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UBC Theses and Dissertations

Semantic studies of intuitionistic logic Criscuolo, Giovanni

Abstract

This thesis is a study of intuitionistic semantics as presented by Beth [2] and Kripke [12], using the usual methods of investigation of classical informal logic. Beth models and Kripke models are presented in a manner which does not depend upon a prior definition of the notion of degree of a formula. It is shown that both classes of intuitionistic models are a generalization of the concept of classical models, i.e. they contain classical models as particular case, and that " branching " is a necessary condition in order that intuitionistic logic be complete with respect to them. Intuitionistic sentential calculus is complete with respect to the strong Beth model, the intersection of Beth models and Kripke models. But, if by analogy with the classical case, we extend them to first order logic we find that they are not adequate because, for example, the sentence [formula omitted]x(B(x) V C) [formula omitted] ([formula omitted]xB(x) V C), where C does not contain x free, is valid in these models but not intuitionistically provable. This observation helps to explain the formal differences between the two classes of models. The simplified Kripke models and the simplified Beth models are then introduced and their equivalence with the Kripke models and the Beth models, respectively, is proved. The first ones allow a better notation and a better understanding of the relation R occurring in the definition of Kripke models. The second ones have the important property that, if the domain is finite, any classically valid sentence is valid in them. Finally a semantic proof of most of the reduction theorems from classical to intuitionistic logic is given.

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