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  1. Proof Theory and Algebra in Logic.Hiroakira Ono - 2019 - Singapore: Springer Singapore.
    This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...)
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  • Constructive Logic with Strong Negation is a Substructural Logic. I.Matthew Spinks & Robert Veroff - 2008 - Studia Logica 88 (3):325-348.
    The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. (...)
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  • Constructive Logic with Strong Negation is a Substructural Logic. II.M. Spinks & R. Veroff - 2008 - Studia Logica 89 (3):401-425.
    The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew. The main result of Part I of this series [41] shows that the equivalent variety semantics of N and the equivalent variety semantics of NFL ew are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive (...)
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  • Subdirectly Irreducible Residuated Semilattices and Positive Universal Classes.Jeffrey S. Olson - 2006 - Studia Logica 83 (1-3):393-406.
    CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)k ≤ (x ⋀ e)k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k).
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  • Nelson’s logic ????Thiago Nascimento, Umberto Rivieccio, João Marcos & Matthew Spinks - 2020 - Logic Journal of the IGPL 28 (6):1182-1206.
    Besides the better-known Nelson logic and paraconsistent Nelson logic, in 1959 David Nelson introduced, with motivations of realizability and constructibility, a logic called $\mathcal{S}$. The logic $\mathcal{S}$ was originally presented by means of a calculus with infinitely many rule schemata and no semantics. We look here at the propositional fragment of $\mathcal{S}$, showing that it is algebraizable, in the sense of Blok and Pigozzi, with respect to a variety of three-potent involutive residuated lattices. We thus introduce the first known algebraic (...)
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  • Stone duality for lattice expansions.Chrysafis Hartonas - 2018 - Logic Journal of the IGPL 26 (5):475-504.
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  • Modal translation of substructural logics.Chrysafis Hartonas - 2020 - Journal of Applied Non-Classical Logics 30 (1):16-49.
    In an article dating back in 1992, Kosta Došen initiated a project of modal translations in substructural logics, aiming at generalising the well-known Gödel–McKinsey–Tarski translation of intuitionistic logic into S4. Došen's translation worked well for (variants of) BCI and stronger systems (BCW, BCK), but not for systems below BCI. Dropping structural rules results in logic systems without distribution. In this article, we show, via translation, that every substructural (indeed, every non-distributive) logic is a fragment of a corresponding sorted, residuated (multi) (...)
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  • Glivenko theorems and negative translations in substructural predicate logics.Hadi Farahani & Hiroakira Ono - 2012 - Archive for Mathematical Logic 51 (7-8):695-707.
    Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFLe. It is shown that there exists the weakest logic over QFLe among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied by using (...)
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