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  1. Towards Incorporating Background Theories Into Quantifier Elimination.Andrzej Szalas - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):325-340.
    In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication and are not applicable to the first-order case. We make a shift to arbitrary relations “ordering” the underlying universe. This allows (...)
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  • A Dichotomy for Some Elementarily Generated Modal Logics.Stanislav Kikot - 2015 - Studia Logica 103 (5):1063-1093.
    In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
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  • Algorithmic Correspondence and Completeness in Modal Logic. V. Recursive Extensions of SQEMA.Willem Conradie, Valentin Goranko & Dimitar Vakarelov - 2010 - Journal of Applied Logic 8 (4):319-333.
    The previously introduced algorithm \sqema\ computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend \sqema\ with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points \mffo. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid mu-calculus. In particular, we prove that (...)
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  • Sahlqvist Theorems for Precontact Logics.Philippe Balbiani & Stanislav Kikot - 2012 - In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic. CSLI Publications. pp. 55-70.
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  • Semantic Characterization of Krancht Formulas.Stanislav Kikot - 2010 - In Lev Beklemishev, Valentin Goranko & Valentin Shehtman (eds.), Advances in Modal Logic, Volume 8. CSLI Publications. pp. 218-234.
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  • An Extension of Kracht's Theorem to Generalized Sahlqvist Formulas.Stanislav Kikot - 2009 - Journal of Applied Non-Classical Logics 19 (2):227-251.
    Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order properties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Kracht's theorem to the class of ‘generalized Sahlqvist formulas' introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.
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  • The Logic of Transitive and Dense Frames: From the Step-Frame Analysis to Full Cut-Elimination.S. Ghilardi & G. Mints - 2014 - Logic Journal of the IGPL 22 (4):585-596.
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  • Algorithmic Correspondence and Completeness in Modal Logic. IV. Semantic Extensions of SQEMA.Willem Conradie & Valentin Goranko - 2008 - Journal of Applied Non-Classical Logics 18 (2-3):175-211.
    In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of (...)
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  • Sahlqvist Correspondence for Modal Mu-Calculus.Johan van Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
    We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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  • Sahlqvist Correspondence for Modal Mu-Calculus.Johan Benthem, Nick Bezhanishvili & Ian Hodkinson - 2012 - Studia Logica 100 (1-2):31-60.
    We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
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  • Modal and Temporal Extensions of Non-Distributive Propositional Logics.Chrysafis Hartonas - 2016 - Logic Journal of the IGPL 24 (2):156-185.
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  • Modal Definability of First-Order Formulas with Free Variables and Query Answering.Stanislav Kikot & Evgeny Zolin - 2013 - Journal of Applied Logic 11 (2):190-216.
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  • On the Strength and Scope of DLS.Willem Conradie - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):279-296.
    We provide syntactic necessary and sufficient conditions on the formulae reducible by the second-order quantifier elimination algorithm DLS. It is shown that DLS is compete for all modal Sahlqvist and Inductive formulae, and that all modal formulae in a single propositional variable on which DLS succeeds are canonical.
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  • Minimal Predicates, Fixed-Points, and Definability.Johan van Benthem - 2005 - Journal of Symbolic Logic 70 (3):696-712.
    Minimal predicates P satisfying a given first-order description φ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ φ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal (...)
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  • Canonicity Results of Substructural and Lattice-Based Logics.Tomoyuki Suzuki - 2011 - Review of Symbolic Logic 4 (1):1-42.
    In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...)
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  • Technical Modal Logic.Marcus Kracht - 2011 - Philosophy Compass 6 (5):350-359.
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  • A Sahlqvist Theorem for Substructural Logic.Tomoyuki Suzuki - 2013 - Review of Symbolic Logic 6 (2):229-253.
    In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.
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  • The Bounded Proof Property Via Step Algebras and Step Frames.Nick Bezhanishvili & Silvio Ghilardi - 2014 - Annals of Pure and Applied Logic 165 (12):1832-1863.
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  • The Ackermann Approach for Modal Logic, Correspondence Theory and Second-Order Reduction.Renate A. Schmidt - 2012 - Journal of Applied Logic 10 (1):52-74.
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  • Algorithmic Correspondence and Canonicity for Distributive Modal Logic.Willem Conradie & Alessandra Palmigiano - 2012 - Annals of Pure and Applied Logic 163 (3):338-376.
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