Rudolf Carnap (1891—1970)

carnapRudolf 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.

1. Life

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.
2. The Structure of Scientific Theories

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.

The Language of Scientific Theories The language of a scientific
theory consists of:

1. a set of symbols and
2. rules to ensure that a sequence of symbols is a well-formed
formula, that is, 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.

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

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.
3. Analytic and Synthetic

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 (that is, all consequences of A are
analytic) while R expresses the empirical portion (that is, 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.

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.
4. Meaning and Verifiability

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.
5. Probability and Inductive Logic

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.
6. Modal Logic and the Philosophy of Language

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."

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.
7. Philosophy of Physics

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 Analiticità, 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.
a. Carnap's Works

* 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 Berücksichtigung 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 Einführung 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

b. 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 Analiticità, 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, Probabilità 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'eredità di Rudolf Carnap: Epistemologia, Filosofia delle
Scienze, Filosofia del Linguaggio, ed. by Alberto Pasquinelli,
Bologna, Italy : CLUEB