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Forcing in Łukasiewicz Predicate Logic

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Abstract

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard models of Ł∀.

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References

  • Barwise J., Robinson A.: ‘Completing theories by forcing’. Annals of Mathematical logic 2, 119–142 (1970)

    Article  Google Scholar 

  • Belluce P., Di Nola A.: ‘The MV-algebra of first order Łukasiewicz logic’. Tatra Mountains Mathematicals Publications 27, 7–22 (2003)

    Google Scholar 

  • Belluce P., Di Nola A.: ‘Simplicial structures in MV-algebras and logic’.. Journal of Symbolic Logic 72(2), 584–600 (2007)

    Article  Google Scholar 

  • Blok,W. J., and D. Pigozzi, ‘Algebraizable logics’, Memoirs of American Mathematics Society 396(77), 1989.

  • Chang, C. C., ‘Algebraic analysis of many valued logic’, Trans. Amer. Math. Soc 88:467–490, 1958.

    Article  Google Scholar 

  • Chang C.C.: ‘A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society 93(1), 74–80 (1959)

    Google Scholar 

  • Chang, C. C., ‘The axiom of comprehension in infinite valued logic’, Math. Scand. 93:9–30, 1963.

    Article  Google Scholar 

  • Chang, C. C., ‘Omitting types of prenex formulas’, Journal of Symbolic Logic 32:61–74, 1967.

    Article  Google Scholar 

  • Cignoli, R., I.M.L. D’Ottaviano,and D. Mundici, Algebraic Foundations of Many-valued Reasoning, volume 7 of Trends in Logic, Studia Logica Library, Kluwer Academic, 2000.

  • Cintula, P., P. Hájek, ‘On theories and models in fuzzy predicate logics’, Journal of Symbolic Logic 71(3):863–880, 2006.

    Article  Google Scholar 

  • Fenstad, J. E., ‘On the consistency of the axiom of comprehension in the Łukasiewicz infinite valued logic’, Math. Scand. 14:65–74, 1964.

    Article  Google Scholar 

  • Hájek,P., Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic, Studia Logica Library, Kluwer Academic, Berlin, 1998.

  • Hájek P.: ‘On arithmetic in the Cantor-Łukasiewicz fuzzy set theory’. Archive for Mathematical Logic 44, 763–782 (2005)

    Article  Google Scholar 

  • Hájek, P., and Z. Haniková, ‘A set theory within fuzzy logic’, in Proceedings of 31st IEEE ISMVL, Warsaw, 2001, pp. 319–324.

  • Hájek, P., and Z. Haniková, ‘A development of set theory in fuzzy logic’, in M. Fitting and E. Orłowska (eds.), Beyond Two: Theory and Applications of Multiple-Valued Logic. Physica-Verlag, Heidelberg, 2003, pp. 273–285.

  • Hájek, P., J. Paris, and J.C. Shepherdson, ‘Rational Pavelka logic is a conservative extension of Łukasiewicz logic’, Journal of Symbolic Logic 65(2):669–682, 2000.

    Article  Google Scholar 

  • Hirschfeld, J., and W. H. Wheeler, Forcing, arithmetic, division rings, Lectures Notes in Mathematics, Springer, 1975.

  • Keisler, H. J., ‘Forcing and the omitting types theories’, in Studies in Model Theory, volume 8 of MAA Studies in Math, Buffalo, N.Y., 1973, pp. 96–133.

  • Keisler, H. J., ‘Forcing in model theory’, Journal of Symbolic Logic 40(4):633–634, December 1975.

    Article  Google Scholar 

  • Lablaniquie, J.-C., ‘Forcing infini et theoreme d’omission des types de Chang’, Ann. Sci. Univ. Clermont, Ser. Math. 16:27–31, 1978.

    Google Scholar 

  • Łukasiewicz, J., and A. Tarski, ‘Untersuchungenüber den Aussagenkalkül’, Comptes Rendus de la Societe de Sciences et des Letters de Varsovie iii(23):1–21, 1930.

  • MCNaughton R.: ‘A theorem about infinite-valued sentential logic’. Journal of Symbolic Logic 16, 1–13 (1951)

    Article  Google Scholar 

  • Mundici D.: ‘Interpretation of AFC? algebras in Łukasiewicz sentential calculus’. Journal of Functional Analysis 65, 15–63 (1986)

    Article  Google Scholar 

  • Murinová, P., and V. Novák, ‘Omitting types in fuzzy predicate logic’, in Proc. of 26th Linz Seminar on Fuzzy Set Theory, 2007.

  • Ragaz, M. E., Arithmetische Klassification von Formelnmenge der unendlichwertigen Logik, PhD thesis, ETH Zrich, 1981.

  • Robinson, A., ‘Forcing in model theory’, in Actes du Congr‘es International des Math´ematiciens, Nice, volume 1, Gauthier-Villars, Paris, 1970, pp. 145–150.

  • Robinson, A., ‘Infinite forcing in model theory’, in Proceedings of the Second Scandinavian Logic Symposium (Oslo, 1970), volume 63 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971, pp. 317–340.

  • Rose, A., and J. B. Rosser, ‘Fragments of many-valued statement calculi’, Transactions of the American Mathematical Society 87(1):1–53, January 1958.

    Article  Google Scholar 

  • Scarpellini B.: ‘Die Nichtaxiomatisierbarkeit des unendlichwertigen Pradikatenkalkuls von Łukasiewicz’. Journal of Symbolic Logic 27(2), 159–170 (1962)

    Article  Google Scholar 

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Authors and Affiliations

  1. Dept of Mathematics and Computer Science, University of Salerno, Via Ponte don Melillo, 84084, Fisciano (SA), Italy

    Antonio Di Nola & Luca Spada

  2. Faculty of Mathematics and Computer Science, Str. Academiei nr.14, sector 1, 010014, Bucuresti, Romania

    George Georgescu

Authors
  1. Antonio Di Nola
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  2. George Georgescu
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  3. Luca Spada
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Correspondence to Luca Spada.

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Nola, A.D., Georgescu, G. & Spada, L. Forcing in Łukasiewicz Predicate Logic. Stud Logica 89, 111–145 (2008). https://doi.org/10.1007/s11225-008-9120-y

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