A logical and algebraic treatment of conditional probability. (English) Zbl 1064.03016

The authors give a logical treatment of conditionals: to deal with the case when the conditioning event has zero probability, they develop a suitable logic over a nonstandard real interval, equipped with an idempotent endomorphism capturing the standard part function. The overall construction is reminiscent of P. Hájek’s approach to probability in his monograph [Metamathematics of fuzzy logic (Trends in Logic – Studia Logica Library 4) Dordrecht: Kluwer (1998; Zbl 0937.03030)]. As a main result, the authors prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over the logic defined in the present paper. For a different treatment of conditionals, along the lines of Carathéodory-von Neumann algebraic probability theory, see the handbook chapter “Probability on MV-algebras” by B. Riečan and the present reviewer [in: E. Pap (ed.), Handbook of measure theory, Vol. II. Amsterdam: North Holland, 869–909 (2002; Zbl 1017.28002)]. In any case, much work must be done to achieve better decision procedures for the coherence of conditional probability assessments.


03B50 Many-valued logic
06D35 MV-algebras
03B48 Probability and inductive logic
03G25 Other algebras related to logic
03B52 Fuzzy logic; logic of vagueness
Full Text: DOI


[1] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes at anneaux reticulés. Lecture Notes in Math. 608, Springer Verlag, Berlin, 1977 · Zbl 0384.06022
[2] Blok, W.J., Ferreirim, I.M.A.: On the structure of hoops. Algebra Universalis 43, 233-257 (2000) · Zbl 1012.06016
[3] Blok, W., Pigozzi, D.: Algebraizable Logics. Mem. Am. Math. Soc. 396 vol. 77 Amer. Math. Soc. Providence 1989 · Zbl 0664.03042
[4] Cignoli, R., D?Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-valued Reasoning. Kluwer, 2000 · Zbl 0937.06009
[5] Cignoli, R., Torrens, A.: An algebraic analysis of Product logic. Multiple Valued Logic 5, 45-65 (2000) · Zbl 0962.03059
[6] Coletti, G.: Numerical and qualitative judgements in probabilistic expert systems. In: R. Scozzafava (ed.), Proc. of Workshop on Probabilistic Methods in Expert Systems, S.I.S. Roma 14 15 October 1993, pp. 37-55
[7] Coletti, G.: Coherent numerical and ordinal probabilistic assessments. IEEE Jour. Trans. on Systems, Men and Cybernetics 24 (12), 1747-1754 (1994) · Zbl 1371.68265
[8] Coletti, G., Scozzafava, R.: Probabilistic Logic in a Coherent Setting. Trends in Logic, Vol. 15, Kluwer, 2002 · Zbl 1005.60007
[9] Coletti, G., Scozzafava, R.: Characterization of coherent comparative probabilities as a tool for their assessment and extention. Jour. of Uncertainty, Fuzziness and Knowledge based Systems, 44 (3), 101-132 (1996) · Zbl 1232.03010
[10] Esteva, F., Godo, L., Hájek, P.: Reasoning about Probability using Fuzzy Logic. Neural Network World, 10 (5), 811-824 (2000)
[11] Esteva, F., Godo, L., Montagna, F.: ?? and two fuzzy logics joining Lukasiewicz and Product logics. Archive for Mathematical Logic 40, 39-67 (2001) · Zbl 0966.03022
[12] Hájek, P.: Metamathematics of Fuzzy Logic, Kluwer, 1998 · Zbl 0937.03030
[13] Hájek, P., Tulipani, S.: Complexity of Fuzzy Probability Logic. Fundamenta Informaticae, 45, 207-213 (2001) · Zbl 0972.03025
[14] Krauss, P.H.: Representation of conditional probability measures on Boolean algebras. Acta Mathematica Academiae Scientiarum Hungaricae, Tomus 19 (3-4), 229-241 (1968) · Zbl 0174.49001
[15] McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, Vol I. Wadsworth and Brooks/Cole, Monterey CA, 1987
[16] Montagna, F.: An algebraic approach to propositional fuzzy logic. J. Logic, Language and Information 9, 91-124 (2000) · Zbl 0942.06006
[17] Montagna, F.: Functorial representation theorems for MV? algebras with additional operators. J. Algebra 238, 99-125 (2001) · Zbl 0987.06012
[18] Mundici, D.: Interpretations of AF C? algebras in ?ukasiewicz sentential calculus. J. Funct. Analysis 65, 15-63 (1986) · Zbl 0597.46059
[19] Mundici, D., Riecan, B.: Probability on MV-algebras. Chapter in Handbook of Measure Theory, E. Pap , Ed., North-Holland, Amsterdam, 2002
[20] Nelson, E.: Radically Elementary Probability Theory. Annals of Math. Studies, Princeton University Press, 1988 · Zbl 0651.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.