Journal of Symbolic Logic

Intensional models for the theory of types

Reinhard Muskens

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Abstract

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.

Article information

Source
J. Symbolic Logic, Volume 72, Issue 1 (2007), 98-118.

Dates
First available in Project Euclid: 23 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1174668386

Digital Object Identifier
doi:10.2178/jsl/1174668386

Mathematical Reviews number (MathSciNet)
MR2298473

Zentralblatt MATH identifier
1116.03008

Citation

Muskens, Reinhard. Intensional models for the theory of types. J. Symbolic Logic 72 (2007), no. 1, 98--118. doi:10.2178/jsl/1174668386. https://projecteuclid.org/euclid.jsl/1174668386


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