A SEMANTIC ANALYSIS OF SOME DISTRIBUTIVE LOGICS WITH NEGATION

In this paper we shall study some extensions of the semilattice based deductive systems S (N) and S (N, 1), where N is the variety of bounded distributive lattices with a negation operator. We shall prove that S (N) and S (N, 1) are the deductive systems generated by the local consequence relation and the global consequence relation associated with (sic)-frames, respectively. Using algebraic and relational methods we will prove that S (N) and some of its extensions are canonical and frame complete.