118 research outputs found

    Falsity Conditions for IF-Sentences

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    We give a procedure to obtain falsity conditions for IF-sentences, using Skolemization. The expressive power of an IF-sentence can then be strongly captured by a pair of Σ11-sentences. A result from [Burgess 2003] shows that, conversely, any pair of incompatible Σ11-sentences corresponds with an IF-sentence.In the second part, we reflect on the influence of the order of the steps (inside-out versus outside-in) in the Skolemization procedures for IF-logic. We also reflect on the nature of game theoretical negation

    A Proof-Theoretic Approach to Certifying Skolemization

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    International audienceWhen presented with a formula to prove, most theorem provers for classical first-order logic process that formula following several steps, one of which is commonly called skolemization. That process eliminates quantifier alternation within formulas by extending the language of the underlying logic with new Skolem functions and by instantiating certain quantifiers with terms built using Skolem functions. In this paper, we address the problem of checking (i.e., certifying) proof evidence that involves Skolem terms. Our goal is to do such certification without using the mathematical concepts of model-theoretic semantics (i.e., preservation of satisfiability) and choice principles (i.e., epsilon terms). Instead , our proof checking kernel is an implementation of Gentzen's sequent calculus, which directly supports quanti-fier alternation by using eigenvariables. We shall describe deskolemization as a mapping from client-side terms, used in proofs generated by theorem provers, into kernel-side terms, used within our proof checking kernel. This mapping which associates skolemized terms to eigenvariables relies on using outer skolemization. We also point out that the removal of Skolem terms from a proof is influenced by the polarities given to propositional connectives

    Correspondences between Classical, Intuitionistic and Uniform Provability

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    Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that embodies a special form of goal-directedness. We determine, first, the circumstances in which the former relations imply the latter. Using this result, we identify the richest versions of the so-called abstract logic programming languages in classical and intuitionistic logic. We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved. Our focus here is on understanding the situations in which this reduction is achieved. However, our discussions indicate the structure of a proof procedure based on the reduction, a matter also considered explicitly elsewhere.Comment: 31 page

    About proof-search in intuitionistic natural deduction calculus using partial Skolemization

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    In this paper, automated proof search in single-conclusion sequential variant of intuitionistic and minimal predicate calculus is considered. In this algorithm, meta-variables and partial Skolemization are used. Theorems of soundness and completeness for the considered algorithm are proved. © Published under licence by IOP Publishing Ltd

    Hilbert's epsilon as an Operator of Indefinite Committed Choice

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    Paul Bernays and David Hilbert carefully avoided overspecification of Hilbert's epsilon-operator and axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the epsilon-operator underspecified. In the meanwhile, there have been several suggestions for semantics of the epsilon as a choice operator. After reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports proof search optimally and is natural in the sense that it does not only mirror some cases of referential interpretation of indefinite articles in natural language, but may also contribute to philosophy of language. Finally, we ask the question whether our epsilon within our free-variable framework can serve as a paradigm useful in the specification and computation of semantics of discourses in natural language.Comment: ii + 73 pages. arXiv admin note: substantial text overlap with arXiv:1104.244

    Representing scope in intuitionistic deductions

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    AbstractIntuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever — rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions
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