An Abstract Algebraic Logic Study of Da Costa’s Logic and Some of Its Paraconsistent Extensions

Abstract

Two famous negative results about da Costa's paraconsistent logic C1 (the failure of the Lindenbaum-Tarski process [44] and its non-algebraizability [39]) have placed C1 seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa's logic C1. On the one hand, we strengthen the negative results about C1 by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, C1 is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of C1 covered in the literature. We prove that for extensions S such as Cilo [26], every algebra in Alg * (S) contains a Boolean subalgebra; and for extensions S such as P 1 , P 2 or P 3 [17,53], every subdirectly irreducible algebra in Alg * (S) has cardinality at most 3. We also characterize the quasivariety Alg * (S) and the intrinsic variety V(S), with