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.
Similar content being viewed by others
References
Andréka, H., Ferenczi, M., & Németi, I. (2012). Cylindric-like algebras and algebraic logic., Bolyai Society Mathematical Studies New York: Springer.
Davis, M. (2005). Applied nonstandard analysis. New York: Dover Publ.
Everett, C. J., & Ulam, S. (1946). Projective algebras. American Journal of Mathematics, 68, 17–28.
Fajardo, S., & Keisler, H. J. (1996). Existence theorems in probability theory. Advanced Mathematics, 120(2), 191–257.
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.
Ferenczi, M. (1983). Measures on cylindric algebras. Acta Mathematica Hungaric, 42, 3–17.
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.
Ferenczi, M. (2009). Non-standard stochastics with a first order algebraization. Studia Logica, 95, 345–354.
Ferenczi, M. (2012). The polyadic generalization of the Boolean axiomatization of fields of sets. Transaction of American Mathematical Society, 364(2), 867–886.
Gaifman, H. (1964). Concerning measures in first order calculi. The Israel Journal of Mathematics, 2(1), 1–18.
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.
Goldblatt, R. (1998). Lectures on the hyperreals. New York: Springer.
Hacking, I. (2001). An introduction to probability and inductive logic. Cambridge: Cambridge University Press.
Henkin, L., Monk, J. D., Tarski, A., Andréka, H., & Németi, I. (1981). Cylindric set algebras., Lectures Notes in Mathematics 883 New York: Springer.
Henkin, L., Monk, J. D., & Tarski, A. (1985). Cylindric algebras. Amsterdam: North-Holland Publishing Co.
Horn, A., & Tarski, A. (1948). Measures in Boolean algebras. Transactions of the American Mathematical Society, 64, 464–497.
Łoś, J. (1955). On the axiomatic treatment of probability. Colloquium Mathematicum, 3, 125–137.
Manzano, M. (1996). Extensions of first-order logic. Cambridge: Cambridge University Press.
Meier, M. (2012). An infinitary probability logic for type spaces. The Israel Journal of Mathematics, 192(1), 1–58.
Milosevic, M., & Ognjanovic, Z. (2012). A first order conditional probability logic. Logic Journal of the IGPL, 20(1), 235–253.
Monk, J. D. (1976). Mathematical logic. New York: Springer.
Raskovic, M., & Djordevic, R. S. (2000). Cylindric probability algebras. Publications de l’Institut Mathématique (Beograd), 68(82), 20–36.
Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland Publishing Co.
Sayed-Ahmed, T. (2005). Algebraic logic, where does it stand today? Bulletin of Symbolic Logic, 11(4), 465–516.
Scott, D., & Krauss, P. (1966). Assigning probabilities to logical formulas. Aspects of inductive logic (pp. 219–264). Amsterdam: North-Holland Publishing Co.
Vladimirov, D. A. (2002). Boolean algebras in analysis. Dordrecht: Kluwer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0444-z