Type theory and functional programming
Type theory and functional programming
Type-theoretical grammar
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The aim of this paper is to build a constructive quantificational semantics for logical modalities. To achieve this goal, two major problems have to be solved: first, how to include varying domains of quantification in the semantics without being driven to conceptual difficulties, and second, how to represent constructively the totality of logically possible worlds.The problem of representing logically possible worlds--classically or constructively--is linked to the first problem mentioned above because logically possible worlds presumably have domains of different cardinalities. It is also argued that the constructive representation of infinite worlds calls for varying domains.On the other hand, the solution to the problem of varying domains is best formalized in a constructive theory with proof-objects, like Martin-Löf's type theory. To enable this, one should be able to give a constructive interpretation for the totality of logically possible worlds.The conceptual background of the paper is Carnap's state description semantics, which is given a new interpretation in Martin-Löf's nonstandard type theory.