Algebraization of Non-structural Logics

In this paper we study some consequences of applying the notion of algebraizable logic [4, 14] to possibly non structural logic. The equivalence between logic and equational logic, where both are allowed to be non structural, defines the class of Possibly Non Structural algebraizable logic (PNS-algebraizable logics). The logic in this class is provided with a semantics that generalizes abstract logics [7, 13]. It is proved that the class Alg(C) of algebraic reduction of reduced models in this semantics determines the equational consequence equivalent to the logic C. Known results about Leibniz operator's behavior for structural logics are proved for non structural logics. The semantics and the algebra provided here for the case of algebraizable structural logic are essentially the same as those obtained using the standard methods of Abstract Algebraic Logic. Annotated Logics [2, 12] are proved to be PNS-algebraizable, and for finite associated lattice, the results are comparable with that obtained by ad-hoc methods in [19, 20, 21].