An abstract algebraic logic study of da costa's logic c-1 and some of its paraconsistent extensions

Two famous negative results about da Costa's paraconsistent logic C-1 (the failure of the Lindenbaum-Tarski process [44] and its non-algebraizability [39]) have placed C-1 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 C-1. On the one hand, we strengthen the negative results about C-1 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, C-1 is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of C-1 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 [16, 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 S = P-1, P-2, and P-3