On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis

Studia Logica 105 (2):253-297 (2017)
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Abstract

Paraconsistent Weak Kleene logic 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. 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, \, generated by the 3-element algebra WK; we also prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ 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 \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ 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 \—and explicitly providing a quasiequational basis for it.

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Author Profiles

Stefano Bonzio
University of Turin
Francesco Paoli
Universita di Cagliari

Citations of this work

Semantical analysis of weak Kleene logics.Roberto Ciuni & Massimiliano Carrara - 2019 - Journal of Applied Non-Classical Logics 29 (1):1-36.
Meaningless Divisions.Damian Szmuc & Thomas Macaulay Ferguson - 2021 - Notre Dame Journal of Formal Logic 62 (3):399-424.

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References found in this work

Saving truth from paradox.Hartry Field - 2008 - New York: Oxford University Press.
The logic of paradox.Graham Priest - 1979 - Journal of Philosophical Logic 8 (1):219 - 241.
An infinity of super-Belnap logics.Umberto Rivieccio - 2012 - Journal of Applied Non-Classical Logics 22 (4):319-335.

View all 7 references / Add more references