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  1. The Algebraic Significance of Weak Excluded Middle Laws.Tomáš Lávička, Tommaso Moraschini & James G. Raftery - 2022 - Mathematical Logic Quarterly 68 (1):79-94.
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  • Selfextensional Logics with a Distributive Nearlattice Term.Luciano J. González - 2019 - Archive for Mathematical Logic 58 (1-2):219-243.
    We define when a ternary term m of an algebraic language \ is called a distributive nearlattice term -term) of a sentential logic \. Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a \-term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras associated with a selfextensional logic with a \-term is a variety, and we obtain (...)
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  • Extension Properties and Subdirect Representation in Abstract Algebraic Logic.Tomáš Lávička & Carles Noguera - 2018 - Studia Logica 106 (6):1065-1095.
    This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters (...)
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  • A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic.Carles Noguera & Tomáš Lávička - 2017 - Studia Logica 105 (3):521-551.
    In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively subdirectly irreducible models. We identify two syntactical notions formulated in terms of intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain (...)
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  • A Note on Natural Extensions in Abstract Algebraic Logic.Petr Cintula & Carles Noguera - 2015 - Studia Logica 103 (4):815-823.
    Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting in (...)
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  • Implicational Logics II: Additional Connectives and Characterizations of Semilinearity.Petr Cintula & Carles Noguera - 2016 - Archive for Mathematical Logic 55 (3-4):353-372.
    This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions (...)
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  • Implicational Logics III: Completeness Properties.Petr Cintula & Carles Noguera - 2018 - Archive for Mathematical Logic 57 (3-4):391-420.
    This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in (...)
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  • Nonassociative Substructural Logics and Their Semilinear Extensions: Axiomatization and Completeness Properties: Nonassociative Substructural Logics.Petr Cintula, Rostislav Horčík & Carles Noguera - 2013 - Review of Symbolic Logic 6 (3):394-423.
    Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form (...)
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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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  • Algebraic Proof Theory: Hypersequents and Hypercompletions.Agata Ciabattoni, Nikolaos Galatos & Kazushige Terui - 2017 - Annals of Pure and Applied Logic 168 (3):693-737.
  • A Study of Truth Predicates in Matrix Semantics.Tommaso Moraschini - 2018 - Review of Symbolic Logic 11 (4):780-804.
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  • A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics.Petr Cintula & Carles Noguera - 2015 - Journal of Symbolic Logic 80 (1):341-358.