Semi-intuitionistic Logic with Strong Negation

There is a well known interplay between the study of algebraic varieties and propositional calculus of various logics. Prime examples of this are boolean algebras and classical logic, and Heyting algebras and intuitionistic logic. After the class of Heyting algebras was generalized to the semi-Heyting algebras by H. Sankappanavar in [San08], its logic counterpart was developed by one of us in [Cor11] and further studied in [CV15]. Nelson algebras, or N-lattices were defined by H. Rasiowa [Ras58] to provide an algebraic semantics to the constructive logic with strong negation proposed by Nelson in [Nel49]. D. Vakarelov in [Vak77] presented a construction of Nelson algebras staring from Heyting ones. Applying this construction to semi-Heyting algebras, we obtained in [CV16] the variety of semi-Nelson algebras as a natural generalization of Nelson algebras. In this variety, the lattice of congruences of an algebra is determined through some of its deductive systems. Furthermore, the class of semi-Nelson algebras is arithmetical, has equationally definable principal congruences and has the congruence extension property. It is the purpose of this work to present a Hilbert-style propositional calculus which is complete with respect to the algebras in this variety. Naming this logic semi-intuitionistic logic with strong negation was a natural choice. We believe that this logic will be of interest from the point of view of Many-Valued Logic, since its algebraic semantics show that it can provide many different interpretations for the implication connective. For example, on a chain with five elements, ten different semi-Nelson algebras may be defined, by changing the implication operation. We present the algebraic motivation, defining semi-Nelson and semi-Heyting algebras. We then introduce the axioms and inference rule for the semi-intuitionistic logic with strong negation, together with some of their consequences. Finally we deal with completeness of the logic with respect to the class of semi-Nelson algebras, and present an axiomatic extension that has the variety of Nelson algebras as its algebraic semantics.