Titlebook: Semantical Investigations in Heyting‘s Intuitionistic Logic; Dov M. Gabbay Book 1981 Springer Science+Business Media B.V. 1981 logic.propo

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Introducing HPC,We introduce first Heyting’s implication as the weakest implication for which the deduction theorem holds. The full Heyting (intuitionistic) predicate calculus is then introduced and some of its basic properties are studied. Some neighbouring systems are also considered.

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The Kripke, Beth and Topological Interpretations for HPC,This chapter gives the basic Kripke, Beth and topological interpretations for a language without equality or function symbols. A later chapter presents the modified Kripke interpretation which is more suitable for languages with equality and function symbols.

Entrancing

,Heyting’s Propositional Calculus and Extensions,A great deal of research has been done on Heyting’s propositional calculus and on intermediate logics, logics between Heyting’s propositional calculus, ., and the classical propositional calculus, .. In this chapter we shall

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Three Intermediate Logics,We saw in previous chapters the following properties of .:

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The Interpolation Theorem,The interpolation theorem for Heyting’s predicate calculus states that if ├.. → . then there is a wff . containing only the constants and predicates appearing in both . and . such that ├.. → . and ├.. → ..

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Modified Kripke Interpretation,We begin this chapter with an example. Consider a language that contains the symbol =. We want to characterize it as the symbol for equality.

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Theories in HPC 1,. is taken as the underlying logic of intuitionistic mathematics. Thus intuitionistic theories are often presented as theories in .. This is particularly useful since there does not exist a large body of intuitionistic mathematics and thus we must take recourse, in many cases, to axiomatic formulations.