Carnap, Paul Rudolf

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(b. Ronsberg [later a part of the city of Wuppertal] Germany, 18 May 1891;

d. Santa Monica, California, 14 September 1970), philosophy, logic, probability theory.

Carnap, usually known as Rudolf Carnap, was a leading exponent of the so-called Vienna Circle—probably the most fertile group of philosopher-scientists since the days of Plato’s Academy—that cooperated closely in Vienna during the 1920s and early 1930s in order to establish a “scientific philosophy.” The stress this group put on the importance of empirically informed ideas, the logical analysis of language in general, and the logical analysis of scientific concepts and theories in particular was instrumental in bringing about “analytical philosophy,” the predominant approach as of 2007 to philosophy (not only) within the English speaking world. Carnap thus became a towering figure of twentieth-century philosophy.

Life and Education . Carnap’s mother, Anna Dörpfeld, was a daughter of the educational refomer Friedrich Wilhelm Dörpfeld (on whom she published a book) and a sister of the archaeologist Wilhelm Dörpfeld, famous for his excavation work in, among others, Olympia, Troy (with Heinrich Schliemann), and Athens. Anna became the second wife of Johann Carnap, who had worked his way up to become a factory owner. Besides his sister Agnes, Carnap had ten stepsiblings his father brought into the marriage; Carnap was seven when his father died. Carnap had a modest but deeply religious family background, which might explain why, although he later became an atheist, he maintained a respectful and tolerant attitude in matters of faith throughout his life. More generally speaking, throughout his life Carnap showed a very open-minded attitude, was quite radical (leftist) in his political views, and embraced enthusiastically everything that had at least some potential to make, one way or the other, the world a better place. This explains the interest he took in Esperanto, or why he welcomed the American civil rights movement.

As the family had moved to Jena, Carnap studied mathematics, physics, and philosophy at the University of Jena (1910–1914, 1918–1919), but spent the year 1911–1912 at the University of Freiburg. In Jena he was fortunate enough not only to attend lectures by Gottlob Frege, the founder of modern mathematical logic, but also to study the philosopher Immanuel Kant (1724–1804) with Bruno Bauch, a leading figure of the neo-Kantian movement (like Jonas Cohn and Heinrich Rickert in Freiburg). Carnap also mentioned the educator Hermann Nohl (who took his doctorate with Wilhelm Dilthey) as an important influence, and Nohl’s student Wilhelm Flit-ner became Carnap’s lifelong friend. Carnap’s attempt to pursue a doctoral degree in experimental physics came to a sudden halt when his advisor was killed in the early days of World War I. After having served in World War I himself, Carnap obtained his teacher certificate for secondary schools in 1920, but his first dissertation proposal—to develop an axiom system for space and time in physics— was not accepted: The physicist Max Wien referred him to the philosopher Bauch and vice versa. In 1921 he thus took his degree with Bauch on the related problem of space. While studying Bertrand Russell’s books, he felt that his mission was to comply with Russell’s request to study logic—for logic is to philosophy what mathematics is to physics—and to create “a school of men with scientific training and philosophical interests, unhampered by the traditions of the past, and not misled by the literary methods of those who copy the ancients in all except their merits” (Russell, quoted in Carnap, 1963, p. 13).

Carnap joined forces with Hans Reichenbach, who introduced him to Moritz Schlick, who in turn secured him a position as instructor in Vienna and thus made him a member of the Vienna Circle. From 1931 to 1935 he held the position as chair for natural philosophy at the University of Prague, which was close enough to allow him to visit Vienna frequently. In 1934 Charles W. Morris from the University of Chicago and Willard V. Quine from Harvard visited him and invited him to the United States. Carnap joined the University of Chicago in the winter of 1936 and stayed there until 1952; he replaced his old friend Reichenbach (who died prematurely) at UCLA in 1954. From 1940 to 1941, he held a visiting position at Harvard; from 1942 to 1944, he went to Santa Fe, New Mexico, on a Rockefeller research grant; from 1952 to 1954 he was a fellow of the Institute of Advanced Study at Princeton. In 1961 he retired from UCLA. He was honored in 1963 by the publication of a volume in Paul A. Schilpp’s Library of Living Philosophers and received many honorary degrees.

Carnap was married twice: from 1917 to 1929 to Elisabeth Schöndube, with whom he had four children, and to Ina von Stoeger from 1935 until her death in 1964. In 1941 he became an American citizen. He died in 1970 from pneumonia that he had contracted while he was at a hospital to cure an ileus.

For purposes of this survey, Carnap’s research is divided into three periods: (1) the “early Carnap” was mostly concerned with empiricism and physics; (2) the “middle Carnap” was mostly concerned with the role and the importance of a theory of language in philosophy and science; and (3) the “later Carnap” was mostly concerned with induction and probability.

The Early Period . The problem Carnap addressed in his dissertation On Space (Der Raum, 1922) was that of conflicting theories about the nature of space among mathematicians, physicists, and philosophers. His solution was to distinguish, define, and apply (accordingly) three meanings of space: (1) Formal space is a mathematical invention that enables the study abstract relational structures of a certain type. Under Russell’s influence, he considered the theory of relations to belong to the province of logic; hence, knowledge of formal space is logical knowledge because of its relational nature. (2) Intuitive space stems from the faculty of “pure intuition,” as neo-Kantians postulated in the wake of Kant’s philosophy. Knowledge of intuitive space is based on pure intuition and is therefore independent of experience. Carnap restricted it however—like Ernst Cassirer before him—to certain topological properties, while regarding other properties (such as metrics, number of dimensions) not as intuitive, but as empirical. (3) Physical space is entirely dependent on experience, and any knowledge about it is hence empirical knowledge. These distinctions allowed Carnap to discuss the role of non-Euclidean geometry in Albert Einstein’s general theory of relativity, without “philosophical blinders.” His other early work is connected to his dissertation topic or deals with the problem of concept formation in physics.

Carnap’s Habilitation (a second “dissertation” required by the German university system in order to qualify for a tenured position as professor), The Logical Structure of the World (Der logische Aufbau der Welt, 1928), was his first attempt at a project of “rational reconstruction.” Given a certain well-established practice, the program of rational reconstruction aims at developing, from scratch, a coherent language and at providing a set of procedures that lead to the same results as the chosen practice; but this time, one proceeds reflectively and methodically and the concepts employed are more clearly defined and more exact than their informal counterparts. Rational reconstruction, therefore, does not give a “genetic” account of the chosen practice (say, psychological, historical, or sociological), but allows for a rigorous logical analysis of it. In The Aufbau, Carnap made the linguistic practice of employing an observational language in everyday and scientific contexts his target for a rational reconstruction. Influenced by Ernst Mach, Russell, and Gestalt psychology, and starting with nothing more than a purely phenomenalistic language—which is quite rich in comparison to mere sense-data—of “total instantaneous experiences” (Elementarerlebnisse), Carnap showed how— by means of what he called Quasi-Analyse—individuals can rationally reconstruct their common practice of speaking about things, other minds, their properties, and their observation and experience. The main technical means of doing this were the logic of relations in general and the relation “recognition-of-similarity” (Ähnlichkeitserinnerung) between Elementarerlebnisse in particular. A little later, because of the criticism of his student Heinrich Neider and influenced by Otto Neurath and his “unity of science” program, he came to favor sentences in an objective physicalistic language (“protocol sentences”)—rather than in a subjective language of Elementarerlebnisse—as the constitutive elements of the reconstruction. Nelson Goodman revised and refined Carnap’s approach in his own Structure of Appearance (1951).

The Middle Period . Whereas the close interaction within the Vienna Circle (Kurt Gödel, Hans Hahn, Karl Menger, Neurath, and Schlick) as well as encounters with its satellites (Karl Popper, Alfred Tarski, and Ludwig Wittgenstein) and kindred spirits elsewhere, such as Reichenbach’s group in Berlin, constantly forced Carnap to further develop his ideas, his next major publication, The Logical Syntax of Language (Die logische Syntax der Sprache, 1934), comes close to a radical transformation in his thought. Neo-Kantians had declared philosophy not to have genuine objects of its own; instead, they had demanded to study reason where reason can be found (to have emerged within the course of civilization), that is, to study theoretical reasoning within the sciences, practical reasoning within law and legislation, and so forth. Carnap took this approach one step further and deprived philosophy even of these “second-hand objects.” The only legitimate project for philosophical inquiry is the logical analysis of language. Philosophy is, according to Carnap, the theory of scientific language (broadly understood so as to include the humanities insofar as they are engaged in research of which the goal is the attainment of objective truth); it is the logic of science. In modern parlance philosophy reduces to philosophy of science—any other kind of philosophy reduces to engagement with mere “pseudo-problems.” The basic assumption here is that, following suggestions from David Hilbert, one has to remove any reference to scientifically dubious objects such as “thought” or “mental content” and refer instead to what provides a firm intersubjective grounding, in this case language and its rules. What formerly was taken to be a concern with “meaning” should now be reconstructed as an exclusive concern with the syntactical properties of sentences. In order to carry through this program, Carnap, drawing heavily on techniques developed by Gödel, devised a general theory of formal languages. According to this theory, formal languages serve as formal counterparts for the scientific idioms of logic and physics. Carnap used formal languages to make his results as precise as possible and focused on logic and physics because he was an adherent of logicism and physicalism. According to the former, mathematics is reducible to (formal) logic, whereas according to the latter, all sciences are ultimately reducible to physics (or, more precisely, that language X is reducible to language Y if X can be translated by purely formal means into Y).

Note, however, that Carnap did not assume that there is just one language for a science, but possibly many that best suit different purposes. This is his famous “principle of tolerance,” that is, the claim that “there are no morals in logic.” Everyone is free to choose the language that seems most appropriate to a particular purpose, while attempts to impose one camp’s scientific idiom onto another’s result in scientific “pseudo-problems.” Accordingly, scientific statements such as “five is a number” should not be rendered contentually as “five is not a thing, but a number”—a pseudo-problem-propagating formulation doomed to lead to endless controversy about the ontology of numbers—but should be understood as a purely syntactical statement, namely, “according to the rules of the chosen language ‘five’ is not an object-word, but a number-word.” In the context of the Logical Syntax, Carnap put forward also his account of the analytic-synthetic distinction (i.e., the difference between those sentences that are “true whatever the empirical facts may be” and those that are not), which later led to a famous controversy with Quine, who denied its meaningfulness.

Under the influence of Tarski and Morris, Carnap later admitted that syntax is not enough; that semantics and pragmatics have to be accounted for as well. Thus, when in Meaning and Necessity (1947) he extended his syntactical treatment to “intensional languages” —that is, languages that include modal talk, say, of possibility and necessity—he gave his treatment a semantic twist. Another aspect of modality, the analysis of possibility that comes with so-called “disposition predicates” such as “brittle,” led Carnap to the important distinction between “observational” and “theoretical” language in the sciences, with the latter’s concepts being declared not to be definable in terms of the observation language.

The Late Period . Carnap devoted more or less the last two decades of his life to issues in the field of probability theory and its applications (The Logical Foundations of Probability, 1950; The Continuum of Inductive Methods, 1952; Studies in Inductive Logic and Probability, vols. 1 and 2, 1971 and 1980, edited with Richard Jeffrey). In distinguishing “logical” or “inductive probability” (also “probability1”

or “credibility”) from “statistical probability,” that is, the relative frequency in the long run (“probability2” and thereby addressing Popper’s doubt about induction, he sought to clarify the first. But also the first notion allows for two readings, one in the sciences and one in practical deliberations. Accordingly, the term c (h,e)—usually written “P (h,e)” and understood as expressing the conditional probability of h given e—can be understood as “the degree to which the hypothesis h is confirmed by the evidence e,” or as “the factor by which the utility of some benefit B for a deliberating agent X must be multiplied.” Using the advances of formal logic and following John Maynard Keynes, Carnap tried to devise a purely logical theory of induction by regarding c (h,e) to be a measure of the extent to which h is entailed by e. For this purpose, Carnap introduced a first-order language with a finite number of monadic predicates and countably many individual constants. For such languages, Tarski’s definition of entailment by “consequence” states that a sentence a entails a sentence b, if every structure A that is a model of a is also a model of b. (Note that Carnap used the related term “state-description” instead of “model.”) Already in Meaning and Necessity Carnap had identified the meaning M(a) of a sentence a with the set (in the finite case the number) of all its models; if one wants to know what the sentence “it’s raining” means, look at all conceivable situations in which it rains. Now, because there is a structure preserving mapping from sentences to their meanings—under which, for example, M(a(b) translates into M(a) (M(b)—conditional probability can be interpreted as partial entailment, in which the ratio of M(h) to M(e) as in the right-hand side of

determines the degree to which e entails h. Any probability assignment to atomic sentences results in a probability measure p as in the right-hand side of the above equation.

Thus, the problem Carnap addressed but according to the majority of his peers failed to solve satisfactorily was to impose certain “natural” constraints on the choice of c (or, equivalently, on the “measure function” m that assigns unconditional probabilities to the atomic sentences) such that from among all other candidates a unique function c(would be singled out that could serve as a basis for adequately modeling our inductive practice. Initially, Carnap discussed two such functions, c† and c*. One arrives at c† by making the plausible assumption that, because in the absence of any data, every situation is equally probable, the same a priori probabilities should be assigned to all state-descriptions. This, however, has the unfortunate consequence that learning from experience would be impossible. Carnap hence proposed to stick to c*. One gets this function by, first, lumping together into a single “structure-description” all those state-descriptions that are isomorphic to one another, and, second, by assigning all structure-descriptions the same a priori probability (for in the absence of any data, every type of situation is equally probable). But all that Carnap could put forward in favor of c* was that it is the only function “not entirely inadequate.” In his later works, Carnap considered a continuum of inductive logics, in which a realvalued parameter λ, with λ∊0,∞, helps one to pick a proper function c, for λ’s value is correlated to one’s willingness to learn from experience. This suggests that Carnap essentially gave up the idea of basing one’s inductive practice on something as objective as logic and leaned toward a subjective interpretation of probability instead; it also suggests that Carnap arrived at the insight that the problem of induction is not susceptible to a “reconstructive” solution but rather requires a “normative” one.

Throughout his career Carnap stood out as the one who actually developed in great detail ideas others were content to pursue informally. To fulfill this ambitious goal, he drew heavily on the formal machinery of both mathematics and mathematical logic. By taking meticulous care to produce results that are either verifiable or refutable, Carnap succeeded in making considerable progress in philosophy, whatever may have been the ultimate success of the particular solutions he advanced.


Most of Carnap’s literary estate is housed at the Archive for Scientific Philosophy, University of Pittsburgh, Pennsylvania, with a mirror at the Philosophisches Archiv, University of Konstanz, Germany. An edition of his collected works, to be published by Open Court, was under preparation in the mid-first decade of the 2000s.


Der Raum: Ein Beitrag zur Wissenschaftslehre. Berlin: Reuther & Reichard, 1922.

Der logische Aufbau der Welt; Scheinprobleme in der Philososphie. Berlin: Weltkreisverlag, 1928. Translated by Rolf A. George as The Logical Structure of the World: Pseudoproblems in Philosophy. Berkeley: University of California Press, 1961.

“Überwindung der Metaphysik durch logische Analyse der Sprache.” Erkenntnis 2 (1932): 219–241. Translated as “The Elimination of Metaphysics through Logical Analysis of Language.” In Logical Positivism, edited by Alfred J. Ayer. Westport, CT: Greenwood, 1978.

“Die physikalische Sprache als Universalsprache der Wissenschaft.” Erkenntnis 2 (1932): 432–465. Translated by Max Black as The Unity of Science. London: Paul, Trench, Trubner, 1934.

Logische Syntax der Sprache. Vienna: Springer, 1934. Translated by Amethe Smeaton as The Logical Syntax of Language. London: Paul, Trench, Trubner, 1937.

Philosophy and Logical Syntax. London: Paul, Trench, Trubner, 1935.

“Testability and Meaning.” Philosophy of Science 3 (1936): 419–471.

Meaning and Necessity: A Study in Semantics and Modal Logic. Chicago: University of Chicago Press, 1947.

“Empiricism, Semantics, and Ontology.” Revue International de Philosophie 4 (1950): 20–40.

Logical Foundations of Probability. Chicago: University of Chicago Press, 1950.

The Continuum of Inductive Methods. Chicago: University of Chicago Press, 1952.

“Intellectual Autobiography” and “Replies and Systematic Expositions.” In The Philosophy of Rudolf Carnap, edited by Paul Arthur Schilpp. La Salle, IL: Open Court, 1963.

Philosophical Foundations of Physics: An Introduction to the Philosophy of Science. Edited by Martin Gardner. New York: Basic Books, 1966.

“Inductive Logic and Inductive Intuition.” In The Problem of Inductive Logic, edited by Imre Lakatos. Amsterdam: North-Holland, 1968.

With Richard Jeffrey, eds. Studies in Inductive Logic and Probability. Berkeley: University of California Press, 1971.

Two Essays on Entropy. Edited by Abner Shimony. Berkeley: University of California Press, 1977.

With Richard Jeffrey, eds. Studies in Inductive Logic and Probability II. Berkeley: University of California Press, 1980.

With W. V. Quine. Dear Carnap, Dear Van: The Quine-Carnap Correspondence and Related Work. Edited by Richard Creath. Berkeley: University of California Press, 1990.

Untersuchung zur allgemeinen Axiomatik. Edited by Thomas Bonk and Jesús Mosterín. Darmstadt: Wissenschaftliche Buchgesellschaft, 2000.


Awodey, Steven, and Carsten Klein, eds. Carnap Brought Home: The View from Jena. Chicago: Open Court, 2004.

Bar-Hillel, Yehousha, ed. Logic and Language: Studies Dedicated to Professor Rudolf Carnap on the Occasion of his Seventieth Birthday. Dordrecht, Netherlands: D. Reidel, 1962.

Cirera, Ramón. Carnap and the Vienna Circle: Empiricism and Logical Syntax. Translated by Dick Edelstein. Amsterdam: Rodopi, 1994.

Coffa, Alberto J. The Semantic Tradition from Kant to Carnap: To the Vienna Station. Edited by Linda Wessels. Cambridge, U.K.: Cambridge University Press, 1991.

Friedman, Michael. Reconsidering Logical Positivism. Cambridge, U.K.: Cambridge University Press, 1999.

———. A Parting of Ways: Carnap, Cassirer, Heidegger. Chicago: Open Court, 2000.

Giere, Richard N., and Alan W. Richardson, eds. Origins of Logical Empiricism. Minneapolis: University of Minnesota Press, 1996. Contains many papers on Carnap and his context.

Goldfarb, Warren, and Tom Ricketts. “Carnap and the Philosophy of Mathematics.” In Wissenschaft und Subjektivität: der Wiener Kreis und die Philosophie des 20. Jahrhunderts, edited by David Bell and Wilhelm Vossenkuhl. Berlin: Akademie-Verlag, 1992.

Hintikka, Jaakko K., ed. Rudolf Carnap, Logical Empiricist: Materials and Perspectives. Dordrecht, The Netherlands: Reidel, 1975.

Proust, Joëlle. Questions of Form: Logic and the Analytic Proposition from Kant to Carnap. Translated by Anastasios Albert Brenner. Minneapolis: University of Minnesota Press, 1989.

Richardson, Alan W. Carnap’s Construction of the World. Cambridge, U.K.: Cambridge University Press, 1998. Contains bibliography, pp. 230–238.

Ricketts, Tom. “Carnap’s Principle of Tolerance, Empiricism, and Conventionalism.” In Reading Putnam, edited by Peter Clark and Bob Hale. Oxford: Blackwell, 1994.

Salmon, Wesley, and Gereon Wolters, eds. Logic, Language, and the Structure of Scientific Theories: Proceedings of the Carnap-Reichenbach Centennial. Pittsburgh, PA: Pittsburgh University Press; Konstanz, Germany: Universitäts-Verlag, 1994.

Sarkar, Sahotra, ed. “Carnap: A Centenary Reappraisal.” Synthese 93, nos. 1–2 (1992).

Schilpp, Paul Arthur, ed. The Philosophy of Rudolf Carnap. La Salle, IL: Open Court, 1963. Contains a bibliography complete up to 1962.

Spohn, Wolfgang, ed. Erkenntnis Orientated: A Centennial Volume for Rudolf Carnap and Hans Reichenbach. Dordrecht, The Netherlands: Kluwer, 1991.

Bernd Buldt