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Logic for Programming, Artificial Intelligence, and Reasoning

LPAR 2015: Logic for Programming, Artificial Intelligence, and Reasoning pp 1–15Cite as

Skolemization for Substructural Logics

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9450)

Abstract

The usual Skolemization procedure, which removes strong quantifiers by introducing new function symbols, is in general unsound for first-order substructural logics defined based on classes of complete residuated lattices. However, it is shown here (following similar ideas of Baaz and Iemhoff for first-order intermediate logics in [1]) that first-order substructural logics with a semantics satisfying certain witnessing conditions admit a “parallel” Skolemization procedure where a strong quantifier is removed by introducing a finite disjunction or conjunction (as appropriate) of formulas with multiple new function symbols. These logics typically lack equivalent prenex forms. Also, semantic consequence does not in general reduce to satisfiability. The Skolemization theorems presented here therefore take various forms, applying to the left or right of the consequence relation, and to all formulas or only prenex formulas.

Keywords

  • Function Symbol
  • Residuated Lattice
  • Intuitionistic Logic
  • Predicate Symbol
  • Sequent Calculus

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

P. Cintula–Supported by RVO 67985807 and Czech Science Foundation GBP202/12/G061.

D. Diaconescu–Supported by Sciex grant 13.192.

G. Metcalfe–Supported by Swiss National Science Foundation grant 200021_146748.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic

    Petr Cintula

  2. Mathematical Institute, University of Bern, Bern, Switzerland

    Denisa Diaconescu & George Metcalfe

  3. Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

    Denisa Diaconescu

Authors
  1. Petr Cintula
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  2. Denisa Diaconescu
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  3. George Metcalfe
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Corresponding author

Correspondence to Denisa Diaconescu .

Editor information

Editors and Affiliations

  1. New York University, New York, New York, USA

    Martin Davis

  2. University of the South Pacific, Suva, Fiji

    Ansgar Fehnker

  3. Macquarie University, Sydney, New South Wales, Australia

    Annabelle McIver

  4. The University of Manchester, Manchester, United Kingdom

    Andrei Voronkov

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Cintula, P., Diaconescu, D., Metcalfe, G. (2015). Skolemization for Substructural Logics. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_1

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  • DOI: https://doi.org/10.1007/978-3-662-48899-7_1

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