Abstract
Herbrand and Skolemization theorems are obtained for a broad family of first-order substructural logics. These logics typically lack equivalent prenex forms, a deduction theorem, and reductions of semantic consequence to satisfiability. The Herbrand and Skolemization theorems therefore take various forms, applying either to the left or right of the consequence relation, and to restricted classes of formulas.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Baaz, M., Ciabattoni, A., Fermüller, C.: Herbrand theorem for prenex Gödel logic and its consequences for theorem proving. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 201–216. Springer, Heidelberg (2001)
Baaz, M., Iemhoff, R.: On Skolemization in constructive theories. Journal of Symbolic Logic 73(3), 969–998 (2008)
Baaz, M., Metcalfe, G.: Herbrand Theorems and Skolemization for Prenex Fuzzy Logics. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 22–31. Springer, Heidelberg (2008)
Baaz, M., Metcalfe, G.: Herbrand’s theorem, Skolemization, and proof systems for first-order Łukasiewicz logic. Journal of Logic and Computation 20(1), 35–54 (2010)
Buss, S. (ed.): Handbook of Proof Theory. Kluwer (1998)
Ciabattoni, A., Galatos, N., Terui, K.: Algebraic proof theory for substructural logics: Cut-elimination and completions. Annals of Pure and Applied Logic 163(3), 266–290 (2012)
Cintula, P., Hájek, P., Noguera, C. (eds.): Handbook of Mathematical Fuzzy Logic (in 2 volumes). College Publications, London (2011)
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, pp. 103–207. College Publications (2011)
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)
García-Cerdaña, À., Armengol, E., Esteva, F.: Fuzzy description logics and t-norm based fuzzy logics. International Journal of Approximate Reasoning 51(6), 632–655 (2010)
Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer, Dordrecht (1998)
Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets and Systems 154(1), 1–15 (2005)
McNaughton, R.: A theorem about infinite-valued sentential logic. Journal of Symbolic Logic 16(1), 1–13 (1951)
Meghini, C., Sebastiani, F., Straccia, U.: A model of multimedia information retrieval. Journal of the ACM 48(5), 909–970 (2001)
Metcalfe, G., Olivetti, N., Gabbay, D.M.: Proof Theory for Fuzzy Logics. Applied Logic Series, vol. 36. Springer (2008)
Minc, G.E.: The Skolem method in intuitionistic calculi. Proceedings of the Steklov Institute of Mathematics 121, 73–109 (1974)
Ono, H.: Algebraic semantics for predicate logics and their completeness. In: Orlowska, E. (ed.) Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pp. 637–650. Physica Verlag, Heidelberg (1999)
Ono, H.: Crawley completions of residuated lattices and algebraic completeness of substructural predicate logics. Studia Logica 100(1-2), 339–359 (2012)
Restall, G.: An Introduction to Substructural Logics. Routledge, New York (2000)
Terui, K.: Herbrand’s theorem via hypercanonical extensions (manuscript)
Vojtáš, P.: Fuzzy logic programming. Fuzzy Sets and Systems 124(3), 361–370 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cintula, P., Metcalfe, G. (2013). Herbrand Theorems for Substructural Logics. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2013. Lecture Notes in Computer Science, vol 8312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45221-5_39
Download citation
DOI: https://doi.org/10.1007/978-3-642-45221-5_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45220-8
Online ISBN: 978-3-642-45221-5
eBook Packages: Computer ScienceComputer Science (R0)