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  1. Harmonious Logic: Craig's Interpolation Theorem and Its Descendants.Solomon Feferman - 2008 - Synthese 164 (3):341 - 357.
    Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, (...)
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  • 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.
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  • An Institution-Independent Proof of Craig Interpolation Theorem.Răzvan Diaconescu - 2004 - Studia Logica 77 (1):59 - 79.
    We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic.We formalise Birkhoff-style axiomatizability within the general abstract model theoretic framework of institution theory by the novel concept of Birkhoff institution.
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  • The Road to Two Theorems of Logic.William Craig - 2008 - Synthese 164 (3):333 - 339.
    Work on how to axiomatize the subtheories of a first-order theory in which only a proper subset of their extra-logical vocabulary is being used led to a theorem on recursive axiomatizability and to an interpolation theorem for first-order logic. There were some fortuitous events and several logicians played a helpful role.
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  • Δ-Core Fuzzy Logics with Propositional Quantifiers, Quantifier Elimination and Uniform Craig Interpolation.Franco Montagna - 2012 - Studia Logica 100 (1-2):289-317.
    In this paper we investigate the connections between quantifier elimination, decidability and Uniform Craig Interpolation in Δ-core fuzzy logics added with propositional quantifiers. As a consequence, we are able to prove that several propositional fuzzy logics have a conservative extension which is a Δ-core fuzzy logic and has Uniform Craig Interpolation.
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  • Failure of Interpolation in Relevant Logics.Alasdair Urquhart - 1993 - Journal of Philosophical Logic 22 (5):449 - 479.
    Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.
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  • A Survey of Abstract Algebraic Logic.J. M. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1-2):13 - 97.
  • Sentential Logics and Maehara Interpolation Property.Janusz Czelakowski - 1985 - Studia Logica 44 (3):265 - 283.
    With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr * (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of (...)
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  • Repairing the Interpolation Theorem in Quantified Modal Logic.Carlos Areces, Patrick Blackburn & Maarten Marx - 2003 - Annals of Pure and Applied Logic 124 (1-3):287-299.
    Quantified hybrid logic is quantified modal logic extended with apparatus for naming states and asserting that a formula is true at a named state. While interpolation and Beth's definability theorem fail in a number of well-known quantified modal logics , their counterparts in quantified hybrid logic have these properties. These are special cases of the main result of the paper: the quantified hybrid logic of any class of frames definable in the bounded fragment of first-order logic has the interpolation property, (...)
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  • Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory.William Craig - 1957 - Journal of Symbolic Logic 22 (3):269-285.
  • Linear Reasoning. A New Form of the Herbrand-Gentzen Theorem.William Craig - 1957 - Journal of Symbolic Logic 22 (3):250-268.
  • Properties of Independently Axiomatizable Bimodal Logics.Marcus Kracht & Frank Wolter - 1991 - Journal of Symbolic Logic 56 (4):1469-1485.
  • Interpolants, Cut Elimination and Flow Graphs for the Propositional Calculus.Alessandra Carbone - 1997 - Annals of Pure and Applied Logic 83 (3):249-299.
    We analyse the structure of propositional proofs in the sequent calculus focusing on the well-known procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to understand why a tautology might be ‘hard to prove’. Given a proof we associate to it a logical graph tracing the flow of formulas in it . We show some general facts about logical graphs such as acyclicity of cut-free proofs and acyclicity of contraction-free proofs , and we give a (...)
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  • Interpolation Via Translations.João Rasga, Walter Carnielli & Cristina Sernadas - 2009 - Mathematical Logic Quarterly 55 (5):515-534.
    A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new translation between (...)
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  • Preservation of Craig Interpolation by the Product of Matrix Logics.C. Sernadas, J. Rasga & A. Sernadas - 2013 - Journal of Applied Logic 11 (3):328-349.
  • Craig Interpolation for Networks of Sentences.H. Jerome Keisler & Jeffrey M. Keisler - 2012 - Annals of Pure and Applied Logic 163 (9):1322-1344.
  • Interpolation in Practical Formal Development.J. Bicarregui, T. Dimitrakos, D. Gabbay & T. Maibaum - 2001 - Logic Journal of the IGPL 9 (2):231-244.
    Interpolation has become one of the standard properties that logicians investigate when designing a logic. In this paper, we provide strong evidence that the presence of interpolants is not only cogent for scientific reasoning but has also important practical implications in computer science. We illustrate that interpolation in general, and uniform splitting interpolants, in particular, play an important role in applications where formality and modularity are invoked. In recognition of the fact that common logical formalisms often lack uniform interpolants, we (...)
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  • A Survey of Abstract Algebraic Logic.J. Font, R. Jansana & D. Pigozzi - 2003 - Studia Logica 74 (1):13-97.
  • Harmonious Logic: Craig’s Interpolation Theorem and its Descendants.Solomon Feferman - 2008 - Synthese 164 (3):341-357.
    Though deceptively simple and plausible on the face of it, Craig's interpolation theorem has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation (...)
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  • Craig Interpolation for Semilinear Substructural Logics.Enrico Marchioni & George Metcalfe - 2012 - Mathematical Logic Quarterly 58 (6):468-481.
    The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R-mingle with unit” logic that have the Craig interpolation property. This latter characterization is obtained using a model-theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids (...)
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  • Some Failures of Interpolation in Modal Logic.George F. Schumm - 1986 - Notre Dame Journal of Formal Logic 27 (1):108-110.