Paraconsistent and paracomplete logics based on \(k\)-cyclic modal pseudocomplemented De Morgan algebras (Q2082260)

This paper introduces \(\mathcal{C}_{k}\)-algebras and two families of logics associated with the class of \(\mathcal{C}_{k}\)-algebras. \(\mathcal{C}_{k}\)-algebras are the class of modal pseudocomplemented De Morgan algebras enriched by a \(k\)-periodic automorphism, which is denoted by \(\lnot_{k}\), where \(k\) is a positive integer. The two families of logics are denoted by \(\mathbb{L}^{\leq}_{k}\) and \(\mathbb{L}_{k}\). The latter consists of 1-assertional logics, while the former is formed by degree-preserving logics. As it is to be expected, \(\mathbb{L}^{\leq}_{k}\) and \(\mathbb{L}_{k}\) share the same set of theorems, but they are distinguished by different properties related to their respective consequence relations. Thus, \(\mathbb{L}^{\leq}_{k}\) is paraconsistent w.r.t. the De Morgan negation \(\sim\) (Lemma 5.3), protoalgebraic and finitely equivalent but not algebraizable (Theorem 5.13). \(\mathbb{L}_{k}\), on its part, is paracomplete (but not paraconsistent) w.r.t. \(\sim\) (Lemma 6.3) and algebraizable (Theorem 6.6). On the other hand, both logics are paraconsistent and paracomplete w.r.t. \(\lnot_{k}\) (Lemmas 5.7, 6.5) and decidable (Theorems 5.15, 6.8). In the conclusions to the paper (Section 7), the authors propose to give syntactic axiomatizations of both \(\mathbb{L}^{\leq}_{k}\) and \(\mathbb{L}_{k}\), as future work.