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Studia Logica

, Volume 105, Issue 2, pp 253–297 | Cite as

On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis

  • Stefano Bonzio
  • José Gil-Férez
  • Francesco PaoliEmail author
  • Luisa Peruzzi
Article

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, \({\mathcal{IBSL}}\), 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 \({\mathcal{IBSL}}\) 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 \({\mathcal{IBSL}}\) and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \({\mathsf{Alg}}\)(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 \({\mathcal{IBSL}}\)—and explicitly providing a quasiequational basis for it.

Keywords

Paraconsistent Weak Kleene Logic Three-valued logics Bisemilattices Płonka sums Abstract algebraic logic 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Stefano Bonzio
    • 1
  • José Gil-Férez
    • 2
  • Francesco Paoli
    • 1
    Email author
  • Luisa Peruzzi
    • 1
  1. 1.Department of Pedagogy, Psychology, PhilosophyUniversity of CagliariCagliariItaly
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

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