Constructive predicate logic with strong negation and model theory.

Abstract

In this paper, we attempt to investigate the so-called constructive predicate logic with strong negation from a model-theoretical point of view. Strong negation was first introduced by Nelson [8] in connection with Kleene's recursive realizability. At the same time, Markov [7] showed independently that intuitionistic (Heyting) negation can be defined by strong negation and implication. Then Vorob'ev [14] formulated constructive propositional logic with strong negation. Polish logicians such as Rasiowa [9],[10] studied in terms of lattice theory. Additionally, we can find a Gentzen-type formulation by Almukdad and Nelson [1] and Ishimoto [5],[6], and a model-theoretic study by Thomason [13] and Routley [11], to cite only a few. Strong negation is a constructive negation different from Heyting negation. For example, in intuitionistic logic, -^(A r\B) is not equivalent to the derivability of at least one formula of -*A or ~^B. And we cannot prove the equivalence between -^VxA(x) and -^A(t). But these are equivalent in constructive logic with strong negation. In this paper, we research this system on the basis of Kripke's many worlds semantics (the so-called Kripke model), and try to provide a Henkin-type proof of the completeness theorem for the system. We will also inquire into the relationships among constructive predicate logic with strong negation, classical logic, and intuitionistic logic.