Proof-theoretic Semantics for Classical Mathematics

Synthese 148 (3):603-622 (2006)
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Abstract

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is definable in the Curry-Howard theory.

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William W. Tait
University of Chicago

References found in this work

Mathematical logic.Willard Van Orman Quine - 1951 - Cambridge,: Harvard University Press.
Combinatory logic.Haskell Brooks Curry - 1958 - Amsterdam,: North-Holland Pub. Co..
Variables Explained Away.Willard V. Quine - 1960 - Journal of Symbolic Logic 32 (1):112-112.
The formulæ-as-types notion of construction.W. A. Howard - 1995 - In Philippe De Groote (ed.), The Curry-Howard isomorphism. Louvain-la-Neuve: Academia.
Combinatory Logic Vol. 1.Haskell Brooks Curry & Robert M. Feys - 1958 - Amsterdam, Netherlands: North-Holland Publishing Company.

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