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  1. Extensions of paraconsistent weak Kleene logic.Francesco Paoli & Michele Pra Baldi - forthcoming - Logic Journal of the IGPL.
    Paraconsistent weak Kleene logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} $. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, (...)
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  • An Algebraic View of Super-Belnap Logics.Hugo Albuquerque, Adam Přenosil & Umberto Rivieccio - 2017 - Studia Logica 105 (6):1051-1086.
    The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of (...)
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  • Paraconsistent Belief Revision: An Algebraic Investigation.Massimiliano Carrara, Davide Fazio & Michele Pra Baldi - 2022 - Erkenntnis 89 (2):725-753.
    This paper offers a logico-algebraic investigation of AGM belief revision based on the logic of paradox ( \(\mathrm {LP}\) ). First, we define a concrete belief revision operator for \(\mathrm {LP}\), proving that it satisfies a generalised version of the traditional AGM postulates. Moreover, we investigate to what extent the Levi and Harper identities, in their classical formulation, can be applied to a paraconsistent account of revision. We show that a generalised Levi-type identity still yields paraconsistent-based revisions that are fully (...)
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  • Cut Elimination, Identity Elimination, and Interpolation in Super-Belnap Logics.Adam Přenosil - 2017 - Studia Logica 105 (6):1255-1289.
    We develop a Gentzen-style proof theory for super-Belnap logics, expanding on an approach initiated by Pynko. We show that just like substructural logics may be understood proof-theoretically as logics which relax the structural rules of classical logic but keep its logical rules as well as the rules of Identity and Cut, super-Belnap logics may be seen as logics which relax Identity and Cut but keep the logical rules as well as the structural rules of classical logic. A generalization of the (...)
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  • Gentzen's cut-free calculus versus the logic of paradox.Alexej P. Pynko - 2010 - Bulletin of the Section of Logic 39 (1/2):35-42.
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  • Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations.Alexej P. Pynko - 2020 - Bulletin of the Section of Logic 49 (4):401-437.
    Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is de_ned by a unique conjunctive matrix ℳ4 with exactly two distinguished values over an expansion.
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  • Extensions of Hałkowska–Zajac's three-valued paraconsistent logic.Alexej P. Pynko - 2002 - Archive for Mathematical Logic 41 (3):299-307.
    As it was proved in [4, Sect. 3], the poset of extensions of the propositional logic defined by a class of logical matrices with equationally-definable set of distinguished values is a retract, under a Galois connection, of the poset of subprevarieties of the prevariety generated by the class of the underlying algebras of the defining matrices. In the present paper we apply this general result to the three-valued paraconsistent logic proposed by Hałkowska–Zajac [2]. Studying corresponding prevarieties, we prove that extensions (...)
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  • The Lattice of Super-Belnap Logics.Adam Přenosil - 2023 - Review of Symbolic Logic 16 (1):114-163.
    We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice (...)
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