed by the imagination of the physicist, while the elementary terms occurring in the empirical laws are “anchored to the ground.” There then remains to connect the theoretical laws with the empirical laws.

This connection is achieved by a kind of reduction sentence that relates the abstract theoretical terms in the theoretical laws with the elementary terms in the empirical laws. This reduction sentence is called the “correspondence rule.” It is a semantical rule that gives a partial and only a partial interpretation to the abstract theoretical terms. Thus the axiomatic system is left open, to make it possible to add new correspondence rules when theories are modified and as physics develops, until one day the theory is completely replaced in a scientific revolution by a newer one with different axioms. The new correspondence rules supply additional empirical meaning to the theoretical terms as the theory is developed, and they also enable the physicist to derive empirical laws from the theoretical laws. The logical connection between the two types of laws enables the theoretical laws to explain known empirical laws. And Carnap maintains that the supreme value of a theory is its power to predict new empirical laws; explaining known laws is of minor importance in his view. He states that every successful revolutionary theory has predicted new empirical laws that are confirmed by experiment.

But there still remains a problem for the logical positivist. In this more complicated relationship between theory and experiment, there is a question of how abstract theoretical terms can be distinguished from “metaphysical nonsense”. Many philosophers of science, such as Popper, maintain that this is a pseudo problem that cannot be solved. But it was resolved to Carnap’s satisfaction by the Ramsey sentence. The Cambridge logician, Frank P. Ramsey, proposed that the combined system of theoretical postulates and correspondence rules constituting the theory be replaced by an equivalent sentence, which does not contain the theoretical terms. In the Ramsey sentence the theoretical terms are eliminated and are replaced by existentially quantified dummy variables. The Ramsey sentence has the same explanatory and predictive power as the original statement of the theory, but without the metaphysical questions that are occasioned by the original formulation with its theoretical terms. Carnap reports that Ramsey did not intend that physicists should abandon their use of theoretical terms; theory is a convenient “short hand” that is useful to the physicist.

Finally mention must be made of another application of
the reductionist logic, the unity-of-science agenda. Both Mach and Duhem
expressed the belief that there is a basic unity to all science. In the Vienna Circle the
principal advocate of using constructional methods for advancing
the unity of science was Otto Neurath, a sociologist who was
interested in the sociology of science as well as its linguistic
analysis. In his
autobiography Carnap stated that Neurath’s interest in this
effort was motivated by the belief that the division between
natural sciences and sociocultural sciences, a division that is
characteristic of the romantic tradition, would be a serious
obstacle to the extension of the empirical-logical method to the
social sciences.
Neurath expressed a preference for the physicalist or thing
language rather than the phenomenalist language, since the
former is easier to apply in social sciences. His own views are given
in his “Foundations of the Social Sciences” in the second volume
of the *International
Encyclopedia of Unified Science* (1944).

But before Neurath had published his views, Carnap had
published his “Logical Foundations of the Unity of Science” in
the first volume of the *
Encyclopedia *(1938), where he set forth the
constructionalist procedures for the logical reduction of the
descriptive vocabulary of the empirical sciences to the
observational thing language.
The use of the thing language presumes in Carnap’s view a
philosophical thesis called physicalism, the view that the whole
of science can be reduced to the physical language, the
language of physical things.
Carnap says that the physiological and behavioristic
approaches in psychology and social science are reducible to the
observational thing language, but that the introspective method
may not be reducible.
The aim of Carnap’s constructionalist program is the
logical reduction of only the descriptive terms in science to
the observational thing language; this effort is not a reduction
of the empirical laws of the sciences to one another. The reduction of laws
occurs as a part of the development of the sciences themselves,
and is the task of the empirical scientist, not of the
philosopher of science.
The constructionalist procedures for the reduction of
descriptive terms for the unity of science are the same as those
that Carnap had developed for the reduction of theoretical
terms.

**
Semantical
Systems: Probability and Induction **

In his article “Testability and Meaning” in
*Philosophy of Science *
(1936) Carnap abandoned the idea of verification, because he
concluded that hypotheses about unobserved events in the
physical world can never be completely verified by observational
evidence. Then he
proposed instead the probabilistic idea of confirmation. He became interested in
the philosophy of probability in 1941, when he considered that
the concept of logical probability might supply an exact
quantitative explication of the concept of confirmation of a
hypothesis with respect to a given body of evidence, such that
it would become possible to speak of a degree of confirmation in
a measurable sense.
Up to that time there were fundamentally two kinds of concepts
of probability, which were proposed by their respective
advocates as alternatives.
The earlier view is the frequency or statistical concept
advanced by Richard von Mises and Hans Reichenbach. The other view is the
logical concept advanced by John Maynard Keynes and by Harold
Jeffreys, and also considered by Ludwig Wittgenstein in his
*Tractatus*, where he
defined probability on the basis of the logical ranges of
propositions.
Wittgenstein’s interpretation construes a probability statement
to be analytic unlike the frequency concept, which construes it
to be synthetic or empirical.
Carnap believed that the logical concept of probability
is the basis for all inductive inference, and therefore he
identifies the concept of logical probability with the concept
of inductive probability.

In 1950 Carnap published
*Logical Foundations of
Probability*. This work on probability is not a development
in either the calculus of probability or the techniques of
statistical inference.
It is Carnap’s contribution to the interpretation of
probability theory with his constructionalist approach and a
further development of his metatheory of semantical systems. Here his distinction
between object language and metalanguage serves as the basis for
his relating the concepts of logical and statistical
probability.
Statements of statistical probability occur in an object
language and are empirical statements about the world.
Statements of logical
probability occur in the metalanguage and are about the degree
of confirmation of statements in the object language. Carnap also refers to the
statements in the metalanguage for scientific theory as
“metascientific” statements.
However, for Carnap metascientific statements are not
empirical, but rather are analytic or L-true; he does not
recognize an empirical metascience. He accepts the frequency
interpretation for the statistical probability asserted by
statements in the object language; statistical probability is
the relative frequency of an occurrence of an event in the long
run. Logical
probability is the estimate of statistical probability, and it
is the measure of the degree of confirmation. Symbolically he expresses
this logical probability as:

**
c(h,e) = r**

which means that hypothesis
** h** is confirmed by evidence

*to the degree*

**e****. The variable**

*r**is the measure of the degree of confirmation, such that*

**r****can take values from 0.0 to 1.0; it is the estimate of the relative frequency and is expressed as:**

*r***
r = m(e*h)/m(e)**

where **
m(e*h)** is the
number of observation sentences describing observed confirming
instances

**of hypothesis**

*e*

*h**,*and

**is the number of observation sentences**

*m(e)*

*e**describing the total number of observed instances, both confirming and disconfirming. He calls*

**a measurement function.**

*m*
In Carnap’s view the logical foundation of probability is
logic in the sense of L-truth, and he therefore draws upon his
metatheory of semantical systems, in which his ideas of state
description and range have a central role. A state description is a
conjunction sentence containing for every atomic sentence that
can be formed in a language, either its affirmation or its
negation but not both.
Thus every L-true sentence is true in every state
description and every L-false or self-contradictory sentence is
false in every state description.
The F-true or factually true sentences are true in only
some state descriptions but are not true in others. When the idea of state
description is related to the concept of logical probability,
the L-true sentences have a degree of confirmation of 1.0, and
the L-false sentences have a degree of confirmation of 0.0. The F-true sentences on
the other hand have a degree of confirmation between 1.0 and
0.0. A closely
related concept is that of the range of a statement. The range is defined as
the class of all state descriptions in which an empirical
statement is true, and it may also be defined as those state
descriptions that L-imply the statement. Using the concept of
range the equation **
r = m(e*h)/m(e)** may be said to be the partial inclusion of the
range of

*e**in*

**as measured by**

*h*

*r**.*Therefore the equation

*c(h,e) = r**is analogous to the statement that*

*e**L-implies*

**except that the range of**

*h***is not completely contained in**

*e*

*h**.*Both types of statements are analytical or L-true statements in the metalanguage, because both are statements in logic, one in inductive logic and the other in deductive logic. In Carnap’s philosophy the logical foundations of probability is logic in the sense of L-truth.

In 1952 Carnap published* The Continuum of Inductive
Methods,* which was to be the volume on the theory of
induction that followed *
Logical Foundations of Probability,* but he became
dissatisfied with this treatment.
For many years he continued to work on induction. At the time of his death
in 1970 he had completed “Inductive Logic and Rational
Decisions” and “A Basic System of Inductive Logic, Part I”,
which were published in S*tudies
in Inductive Logic and Probability,* Volume I (ed. Carnap
and Jeffrey, 1971).
Carnap did not complete Part II of “A Basic System”, and it was
edited for publication in 1980 by Jeffrey in
*Studies in Inductive Logic
and Probability*, Volume II.
In “Inductive Logic and Rational Decisions” Carnap is
concerned with Bayesian decision theory.

In this context the term “probability” does not mean relative frequency, but rather means degree of belief. He distinguishes the psychological concept of actual degree of belief from the logical concept of rational degree of belief. The former is empirical and descriptive, while the latter is normative for rational decision making. Carnap considers the former to be subjective, since it differs from one individual person to another, while the latter is objective. Carnap maintains that contrary to prevailing opinion relative frequency is not the only kind of objective probability. He also calls the former “actual credence” and the latter “rational credence”. Rational credence is the link between descriptive theory and inductive logic, and like inductive logic it is formal, deductive and axiomatic. The concepts of inductive logic and of normative decision theory are similar but not identical. The latter are quasi psychological, while the former have nothing to do with observers and agents, even as these are generalized so that the decision theory is not subjective. Hence there are separate measure functions and confirmation functions for rational decision theory and for inductive logic. In his “A Basic System of Inductive Logic” Carnap develops a set-theoretic axiomatic system, which uses set connectives instead of sentence connectives, and which is equivalent to the customary axiom systems for conditional probability.

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**NOTE: Pages do not corresponds
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