Logic for Programming, Artificial Intelligence, and Reasoning

Logic for Programming, Artificial Intelligence, and Reasoning pp 1-15

Skolemization for Substructural Logics

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Petr Cintula
    • 1
  • Denisa Diaconescu
    • 2
    • 3
  • George Metcalfe
    • 2
  1. 1.Institute of Computer ScienceCzech Academy of SciencesPragueCzech Republic
  2. 2.Mathematical InstituteUniversity of BernBernSwitzerland
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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