Rudolf Carnap, a German-born
philosopher and naturalized U.S. citizen, was a leading
exponent of logical positivism and was one of the major
philosophers of the twentieth century. He made
significant contributions to philosophy of science,
philosophy of language, the theory of probability, and
classical, inductive and modal logic. He rejected
metaphysics as meaningless because metaphysical
statements cannot be proved or disproved by experience.
He asserted that many philosophical problems are indeed
pseudo-problems, the outcome of a misuse of language.
Some of them can be resolved when we recognize that they
are not expressing matters of fact, but rather concern
the choice between different linguistic frameworks. Thus
the logical analysis of language becomes the principal
instrument in resolving philosophical problems. Since
ordinary language is ambiguous, Carnap asserted the
necessity of studying philosophical issues in artificial
languages, which are governed by the rules of logic and
mathematics. In such languages, he dealt with the
problems of the meaning of a statement, the different
interpretations of probability, the nature of
explanation, and the distinctions between analytic and
synthetic, a priori and a posteriori, and necessary and
contingent statements.
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Table of Contents
(Clicking on the links below will take you to those
parts of this article)
1. Life
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Rudolf Carnap was born on May 18, 1891, in Ronsdorf,
Germany. In 1898, after his father's death, his family
moved to Barmen, where Carnap studied at the Gymnasium.
From 1910 to1914 he studied philosophy, physics and
mathematics at the universities of Jena and Freiburg. He
studied Kant under Bruno Bauch and later recalled how a
whole year was devoted to the discussion of The
Critique of Pure Reason. Carnap became especially
interested in Kant¡¯s theory of space. Carnap took three
courses from Gottlob Frege in 1910, 1913 and 1914. Frege
was professor of mathematics at Jena. During those
courses, Frege expounded his system of logic and its
applications in mathematics. However, Carnap¡¯s principal
interest at that time was in physics, and by 1913 he was
planning to write his dissertation on thermionic
emission. His studies were interrupted by World War I
and Carnap served at the front until 1917. He then moved
to Berlin and studied the theory of relativity. (At that
time, Albert Einstein was professor of physics at the
University of Berlin.)
After the war, Carnap developed a new dissertation,
this time on an axiomatic system for the physical theory
of space and time. He submitted a draft to physicist Max
Wien, director of the Institute of Physics at the
University of Jena, and to Bruno Bauch. Both found the
work interesting, but Wien told Carnap the dissertation
was pertinent to philosophy, not to physics, while Bauch
said it was relevant to physics. Carnap then chose to
write a dissertation under the direction of Bauch on the
theory of space from a philosophical point of view.
Entitled Der Raum (Space), the work was clearly
influenced by Kantian philosophy. Submitted in 1921, it
was published the following year in a supplemental issue
of Kant-Studien.
Carnap's involvement with the Vienna Circle developed
over the next few years. He met Hans Reichenbach at a
conference on philosophy held at Erlangen in 1923.
Reichenbach introduced him to Moritz Schlick, then
professor of the theory of inductive science at Vienna.
Carnap visited Schlick - and the Vienna Circle - in 1925
and the following year moved to Vienna to become
assistant professor at the University of Vienna. He
became a leading member of the Vienna Circle and, in
1929, with Hans Hahn and Otto Neurath, he wrote the
manifesto of the Circle.
In 1928, Carnap published The Logical Structure of
the World, in which he developed a formal version of
empiricism arguing that all scientific terms are
definable by means of a phenomenalistic language. The
great merit of the book was the rigor with which Carnap
developed his theory. In the same year he published
Pseudoproblems in Philosophy asserting the
meaninglessness of many philosophical problems. He was
closely involved in the First Conference on
Epistemology, held in Prague in 1929 and organized by
the Vienna Circle and the Berlin Circle (the latter
founded by Reichenbach in 1928). The following year, he
and Reichenbach founded the journal Erkenntnis.
At the same time, Carnap met Alfred Tarski, who was
developing his semantical theory of truth. Carnap was
also interested in mathematical logic and wrote a manual
of logic, entitled Abriss der Logistik (1929).
In 1931, Carnap moved to Prague to become professor
of natural philosophy at the German University. It was
there that he made his important contribution to logic
with The Logical Syntax of Language (1934). His
stay in Prague, however, was cut short by the Nazi rise
to power. In 1935, with the aid of the American
philosophers Charles Morris and Willard Van Orman Quine,
whom he had met in Prague the previous year, Carnap
moved to the United States. He became an American
citizen in 1941.
From 1936 to1952, Carnap was a professor at the
University of Chicago (with the year 1940-41 spent as a
visiting professor at Harvard University). He then spent
two years at the Institute for Advanced Study at
Princeton before taking an appointment at the University
of California at Los Angeles.
In the 1940s, stimulated by Tarskian model theory,
Carnap became interested in semantics. He wrote several
books on semantics: Introduction to Semantics
(1942), Formalization of Logic (1943), and
Meaning and Necessity: A Study in Semantics and Modal
Logic (1947). In Meaning and Necessity,
Carnap used semantics to explain modalities.
Subsequently he began to work on the structure of
scientific theories. His main concerns were (i) to give
an account of the distinction between analytic and
synthetic statements and (ii) to give a suitable
formulation of the verifiability principle; that is, to
find a criterion of significance appropriate to
scientific language. Other important works were "Meaning
Postulates" (1952) and "Observation Language and
Theoretical Language" (1958). The latter sets out
Carnap's definitive view on the analytic-synthetic
distinction. "The Methodological Character of
Theoretical Concepts" (1958) is an attempt to give a
tentative definition of a criterion of significance for
scientific language. Carnap was also interested in
formal logic (Introduction to Symbolic Logic,
1954) and in inductive logic (Logical Foundations of
Probability, 1950; The Continuum of Inductive
Methods, 1952). The Philosophy of Rudolf Carnap,
ed. by Paul Arthur Schilpp, was published in 1963 and
includes an intellectual autobiography. Philosophical
Foundations of Physics, ed. by Martin Gardner, was
published in 1966. Carnap was working on the theory of
inductive logic when he died on September 14, 1970, at
Santa Monica, California.
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2. The Structure of Scientific
Theories
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In Carnap's opinion, a scientific theory is an
interpreted axiomatic formal system. It consists of:
- a formal language, including logical and
non-logical terms;
- a set of logical-mathematical axioms and
rules of inference;
- a set of non-logical axioms, expressing the
empirical portion of the theory;
- a set of meaning postulates stating the
meaning of non-logical terms, which formalize
the analytic truths of the theory;
- a set of rules of correspondence, which give
an empirical interpretation of the theory.
The sets of meaning postulates and rules of
correspondence may be included in the set of non-logical
axioms. Indeed, meaning postulates and rules of
correspondence are not usually explicitly distinguished
from non-logical axioms; only one set of axioms is
formulated. One of the main purposes of the philosophy
of science is to show the difference between the various
kinds of statements. Back to Table of Contents The
Language of Scientific Theories The language of a
scientific theory consists of (i) a set of symbols and
(ii) rules to ensure that a sequence of symbols is a
well-formed formula, i.e., correct with respect to
syntax. Among the symbols of the language are logical
and non-logical terms. The set of logical terms include
logical symbols, e.g., connectives and quantifiers, and
mathematical symbols, e.g., numbers, derivatives, and
integrals. Non-logical terms are divided into
observational and theoretical. They are symbols denoting
physical entities, properties or relations such as
'blue', 'cold', ' warmer than', 'proton',
'electromagnetic field'. Formulas are divided into: (i)
logical statements, which do not contain non-logical
terms; (ii) observational statements, which contain
observational terms but no theoretical terms; (iii)
purely theoretical statements, which contain theoretical
terms but no observational terms and (iv) rules of
correspondence, which contain both observational and
theoretical terms.
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Classification of statements in a
scientific language
|
type of statement
|
observational terms
|
theoretical terms
|
logical statements |
No |
No |
observational statements |
Yes |
No |
purely theoretical
statements |
No |
Yes |
rules of correspondence |
Yes |
Yes |
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Observational language contains only logical and
observational statements; theoretical language contains
logical and theoretical statements and rules of
correspondence.
The distinction between observational and theoretical
terms is a central tenet of logical positivism and at
the core of Carnap's view on scientific theories. In his
book Philosophical Foundations of Physics (1966), Carnap
bases the distinction between observational and
theoretical terms on the distinction between two kinds
of scientific laws, namely empirical laws and
theoretical laws.
An empirical law deals with objects or properties
that can be observed or measured by means of simple
procedures. This kind of law can be directly confirmed
by empirical observations. It can explain and forecast
facts and be thought of as an inductive generalization
of such factual observations. Typically, an empirical
law which deals with measurable physical quantities, can
be established by means of measuring such quantities in
suitable cases and then interpolating a simple curve
between the measured values. For example, a physicist
could measure the volume V, the temperature T and the
pressure P of a gas in diverse experiments, and he could
find the law PV=RT, for a suitable constant R.
A theoretical law, on the other hand, is concerned
with objects or properties we cannot observe or measure
but only infer from direct observations. A theoretical
law cannot be justified by means of direct observation.
It is not an inductive generalization but a hypothesis
reaching beyond experience. While an empirical law can
explain and forecast facts, a theoretical law can
explain and forecast empirical laws. The method of
justifying a theoretical law is indirect: a scientist
does not test the law itself but, rather, the empirical
laws that are among its consequences.
The distinction between empirical and theoretical
laws entails the distinction between observational and
theoretical properties, and hence between observational
and theoretical terms. The distinction in many
situations is clear, for example: the laws that deal
with the pressure, volume and temperature of a gas are
empirical laws and the corresponding terms are
observational; while the laws of quantum mechanics are
theoretical. Carnap admits, however, that the
distinction is not always clear and the line of
demarcation often arbitrary. In some ways the
distinction between observational and theoretical terms
is similar to that between macro-events, which are
characterized by physical quantities that remain
constant over a large portion of space and time, and
micro-events, where physical quantities change rapidly
in space or time.
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3. Analytic and Synthetic
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To the logical empiricist, all statements can be
divided into two classes: analytic a priori and
synthetic a posteriori. There can be no synthetic a
priori statements. A substantial aspect of Carnap's work
was his attempt to give precise definition to the
distinction between analytic and synthetic statements.
In The Logical Syntax of Language (1934),
Carnap studied a formal language that could express
classical mathematics and scientific theories, for
example, classical physics. Carnap would have known Kurt
Gödel¡¯s 1931 article on the incompleteness of
mathematics. He was, therefore, aware of the substantial
difference between the two concepts of proof and
consequence: some statements, despite being a
logical consequence of the axioms of mathematics, are
not provable by means of these axioms. He would not,
however, have been able to take account of Alfred
Tarski¡¯s essay on semantics, first published in Polish
in 1933. Tarski¡¯s essay led to the notion of logical
consequence being regarded as a semantic concept and
defined by means of model theory. These circumstances
explain how Carnap, in The Logical Syntax of Language,
gave a purely syntactic formulation of the concept of
logical consequence. However, he did define a new rule
of inference, now called the omega-rule, but
formerly called the Carnap rule:
From the infinite series of premises A(1), A(2), ...
, A(n), A(n+1) ,..., we can infer the conclusion
(x)A(x)
Carnap defines the notion of logical consequence
in the following way: a statement A is a logical
consequence of a set S of statements if and only if
there is a proof of A based on the set S; it is
admissible to use the omega-rule in the proof of
A. In the definition of the notion of provable,
however, a statement A is provable by means of a set S
of statements if and only if there is a proof of A based
on the set S, but the omega-rule is not
admissible in the proof of A. (A formal system which
admits the use of the omega-rule is complete, so
Gödel's incompleteness theorem does not apply to such
formal systems.
Carnap then proceeded to define some kinds of
statements: (i) a statement is L-true if and only if it
is a logical consequence of the empty set of statements;
(ii) a statement is L-false if and only if all
statements are a logical consequence of it; (iii) a
statement is analytic if and only if it is L-true or
L-false; (iv) a statement is synthetic if and only if is
not analytic. Carnap thus defines analytic statements as
logically determined statements: their truth depends on
logical rules of inference and is independent of
experience. Thus, analytic statements are a priori while
synthetic statements are a posteriori, because they are
not logically determined.
Carnap maintained his definitions of statements in
his article ¡°Testability and Meaning¡± (1936) and
his book Meaning and Necessity (1947). In ¡°Testability
and Meaning,¡± he introduced semantic concepts: a
statement is analytic if and only if it is logically
true; it is self-contradictory if and only if it is
logically false. In any other case, the statement is
synthetic. In Meaning and Necessity. Carnap first
defines the notion of L-true (a statement is L-true if
its truth depends on semantic rules) and then defines
the notion of L-false (a statements if L-false if its
negation is L-true). A statement is L-determined if it
is L-true or L-false; analytic statements are
L-determined, while synthetic statements are not
L-determined. This is very similar to the definitions
Carnap gave in The Logical Syntax of Language but
with the change from syntactic to semantic concepts.
In 1951, Quine published the article "Two Dogmas of
Empiricism," in which he disputed the distinction made
between analytic and synthetic statements. In response,
Carnap partially changed his point of view on this
problem. His first response to Quine came in "Meaning
postulates" (1952) where Carnap suggested that analytic
statements are those which can be derived from a set of
appropriate sentences that he called meaning postulates.
Such sentences define the meaning of non logical terms
and thus the set of analytic statements is not equal to
the set of logically true statements. Later, in
"Observation language and theoretical language" (1958),
he expressed a general method for determining a set of
meaning postulates for the language of a scientific
theory. He further expounded on this method in his reply
to Carl Gustav Hempel in The Philosophy of Rudolf
Carnap (1963), and in Philosophical Foundations
of Physics (1966). Suppose the number of non-logical
axioms is finite. Let T be the conjunction of all purely
theoretical axioms, and C the conjunction of all
correspondence postulates and TC the conjunction of T
and C. The theory is equivalent to the single axiom TC.
Carnap formulates the following problems: how can we
find two statements, say A and R, so that A expresses
the analytic portion of the theory (i.e., all
consequences of A are analytic) while R expresses the
empirical portion (i.e., all consequences of R are
synthetic)? The empirical content of the theory is
formulated by means of a Ramsey sentence (a discovery of
the English philosopher Frank Ramsey). Carnap¡¯s solution
to the problem builds a Ramsey sentence on the following
instructions:
1.Replace every theoretical term in TC with a
variable.
2.Add an appropriate number of existential
quantifiers at the beginning of the sentence.
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Look at the following example. Let TC(O
1 ,..,O n
,T 1 ,...,T m
) be the conjunction of T and C; in TC there are
observational terms O 1 ...O
n and theoretical terms T
1 ...T m .
The Ramsey sentence (R) is
EX 1 ...EX m
TC(O 1 ,...,O
n ,X 1
,...,X m )
Every observational statement which is derivable from
TC is also derivable from R and vice versa so that, R
expresses exactly the empirical portion of the theory.
Carnap proposes the statement R TC as the only meaning
postulate; this became known as the Carnap sentence.
Note that every empirical statement that can be derived
from the Carnap sentence is logically true, and thus the
Carnap sentence lacks empirical consequences. So, a
statement is analytic if it is derivable from the Carnap
sentence; otherwise the statement is synthetic. The
requirements of Carnap's method can be summarized as
follows : (i) non-logical axioms must be explicitly
stated, (ii) the number of non-logical axioms must be
finite and (iii) observational terms must be clearly
distinguished from theoretical terms.
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4. Meaning and Verifiability
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Perhaps the most famous tenet of logical empiricism
is the verifiability principle, according to
which a synthetic statement is meaningful only if it is
verifiable. Carnap sought to give a logical formulation
of this principle. In The Logical Structure of the
World (1928) he asserted that a statement is
meaningful only if every non-logical term is explicitly
definable by means of a very restricted phenomenalistic
language. A few years later, Carnap realized that this
thesis was untenable because a phenomenalistic language
is insufficient to define physical concepts. Thus he
choose an objective language ("thing language") as the
basic language, one in which every primitive term is a
physical term. All other terms (biological,
psychological, cultural) must be defined by means of
basic terms. To overcome the problem that an explicit
definition is often impossible, Carnap used
dispositional concepts, which can be introduced by means
of reduction sentences. For example, if A, B, C and D
are observational terms and Q is a dispositional
concept, then
(x)[Ax ® (Bx
« Qx)]
(x)[Cx ® (Dx
« ~Qx)]
are reduction sentences for Q. In ¡°Testability and
Meaning¡± (1936) Carnap revised the new verifiability
principle in this way: all terms must be reducible,
by means of definitions or reduction sentences, to the
observational language. But this proved to be
inadequate. K. R. Popper showed not only that some
metaphysical terms can be reduced to the observational
language and thus fulfill Carnap's requirements, but
also that some genuine physical concepts are forbidden.
Carnap acknowledged that criticism and in "The
Methodological Character of Theoretical Concepts" (1956)
sought to develop a further definition. The main
philosophical properties of Carnap's new principle can
be outlined under three headings. First, of all, the
significance of a term becomes a relative concept: a
term is meaningful with respect to a given theory and a
given language. The meaning of a concept thus depends
on the theory in which that concept is used. This
represents a significant modification in empiricism's
theory of meaning. Secondly, Carnap explicitly
acknowledges that some theoretical terms cannot be
reduced to the observational language: they acquire an
empirical meaning by means of the links with other
reducible theoretical terms. Third, Carnap realizes that
the principle of operationalism is too restrictive.
Operationalism was formulated by the American physicist
Percy Williams Bridgman (1882-1961) in his book The
Logic of Modern Physics (1927). According to
Bridgman, every physical concept is defined by the
operations a physicist uses to apply it. Bridgman
asserted that the curvature of space-time, a concept
used by Einstein in his general theory of relativity, is
meaningless, because it is not definable by means of
operations., Bridgman subsequently changed his
philosophical point of view, and admitted there is an
indirect connection with observations. Perhaps
influenced by Popper's criticism, or by the problematic
consequences of a strict operationalism, Carnap changed
his earlier point of view and freely admitted a very
indirect connection between theoretical terms and the
observational language.
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5. Probability and Inductive Logic
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A variety of interpretations of probability have been
proposed:
•Classical interpretation. The probability of an
event is the ratio of the favorable outcomes to the
possible outcomes. For example: a die is thrown with
the result that "the score is five". There are six
possible outcomes with only one favorable; thus the
probability of "the score is five" is one sixth.
•Axiomatic interpretation. The probability is
whatever fulfils the axioms of the theory of
probability. In the early 1930s, the Russian
mathematician Andrei Nikolaevich Kolmogorov
(1903-1987) formulated the first axiomatic system
for probability.
•Frequency interpretation, now the favored
interpretation in empirical science. The probability
of an event in a sequence of events is the limit of
the relative frequency of that event. Example: throw
a die several times and record the scores; the
relative frequency of "the score is five" is about
one sixth; the limit of the relative frequency is
exactly one sixth.
•Probability as a degree of confirmation. This was
an approach supported by Carnap and students of
inductive logic. The probability of a statement is
the degree of confirmation the empirical evidence
gives to the statement. Example: the statement "the
score is five" receives a partial confirmation by
the evidence; its degree of confirmation is one
sixth.
•Subjective interpretation. The probability is a
measure of the degree of belief. A special case is
the theory that the probability is a fair betting
quotient - this interpretation was supported by
Carnap. Example: suppose you bet that the score
would be five; you bet a dollar and, if you win, you
will receive six dollars: this is a fair bet.
•Propensity interpretation. This is a proposal of K.
R. Popper. The probability of an event is an
objective property of the event. For example: the
physical properties of a die (the die is
homogeneous; it has six sides; on every side there
is a different number between one and six; etc.)
explain the fact that the limit of the relative
frequency of "the score is five" is one sixth.
Carnap devoted himself to giving an account of the
probability as a degree of confirmation. The
philosophically most significant consequences of his
research arise from his assertion that the probability
of a statement, with respect to a given body of
evidence, is a logical relation between the statement
and the evidence. Thus it is necessary to build an
inductive logic; that is, a logic which studies the
logical relations between statements and evidence.
Inductive logic would give us a mathematical method of
evaluating the reliability of an hypothesis. In this way
inductive logic would answer the problem raised by David
Hume's analysis of induction. Of course, we cannot be
sure that an hypothesis is true; but we can evaluate its
degree of confirmation and we can thus compare
alternative theories.
In spite of the abundance of logical and mathematical
methods Carnap used in his own research on the inductive
logic, he was not able to formulate a theory of the
inductive confirmation of scientific laws. In fact, in
Carnap's inductive logic, the degree of confirmation
of every universal law is always zero.
Carnap tried to employ the physical-mathematical
theory of thermodynamic entropy to develop a
comprehensive theory of inductive logic, but his plan
never progressed beyond an outline stage. His works on
entropy were published posthumously.
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6. Modal Logic and the Philosophy
of Language
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The following table, which is an adaptation of a
similar table Carnap used in Meaning and Necessity,
shows the relations between modal properties such as
necessary and impossible and logical properties such
as L-true, L-false, analytic, synthetic. The
symbol N means "necessarily", so that Np means
"necessarily p" or ¡°p is necessary.¡±
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Modal and logical properties of
statements
|
Modalities
|
Formalization
|
Logical status
|
p is necessary |
Np L |
true, analytic |
p is impossible |
N~p L |
false, contradictory |
p is contingent |
~Np & ~N~p |
factual, synthetic |
p is not necessary |
~Np Not L |
true |
p is possible |
~N~p Not L |
false |
p is not contingent |
Np v N~p L |
determined, not synthetic |
Carnap identifies the necessity of a statement p with
its logical truth: a statement is necessary if and only
if it is logically true. Thus modal properties can be
defined by means of the usual logical properties of
statements. Np, i.e., "necessarily p", is true if and
only if p is logically true. He defines the possibility
of p as "it is not necessary that not p". That is,
"possibly p" is defined as ~N~p. The impossibility of p
means that p is logically false. It must be stressed
that, in Carnap's opinion, every modal concept is
definable by means of the logical properties of
statements. Modal concepts are thus explicable from a
classical point of view (meaning "using classical
logic", e.g., first order logic). Carnap was aware that
the symbol N is definable only in the meta-language, not
in the object language. Np means "p is logically true",
and the last statement belongs to the meta-language;
thus N is not explicitly definable in the language of a
formal logic, and we cannot eliminate the term N. More
precisely, we can define N only by means of another
modal symbol we take as a primitive symbol, so that at
least one modal symbol is required among the primitive
symbols.
Carnap's formulation of modal logic is very important
from a historical point of view. Carnap gave the first
semantic analysis of a modal logic, using Tarskian model
theory to explain the conditions in which "necessarily
p" is true. He also solved the problem of the meaning of
the statement (x)N[Ax], where Ax is
a sentence in which the individual variable x
occurs. Carnap showed that (x)N[Ax] is
equivalent to N[(x)Ax] or, more precisely,
he proved we can assume its equivalence without
contradictions.
From a broader philosophical point of view, Carnap
believed that modalities did not require a new
conceptual framework; a semantic logic of language can
explain the modal concepts. The method he used in
explaining modalities was a typical example of his
philosophical analysis. Another interesting example is
the explanation of belief-sentences which Carnap
gave in Meaning and Necessity. Carnap asserts
that two sentences have the same extension if
they are equivalent, i.e., if they are both true or both
false. On the other hand, two sentences have the same
intension if they are logically equivalent, i.e.,
their equivalence is due to the semantic rules of the
language. Let A be a sentence in which another sentence
occurs, say p. A is called "extensional with respect to
p" if and only if the truth value of A does not change
if we substitute the sentence p with an equivalent
sentence q. A is called "intensional with respect to p"
if and only if (i) A is not extensional with respect to
p and (ii) the truth of A does not change if we
substitute the sentence p with a logically equivalent
sentence q. The following examples arise from Carnap¡¯s
assertions:
•The sentence A v B is extensional with respect to
both A and B; we can substitute A and B with
equivalent sentences and the truth value of A v B
does not change.
•Suppose A is true but not L-true; therefore the
sentences A v ~A and A are equivalent (both are
true) and, of course, they are not L-equivalent. The
sentence N(A v ~A) is true and the sentence N(A) is
false; thus N(A) is not extensional with respect to
A. On the contrary, if C is a sentence L-equivalent
to A v ~A, then N(A v ~A) and N(C) are both true:
N(A) is intensional with respect to A.
There are sentences which are neither extensional not
intensional; for example, belief-sentences. Carnap's
example is "John believes that D". Suppose that "John
believes that D" is true; let A be a sentence equivalent
to D and let B be a sentence L-equivalent to D. It is
possible that the sentences "John believes that A" and
"John believes that B" are false. In fact, John can
believe that a sentence is true, but he can believe that
a logically equivalent sentence is false. To explain
belief-sentences, Carnap defines the notion of
intensional isomorphism. In broad terms, two sentences
are intensionally isomorphic if and only if their
corresponding elements are L-equivalent. In the
belief-sentence "John believes that D" we can substitute
D with an intensionally isomorphic sentence C.
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7. Philosophy of Physics
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The first and the last books Carnap published during
his lifetime were concerned with the philosophy of
physics: his doctoral dissertation (Der Raum,
1922) and Philosophical Foundations of Physics,
ed. by Martin Gardner, 1966. Der Raum deals with
the philosophy of space. Carnap recognizes the
difference between three kinds of theories of space:
formal, physical and intuitive s. Formal space is
analytic a priori; it is concerned with the formal
properties of the space that is with those properties
which are a logical consequence of a definite set of
axioms. Physical space is synthetic a posteriori; it is
the object of natural science, and we can know its
structure only by means of experience. Intuitive space
is synthetic a priori, and is known via a priori
intuition. According to Carnap, the distinction between
three different kinds of space is similar to the
distinction between three different aspects of geometry:
projective, metric and topological respectively.
Some aspects of Der Raum remain very
interesting. First, Carnap accepts a neo-Kantian
philosophical point of view. Intuitive space, with its
synthetic a priori character, is a concession to Kantian
philosophy. Second, Carnap uses the methods of
mathematical logic; for example, the characterization of
intuitive space is given by means of Hilbert's axioms
for topology. Thirdly, the distinction between formal
and physical space is similar to the distinction between
mathematical and physical geometry. This distinction,
first proposed by Hans Reichenbach and later accepted by
Carnap, and became the official position of logical
empiricism on the philosophy of space.
Carnap also developed a formal system for space-time
topology. He asserted (1925) that space relations are
based on the causal propagation of a signal, while the
causal propagation itself is based on the time order.
Philosophical Foundations of Physics is a clear and
approachable survey of topics from the philosophy of
physics based on Carnap's university lectures. Some
theories expressed there are not those of Carnap alone,
but they belong to the common heritage of logical
empiricism. The subjects dealt with in the book include:
- The structure of scientific explanation:
deductive and probabilistic explanation.
- The philosophical and physical significance
of non-Euclidean geometry; the theory of space
in the general theory of relativity. Carnap
argues against Kantian philosophy, especially
against the synthetic a priori, and against
conventionalism. He gives a clear explanation of
the main properties of non-Euclidean geometry.
- Determinism and quantum physics.
- The nature of scientific language. Carnap
deals with (i) the distinction between
observational and theoretical terms, (ii) the
distinction between analytic and synthetic
statements and (iii) quantitative concepts.
As a sample of the content of Philosophical
Foundations of Physics we can briefly look at
Carnap's thought on scientific explanation. Carnap
accepts the classical theory developed by Carl Gustav
Hempel. Carnap gives the following example to explain
the general structure of a scientific explanation:
(x)(Px ® Qx)
Pa
---------
Qa
where the first statement is a scientific law; the
second, is a description of the initial conditions; and
the third, is the description of the event we want to
explain. The last statement is a logical consequence of
the first and the second, which are the premises of the
explanation. A scientific explanation is thus a logical
derivation of an appropriate statement from a set of
premises, which state universal laws and initial
conditions. According to Carnap, there is another kind
of scientific explanation, probabilistic explanation, in
which at least one universal law is not a deterministic
law, but a probabilistic law. Again Carnap¡¯s example is:
fr(Q,P) = 0.8
Pa
----------
Qa
where the first sentence means "the relative
frequency of Q with respect to P is 0.8". Qa is not a
logical consequence of the premises; therefore this kind
of explanation determines only a certain degree of
confirmation for the event we want to explain.
¡¡
8. Carnap's Heritage
¡¡
Carnap's work has stimulated much debate. A
substantial scholarly literature, both critical and
supportive, has developed from examination of his
thought. With respect to the analytic-synthetic
distinction, Ryszard Wojcicki and Marian Przelecki - two
Polish logicians - formulated a semantic definition of
the distinction between analytic and synthetic. They
proved that the Carnap sentence is the weakest meaning
postulate, i.e., every meaning postulate entails the
Carnap sentence. As a result, the set of analytic
statements which are a logical consequence of the Carnap
sentence is the smallest set of analytic statements.
Wojcicki and Przelecki's research is independent of the
distinction between observational and theoretical terms,
i.e., their suggested definition also works in a purely
theoretical language. They also dispense with the
requirement for a finite number of non-logical axioms.
The tentative definition of meaningfulness that
Carnap proposed in "The Methodological Character of
Theoretical Concepts" has been proved untenable. See,
for example, David Kaplan, "Significance and
Analyticity" in Rudolf Carnap, Logical Empiricist
and Marco Mondadori's introduction to Analiticit¨¤,
Significanza, Induzione, in which Mondadori suggests
a possible correction of Carnap's definition.
With respect to inductive logic, I mention only
Jaakko Hintikka's generalization of Carnap's continuum
of inductive methods. In Carnap's inductive logic, the
probability of every universal law is always zero.
Hintikka succeeded in formulating an inductive logic in
which universal laws can obtain a positive degree of
confirmation.
In Meaning and Necessity, 1947, Carnap was the
first logician to use a semantic method to explain
modalities. However, he used Tarskian model theory, so
that every model of the language is an admissible model.
In 1972 the American philosopher Saul Kripke was able to
prove that a full semantics of modalities can be
attained by means of possible-worlds semantics.
According to Kripke, not all possible models are
admissible. J. Hintikka's essay "Carnap's heritage in
logical semantics" in Rudolf Carnap, Logical
Empiricist, shows that Carnap came extremely close
to possible-worlds semantics, but was not able to go
beyond classical model theory.
The omega-rule, which Carnap proposed in
The Logical Syntax of Language, has come into
widespread use in metamathematical research over a broad
range of subjects.
¡¡
9. References and Further Reading
¡¡
The Philosophy of Rudolf Carnap (1963)
contains the most complete bibliography of Carnap's
work. Listed below are Carnap's most important works,
arranged in chronological order.
- 1922
Der Raum: Ein Beitrag zur Wissenschaftslehre,
dissertation, in Kant-Studien,
Ergänzungshefte, n. 56
- 1925 "Über
die Abhängigkeit der Eigenschaften der Raumes
von denen der Zeit" in Kant-Studien, 30
- 1926
Physikalische Begriffsbildung, Karlsruhe :
Braun, (Wissen und Wirken ; 39)
- 1928
Scheinprobleme in der Philosophie, Berlin :
Weltkreis-Verlag
- 1928 Der Logische
Aufbau der Welt, Leipzig : Felix Meiner
Verlag (English translation The Logical
Structure of the World; Pseudoproblems in
Philosophy, Berkeley : University of
California Press, 1967)
- 1929 (with
Otto Neurath and Hans Hahn) Wissenschaftliche
Weltauffassung der Wiener Kreis, Vienna : A.
Wolf
- 1929
Abriss der Logistik, mit besonderer
Ber¨¹cksichtigung der Relationstheorie und ihrer
Anwendungen, Vienna : Springer
- 1932 "Die
physikalische Sprache als Universalsprache der
Wissenschaft" in Erkenntnis, II (English
translation The Unity of Science, London
: Kegan Paul, 1934)
- 1934 Logische Syntax
der Sprache (English translation The
Logical Syntax of Language, New York :
Humanities, 1937)
- 1935 Philosophy and
Logical Syntax, London : Kegan Paul
- 1936 "Testability and
meaning" in Philosophy of Science, III
(1936) and IV (1937)
- 1938 "Logical Foundations
of the Unity of Science" in International
Encyclopaedia of Unified Science, vol. I n.
1, Chicago : University of Chicago Press
- 1939 "Foundations of Logic
and Mathematics" in International
Encyclopaedia of Unified Science, vol. I n.
3, Chicago : University of Chicago Press
- 1942 Introduction to
Semantics, Cambridge, Mass. : Harvard
University Press
- 1943 Formalization of
Logic, Cambridge, Mass. : Harvard University
Press
- 1947 Meaning and
Necessity: a Study in Semantics and Modal Logic,
Chicago : University of Chicago Press
- 1950 Logical
Foundations of Probability, Chicago :
University of Chicago Press
- 1952 "Meaning postulates"
in Philosophical Studies, III (now in
Meaning and Necessity, 1956, 2nd edition)
- 1952 The Continuum of
Inductive Methods, Chicago : University of
Chicago Press
- 1954 Einf¨¹hrung in die
Symbolische Logik, Vienna : Springer
(English translation Introduction to Symbolic
Logic and its Applications, New York :
Dover, 1958)
- 1956 "The Methodological
Character of Theoretical Concepts" in
Minnesota Studies in the Philosophy of Science,
vol. I, ed. by H. Feigl and M. Scriven,
Minneapolis : University of Minnesota Press
- 1958 "Beobacthungssprache
und theoretische Sprache" in Dialectica,
XII (English translation "Observation Language
and Theoretical Language" in Rudolf Carnap,
Logical Empiricist, Dordrecht, Holl. : D.
Reidel Publishing Company, 1975)
- 1966 Philosophical
Foundations of Physics, ed. by Martin
Gardner, New York : Basic Books
- 1977 Two Essays on
Entropy, ed. by Abner Shimony, Berkeley :
University of California Press
Other Sources
- 1962 Logic and
Language: Studies Dedicated to Professor Rudolf
Carnap on the Occasion of his Seventieth
Birthday, Dordrect, Holl. : D. Reidel
Publishing Company
- 1963 The Philosophy of
Rudolf Carnap, ed. by Paul Arthur Schillp,
La Salle, Ill. : Open Court Pub. Co.
- 1970 PSA 1970:
Proceedings of the 1970 Biennial Meeting of the
Philosophy of Science Association: In Memory of
Rudolf Carnap, Dordrect, Holl. : D. Reidel
Publishing Company
- 1971 Analiticit¨¤,
Significanza, Induzione, ed. by Alberto
Meotti e Marco Mondadori, Bologna, Italy : il
Mulino
- 1975 Rudolf Carnap,
Logical Empiricist. Materials and Perspectives,
ed. by Jaakko Hintikka, Dordrecht, Holl. : D.
Reidel Publishing Company
- 1986 Joëlle Proust,
Questions de Forme: Logique at Proposition
Analytique de Kant a Carnap, Paris, France:
Fayard (English translation Questions of
Forms: Logic and Analytic Propositions from Kant
to Carnap, Minneapolis : University of
Minnesota Press)
- 1990 Dear Carnap, Dear
Van: The Quine-Carnap Correspondence and Related
Work, ed. by Richard Creath, Berkeley :
University of California Press
- 1991 Maria Grazia
Sandrini, Probabilit¨¤ e Induzione: Carnap e
la Conferma come Concetto Semantico, Milano,
Italy : Franco Angeli
- 1991 Erkenntnis
Orientated: A Centennial Volume for Rudolf
Carnap and Hans Reichenbach, ed. by Wolfgang
Spohn, Dordrecht; Boston : Kluwer Academic
Publishers
- 1991 Logic, Language,
and the Structure of Scientific Theories:
Proceedings of the Carnap-Reichenbach
Centennial, University of Konstanz, 21-24 May
1991 Pittsburgh : University of Pittsburgh
Press; [Konstanz] : Universitasverlag Konstanz
- 1995 L'eredit¨¤ di
Rudolf Carnap: Epistemologia, Filosofia delle
Scienze, Filosofia del Linguaggio, ed. by
Alberto Pasquinelli, Bologna, Italy : CLUEB
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