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 N4^{\bot} and Wansing's connexive logic C^{\bot}. We also prove that sixteen expansions of Heyting-Brouwer logic HB via strong negation suggested by H. Wansing in [28] 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.

Cite: Odintsov S.P.
Definability in logics with strong negation
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2026. P.144-162.

Dates:

Submitted:	Feb 13, 2026
Accepted:	Apr 18, 2026

Identifiers: No identifiers