On a Semantical Language Hierarchy in a Constructive Mathematical Logic

Abstract

The subject of this paper is a constructive mathematical logic, i.e. a logic suitable for constructive mathematics. Constructive mathematics differs radically from classical mathematics in the understanding of existence and disjunction. We say in constructive mathematics that there exists an object satisfying some condition iff we possess a construction of such an object. We say that the disjunction ‘A or B’ takes place (A and B are propositions) iff we possess a construction of a true proposition which coincides either with A or with B. Whatever might be the classical understanding of existence and disjunction, it differs from the constructive one inasmuch as such constructions are not needed according to the classicist’s point of view.