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Partially undetermined many-valued events and their conditional probability. (English) Zbl 1261.03096
In [D. Dubois and H. Prade, “Conditional objects as nonmonotonic consequence relations”, IEEE Trans. Syst. Man Cybern. 24, No. 12, 1724–1740 (1994)], logic for classical conditional events was investigated. In their approach, the truth value of a conditional event may be undetermined. Now this theory is extended to many-valued events; both events in a conditional statement can be many-valued and can be undetermined. Even the value of undeterminacy can be expressed in terms of many-valued logic. In contrast to D. Mundici’s approach [Trends Log. Stud. Log. Libr. 28, 213–232 (2009; Zbl 1163.03016)], the approach presented here leads to an algebra of conditionals. Compared to [G. Coletti and R. Scozzafava, Probabilistic logic in a coherent setting. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1040.03017)], many-valued statements are considered and, moreover, the author gives an interpretation in terms of bets in the style of de Finetti. On the other hand, this approach does not allow to define conditional probabilities if the conditioning event is determined with probability zero.
The author shows that the whole investigation can be carried out in a logical and algebraic setting, and he finds a logical characterization of coherence for assessments of partially undetermined events.

MSC:
03B50 Many-valued logic
06D35 MV-algebras
03B48 Probability and inductive logic
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