Topos Based Semantic for Constructive Logic with Strong Negation

The theory of elementary toposes plays the fundamental role in the categorial analysis of the intuitionistic logic. The main theorem of this theory uses the fact that sets E(A,Ω) (for any object A of a topos E) are Heyting algebras with operations defined in categorial terms. More exactly, subobject classifier true: 1 → Ω permits us define truth-morphism on Ω and operations in E(A,Ω) are defined by them uniformly. The aim of this paper is to show usefulness of toposes in the categorial analysis of the constructive logic with strong negation (CLSN , for short) too. In any topos E we distinguish an object Λ and its truth-arrows that the sets E(A,Λ) have the structure of a Nelson algebra. The object Λ (internal Nelson algebra) in E is defined as a result of an application, to the internal Heyting algebra Ω, the topos counterpart of the well-known classical construction of the Nelson algebra N(B) for a given Heyting algebra B, (see [1], [4] for a generalization). We denote by HA the variety of all Heyting algebras and by NA the variety of all Nelson algebras. Explanations of definitions and notations of used notions from topos theory are in [2]. Truth-morphisms are denoted like respective connectives.