Models and interpretations in the
fifties: Carnap and Kemeny
Pierre Wagner
Université Paris 1 Panthéon-Sorbonne
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, 2009.
Robinson, “On the application of symbolic
logic to algebra”, 1950
Tarski, “Contributions to the theory of
models”, 1954.
2. Carnap’s semantics: a forerunner of model theory?
Carnap, Introduction to Semantics, 1942.
2. Carnap’s semantics: a forerunner of model theory?
Carnap, Introduction to Semantics, 1942.
“Carnap had long since rejected the view that
mathematics consists of ‘statements endowed
with meaning’” (Hodges 2009, p. 483)
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, uninterpreted calculi, while for Tarski there seems to be
no sharp demarcation.”
(Carnap, Introduction to Semantics, 1942, p. xi)
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)
3.Did Carnap miss the model-theoretical turn?
3.Did Carnap miss the model-theoretical turn?
J. Hintikka, “Carnap’s Heritage in Logical
Semantics”, in Hintikka, ed., Rudolf Carnap,
Logical Empiricist, D. Reidel, 1975.
3.Did Carnap miss the model-theoretical turn?
“we seem to have here a full-fledged possibleworlds semantics explicitly outlined by
Carnap” (Hintikka 1975, p. 223)
3.Did Carnap miss the model-theoretical turn?
“we seem to have here a full-fledged possibleworlds 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 reinterpreted in a model.” (Hintikka 1975)
3.Did Carnap miss the model-theoretical turn?
“we seem to have here a full-fledged possibleworlds 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.”
“it is this apparently small point that precludes
Carnap from some of the most promising uses of
possible-worlds semantics” (Hintikka 1975)
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).
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).
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).
4. Carnap and Kemeny: interactions
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
• Carnap, Meaning and Necessity, 1947.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
• Carnap, Meaning and Necessity, 1947.
• Kemeny, “Models and Logical Systems”, JSL, 1948.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
• Carnap, Meaning and Necessity, 1947.
• Kemeny, “Models and Logical Systems”, JSL, 1948.
• Carnap, Logical Foundations of Probability, 1950.
• Kemeny, review of Carnap’s Log. Found. of Proba., 1951.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
• Carnap, Meaning and Necessity, 1947.
• Kemeny, “Models and Logical Systems”, JSL, 1948.
• Carnap, Logical Foundations of Probability, 1950.
• Kemeny, review of Carnap’s Log. Found. of Proba., 1951.
• 1952-1953: Carnap and Kemeny at the Inst. Adv. Study.
4. Carnap and Kemeny: interactions
• Carnap, Foundations of Logic and Mathematics, 1939.
• Carnap, Introduction to Semantics, 1942.
• Carnap, Formalization of Logic, 1943 (first draft written in
1938).
• Carnap, Meaning and Necessity, 1947.
• Kemeny, “Models and Logical Systems”, JSL, 1948.
• Carnap, Logical Foundations of Probability, 1950.
• Kemeny, review of Carnap’s Log. Found. of Proba., 1951.
• 1952-1953: Carnap and Kemeny at the Inst. Adv. Study.
• Kemeny, “A New Approach to Semantics”, JSL, 1956.
4. Carnap and Kemeny: interactions
•
•
•
•
•
•
•
•
•
•
Carnap, Foundations of Logic and Mathematics, 1939.
Carnap, Introduction to Semantics, 1942.
Carnap, Formalization of Logic, 1943 (first draft written in 1938).
Carnap, Meaning and Necessity, 1947.
Kemeny, “Models and Logical Systems”, JSL, 1948.
Carnap, Logical Foundations of Probability, 1950.
Kemeny, review of Carnap’s Log. Found. of Proba., 1951.
1952-1953: Carnap and Kemeny at the Inst. Adv. Study.
Kemeny, “A New Approach to Semantics”, JSL, 1956.
Carnap, “Notes on Semantics” (52 pages; first draft in 1955, revised in
1959, published in 1972).
4. Carnap and Kemeny: interactions
•
•
•
•
•
•
•
•
•
•
Carnap, Foundations of Logic and Mathematics, 1939.
Carnap, Introduction to Semantics, 1942.
Carnap, Formalization of Logic, 1943 (first draft written in 1938).
Carnap, Meaning and Necessity, 1947.
Kemeny, “Models and Logical Systems”, JSL, 1948.
Carnap, Logical Foundations of Probability, 1950.
Kemeny, review of Carnap’s Log. Found. of Proba., 1951.
1952-1953: Carnap and Kemeny at the Inst. Adv. Study.
Kemeny, “A New Approach to Semantics”, JSL, 1956.
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)
5. The peculiarities of Carnap’s semantics
5. The peculiarities of Carnap’s semantics
• No completeness theorem
5. The peculiarities of Carnap’s semantics
• No completeness theorem
• No separate study of first order logic
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
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
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
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.
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).
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.
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.
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.
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.
6. Carnap’s philosophical agenda
• No sharp line between logic and mathematics
6. Carnap’s philosophical agenda
• No sharp line between logic and mathematics
• Explication of: formal science vs. empirical science
6. Carnap’s philosophical agenda
• No sharp line between logic and mathematics
• Explication of: formal science vs. empirical science
• Explication of ‘analytic’
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
7. Kemeny’s work in semantics
Kemeny, “Models of Logical Systems” JSL, 1948.
Kemeny, “A New Approach to Semantics”, JSL, 1956.
7. Kemeny’s work in semantics
Kemeny, “Models of Logical Systems” JSL, 1948.
Kemeny, “A New Approach to Semantics”, JSL, 1956.
• “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.)
7. Kemeny’s work in semantics
Kemeny, “Models of Logical Systems” JSL, 1948.
Kemeny, “A New Approach to Semantics”, JSL, 1956.
“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).
Descargar

Models and interpretations in the fifties: Carnap and …