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  1. On the Algebraizability of the Implicational Fragment of Abelian Logic.Sam Butchart & Susan Rogerson - 2014 - Studia Logica 102 (5):981-1001.
    In this paper we consider the implicational fragment of Abelian logic \ . We show that although the Abelian groups provide an semantics for the set of theorems of \ they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi . In the second part of the paper, we investigate an extension of \ in the same language and having the same set of theorems and (...)
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  • Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties.Francesco Paoli, Matthew Spinks & Robert Veroff - 2008 - Logica Universalis 2 (2):209-233.
    We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, (...)
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  • Structural Completeness of the Propositional Calculus.W. A. Pogorzelski - 1975 - Journal of Symbolic Logic 40 (4):604-605.
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  • BCK is not Structurally Complete.Tomasz Kowalski - 2014 - Notre Dame Journal of Formal Logic 55 (2):197-204.
    We exhibit a simple inference rule, which is admissible but not derivable in BCK, proving that BCK is not structurally complete. The argument is proof-theoretical.
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  • Logic and groups.Francesco Paoli - 2001 - Logic and Logical Philosophy 9:109.
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  • Structural Completeness in Fuzzy Logics.Petr Cintula & George Metcalfe - 2009 - Notre Dame Journal of Formal Logic 50 (2):153-182.
    Structural completeness properties are investigated for a range of popular t-norm based fuzzy logics—including Łukasiewicz Logic, Gödel Logic, Product Logic, and Hájek's Basic Logic—and their fragments. General methods are defined and used to establish these properties or exhibit their failure, solving a number of open problems.
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  • Self-implications in BCI.Tomasz Kowalski - 2008 - Notre Dame Journal of Formal Logic 49 (3):295-305.
    Humberstone asks whether every theorem of BCI provably implies $\phi\to\phi$ for some formula $\phi$. Meyer conjectures that the axiom $\mathbf{B}$ does not imply any such "self-implication." We prove a slightly stronger result, thereby confirming Meyer's conjecture.
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  • BCI-algebras from the point of view of logic.Jacek K. Kabzinski - 1983 - Bulletin of the Section of Logic 12 (3):126-128.
    The following logics are the most noteworthy from the perspective of the calculus of combinators: the Hilbert’s positive implicational logic , the Church’s weak theory of implication , the BCK-logic, and the BCI-logic. Their significance is due to a certain correspondence between combinators and implicational formulas . The first three logics mentioned have been immensely investigated but it was not so in case of the remaining one. The BCI-logics was mentioned by A. N. Prior in the second edition of his (...)
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  • A Note on Monothetic BCI.Tomasz Kowalski & Sam Butchart - 2006 - Notre Dame Journal of Formal Logic 47 (4):541-544.
    In "Variations on a theme of Curry," Humberstone conjectured that a certain logic, intermediate between BCI and BCK, is none other than monothetic BCI—the smallest extension of BCI in which all theorems are provably equivalent. In this note, we present a proof of this conjecture.
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  • Variations on a Theme of Curry.Lloyd Humberstone - 2006 - Notre Dame Journal of Formal Logic 47 (1):101-131.
    After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which closes with (...)
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  • Extensions of Intuitionistic Logic Without the Deduction Theorem: Some Simple Examples.Lloyd Humberstone - 2006 - Reports on Mathematical Logic:45-82.
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  • Structural Completeness in Substructural Logics.J. S. Olson, J. G. Raftery & C. J. Van Alten - 2008 - Logic Journal of the IGPL 16 (5):453-495.
    Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.
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  • Corrigendum: Residuation in Commutative Ordered Monoids with Minimal Zero.J. Raftery & C. van Alten - 2005 - Reports on Mathematical Logic:133-135.
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  • Residuation in Commutative Ordered Monoids with Minimal Zero.James Raftery & Clint Van Alten - 2000 - Reports on Mathematical Logic:23-57.
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