Published online by Cambridge University Press: 04 October 2022
Two famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed
${\mathscr {C}}_1$ 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
${\mathscr {C}}_1$. On the one hand, we strengthen the negative results about
${\mathscr {C}}_1$ 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,
${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of
${\mathscr {C}}_1$ covered in the literature. We prove that for extensions
${\mathcal {S}}$ such as
${\mathcal {C}ilo}$ [26], every algebra in
${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions
${\mathcal {S}}$ such as
,
, or
[16, 53], every subdirectly irreducible algebra in
${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety
${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety
$\mathbb {V}({\mathcal {S}})$, with
,
, and
.