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  1. The Disjunction Property of Intermediate Propositional Logics.Alexander Chagrov & Michael Zakharyashchev - 1991 - Studia Logica 50 (2):189 - 216.
    This paper is a survey of results concerning the disjunction property, Halldén-completeness, and other related properties of intermediate prepositional logics and normal modal logics containing S4.
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  • From Interior Algebras to Unital ℓ-Groups: A Unifying Treatment of Modal Residuated Lattices.William Young - 2015 - Studia Logica 103 (2):265-286.
    Much work has been done on specific instances of residuated lattices with modal operators . In this paper, we develop a general framework that subsumes three important classes of modal residuated lattices: interior algebras, Abelian ℓ-groups with conuclei, and negative cones of ℓ-groups with nuclei. We then use this framework to obtain results about these three cases simultaneously. In particular, we show that a categorical equivalence exists in each of these cases. The approach used here emphasizes the role played by (...)
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  • Weak Arithmetical Interpretations for the Logic of Proofs.Roman Kuznets & Thomas Studer - 2016 - Logic Journal of the IGPL 24 (3):424-440.
  • A Semantic Hierarchy for Intuitionistic Logic.Guram Bezhanishvili & Wesley H. Holliday - 2019 - Indagationes Mathematicae 30 (3):403-469.
    Brouwer's views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke (...)
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  • Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic.Corey Edward Mulvihill - 2015 - Dissertation, University of Waterloo
    Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...)
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  • Multimodal and Intuitionistic Logics in Simple Type Theory.Christoph Benzmueller & Lawrence Paulson - 2010 - Logic Journal of the IGPL 18 (6):881-892.
    We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational investigations of various non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.
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  • The Genesis of Possible Worlds Semantics.B. Jack Copeland - 2002 - Journal of Philosophical Logic 31 (2):99-137.
    This article traces the development of possible worlds semantics through the work of: Wittgenstein, 1913-1921; Feys, 1924; McKinsey, 1945; Carnap, 1945-1947; McKinsey, Tarski and Jónsson, 1947-1952; von Wright, 1951; Becker, 1952; Prior, 1953-1954; Montague, 1955; Meredith and Prior, 1956; Geach, 1960; Smiley, 1955-1957; Kanger, 1957; Hintikka, 1957; Guillaume, 1958; Binkley, 1958; Bayart, 1958-1959; Drake, 1959-1961; Kripke, 1958-1965.
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  • Lattice Logic as a Fragment of Residuated Modal Logic.Chrysafis Hartonas - 2018 - Journal of Applied Non-Classical Logics 29 (2):152-170.
    ABSTRACTCorrespondence and Shalqvist theories for Modal Logics rely on the simple observation that a relational structure is at the same time the basis for a model of modal logic and for a model of first-order logic with a binary predicate for the accessibility relation. If the underlying set of the frame is split into two components,, and, then frames are at the same time the basis for models of non-distributive lattice logic and of two-sorted, residuated modal logic. This suggests that (...)
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  • Analogues of Bull’s Theorem for Hybrid Logic.Willem Conradie & Claudette Robinson - 2019 - Logic Journal of the IGPL 27 (3):281-313.
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  • Almost Structural Completeness; an Algebraic Approach.Wojciech Dzik & Michał M. Stronkowski - 2016 - Annals of Pure and Applied Logic 167 (7):525-556.
  • Natural Factors of the Medvedev Lattice Capturing IPC.Rutger Kuyper - 2014 - Archive for Mathematical Logic 53 (7-8):865-879.
  • Tarski's Theorem on Intuitionistic Logic, for Polyhedra.Nick Bezhanishvili, Vincenzo Marra, Daniel McNeill & Andrea Pedrini - 2018 - Annals of Pure and Applied Logic 169 (5):373-391.
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  • Explicit Provability and Constructive Semantics. [REVIEW]Jeremy Avigad - 2002 - Bulletin of Symbolic Logic 8 (3):432-432.
  • Proof Analysis in Intermediate Logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1-2):71-92.
    Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...)
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  • On Friedman's Problem in Mathematical Logic.Tadeusz Prucnal - 1978 - Bulletin of the Section of Logic 7 (3):137-140.
  • Natural Factors of the Muchnik Lattice Capturing IPC.Rutger Kuyper - 2013 - Annals of Pure and Applied Logic 164 (10):1025-1036.
    We give natural examples of factors of the Muchnik lattice which capture intuitionistic propositional logic , arising from the concepts of lowness, 1-genericity, hyperimmune-freeness and computable traceability. This provides a purely computational semantics for IPC.
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  • Fatal Heyting Algebras and Forcing Persistent Sentences.Leo Esakia & Benedikt Löwe - 2012 - Studia Logica 100 (1-2):163-173.
    Hamkins and Löwe proved that the modal logic of forcing is S4.2 . In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra H ZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
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  • Intermediate Logics and Factors of the Medvedev Lattice.Andrea Sorbi & Sebastiaan A. Terwijn - 2008 - Annals of Pure and Applied Logic 155 (2):69-85.
    We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.
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  • Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤Ω.Richard Zach - 2004 - Journal of Philosophical Logic 33 (2):155-164.
    Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics are decidable. (...)
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  • Erdős Graphs Resolve Fine's Canonicity Problem.Robert Goldblatt, Ian Hodkinson & Yde Venema - 2004 - Bulletin of Symbolic Logic 10 (2):186-208.
    We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...)
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  • Frontiers of Conditional Logic.Yale Weiss - 2019 - Dissertation, The Graduate Center, City University of New York
    Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...)
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  • Hedges: A Study in Meaning Criteria and the Logic of Fuzzy Concepts. [REVIEW]George Lakoff - 1973 - Journal of Philosophical Logic 2 (4):458 - 508.
  • Spatial Logic of Tangled Closure Operators and Modal Mu-Calculus.Robert Goldblatt & Ian Hodkinson - 2017 - Annals of Pure and Applied Logic 168 (5):1032-1090.
  • On an Intuitionistic Logic for Pragmatics.Gianluigi Bellin, Massimiliano Carrara & Daniele Chiffi - 2018 - Journal of Logic and Computation 50 (28):935–966..
    We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justi cations and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the (...)
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  • Willem Blok and Modal Logic.W. Rautenberg, M. Zakharyaschev & F. Wolter - 2006 - Studia Logica 83 (1-3):15-30.
    We present our personal view on W.J. Blok's contribution to modal logic.
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  • Autoreferential Semantics for Many-Valued Modal Logics.Zoran Majkic - 2008 - Journal of Applied Non-Classical Logics 18 (1):79-125.
    In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices is based on relative pseudo-complements as in intuitionistic logic. (...)
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  • Foreword.Philippe Balbiani - 2002 - Journal of Applied Non-Classical Logics 12 (3-4):317-318.
  • The Defective Conditional in Mathematics.Mathieu Vidal - 2014 - Journal of Applied Non-Classical Logics 24 (1-2):169-179.
    This article focuses on defective conditionals ? namely indicative conditionals whose antecedents are false and whose truth-values therefore cannot be determined. The problem is to decide which formal connective can adequately represent this usage. Classical logic renders defective conditionals true whereas traditional mathematics dismisses them as irrelevant. This difference in treatment entails that, at the propositional level, classical logic validates some sentences that are intuitively false in plane geometry. With two proofs, I show that the same flaw is shared by (...)
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  • A Modal Theorem-Preserving Translation of a Class of Three-Valued Logics of Incomplete Information.D. Ciucci & D. Dubois - 2013 - Journal of Applied Non-Classical Logics 23 (4):321-352.
    There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...)
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  • Realization of Intuitionistic Logic by Proof Polynomials.Sergei N. Artemov - 1999 - Journal of Applied Non-Classical Logics 9 (2-3):285-301.
    ABSTRACT In 1933 Gödel introduced an axiomatic system, currently known as S4, for a logic of an absolute provability, i.e. not depending on the formalism chosen ([God 33]). The problem of finding a fair provability model for S4 was left open. The famous formal provability predicate which first appeared in the Gödel Incompleteness Theorem does not do this job: the logic of formal provability is not compatible with S4. As was discovered in [Art 95], this defect of the formal provability (...)
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  • A Strange Remark Attributed to Gödel.Lloyd Humberstone - 2003 - History and Philosophy of Logic 24 (1):39-44.
    We assemble material from the literature on matrix methodology for sentential logic?without claiming to present any new logical results?in order to show that Gödel once made (or at least, is quoted as having made) an uncharacteristically ill-considered remark in this area.
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  • Basic Intuitionistic Conditional Logic.Yale Weiss - 2019 - Journal of Philosophical Logic 48 (3):447-469.
    Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas’ important classical system CK. I give ICK both worlds (...)
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  • Explicit Provability and Constructive Semantics.Sergei N. Artemov - 2001 - Bulletin of Symbolic Logic 7 (1):1-36.
    In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...)
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  • Conservative Translations.Hércules A. Feitosa & Itala M. Loffredo D'Ottaviano - 2001 - Annals of Pure and Applied Logic 108 (1-3):205-227.
    In this paper we introduce the concept of conservative translation between logics. We present some necessary and sufficient conditions for a translation to be conservative and study some general properties of logical systems, these properties being characterized by the existence of conservative translations between the systems. We prove that the class constituted by logics and conservative translations between them determines a co-complete subcategory of the bi-complete category constituted by logics and translations.
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  • Problems of Substitution and Admissibility in the Modal System Grz and in Intuitionistic Propositional Calculus.V. V. Rybakov - 1990 - Annals of Pure and Applied Logic 50 (1):71-106.
    Questions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra imageω extended in signature by adding constants for free generators. As corollaries we obtain: there exists an algorithm for the recognition of admissibility of rules with parameters in the modal system Grz, the substitution problem for Grz and for (...)
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  • Completeness Results for Intuitionistic and Modal Logic in a Categorical Setting.M. Makkai & G. E. Reyes - 1995 - Annals of Pure and Applied Logic 72 (1):25-101.
    Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere . The completeness theorems are used to conclude results asserting that certain (...)
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  • Characterization of Classes of Frames in Modal Language.Kazimierz Trzęsicki - 2012 - Studies in Logic, Grammar and Rhetoric 27 (40).
    No categories
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  • Epistemic and Intuitionistic Formal Systems.R. C. Flagg & H. Friedman - 1986 - Annals of Pure and Applied Logic 32 (1):53-60.
  • Duality Results for Residuated Lattices.Chrysafis Hartonas - 2019 - Logica Universalis 13 (1):77-99.
    We present dualities for implicative and residuated lattices. In combination with our recent article on a discrete duality for lattices with unary modal operators, the present article contributes in filling in a gap in the development of Orłowska and Rewitzky’s research program of discrete dualities, which seemed to have stumbled on the case of non-distributive lattices with operators. We discuss dualities via truth, which are essential in relating the non-distributive logic of two-sorted frames with their sorted, residuated modal logic, as (...)
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  • Intuitionistic Logic and Modality Via Topology.Leo Esakia - 2004 - Annals of Pure and Applied Logic 127 (1-3):155-170.
    In the pioneering article and two papers, written jointly with McKinsey, Tarski developed the so-called algebraic and topological frameworks for the Intuitionistic Logic and the Lewis modal system. In this paper, we present an outline of modern systems with a topological tinge. We consider topological interpretation of basic systems GL and G of the provability logic in terms of the Cantor derivative and the Hausdorff residue.
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  • From the Knowability Paradox to the Existence of Proofs.W. Dean & H. Kurokawa - 2010 - Synthese 176 (2):177 - 225.
    The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed (...)
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  • Modal Multilattice Logic.Norihiro Kamide & Yaroslav Shramko - 2017 - Logica Universalis 11 (3):317-343.
    A modal extension of multilattice logic, called modal multilattice logic, is introduced as a Gentzen-type sequent calculus \. Theorems for embedding \ into a Gentzen-type sequent calculus S4C and vice versa are proved. The cut-elimination theorem for \ is shown. A Kripke semantics for \ is introduced, and the completeness theorem with respect to this semantics is proved. Moreover, the duality principle is proved as a characteristic property of \.
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  • Bivalence: Meaning Theory Vs Metaphysics.Peter Pagin - 1998 - Theoria 64 (2-3):157-186.
    This paper is an attack on the Dummett-Prawitz view that the principle of bivalence has a crucial double significance, metaphysical and meaning theoretical. On the one hand it is said that holding bivalence valid is what characterizes a realistic view, i.e. a view in metaphysics, and on the other hand it is said that there are meaning theoretical arguments against its acceptability. I argue that these two aspects are incompatible. If the failure of validity of bivalence depends on properties of (...)
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  • Interpretations of Intuitionist Logic in Non-Normal Modal Logics.Colin Oakes - 1999 - Journal of Philosophical Logic 28 (1):47-60.
    Historically, it was the interpretations of intuitionist logic in the modal logic S4 that inspired the standard Kripke semantics for intuitionist logic. The inspiration of this paper is the interpretation of intuitionist logic in the non-normal modal logic S3: an S3 model structure can be 'looked at' as an intuitionist model structure and the semantics for S3 can be 'cashed in' to obtain a non-normal semantics for intuitionist propositional logic. This non-normal semantics is then extended to intuitionist quantificational logic.
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  • Provability as a Deontic Notion.Charles F. Kielkopf - 1971 - Theory and Decision 2 (1):1-15.
  • A Pragmatic Interpretation of Intuitionistic Propositional Logic.Carlo Dalla Pozza & Claudio Garola - 1995 - Erkenntnis 43 (1):81-109.
    We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with atruth value defined classically, (...)
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  • On Non-Self-Referential Fragments of Modal Logics.Junhua Yu - 2017 - Annals of Pure and Applied Logic 168 (4):776-803.
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  • Two Classes of Intermediate Propositional Logics Without Disjunction Property.Fabio Bellissima - 1989 - Archive for Mathematical Logic 28 (1):23-33.
  • Carnapian Extensions of S.Herbert E. Hendry & M. L. Pokriefka - 1985 - Journal of Philosophical Logic 14 (2):111 - 128.
  • Unified Quantum Logic.Mladen Pavičić - 1989 - Foundations of Physics 19 (8):999-1016.
    Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.
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