The Logic of Generalized Truth Values and the Logic of Bilattices

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, $${\models_t}$$ ? t and $${\models_f}$$ ? f , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 (Shramko and Wansing, J Philos Logic, 34:121---153, 2005). The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive bilattices with conflation (Rivieccio, 2010).