11 found
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  1.  35
    Intuitionistic Logic is a Connexive Logic.Davide Fazio, Antonio Ledda & Francesco Paoli - 2023 - Studia Logica 112 (1):95-139.
    We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic ($$\textrm{CHL}$$ CHL ), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: $$\textrm{CHL}$$ CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for $$\textrm{CHL}$$ CHL ; moreover, we (...)
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  2.  10
    Connexive Implications in Substructural Logics.Davide Fazio & Gavin St John - forthcoming - Review of Symbolic Logic:1-32.
    This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we (...)
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  3.  14
    An Algebraic Investigation of the Connexive Logic $$\textsf{C}$$.Davide Fazio & Sergei P. Odintsov - 2023 - Studia Logica 112 (1):37-67.
    In this paper we show that axiomatic extensions of H. Wansing’s connexive logic $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ ) are algebraizable (in the sense of J.W. Blok and D. Pigozzi) with respect to sub-varieties of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras. We develop the structure theory of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras, and we prove their representability in terms of twist-like constructions over implicative lattices (Heyting algebras). As a consequence, we further clarify the relationship between the aforementioned classes. Finally, taking advantage of (...)
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  4.  14
    Algebraic Properties of Paraorthomodular Posets.Ivan Chajda, Davide Fazio, Helmut Länger, Antonio Ledda & Jan Paseka - 2022 - Logic Journal of the IGPL 30 (5):840-869.
    Paraorthomodular posets are bounded partially ordered sets with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from algebraic and order-theoretic perspectives. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features (...)
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  5.  14
    On J. Czelakowski’s Contributions to Quantum Logic and the Foundation of Quantum Mechanics.Davide Fazio - 2024 - In Jacek Malinowski & Rafał Palczewski (eds.), Janusz Czelakowski on Logical Consequence. Springer Verlag. pp. 233-264.
    This paper provides an overview of Janusz Czelakowski’s contributions to the theory of partial Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and, more in general, to the foundation of Quantum Mechanics. Particular attention is paid to the logic of partial Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-algebras, to characterizations of PBAs embeddable into Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and their representation as self-adjoint idempotent (...)
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  6.  7
    Implication in Sharply Paraorthomodular and Relatively Paraorthomodular Posets.Ivan Chajda, Davide Fazio, Helmut Länger, Antonio Ledda & Jan Paseka - 2024 - In Jacek Malinowski & Rafał Palczewski (eds.), Janusz Czelakowski on Logical Consequence. Springer Verlag. pp. 419-446.
    In this paper we show that several classes of partially ordered structures having paraorthomodular reducts, or whose sections may be regarded as paraorthomodular posets, admit a quite natural notion of implication, that admits a suitable notion of adjointness. Within this framework, we propose a smooth generalization of celebrated Greechie’s theorems on amalgams of finite Boolean algebras to the realm of Kleene lattices.
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  7.  9
    Some Remarks on the Logic of Probabilistic Relevance.Davide Fazio & Raffaele Mascella - forthcoming - Logic and Logical Philosophy:1-44.
    In this paper we deepen some aspects of the statistical approach to relevance by providing logics for the syntactical treatment of probabilistic relevance relations. Specifically, we define conservative expansions of Classical Logic endowed with a ternary connective ⇝ - indeed, a constrained material implication - whose intuitive reading is “x materially implies y and it is relevant to y under the evidence z”. In turn, this ensures the definability of a formula in three-variables R(x, z, y) which is the representative (...)
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  8.  25
    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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  9.  25
    A Substructural Gentzen Calculus for Orthomodular Quantum Logic.Davide Fazio, Antonio Ledda, Francesco Paoli & Gavin St John - 2023 - Review of Symbolic Logic 16 (4):1177-1198.
    We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers (...)
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  10.  23
    On the structure theory of Łukasiewicz near semirings.Ivan Chajda, Davide Fazio & Antonio Ledda - 2018 - Logic Journal of the IGPL 26 (1):14-28.
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  11.  31
    Algebraic Perspectives on Substructural Logics.Davide Fazio, Antonio Ledda & Francesco Paoli (eds.) - 2020 - Springer International Publishing.
    This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. -/- Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over (...)
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