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Models and interpretations in the fifties: Carnap and Kemeny Pierre Wagner Université Paris 1 Panthéon-Sorbonne

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1. The emergence of model theory in the 1950s Hodges, “Set Theory, Model Theory, and Computability Theory”, in L. Haaparanta, ed., The Development of Modern Logic, OUP, Robinson, “On the application of symbolic logic to algebra”, 1950 Tarski, “Contributions to the theory of models”, 1954.

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2. Carnap’s semantics: a forerunner of model theory? Carnap, Introduction to Semantics, 1942.

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2. Carnap’s semantics: a forerunner of model theory? Carnap, Introduction to Semantics, “Carnap had long since rejected the view that mathematics consists of ‘statements endowed with meaning’” (Hodges 2009, p. 483)

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2. Carnap’s semantics: a forerunner of model theory? “the present book owes very much to Tarski, more indeed than to any other single influence. On the other hand, our conceptions of semantics seem to diverge at certain points. First (…) I emphasize the distinction between semantics and syntax, i.e. between semantical systems as interpreted language systems and purely formal, uninter- preted calculi, while for Tarski there seems to be no sharp demarcation.” (Carnap, Introduction to Semantics, 1942, p. xi)

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2. Carnap’s semantics: a forerunner of model theory? “Incidentally Robinson also took from Carnap’s semantics the idea of adding constant symbols to the formal language as names of the elements of a structure.” (Hodges, 2009, p. 483)

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3.Did Carnap miss the model-theoretical turn?

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J. Hintikka, “Carnap’s Heritage in Logical Semantics”, in Hintikka, ed., Rudolf Carnap, Logical Empiricist, D. Reidel, 1975.

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3.Did Carnap miss the model-theoretical turn? “we seem to have here a full-fledged possible- worlds semantics explicitly outlined by Carnap” (Hintikka 1975, p. 223)

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3.Did Carnap miss the model-theoretical turn? “we seem to have here a full-fledged possible- worlds semantics explicitly outlined by Carnap” “Yet this impression is definitely misleading. (…) His notion of a model is not that of a possible world, for he is, e.g., allowing descriptive predicates to be arbitrarily re- interpreted in a model.” (Hintikka 1975)

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3.Did Carnap miss the model-theoretical turn? “we seem to have here a full-fledged possible- worlds semantics explicitly outlined by Carnap” “Yet this impression is definitely misleading. (…) His notion of a model is not that of a possible world, for he is, e.g., allowing descriptive predi- cates to be arbitrarily re-interpreted in a model.” “it is this apparently small point that precludes Carnap from some of the most promising uses of possible-worlds semantics” (Hintikka 1975)

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3.Did Carnap miss the model-theoretical turn? “The idea of defining logical truth for a given language by considering the range of all possible semantic interpretations was simply absent from logic in the 1940s. Although this conception is now taken for granted, it was not present in Tarski’s famous paper on truth” (Awodey, “Carnap’s Quest for Analyticity”, in Friedman and Creath, eds. Cambridge Companion to Carnap, 2007).

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3.Did Carnap miss the model-theoretical turn? “The idea of defining logical truth for a given language by considering the range of all possible semantic interpretations was simply absent from logic in the 1940s. Although this conception is now taken for granted, it was not present in Tarski’s famous paper on truth” “Similarly, Carnap’s semantic systems in the 1940s consist always of a single fixed interpretation.” (Awodey 2007).

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3.Did Carnap miss the model-theoretical turn? “The idea of defining logical truth for a given language by considering the range of all possible semantic interpretations wa simply absent from logic in the 1940s. Although this conception is now taken for granted, it was not present in Tarski’s famous paper on truth” “Similarly, Carnap’s semantic systems in the 1940s consist always of a single fixed interpretation.” “In fact, the idea seems first to have been suggested by Kemeny (1948), who was an associate of Carnap and was explicitely responding to Carnap’s semantic work.” (Awodey 2007).

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4. Carnap and Kemeny: interactions

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Carnap, Foundations of Logic and Mathematics, 1939.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, 1942.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938).

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, 1947.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, 1948.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, Carnap, Logical Foundations of Probability, Kemeny, review of Carnap’s Log. Found. of Proba., 1951.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, Carnap, Logical Foundations of Probability, Kemeny, review of Carnap’s Log. Found. of Proba., : Carnap and Kemeny at the Inst. Adv. Study.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, Carnap, Logical Foundations of Probability, Kemeny, review of Carnap’s Log. Found. of Proba., : Carnap and Kemeny at the Inst. Adv. Study. Kemeny, “A New Approach to Semantics”, JSL, 1956.

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, Carnap, Logical Foundations of Probability, Kemeny, review of Carnap’s Log. Found. of Proba., : Carnap and Kemeny at the Inst. Adv. Study. Kemeny, “A New Approach to Semantics”, JSL, Carnap, “Notes on Semantics” (52 pages; first draft in 1955, revised in 1959, published in 1972).

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4. Carnap and Kemeny: interactions Carnap, Foundations of Logic and Mathematics, Carnap, Introduction to Semantics, Carnap, Formalization of Logic, 1943 (first draft written in 1938). Carnap, Meaning and Necessity, Kemeny, “Models and Logical Systems”, JSL, Carnap, Logical Foundations of Probability, Kemeny, review of Carnap’s Log. Found. of Proba., : Carnap and Kemeny at the Inst. Adv. Study. Kemeny, “A New Approach to Semantics”, JSL, Carnap, “Notes on Semantics” (52 pages; first draft in 1955, revised in 1959, published in 1972). Carnap, “My conception of Semantics”, in Schilpp 1963 (wr. late 1950s)

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5. The peculiarities of Carnap’s semantics

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No completeness theorem

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5. The peculiarities of Carnap’s semantics No completeness theorem No separate study of first order logic

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5. The peculiarities of Carnap’s semantics No completeness theorem No separate study of first order logic The contention that no full formalization of propositional logic have been given so far

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5. The peculiarities of Carnap’s semantics No completeness theorem No separate study of first order logic The contention that no full formalization of propositional logic have been given so far No single semantic method

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5. The peculiarities of Carnap’s semantics No completeness theorem No separate study of first order logic The contention that no full formalization of propositional logic have been given so far No single semantic method Separate construction of two kinds of systems with their own languages: – Semantic systems – Calculi

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth.

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth. -Calculus K. Typically, it is defined by rules of formation, axioms, and rules of transformation (finite or infinite).

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth. -Calculus K. Typically, it is defined by rules of formation, axioms, and rules of transformation (finite or infinite). -S is an interpretation of K if all the sentences of K are sentences of S.

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth. -Calculus K. Typically, it is defined by rules of formation, axioms, and rules of transformation (finite or infinite). -S is an interpretation of K if all the sentences of K are sentences of S. -S is a true interpretation of K only if all the theorems of K are true sentences of S.

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth. -Calculus K. Typically, it is defined by rules of formation, axioms, and rules of transformation (finite or infinite). -S is an interpretation of K if all the sentences of K are sentences of S. -S is a true interpretation of K only if all the theorems of K are true sentences of S. -A sentence of S is L-true is it holds in every state description in S.

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5. The peculiarities of Carnap’s semantics -Semantic system S. Typically, it is defined by rules of formation, rules of designation, and rules of truth. -Calculus K. Typically, it is defined by rules of formation, axioms, and rules of transformation (finite or infinite). -S is an interpretation of K if all the sentences of K are sentences of S. -S is a true interpretation of K only if all the theorems of K are true sentences of S. -A sentence of S is L-true is it holds in every state description in S. -S is an L-true interpretation of K only if all the theorems of K are L-true sentences of S.

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6. Carnap’s philosophical agenda No sharp line between logic and mathematics

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6. Carnap’s philosophical agenda No sharp line between logic and mathematics Explication of: formal science vs. empirical science

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6. Carnap’s philosophical agenda No sharp line between logic and mathematics Explication of: formal science vs. empirical science Explication of ‘analytic’

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6. Carnap’s philosophical agenda No sharp line between logic and mathematics Explication of: formal science vs. empirical science Explication of ‘analytic’ Completeness: each logical (including mathematical) sentence is analytic or contradictory

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7. Kemeny’s work in semantics Kemeny, “Models of Logical Systems” JSL, Kemeny, “A New Approach to Semantics”, JSL, 1956.

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7. Kemeny’s work in semantics Kemeny, “Models of Logical Systems” JSL, Kemeny, “A New Approach to Semantics”, JSL, “I hope to show that by means of this approach a satisfactory definition can be given for such controversial concepts as analyticity, and at the same time the approach leads to a unified foundation for formalized Semantics” (Kemeny 1956). “As a matter of fact, while there is no difficulty with the concept of analytic truth, there are reasons to doubt the fruitfulness of the concept of logical truth” (ibid.)

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7. Kemeny’s work in semantics Kemeny, “Models of Logical Systems” JSL, Kemeny, “A New Approach to Semantics”, JSL, “Historically speaking, mathematical logic developed as an attempt to formalize certain mathematical systems. These mathematical systems were to serve as models for the formalized logical systems. In this case, the models were given and formal systems were built which have these models. (…) There is general agreement that the original mathematical system, or the intuitively conceived ideas, should form a model of the formal system – whatever that may mean. This question cannot be answered by first building a system and then defining what we mean by a model for that system. (…) That’s clearly not the right way to do it. Just as we need one definition of continuity for all functions, we need a definition of what constitutes a model of any given system.” (Kemeny 1948, p. 19).

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