Abstract

AbstractIn the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic.All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logic.

Highlights

  • Is a list of propositions we will prove

  • Completeness seems a relevant issue, in order to have an intuitionistic interpretation of A close to the original classical meaning of A

  • We must choose if we prefer the interpretations with an intuitionistic negation weaker or stronger than the negation in L

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Summary

Introduction

Is a list of propositions we will prove. Let T be any classical theory.PROPOSITION (a). Is a list of propositions we will prove. IfT is recursive, there are quite natural classes of models ofT, like Tarski models, which are intuitionistically complete only in the case T is decidable {not very often)}. The main result of the paper is: if we want a class of model ofT to be complete, and each model of the class to be consistent, in general, we have to interpret classical ->, —> by connectives intuitionistically strictly stronger than ->, —>; and V, 3, by connectives intuitionistically strictly weaker than V, 3. Many classes of models of this kind are known, all more or less inspired by KripkeBeth models (for which we refer to [11])

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