# zbMATH — the first resource for mathematics

The logical view of conditioning and its application to possibility and evidence theories. (English) Zbl 0696.03006
Summary: The concept of conditioning is well known in probability theory, where it is used in artificial intelligence to quantify the uncertainty of rules, but is totally absent from logic, where material implication is widely used to express conditional statements. This problem has intrigued several researchers in the past. This paper is both a survey of works pertaining to the introduction of conditioning relations in logic and a discussion about how these conditioning relations leave some room for non-monotonicity and might be useful in formalizing the concept of production rules in expert systems. Moreover, it is shown how this formal setting leads to generalizing Cox’s axiomatic approach to conditional probability. Conditional probability, possibility, and belief functions are then obtained by solving functional equations. They obey the natural requirement that a conditional uncertainty measure be the measure of a conditional, such conditional being expressed as a conditioning relation between propositions that may fail to coincide with material implication.

##### MSC:
 03B48 Probability and inductive logic 68T99 Artificial intelligence
Full Text:
##### References:
 [1] Nilsson, N., Probabilistic logic, Ai, 28, 71-87, (1986) · Zbl 0589.03007 [2] Schay, G., An algebra of conditional events, J. math. anal. appl., 24, 334-344, (1968) · Zbl 0211.20302 [3] Stalnaker, R., A theory of conditionals, (), 41-55, Also [4] Lewis, D., Probabilities of conditionals and conditional probabilities, (), 85, 129-150, (1976), Also [5] () [6] Ca;abrese, P., An algebraic synthesis of the foundations of logic and probability, Inf. sci., 42, 187-237, (1987) [7] Nguyen, H.T., On representation and combinability of uncertainty, (), 506-509 [8] Goodman, I.R., A measure-free approach to conditioning, (), 270-277 [9] Goodman, I.R.; Nguyen, H.T., Conditional objects and the modeling of uncertainties, (), 119-138 [10] Goodman, I. R., and Nguyen, H. T., Foundations for an algebraic theory of conditioning, Fuzzy Sets Syst., to appear. [11] Dubois, D.; Prade, H.; Dubois, D.; Prade, H., Théorie des possibilités, applications à la représentation des connaissances en informatique, (), 140-143 [12] De Finetti, B.; Kyburg, H., Studies in subjective probability, (), 7, 95-158, (1937), translated by [13] Cox, R.T., Probability, frequency and reasonable expectation, Am. J. phys., 14, 1-13, (1946) · Zbl 0063.01001 [14] Cheeseman, P., In defense of probability, (), 1002-1009 [15] Cheeseman, P., An inquiry into computer understanding (with 23 discussions and a reply), Comput. intell., 4, 58-142, (1988) [16] Horvitz, E.J.; Heckerman, D.E.; Langlotz, C.P., A framework for comparing alternative formalisms for plausible reasoning, (), 210-214 [17] () [18] Sanchez, E., Importance in knowledge systems, Information systems, 14, 6, (1989), to appear in [19] Rescher, N., Many-valued logic, (1969), McGraw-Hill New York · Zbl 0248.02023 [20] Dubois, D.; Prade, H., An introduction to possibilistic and fuzzy logics, (), 287-315 [21] Heckerman, D.E., An axiomatic framework for belief updates, (), 11-22 [22] Aczel, J., () [23] Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. math. (debrecen), 10, 69-81, (1963) · Zbl 0119.14001 [24] Trillas, E., Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica, III, 1, 47-59, (1979) [25] Alsina, C., On a family of connectives for fuzzy sets, Fuzzy sets syst., 16, 231-235, (1985) · Zbl 0603.39005 [26] Zadeh, L.A., Fuzzy sers as a basis for a theory of possibility, Fuzzy sets syst., 1, 3-28, (1978) · Zbl 0377.04002 [27] Shafer, G., A mathematical theory of evidence, (1976), Princeton Univ. Press Princeton, N.J · Zbl 0359.62002 [28] Kyburg, H., Bayesian and non-Bayesian evidential updating, Ai, 31, 271-293, (1987) · Zbl 0622.68069 [29] Suppes, P.; Zanotti, M., On using random relations to generate upper and lower probabilities, Synthese, 36, 427-440, (1977) · Zbl 0382.60004 [30] Dubois, D.; Prade, H., Possibilistic inference under matrix form, (), 112-126 [31] Dubois, D.; Prade, H., Properties of measures of information in evidence and possibility theories, Fuzzy sets syst., 24, 161-182, (1987) · Zbl 0633.94009 [32] Hisdal, E., Conditional possibilities: independence and non-interaction, Fuzzy sets syst., 1, 283-297, (1978) · Zbl 0393.94050 [33] Smets, P., Belief functions, (), 253-286 [34] Weber, S., Conditional measures and their applications to fuzzy sets, (), 412-415 [35] Weber, S., Conditional measures based on Archimedean semi-groups, Fuzzy sets syst., 27, 63-72, (1988) · Zbl 0648.28015 [36] Fine, T.L., Theories of probability, (1973), Academic Press New York [37] Krantz, D.H.; Luce, R.D.; Suppes, P.; Tversky, A., ()
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.