Connexive logics and connexive semi-Heyting algebras

TL;DR

The paper develops a theory of connexive logic anchored in semi-Heyting algebras by introducing Connexive Semi-Heyting Logic (CSH) and a parallel suite of axiomatisations AT1/AT2/BT1/BT2. It shows SH is implicative and algebraizable, and it identifies CSH as a natural connexive extension with equivalent algebraic semantics in the variety \mathbb{CSH}; it further studies three-valued connexive algebras and their equivalence to 3-valued Heyting logic, as well as the roles of commutativity and anti-Boolean structures. A detailed lattice analysis reveals tight interrelations: AT1 = AT2, BT1 = CSH, BT2 ⊃ BT1 ⊃ AT1, and BT2 ⊂ AT1, with additional links to Ex and 0 to 1 identities. The results illuminate how connexive theses interact with semi-Heyting algebraic structure and provide new directions for exploring amalgamation, decidability, and extensions within this non-classical logical landscape.

Abstract

In this paper, we define and investigate a connexive logic, called 'Connexive semi-Heyting logic' (\mathcal{CSH} for short) and a new subvariety CSH of the variety SH of semi-Heyting algebras. It is shown that the logic \mathcal{CSH} is implicative in the sense of Rasiowa, and is algebraizable with CSH as an equivalent algebraic semantics (in the sense of Blok and Pigozzi). We also introduce the logics \mathcal{AT}i and \mathcal{BT}i, i = 1, 2, along with the subvarieties ATi and BTi, i = 1, 2, of SH. It is then shown that AT1 = AT2 and CSH = BT1 \subset BT2 \subset AT1. A 3-valued connexive semi-Heyting logic \mathcal{CSH}3 and its equivalent algebraic semantics CSH3 are introduced and axiomatized; and it is then shown that CSH3 is deductively equivalent to the 3-valued intuitionistic logic. New characterizations of anti-Boolean semi-Heyting algebras are given. We show that BT2 \cap SHc = V(2), and SHc \subset AT1, where SHc is defined by x \to y = y \to x. It is proved that the identity (AT1) is equivalent to the identity x* \to y* = y* \to x* (* being the pseudocomplement) in StSH and also is equivalent to 0 \to 1 = 0 in SH. We show that AT1 \cap EX \subset BT1, where EX is defined by x \to (y \to z) = y \to (x \to z). The paper concludes with some further remarks, mentions some open problems for future research and proposes two new principles to be considered as Connexive Theses.