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Probabilities defined on standard and non-standard cylindric set algebras

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Abstract

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

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References

  • Andréka, H., Ferenczi, M., & Németi, I. (2012). Cylindric-like algebras and algebraic logic., Bolyai Society Mathematical Studies New York: Springer.

    Google Scholar 

  • Davis, M. (2005). Applied nonstandard analysis. New York: Dover Publ.

    Google Scholar 

  • Everett, C. J., & Ulam, S. (1946). Projective algebras. American Journal of Mathematics, 68, 17–28.

    Article  Google Scholar 

  • Fajardo, S., & Keisler, H. J. (1996). Existence theorems in probability theory. Advanced Mathematics, 120(2), 191–257.

    Article  Google Scholar 

  • Fenstad, J. E. (1976). Representations of probabilities defined on first order languages. In Jn Crossley (Ed.), Sets, models and recursion theory (pp. 156–172). North Holland: North-Holland Publishing Co.

    Google Scholar 

  • Ferenczi, M. (1983). Measures on cylindric algebras. Acta Mathematica Hungaric, 42, 3–17.

    Article  Google Scholar 

  • Ferenczi, M. (1991). Measures on free products of formula algebras and the analogies with homomorphisms. Algebraic Logic (Proc. Conf. Budapest) Coll. Math. Soc. J. Bolyai, (pp. 173–181). North Holland.

  • Ferenczi, M. (2005). Probabilities on first order models. Publications de l’Institut Mathematique, 78(92), 107–115.

    Article  Google Scholar 

  • Ferenczi, M. (2009). Non-standard stochastics with a first order algebraization. Studia Logica, 95, 345–354.

    Article  Google Scholar 

  • Ferenczi, M. (2012). The polyadic generalization of the Boolean axiomatization of fields of sets. Transaction of American Mathematical Society, 364(2), 867–886.

    Article  Google Scholar 

  • Gaifman, H. (1964). Concerning measures in first order calculi. The Israel Journal of Mathematics, 2(1), 1–18.

    Article  Google Scholar 

  • Georgescu, G. (1978). Extension of probabilities defined on polyadic algebras. Bulletin Mathématique de la Société des Sciences Mathématique de la République Socialiste Roumanie, 22(70), 15–26.

    Google Scholar 

  • Goldblatt, R. (1998). Lectures on the hyperreals. New York: Springer.

    Book  Google Scholar 

  • Hacking, I. (2001). An introduction to probability and inductive logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Henkin, L., Monk, J. D., Tarski, A., Andréka, H., & Németi, I. (1981). Cylindric set algebras., Lectures Notes in Mathematics 883 New York: Springer.

    Book  Google Scholar 

  • Henkin, L., Monk, J. D., & Tarski, A. (1985). Cylindric algebras. Amsterdam: North-Holland Publishing Co.

    Google Scholar 

  • Horn, A., & Tarski, A. (1948). Measures in Boolean algebras. Transactions of the American Mathematical Society, 64, 464–497.

    Article  Google Scholar 

  • Łoś, J. (1955). On the axiomatic treatment of probability. Colloquium Mathematicum, 3, 125–137.

    Article  Google Scholar 

  • Manzano, M. (1996). Extensions of first-order logic. Cambridge: Cambridge University Press.

    Google Scholar 

  • Meier, M. (2012). An infinitary probability logic for type spaces. The Israel Journal of Mathematics, 192(1), 1–58.

    Article  Google Scholar 

  • Milosevic, M., & Ognjanovic, Z. (2012). A first order conditional probability logic. Logic Journal of the IGPL, 20(1), 235–253.

    Article  Google Scholar 

  • Monk, J. D. (1976). Mathematical logic. New York: Springer.

    Book  Google Scholar 

  • Raskovic, M., & Djordevic, R. S. (2000). Cylindric probability algebras. Publications de l’Institut Mathématique (Beograd), 68(82), 20–36.

    Google Scholar 

  • Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.

    Google Scholar 

  • Sayed-Ahmed, T. (2005). Algebraic logic, where does it stand today? Bulletin of Symbolic Logic, 11(4), 465–516.

    Article  Google Scholar 

  • Scott, D., & Krauss, P. (1966). Assigning probabilities to logical formulas. Aspects of inductive logic (pp. 219–264). Amsterdam: North-Holland Publishing Co.

    Chapter  Google Scholar 

  • Vladimirov, D. A. (2002). Boolean algebras in analysis. Dordrecht: Kluwer.

    Book  Google Scholar 

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  1. Department of Algebra, Budapest University of Technology, Egry J. u. 1, Budapest , 1111, Hungary

    Miklós Ferenczi

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  1. Miklós Ferenczi
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Correspondence to Miklós Ferenczi.

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Ferenczi, M. Probabilities defined on standard and non-standard cylindric set algebras. Synthese 192, 2025–2033 (2015). https://doi.org/10.1007/s11229-014-0444-z

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  • DOI: https://doi.org/10.1007/s11229-014-0444-z

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