Skolemization for Substructural Logics
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 ) 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.
- 1.Baaz, M., Lemhoff, R.: Skolemization in intermediate logics with the finite model property. SubmittedGoogle Scholar
- 5.Cintula, P., Hájek, P., Noguera, C. (eds).: Handbook of Mathematical Fuzzy Logic (in 2 volumes), volume 37, 38 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London (2011)Google Scholar
- 7.Cintula, P., Noguera, C.: A general framework for mathematical fuzzy logic. In: Cintula, P., Hájek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic. vol. 1, vol. 37 of Studies in Logic, Mathematical Logic and Foundations, pp. 103–207. College Publications, London (2011)Google Scholar
- 9.Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Amsterdam (2007)Google Scholar
- 14.Metcalfe, G., Olivetti, N., Gabbay, D.M.: Proof Theory for Fuzzy Logics. vol. 36 of Applied Logic Series. Springer, Heidelberg (2008)Google Scholar