Intensional models for the theory of types

Abstract

AbstractIn this paper we defineintensionalmodels 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, sinceITLis truly intensional, it can be used to model ascriptions of propositional attitude without predictinglogical omniscience. In order to illustrate this a small fragment of English is defined and provided with anITLsemantics. Secondly, it is shown thatITLmodels contain certain objects that can be identified withpossible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.