Podsumowanie ogólne dorobku
Główne dzieła – Logiczna struktura świata, Filozofia jako analiza języka nauki.
1. Strukturalne deskrypcje określone
Sposób funkcjonowania ludzkiej świadomości i nauki.
„Ślepa mapa”.
Zagadnienia nauki są strukturalne, nie dotyczą konkretnych przedmiotów.
Każda nazwa przedmiotu, która pojawia się w twierdzeniach naukowych może być zastąpiona przez strukturalną deskrypcję określoną tego przedmiotu ze wskazaniem, do jakiej dziedziny przedmiotów deskrypcja ta się odnosi.
Przedmiot jako struktura charakteryzowana przez relacje.
Teoria konstrukcji – istnieje tylko jedna dziedzina przedmiotów i każde twierdzenie nauki do niej się odnosi
Wskazanie dziedziny przedmiotów zbędne – każde twierdzenie naukowe może być tak przekształcone, że będzie jedynie twierdzeniem strukturalnym – Jedność języka nauki
Nauka ma mówić o tym, co jest obiektywne, a wszystko, co nie należy do struktury, lecz do materiału (czyli da się wskazać przy użyciu konkretnej definicji ostensywnej) jest subiektywne (knowledge by acquaintance).
Każda deskrypcja musi być określona i strukturalna (wpływ Russella).
2. Kręgi podobieństwa
Definiujemy ekstensjonalną relację podobieństwa (np. przynajmniej 1 wspólna barwa).
Opis ma się opierać na relacji (kręgach podobieństwa), a nie konkretnej interpretacji
Przestrzenna symbolizacja elementarnych doznań i ich składników (bliskość dwóch punktów w przestrzeni będzie reprezentowana przez relację podobieństwa między wybranymi jakościami)
Czysta świadomość działa przez budowę strukturalnych deskrypcji określonych i ze względu na wyróżnione jakości buduje kręgi podobieństwa. Emocje są nieistotne, intersubiektywnie niekomunikowalne.
3. Teoria konstrukcyjna
Celem – stworzenie systemu konstrukcyjnego – warstwowo uporządkowanego systemu przedmiotów (pojęć). Przedmioty na każdym poziomie są „konstruowane”z przedmiotów z niższych poziomów.
Dwa wielkie kręgi podobieństwa – krąg przedmiotów fizycznych i krąg przedmiotów psychicznych.
(trzeba zredukować)
Cztery podstawowe dziedziny (ze względu na pierwszeństwo epistemiczne):
autopsychiczna
fizyczna
heteropsychologiczna
kulturowa
4. Logiczna struktura języka – formalna teoria form lingwistycznych danego języka
Opis formalnych reguł oraz wskazanie ich konsekwencji, bez odwoływania się do znaczenia symboli (wyrażeń), lecz wyłącznie do ich rodzaju i porządku
Struktura jest wszystkim, treść niczym. Znaczenie słów jest nieistotne.
5. Jedność nauki. Język fizykalny
Predykaty pierwotne języka fizykalnego J muszą być obserwacyjne
Język J zbudowany zgodnie z wymogami empiryzmu, ale bez zdań spostrzeżeniowych – interubiektywny
Podstawowy język predykatów obserwacyjnych jest językiem prywatnym, ale musimy posługiwać się językiem fizykalnym, bo jest intersubiektywny.
Terminy wszystkich gałęzi nauk logicznie jednorodne
Fizykalizm i teza o jedności języka nauki nie mówią o świecie, lecz o języku.
Spór o zdania protokolarne (SchlickNeurath) – atomizm konstrukcyjny Neuratha jest niewykonalny.
Zdania fizyki sprowadzone przez wyprowadzenie ich konsekwencji za pomocą reguł transformacji do zdań o formie zdań protokolarnych, które zostały porównane ze zdaniami protokolarnymi faktycznie ustanowionymi i potwierdzone lub obalone.
(Ale same zdania protokolarne nie wystarczą – stąd warstwy)
Niektóre pojęcia są nienaukowe nie dlatego, że są nieredukowalne do terminów zdań obserwacyjnych (bo taki charakter ma wiele naukowych pojęć teoretycznych, np. grawitacja), ale dlatego, że nie podają praw umożliwiających ich sprawdzenie
6. Język etyki
 Wypowiedzi wartościujące to tylko nakazy podane w zwodniczej formie gramatycznej.
 Nie są sprawdzalne i nie mają żadnego sensu poznawczego (nie są prawdziwe ani fałszywe) – metafizyczne.
Opracowanie po angielsku
Table of Contents (Clicking on the links below
will take you to that part of this article)
Life
Carnap wrote an intellectual autobiography published in The Philosophy of
Rudolf Carnap, ed. by Paul Arthur Schillp, La Salle, Ill. : Open Court Pub.
Co., 1963. That autobiography is the main source of the following biographical
notes.
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. During the years between 1910 and 1914 he studied philosophy,
physics and mathematics at the University of Jena and Freiburg. Among his teachers
was neoKantian philosopher Bruno Bauch, with whom he studied Kantian philosophy.
In his intellectual autobiography, Carnap remembers that The Critique of
Pure Reason was carefully discussed through a whole year. Carnap was especially
interested in the Kantian theory of space. In 1910, Carnap attended Gottlob
Frege's lectures on logic (Frege was professor of mathematics at Jena). Carnap
attended a second course by Frege in 1913 – there were only three students at
that course – and a third course in 1914. During those courses, Frege explained
his system of logic and some applications in mathematics. However, during those
years, Carnap was mainly interested in physics; in 1913 he planned to write
his dissertation on a problem of experimental physics, namely thermionic emission.
World War I frustrated the project. Carnap served at the front until 1917, when
he was moved to Berlin. There he studied the theory of relativity. At the time,
Albert Einstein was professor of physics at the University of Berlin.
After the war, Carnap sketched a dissertation on an axiomatic system for the
physical theory of space and time. He submitted the 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. Eventually, in 1921, Carnap wrote his dissertation under the direction
of Bauch. His work dealt with the theory of space from a philosophical point
of view. The work – entitled Der Raum (Space) – is evidently influenced
by Kantian philosophy. Der Raum was published in 1922 in a supplemental
issue of KantStudien.
Carnap's first works were concerned with the foundations of physics; he wrote
essays on causality and the theory of spacetime. In 1923 he met Hans Reichenbach
at a conference on philosophy held at Erlangen. Reichenbach introduced him to
Moritz Schlick, professor of the theory of inductive science at Vienna. Carnap
visited Schlick – and the Vienna Circle – in 1925. The following year he moved
to Vienna and became assistant professor at the University of Vienna. He took
part in the Vienna Circle's meetings, where he met Hans Hahn, Otto Neurath,
Kurt Gödel and, in 1926, Ludwig Wittgenstein; he also met Karl Popper.
He became one of the leading members of the Vienna Circle – and, of course,
of logical positivism – and, in 1929, he wrote, with Hahn and Neurath, the manifesto
of the Circle. In 1928 Carnap published The Logical Structure of the World,
in which he developed a formal version of empiricism: according to him, all
scientific terms are definable by means of a phenomenalistic language. The great
merit of that work is the rigor with which Carnap developed his theory. In the
same year he published Pseudoproblems in Philosophy, in which he asserted
that many alleged philosophical problems are meaningless. In 1929 the Vienna
Circle and the Berlin Circle – the latter was founded in 1928 by Reichenbach – organized the First Conference on Epistemology, held in Prague. In 1930, Carnap
and Reichenbach founded the journal Erkenntnis. In the same year Carnap
met 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, where he became professor of natural philosophy
at the German University. In those years, his most important contribution to
logic was The Logical Syntax of Language (1934). In 1933, Adolf Hitler
became Chancellor of Germany; two years later – in 1935 – Carnap moved to the
United States, helped by Charles Morris and Willard Van Orman Quine, whom he
had met in Prague in 1934. He became an American citizen in 1941.
In the years between 1936 and 1952, he was a professor at the University of
Chicago (during 194041 he was a visiting professor at Harvard University);
in 195254 he was a professor at the Institute for Advanced Study at Princeton
and, from 1954, professor at the University of California at Los Angeles.
In the 1940s, stimulated by Tarskian model theory, Carnap became interested
in semantics. During those years he wrote several books on semantics: Introduction
to Semantics (1942), Formalization of Logic (1943), Meaning and
Necessity: A Study in Semantics and Modal Logic (1947). In Meaning and
Necessity, Carnap used semantics to explain modalities. Afterwards he thought
about the structure of scientific theories: his main interests 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. Two
other important works are „Meaning postulates”(1952) and „Observation Language
and Theoretical Language”(1958). The latter states Carnap's definitive view
on the analyticsynthetic 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 Schillp, was
published in 1963; and 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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The
Structure of Scientific Theories
A scientific theory – in Carnap's opinion – is an interpreted axiomatic formal
system. It consists of:

a formal language, including logical and nonlogical terms;
a set of logicalmathematical axioms and rules of inference;
a set of nonlogical axioms, expressing the empirical portion of the theory;
a set of meaning postulates, stating the meaning of nonlogical terms; they
formalize the analytic truths of the theory;
 a set of rules of correspondence; they give an empirical interpretation
of the theory.
Note that the set of meaning postulates and the set of rules of correspondence
may be included in the set of nonlogical axioms, i.e., it is not necessary that
meaning postulates and rules of correspondence be explicitly stated. Indeed, meaning
postulates and rules of correspondence usually are not explicitly distinguished
from nonlogical axioms; only one set of axioms is formulated and one of the main
purposes of the philosophy of science is to show the difference between the various
kinds of statements. Now I shall examine Carnap's view on different constituents
of a theory.
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The
Language of Scientific Theories
The language consists of (i) a set of symbols and (ii) effective rules that
determine whether a sequence of symbols is a wellformed formula, i.e., correct
with respect to syntax. Among the symbols of the language of a scientific theory
are logical and nonlogical terms. The set of logical terms contains both logical
symbols, e.g., connectives and quantifiers, and mathematical symbols, e.g.,
numbers, derivatives, and integrals. Nonlogical terms are symbols denoting
physical entities or properties or relations, e.g., 'blue', 'cold', 'more warm
than', 'proton', 'electromagnetic field'. Nonlogical terms are divided into
observational terms and theoretical terms. Formulas are divided into: (i) logical
statements, which do not contain nonlogical 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 statementobservational terms 
theoretical terms 
logical statementsNo 
No 
observational statementsYes 
No 
purely theoretical statementsNo 
Yes 
rules of correspondenceYes 
Yes 
The observational language contains only logical and observational statements;
the theoretical language contains logical and theoretical statements and rules
of correspondence.
The distinction between observational terms and theoretical terms is a main
principle of logical positivism; Carnap's view on scientific theories depends
on this distinction. 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. Empirical laws can receive a direct
confirmation by empirical observations. That is, they can be justified by observations
of facts, and can be thought to be an inductive generalization of such observations.
This kind of law can explain and forecast facts; it deals with facts and joins
facts to facts. Ideally, an empirical law which deals with measurable physical
quantities, can be discovered 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.
On the contrary, a theoretical law is concerned with objects or properties
we cannot observe or measure but we can only infer from direct observations.
There is no way of justifying a theoretical law by means of direct observation,
and theoretical laws are not inductive generalizations: they are hypotheses
that go far beyond the 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 he tests the empirical laws that are among its consequences.
The distinction between empirical and theoretical laws entails the distinction
between observational and theoretical properties, and thus also the distinction
between observational and theoretical terms. Carnap admits that the distinction
is not always clear and the line of demarcation between the two kinds of terms
is often arbitrary. To some extent, the distinction between observational and
theoretical terms is similar to the distinction between macroevents, which
are characterized by physical quantities that are constant in a large portion
of space and time, and microevents, where physical quantities change rapidly
in space or time. However, in many situations, the distinction between observational
and theoretical terms is clear; for example, the laws that deal with the pressure,
the volume and the temperature of a gas are empirical laws and the corresponding
terms are observational, while the laws of quantum mechanics are theoretical.
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Analytic and Synthetic
One of the main principles of the logical empiricism is the disintegration
of the synthetic a priori. All statements can be divided into two classes: analytic
a priori statements and synthetic a posteriori statements. Thus synthetic a
priori statements do not exist. Now I shall briefly trace the history of Carnap's
efforts to give a precise definition of the distinction between analytic and
synthetic statements.
In his book The Logical Syntax of Language, published in 1934, Carnap
studies a formal language which can express classical mathematics and scientific
theories. For example, classical physics can be formulated in that language.
When Carnap published The Logical Syntax of Language, Gödel had
already published (in 1931) his work on the incompleteness of mathematics; thus
Carnap was aware of the substantial difference between the two concepts of proof and consequence: some statements, in spite of being a logical consequence
of the axioms of mathematics, are not provable by means of these axioms. The
English version of Tarski's essay on semantics was published in 1935 (the Polish
original was published in 1933); so Carnap did not know the logical theory of
the semantics of a formal language. These circumstances explain the fact that
Carnap, in The Logical Syntax of Language, gives a purely syntactic formulation
of the concept of logical consequence (after the publication of Tarski's essay,
the notion of logical consequence is regarded as a semantic concepts and is
defined by means of model theory). However, Carnap defines a new rule of inference,
now called the omegarule, but formerly called the Carnap rule:
from premises A(1), A(2), ... , A(n), A(n+1) ,... we can infer the
conclusion (x)Ax
Carnap defines the notion of logical consequence: 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 omegarule in the
proof of A. The definition of the notion of provable is: 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 omegarule is not admissible in the
proof of A. Note that a formal system which admits the use of the omegarule
is complete, that is Gödel's incompleteness theorem does not apply to such
formal systems.
Finally, Carnap defines some kinds of statements: (i) a statement is Ltrue
if and only if is a logical consequence of the empty set of statements; (ii)
a statement is Lfalse if and only if all statements are a logical consequence
of it; (iii) a statement is analytic if and only if is Ltrue or Lfalse; (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. That is, analytic
statements are a priori; on the contrary, synthetic statements are a posteriori,
because they are not logically determined.
In Testability and Meaning (1936), Carnap gave a very similar definition.
A statement is analytic if and only if it is logically true; is selfcontradictory
if and only if it is logically false; otherwise the statement is synthetic.
Note the fact that Carnap, in Testability and Meaning, used the notion
of true and false; that is, he used semantic notions. Meaning and Necessity was published in 1947. In this work Carnap gave
a similar definition. He first defines the notion of Ltrue (a statement is
Ltrue if its truth depends on semantic rules) and then defines the notion of
Lfalse (a statements if Lfalse if its negation is Ltrue). A statement is
Ldetermined if it is Ltrue or Lfalse; analytic statements are Ldetermined,
while synthetic statements are not Ldetermined. This definition is very similar
to the definition Carnap gives in The Logical Syntax of Language; however,
in The Logical Syntax of Language Carnap uses only syntactic concepts,
while in Meaning and Necessity he uses semantic concepts.
In 1951, the American philosopher Quine published the article „Two dogmas
of empiricism,”in which Quine criticizes the distinction between analytic and
synthetic statements. As a consequence of Quine's criticism, Carnap partially
changed his point of view on this problem. Carnap's reply to Quine was first
expressed in „Meaning postulates”(1952), in which Carnap suggests that analytic
statements are those which are derivable from a set of appropriate sentences
that he called meaning postulates – those sentences define the meaning of non
logical terms; thus the set of analytic statements is not equal to the set of
logically true statements. Afterwards he wrote „Observation language and theoretical
language”(1958), in which he expressed a general method of determining a set
of meaning postulates for the language of a scientific theory. Carnap expressed
the very same method also in his reply to Carl Gustav Hempel in The Philosophy
of Rudolf Carnap (1963), and subsequently in Philosophical Foundations
of Physics (1966). Now I briefly explain Carnap's method. Suppose the number
of nonlogical axioms is finite; let T be the conjunction of all purely theoretical
axioms, let C be the conjunction of all correspondence postulates and let TC
be 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, named after Frank Plumpton Ramsey
(19031930), English philosopher, who discovered it. A Ramsey sentence is built
by means of the following instructions:

Replace every theoretical term in TC with a variable.
 Add at the beginning of the sentence an appropriate number of existential
quantifiers.
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; that is, R expresses exactly the empirical portion of the theory.
Carnap proposes the statement R
TC as the only meaning postulate; this statement is known as the Carnap sentence.
Note that every empirical statement which is derivable from the Carnap sentence
is logically true, and thus the Carnap sentence lacks empirical consequences.
So – according to Carnap – a statement is analytic if it is derivable from the
Carnap sentence; otherwise the statement is synthetic. I list the requirements
of Carnap's method: (i) nonlogical axioms must be explicitly stated, (ii) the
number of nonlogical axioms must be finite and (iii) observational terms must
be clearly distinguished from theoretical terms.
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Meaning and
Verifiability
Perhaps the most famous tenet of the logical empiricism is the verifiability
principle, according to which a synthetic statement is meaningful only
if it is verifiable. It is very interesting to trace Carnap's effort to
give a logical formulation of this principle. In The Logical Structure of
the World (1928) Carnap asserts that a statement is meaningful only if every
nonlogical term is explicitly definable by means of a very restricted phenomenalistic
language. A few years later, Carnap realized that this thesis is untenable;
a phenomenalistic language is too poor to define physical concepts. Thus he
choose an objective language ("thing language”) as the basic language; in this
language every primitive term is a physical term. All other terms (biological,
psychological, cultural) must be defined by means of basic terms. Carnap also
realized that an explicit definition is often impossible. There are 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
gives an account of the new verifiability principle: all terms must be reducible,
by means of definitions or reduction sentences, to the observational language.
This principle was proved inadequate: K. R. Popper proved not only that some metaphysical
terms can be reduced to the observational language, so they fulfil Carnap's requirements,
but also that some genuine physical concepts are forbidden by Carnap's version
of the verifiability principle. Carnap acknowledged that criticism. In „The Methodological
Character of Theoretical Concepts”(1956) Carnap gives a new criterion of significance.
The definition is rather involved, so I will mention only the main philosophical
properties of Carnap's new principle. 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 is a very important modification in empiricism's
theory of meaning. Secondly, Carnap explicitly acknowledges that some theoretical
terms can be not reduced to the observational language: they acquire an empirical
meaning by means of the links with other theoretical terms which are reducible.
Thirdly, Carnap realizes that the principle of operationalism is too restrictive.
The operationalism was formulated by Nobelprizewinning American physicist Percy
Williams Bridgman (18821961) 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 spacetime, a concept
used by Einstein in his general theory of relativity, is meaningless, because
it is not definable by means of operations. However, Bridgman subsequently changed
his philosophical point of view, and he admitted there is an indirect connection
with observations. Perhaps moved by Popper's criticism, or moved by the unreasonable
consequence of a strict operationalism (the exclusion of Einstein's theory of
curvature of spacetime from legitimate physics), 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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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. Example: a die is cast; the event
is „the score is five”; there are six outcomes and 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 (19031987) formulated the first axiomatic system
for probability.
Frequency interpretation, which is now the favourite 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, supported by Carnap and by 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, due to K. R. Popper. The probability of an event
is an objective property of the event. 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 technical details of Carnap's works are very involved, so I
shall only mention the most philosophically significant consequences of his research.
He asserted 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. The inductive logic would give
us a mathematical method of evaluating the reliability of an hypothesis; therefore
the inductive logic would give an answer to 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 physicalmathematical theory of thermodynamics
entropy to develop a comprehensive theory of the inductive logic, but his plan
remained in a sketchy state. His works on entropy were published posthumously.
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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, impossible, and logical properties such as Ltrue, Lfalse, analytic, synthetic. The symbol
N means „necessarily”, so that Np means „necessarily p”.
Modal and logical properties of statements
ModalitiesFormalization 
Logical status 
p is necessary 
Np 
Ltrue, analytic 
p is impossible 
N~p 
Lfalse, contradictory 
p is contingent 
~Np & ~N~p 
Factual, synthetic 
p is not necessary 
~Np 
Not Ltrue 
p is possible 
~N~p 
Not Lfalse 
p is not contingent 
Np v N~p 
Ldetermined, 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, Carnap
asserts. 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. I stress that, in Carnap's opinion, every modal concept
is definable by means of logical properties of statements so that modal concepts
are explicable from a classical point of view (classical means „using classical
logic”, e.g., first order logic). Note that Carnap was aware of the fact that
the symbol N is definable in the metalanguage, not in the object language.
Np means „p is logically true”, and the last statement belongs to the metalanguage;
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 assume 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. Carnap
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 that equivalence without contradictions.
From a more general philosophical point of view, Carnap believes that modalities
do not require a new conceptual framework; a semantic logic of language can
explain the modal concepts.
The method Carnap uses in explaining modalities is a typical example of Carnap's
philosophical analysis. Another interesting example is the explanation of beliefsentences 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
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. Look at the following examples,
due to Carnap.

First example. 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.
 Second example. Suppose A is true but not Ltrue; therefore the sentences
A v ~A and A are equivalent (both are true)
and, of course, they are not Lequivalent. 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 Lequivalent 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,
beliefsentences. 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
Lequivalent 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
beliefsentences, Carnap defines the notion of intensional isomorphism.
Roughly speaking, two sentences are intensionally isomorphic if and only if their
corresponding elements are Lequivalent. In the beliefsentence „John believes
that D”we can substitute D with an intensionally isomorphic sentence C.
Back to Table of Contents
Philosophy of Physics
The first and the last of the books Carnap published during his life are concerned
with the philosophy of physics; they are respectively the dissertation written
for his doctorate (Der Raum, 1921, published in the following year in
a supplemental issue of KantStudien) and Philosophical Foundations
of Physics, ed. by Martin Gardner, 1966. In 1977, Two Essays on Entropy,
ed. by Abner Shimony, was published posthumously. Der Raum deals with the philosophy of space. Carnap recognizes the
difference between three kinds of theories of space: formal, physical and intuitive
space. 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 geometry, respectively.
Some aspects of Der Raum are very interesting. First of all, Carnap
accepts a neoKantian philosophical point of view. Intuitive space, with its
synthetic a priori character, is a concession to Kantian philosophy. Secondly,
in this work Carnap uses the methods of mathematical logic; for example, the
characterization of the 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, proposed by Hans Reichenbach during those years, was later accepted
by Carnap and became the official position of the logical empiricism on the
philosophy of space.
Carnap also developed a formal system for spacetime 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 survey on many aspects of
the philosophy of physics; it is an excerpt from Carnap's university lessons.
Some theories expressed there are not due to Carnap, but they belong to the
common heritage of logical empiricism. This book is very clear and easy to understand.
It employs few logical and mathematical formulas, and it is rich in examples.
The following is a brief list of the subjects it deals with.

The structure of scientific explanation: deductive and probabilistic explanation.
Philosophical and physical significance of nonEuclidean 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 nonEuclidean 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 an example of the content of Philosophical Foundations of Physics I
shall briefly examine Carnap's thought on scientific explanation. Carnap accepts
the classical theory due to Carl Gustav Hempel. The following example of Carnap's
explains 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
the general laws and the 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. An example – due to Carnap – 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.
Back to Table of Contents
Carnap's Heritage
Carnap's works have raised many debates. A large number of articles is devoted
to a careful examination of his thought, sometimes criticizing his point of
view, sometimes in defense of his philosophy. I shall mention some researches
dealing with developments of Carnap's philosophy.
With respect to the analyticsynthetic distinction, Ryszard Wojcicki and Marian
Przelecki – two Polish logicians – formulated a semantic definition of the distinction
between analytic and synthetic; they proved Carnap sentence is the weakest meaning
postulate, i.e., every meaning postulate entails the Carnap sentence. Therefore
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.
The requirement of a finite number of nonlogical axioms is also removed.
The tentative definition of meaningfulness that Carnap proposed in „The Methodological
Character of Theoretical Concepts”was proved to be untenable. See, for example,
David Kaplan, „Significance and Analyticity”in Rudolf Carnap, Logical Empiricist or 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 attainable by means of possibleworlds semantics. According to
Kripke, not all possible models are admissible. You can read J. Hintikka's essay „Carnap's heritage in logical semantics”in Rudolf Carnap, Logical Empiricist,
which explains that Carnap came extremely close to possibleworlds semantics
but was not able to go beyond classical model theory.
I must stress that the omegarule, which Carnap proposed in The
Logical Syntax of Language, is now widespreadly used in metamathematical
research – usually very involved – on many different subjects.
Back to Table of Contents
Suggestions for Further Reading
In The Philosophy of Rudolf Carnap (1963) there is the most complete
bibliography of Carnap's work. I will only mention Carnap's main works, arranged
in chronological order.

1922 Der Raum: Ein Beitrag zur Wissenschaftslehre, dissertation,
in KantStudien, Ergänzungshefte, n. 56
1925 „Über die Abhängigkeit der Eigenschaften der Raumes
von denen der Zeit”in KantStudien, 30
1926 Physikalische Begriffsbildung, Karlsruhe : Braun, (Wissen
und Wirken ; 39)
1928 Scheinprobleme in der Philosophie, Berlin : WeltkreisVerlag
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

1962 Logic and Language: Studies Dedicated to Professor Rudolf Carnap
on the Occasion of his Seventieth Birthday, Dordrecth, 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, Dordrecth, 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 QuineCarnap 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 CarnapReichenbach Centennial, University of Konstanz,
2124 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
Author Information:
Mauro Murzi