Possibilistic conditioning framed in fuzzy logics
type: Item
instance of: scholarly article
title: Possibilistic conditioning framed in fuzzy logics
zbMATH Open document ID: 1122.03022
author: Enrico Marchioni
published in: International Journal of Approximate Reasoning
publication date: 2006-12-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03B52
Mathematics Subject Classification ID: 94D05
zbMATH DE Number: 5078208
zbMATH Keywords: possibility theory
zbMATH Keywords: conditional possibility
zbMATH Keywords: conditional necessity
zbMATH Keywords: fuzzy logic
zbMATH Keywords: coherence
zbMATH Keywords: compactness
zbMATH Keywords: completness
zbMATH Keywords: \(K\)-satisfiability
zbMATH Keywords: Rational Pavelka Logic
zbMATH Keywords: fuzzy implication
zbMATH Keywords: fuzzy modal logic
reviewed by: Mirko Navara
MaRDI profile type: Publication
full work available at URL: j.ijar.2006.03.002
OpenAlex ID: W2063518451
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DOI: 10.1016/J.IJAR.2006.03.002
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description: scientific article; zbMATH DE number 5078208