Measure-Free Conditioning, Probability and Non-Monotonic Reasoning

Recent results in the foundations of probability theory indicate that a conditional probability can be viewed as a probability attached to a mathematical entity called a measure-free conditional. Such a measure-free conditional can receive a semantics in terms of a trivalent logic and logical operations are defined on conditionals in terms of truth-tables. It is shown that these results can be useful to justify Cox's axiomatic framework for probability as well as its application to other theories of uncertainty (Shafer's plausibility functions and Zadeh's possibility measures). Moreover it is shown that measure-free conditionals have the properties of well-behaved non-monotonic inference rules.

[1]  Peter C. Cheeseman,et al.  In defense of An inquiry into computer understanding , 1988, Comput. Intell..

[2]  G. Schay.  An algebra of conditional events , 1968 .

[3]  I. Goodman.  Conditional Objects and the Modeling of Uncertainties , 1988 .

[4]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[5]  Didier Dubois,et al.  The logical view of conditioning and its application to possibility and evidence theories , 1990, Int. J. Approx. Reason..

[6]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[7]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[8]  V. G. Troitsky,et al.  Journal of Mathematical Analysis and Applications , 1960 .

[9]  Dov M. Gabbay,et al.  Theoretical Foundations for Non-Monotonic Reasoning in Expert Systems , 1989, Logics and Models of Concurrent Systems.

[10]  Glenn Shafer,et al.  Comments on An inquiry into computer understanding by Peter Cheeseman , 1988, Comput. Intell..

[11]  Philip G. Calabrese,et al.  An algebraic synthesis of the foundations of logic and probability , 1987, Inf. Sci..

[12]  William Harper,et al.  Ifs. Conditionals, Belief, Decision, Chance, and Time , 1981 .

[13]  Lotfi A. Zadeh,et al.  Please Scroll down for Article International Journal of General Systems Fuzzy Sets and Systems* Fuzzy Sets and Systems* , 2022 .

[14]  J. Aczél,et al.  Lectures on Functional Equations and Their Applications , 1968 .

[15]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

[16]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[17]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[18]  R. T. Cox.  Probability, frequency and reasonable expectation , 1990 .

[19]  Romas Aleliunas,et al.  A New Normative Theory of Probabilistic Logic , 1990 .

[20]  N. Rescher.  Many Valued Logic , 1969 .

[21]  E. W. Adams,et al.  The logic of conditionals , 1975 .

[22]  O. H. Lowry.  Academic press. , 1972, Analytical chemistry.