Switch to: References

Add citations

You must login to add citations.
  1. Lyndon interpolation theorem of instantial neighborhood logic – constructively via a sequent calculus.Junhua Yu - 2020 - Annals of Pure and Applied Logic 171 (1):102721.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • The Automorphism Group of the Fraïssé Limit of Finite Heyting Algebras.Kentarô Yamamoto - 2023 - Journal of Symbolic Logic 88 (3):1310-1320.
    Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Substitutions of< i> Σ_< sub> 1< sup> 0-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. [REVIEW]Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1):227-271.
    This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  • Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic.Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1-3):227-271.
    This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   15 citations  
  • Uniform interpolation and compact congruences.Samuel J. van Gool, George Metcalfe & Constantine Tsinakis - 2017 - Annals of Pure and Applied Logic 168 (10):1927-1948.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  • The many faces of interpolation.Johan van Benthem - 2008 - Synthese 164 (3):451-460.
    We present a number of, somewhat unusual, ways of describing what Craig’s interpolation theorem achieves, and use them to identify some open problems and further directions.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • Interpolation property for bicartesian closed categories.Djordje Čubrić - 1994 - Archive for Mathematical Logic 33 (4):291-319.
    We show that proofs in the intuitionistic propositional logic factor through interpolants-in this way we prove a stronger interpolation property than the usual one which gives only the existence of interpolants.Translating that to categorical terms, we show that Pushouts (bipushouts) of bicartesian closed categories have the interpolation property (Theorem 3.2).
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Combining Intuitionistic and Classical Propositional Logic: Gentzenization and Craig Interpolation.Masanobu Toyooka & Katsuhiko Sano - forthcoming - Studia Logica:1-31.
    This paper studies a combined system of intuitionistic and classical propositional logic from proof-theoretic viewpoints. Based on the semantic treatment of Humberstone (J Philos Log 8:171–196, 1979) and del Cerro and Herzig (Frontiers of combining systems: FroCoS, Springer, 1996), a sequent calculus $$\textsf{G}(\textbf{C}+\textbf{J})$$ is proposed. An approximate idea of obtaining $$\textsf{G}(\textbf{C}+\textbf{J})$$ is adding rules for classical implication on top of the intuitionistic multi-succedent sequent calculus by Maehara (Nagoya Math J 7:45–64, 1954). However, in the semantic treatment, some formulas do not (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  • Mereotopology in 2nd-Order and Modal Extensions of Intuitionistic Propositional Logic.Paolo Torrini, John G. Stell & Brandon Bennett - 2002 - Journal of Applied Non-Classical Logics 12 (3-4):495-525.
    We show how mereotopological notions can be expressed by extending intuitionistic propositional logic with propositional quantification and a strong modal operator. We first prove completeness for the logics wrt Kripke models; then we trace the correspondence between Kripke models and topological spaces that have been enhanced with an explicit notion of expressible region. We show how some qualitative spatial notions can be expressed in topological terms. We use the semantical and topological results in order to show how in some extensions (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Pitts' Quantifiers Are Not Topological Quantification.Tomasz Połacik - 1998 - Notre Dame Journal of Formal Logic 39 (4):531-544.
    We show that Pitts' modeling of propositional quantification in intuitionistic logic (as the appropriate interpolants) does not coincide with the topological interpretation. This contrasts with the case of the monadic language and the interpretation over sufficiently regular topological spaces. We also point to the difference between the topological interpretation over sufficiently regular spaces and the interpretation of propositional quantifiers in Kripke models.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  • A Syntactic Embedding of Predicate Logic into Second-Order Propositional Logic.Morten H. Sørensen & Paweł Urzyczyn - 2010 - Notre Dame Journal of Formal Logic 51 (4):457-473.
    We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic propositional logic. We also show (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  • Second order propositional operators over Cantor space.Tomasz Połacik - 1994 - Studia Logica 53 (1):93 - 105.
  • Propositional Quantification in the Topological Semantics for S.Philip Kremer - 1997 - Notre Dame Journal of Formal Logic 38 (2):295-313.
    Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  • Description of all functions definable by formulæ of the 2nd order intuitionistic propositional calculus on some linear Heyting algebras.Dimitri Pataraia - 2006 - Journal of Applied Non-Classical Logics 16 (3-4):457-483.
    Explicit description of maps definable by formulæ of the second order intuitionistic propositional calculus is given on two classes of linear Heyting algebras—the dense ones and the ones which possess successors. As a consequence, it is shown that over these classes every formula is equivalent to a quantifier free formula in the dense case, and to a formula with quantifiers confined to the applications of the successor in the second case.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  • On flattening elimination rules.Grigory K. Olkhovikov & Peter Schroeder-Heister - 2014 - Review of Symbolic Logic 7 (1):60-72.
  • Model Completions for Universal Classes of Algebras: Necessary and Sufficient Conditions.George Metcalfe & Luca Reggio - 2023 - Journal of Symbolic Logic 88 (1):381-417.
    Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Uniform Lyndon interpolation property in propositional modal logics.Taishi Kurahashi - 2020 - Archive for Mathematical Logic 59 (5-6):659-678.
    We introduce and investigate the notion of uniform Lyndon interpolation property which is a strengthening of both uniform interpolation property and Lyndon interpolation property. We prove several propositional modal logics including \, \, \ and \ enjoy ULIP. Our proofs are modifications of Visser’s proofs of uniform interpolation property using layered bisimulations Gödel’96, logical foundations of mathematics, computer science and physics—Kurt Gödel’s legacy, Springer, Berlin, 1996). Also we give a new upper bound on the complexity of uniform interpolants for \ (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Uniform interpolation and coherence.Tomasz Kowalski & George Metcalfe - 2019 - Annals of Pure and Applied Logic 170 (7):825-841.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Logic-based ontology comparison and module extraction, with an application to DL-Lite.Roman Kontchakov, Frank Wolter & Michael Zakharyaschev - 2010 - Artificial Intelligence 174 (15):1093-1141.
  • Computing interpolants in implicational logics.Makoto Kanazawa - 2006 - Annals of Pure and Applied Logic 142 (1):125-201.
    I present a new syntactical method for proving the Interpolation Theorem for the implicational fragment of intuitionistic logic and its substructural subsystems. This method, like Prawitz’s, works on natural deductions rather than sequent derivations, and, unlike existing methods, always finds a ‘strongest’ interpolant under a certain restricted but reasonable notion of what counts as an ‘interpolant’.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Uniform interpolation and the existence of sequent calculi.Rosalie Iemhoff - 2019 - Annals of Pure and Applied Logic 170 (11):102711.
  • The G4i Analogue of a G3i Sequent Calculus.Rosalie Iemhoff - 2022 - Studia Logica 110 (6):1493-1506.
    This paper provides a method to obtain terminating analytic calculi for a large class of intuitionistic modal logics. For a given logic L with a cut-free calculus G that is an extension of G3ip the method produces a terminating analytic calculus that is an extension of G4ip and equivalent to G. G4ip was introduced by Roy Dyckhoff in 1992 as a terminating analogue of the calculus G3ip for intuitionistic propositional logic. Thus this paper can be viewed as an extension of (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  • Uniform interpolation and sequent calculi in modal logic.Rosalie Iemhoff - 2019 - Archive for Mathematical Logic 58 (1-2):155-181.
    A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic \. New is the result that \ has uniform interpolation. The results imply that for modal logics \ and \, which are known not to have uniform interpolation, certain sequent calculi cannot exist.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Doing logic by computer: interpolation in fragments of intuitionistic propositional logic.Lex Hendriks - 2000 - Annals of Pure and Applied Logic 104 (1-3):97-112.
    In this paper we study the interpolation property in fragments of intuitionistic and propositional logic, using both proof theoretic and semantic techniques. We will also sketch some computational methods, based on the semantical techniques introduced, to obtain counterexamples in fragment where interpolation does not hold.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Decidability of admissibility: On a problem by Friedman and its solution by Rybakov.Jeroen P. Goudsmit - 2021 - Bulletin of Symbolic Logic 27 (1):1-38.
    Rybakov proved that the admissible rules of $\mathsf {IPC}$ are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Undefinability of propositional quantifiers in the modal system S.Silvio Ghilardi & Marek Zawadowski - 1995 - Studia Logica 55 (2):259 - 271.
    We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   21 citations  
  • Model completions and r-Heyting categories.Silvio Ghilardi & Marek Zawadowski - 1997 - Annals of Pure and Applied Logic 88 (1):27-46.
    Under some assumptions on an equational theory S , we give a necessary and sufficient condition so that S admits a model completion. These assumptions are often met by the equational theories arising from logic. They say that the dual of the category of finitely presented S-algebras has some categorical stucture. The results of this paper combined with those of [7] show that all the 8 theories of amalgamable varieties of Heyting algebras [12] admit a model completion. Further applications to (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • An algebraic theory of normal forms.Silvio Ghilardi - 1995 - Annals of Pure and Applied Logic 71 (3):189-245.
    In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   18 citations  
  • On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond.Luck Darnière & Markus Junker - 2010 - Archive for Mathematical Logic 49 (7-8):743-771.
    We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  • Interpolation in non-classical logics.Giovanna D’Agostino - 2008 - Synthese 164 (3):421 - 435.
    We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  • Questions and Dependency in Intuitionistic Logic.Ivano Ciardelli, Rosalie Iemhoff & Fan Yang - 2020 - Notre Dame Journal of Formal Logic 61 (1):75-115.
    In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • Proof Theory for Positive Logic with Weak Negation.Marta Bílková & Almudena Colacito - 2020 - Studia Logica 108 (4):649-686.
    Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Uniform Interpolation and Propositional Quantifiers in Modal Logics.Marta Bílková - 2007 - Studia Logica 85 (1):1-31.
    We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   10 citations  
  • Extendible Formulas in Two Variables in Intuitionistic Logic.Nick Bezhanishvili & Dick Jongh - 2012 - Studia Logica 100 (1-2):61-89.
    We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n -universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Extendible Formulas in Two Variables in Intuitionistic Logic.Nick Bezhanishvili & Dick de Jongh - 2012 - Studia Logica 100 (1-2):61-89.
    We give alternative characterizations of exact, extendible and projective formulas in intuitionistic propositional calculus IPC in terms of n-universal models. From these characterizations we derive a new syntactic description of all extendible formulas of IPC in two variables. For the formulas in two variables we also give an alternative proof of Ghilardi’s theorem that every extendible formula is projective.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Interpolation in fuzzy logic.Matthias Baaz & Helmut Veith - 1999 - Archive for Mathematical Logic 38 (7):461-489.
    We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   13 citations  
  • Uniform interpolation in substructural logics.Majid Alizadeh, Farzaneh Derakhshan & Hiroakira Ono - 2014 - Review of Symbolic Logic 7 (3):455-483.
  • Remarks on uniform interpolation property.Majid Alizadeh - forthcoming - Logic Journal of the IGPL.
    A logic |$\mathcal{L}$| is said to satisfy the descending chain condition, DCC, if any descending chain of formulas in |$\mathcal{L}$| with ordering induced by |$\vdash _{\mathcal{L}};$| eventually stops. In this short note, we first establish a general theorem, which states that if a propositional logic |$\mathcal{L}$| satisfies both DCC and has the Craig Interpolation Property, CIP, then it satisfies the Uniform Interpolation Property, UIP, as well. As a result, by using the Nishimura lattice, we give a new simply proof of (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Interpolation Property on Visser's Formal Propositional Logic.Majid Alizadeh & Masoud Memarzadeh - 2022 - Bulletin of the Section of Logic 51 (3):297-316.
    In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  • Rules and Arithmetics.Albert Visser - 1999 - Notre Dame Journal of Formal Logic 40 (1):116-140.
    This paper is concerned with the logical structure of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   18 citations  
  • Lyndon’s interpolation property for the logic of strict implication.Narbe Aboolian & Majid Alizadeh - 2022 - Logic Journal of the IGPL 30 (1):34-70.
    The main result proves Lyndon’s and Craig’s interpolation properties for the logic of strict implication ${\textsf{F}}$, with a purely syntactical method. A cut-free G3-style sequent calculus $ {\textsf{GF}} $ and its single-succedent variant $ \textsf{GF}_{\textsf{s}} $ are introduced. $ {\textsf{GF}} $ can be extended to a G3-variant of the sequent calculus GBPC3 for Visser’s basic logic. Also a simple syntactic proof of known embedding result of $ {\textsf{F}} $ into $ {\textsf{K}} $ is provided. An extension of $ {\textsf{F}} $, (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  • Dag Prawitz on Proofs and Meaning.Heinrich Wansing (ed.) - 2015 - Cham, Switzerland: Springer.
    This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Necessity of Thought.Cesare Cozzo - 2015 - In Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning. Springer. pp. 101-20.
    The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...)
    Direct download  
     
    Export citation  
     
    Bookmark   6 citations  
  • Some Formal Semantics for Epistemic Modesty.Christopher Steinsvold - 2020 - Logic and Logical Philosophy 29 (3):381-413.
    Given the frequency of human error, it seems rational to believe that some of our own rational beliefs are false. This is the axiom of epistemic modesty. Unfortunately, using standard propositional quantification, and the usual relational semantics, this axiom is semantically inconsistent with a common logic for rational belief, namely KD45. Here we explore two alternative semantics for KD45 and the axiom of epistemic modesty. The first uses the usual relational semantics and bisimulation quantifiers. The second uses a topological semantics (...)
    Direct download  
     
    Export citation  
     
    Bookmark