Possibilistic conditioning framed in fuzzy logics
The author studies conditional possibility, introduced by \textit{B. Bouchon-Meunier, G. Coletti} and \textit{C. Marsala} [``Independence and possibilistic conditioning'', Ann. Math. Artif. Intell. 35, 107--123 (2002; Zbl 1004.60001)] as a generalization of possibility measures. The paper follows the approach already adopted by the author for conditional probability. He considers conditional possibility in the form \(\Pi(\varphi| \chi)=I(\Pi(\chi),\Pi(\varphi\land\chi))\), where \(I\) is an appropriate (residuated) fuzzy implication. Arguments are given why Gödel implication is not as convenient as the product (Goguen) implication in this formula. Many fuzzy logics are considered and compared as the basis for conditional probability, among them, the Rational Pavelka Logic (RPL) is found to be the most promising one. Further, the author develops a fuzzy modal logic of conditional possibility and studies its Kripke structure. The logic (called FCP) is shown to be strongly complete for finite theories (a modified definition of a theorem is needed). Consequences for \(K\)-satisfiability (\(K\subseteq [0,1]\)) are derived. The results are put into the context of coherent conditional possibilities (an analogue of coherent conditional probabilities of [\textit{G. Coletti} and \textit{R. Scozzafava}, Probabilistic logic in a coherent setting. Dordrecht: Kluwer Academic Publishers (2002; Zbl 1040.03017)]). Similar results for conditional necessity and generalized conditional possibility/necessity are discussed. Finally, qualitative comparative relations are studied in this logical framework.
scientific article; zbMATH DE number 5078208
scientific article