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Skolemization for Substructural Logics

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


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.


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

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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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Correspondence to Denisa Diaconescu .

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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.

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