Bilattice Logics and Demi-Negation

Abstract

The logic of distributive bilattices, an offshoot of Belnap and Dunn’s four-valued logic, is presented as an appropriate logic for deductive reasoning in situations where (a) sentences can be accepted and rejected at the same time, and (b) the semantic value of a sentence α is given, depending on the dominant connective in α, in terms of mutually independent acceptance and rejection conditions. Taking our cue from some writings by Lloyd Humberstone, we expand this logic by a demi-negation connective, whose behaviour is intermediate between affirmation and negation. The resulting logic is introduced semantically and axiomatised using methods from abstract algebraic logic.