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On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis

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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.

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Authors and Affiliations

  1. Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Cagliari, Italy

    Stefano Bonzio, Francesco Paoli & Luisa Peruzzi

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    José Gil-Férez

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  1. Stefano Bonzio
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  2. José Gil-Férez
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  3. Francesco Paoli
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  4. Luisa Peruzzi
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Correspondence to Francesco Paoli.

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Bonzio, S., Gil-Férez, J., Paoli, F. et al. On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis. Stud Logica 105, 253–297 (2017). https://doi.org/10.1007/s11225-016-9689-5

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