4,548 research outputs found

    Intuitionistic Completeness of First-Order Logic

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    We establish completeness for intuitionistic first-order logic, iFOL, showing that is a formula is provable if and only if it is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the intended semantics of iFOL. Our proof is intuitionistic and provides an effective procedure Prf that converts uniform evidence into a formal first-order proof. We have implemented Prf . Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuitionistically valid, but not conversely. Our strongest result requires the Fan Theorem; it can also be proved classically by showing that Prf terminates using K¨onig’s Theorem. The fundamental idea behind our completeness theorem is that a single evidence term evd witnesses the uniform validity of a minimal logic formula F. Finding even one uniform realizer guarantees validity because Prf (F, evd) builds a first-order proof of F, establishing its uniform validity and providing a purely logical normalized realizer. We establish completeness for iFOL as follows. Friedman showed that iFOL can be embedded in minimal logic (mFOL). By his transformation, mapping formula A to F r(A). If A is uniformly valid, then so is F r(A), and by our Basic Completeness result, we can find a proof of F r(A) in minimal logic. Then we prove A from F r(A) in intuitionistic logic by a proof procedure fixed in advance. Our result resolves an open question posed by Beth in 1947

    Intuitionistic completeness for first order classical logic

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    In the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic. All results included here, but a part of the proposition (a) below, are new. This work is, ideally, a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logi

    Simple proof of the completeness theorem for second order classical and intuitionictic logic by reduction to first-order mono-sorted logic

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    International audienceWe present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic

    Kripke Semantics and Proof Systems for Combining Intuitionistic Logic and Classical Logic

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    International audienceWe combine intuitionistic logic and classical logic into a new, first-order logic called Polarized Intuitionistic Logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzen's intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girard's LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalin's multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable

    Sequent calculi and decidability for intuitionistic hybrid logic

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    AbstractIn this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability

    Categories with families and first-order logic with dependent sorts

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    First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-L\"of type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.Comment: 83 page

    Kripke Models for Classical Logic

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    We introduce a notion of Kripke model for classical logic for which we constructively prove soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications
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