On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis

Abstract

Paraconsistent Weak Kleene logic (PWK) is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic (AAL). We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, $${\mathcal{IBSL}}$$IBSL, generated by the 3-element algebra WK; we also prove that every involutive bisemilattice is representable as the Plonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that $${\mathcal{IBSL}}$$IBSL is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of $${\mathcal{IBSL}}$$IBSL and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL--we describe the class $${\mathsf{Alg}}$$Alg(PWK) of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK--which differs from $${\mathcal{IBSL}}$$IBSL--and explicitly providing a quasiequational basis for it.