Fuzzy logic and probability

In this paper we deal with a new approach to probabilistic reasoning in a logical frame work. Nearly allmost all logics of probability that have been proposed in the literature are based on classical two-valued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a fuzy logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truth-values of some suitable fuzzy propositions associated to the crisp ones. Moreover, suggestions and examples of how to extend the formalism to cope with conditional probabilities and with other uncertainty formalisms are also provided.

[1]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.

[2]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[3]  Marc Snir,et al.  Probabilities over rich languages, testing and randomness , 1982, Journal of Symbolic Logic.

[4]  Fahiem Bacchus,et al.  Representing and reasoning with probabilistic knowledge - a logical approach to probabilities , 1991 .

[5]  Nils J. Nilsson,et al.  Probabilistic Logic * , 2022 .

[6]  Petr Hájek,et al.  On Logics of Approximate Reasoning , 1992, Logic at Work.

[7]  Henri Prade,et al.  Fuzzy sets and probability: misunderstandings, bridges and gaps , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[8]  T. Havránek,et al.  Mechanizing Hypothesis Formation: Mathematical Foundations for a General Theory , 1978 .

[9]  Petr Hájek,et al.  Medical Fuzz;y Expert Systems and Reasoning about Beliefs , 1995, AIME.

[10]  H. Zimmermann,et al.  Fuzzy Set Theory and Its Applications , 1993 .

[11]  Nic Wilson,et al.  A Logical View of Probability , 1994, ECAI.

[12]  Fahiem Bacchus,et al.  Representing and reasoning with probabilistic knowledge , 1988 .

[13]  Giangiacomo Gerla,et al.  Inferences in Probability Logic , 1994, Artif. Intell..

[14]  P. Hájek.  Fuzzy logic and arithmetical hierarchy , 1995 .

[15]  Petr Hájek,et al.  On Logics of Approximate Reasoning , 1992, Logic at Work.

[16]  Dov M. Gabbay,et al.  Handbook of Logic in Artificial Intelligence and Logic Programming: Volume 3: Nonmonotonic Reasoning and Uncertain Reasoning , 1994 .

[17]  Lotfi A. Zadeh,et al.  Is Probability Theory Sufficient for Dealing with Uncertainty in AI: A Negative View , 1985, UAI.

[18]  D. Scott,et al.  Assigning Probabilities to Logical Formulas , 1966 .