Abstract:

In this paper, we study the logics with strong negation from the point of view of abstract algebraic logic. We define the notion of implicative logic with strong negation (sn-implicative logic), which can be considered as an abstract notion of a logic with strong negation, and the notion of weak  definitional equivalence  of logics,  based on  that of weakly structural translation. We also distinguish among sn-implicative logics the subclass of so called bl-logics and prove that for bl-logics weak definitional equivalence  implies definitional equivalence. As examples of weakly definitionally equivalent logics we consider  Nelson's constructive logic $\mathsf{N4}^{\bot}$ and Wansing's connexive logic $\mathsf{C}^{\bot}$. We also prove that sixteen expansions of Heyting-Brouwer logic {\sf HB} via strong negation suggested by H.~Wansing in {Wansing08} are weakly definitionally equivalent. Finally, we prove that all implicative logics satisfying the deduction theorem possess the Beth property B2 and that all explosive sn-implicative logics satisfying the deduction theorem possess an analog of Beth property adopted to the language with strong negation.