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Coherent conditional probability in a fuzzy logic setting

Coherent conditional probability in a fuzzy logic setting
Coherent conditional probability in a fuzzy logic setting
Very recently, a (fuzzy modal) logic to reason about coherent conditional probability, in the sense of de Finetti, has been introduced by the authors. Under this approach, a conditional probability μ (· ·) is taken as a primitive notion defined over conditional events of the form "phi; given Ψ", φ Ψ for short, where Ψ is not the impossible event. The logic, called FCP(Ł∏), exploits an idea already used by Hájek and colleagues to define a logic for (unconditional) probability in the framework of fuzzy logics. Namely, we take the probability of the conditional event "φ Ψ" as the truth-value of the (fuzzy) modal proposition P (φ Ψ), read as "φ Ψ is probable". The logic FCP(Ł∏), which is built up over the many-valued logic Ł∏1/2 (a logic which combines the well-known Łukasiewicz and Product fuzzy logics), was shown to be complete for modal theories with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Indeed, checking coherence of a (generalized) probability assessment to an arbitrary family of conditional events becomes tantamount to checking consistency of a suitably defined theory over the logic FCP(Ł∏). In this paper we provide further results for the logic FCP(Ł∏). In particular, we extend the previous completeness result by allowing the presence of non-modal formulas in the theories, which are used to describe logical relationships among events. This increases the knowledge modelling power of FCP(Ł∏). Then, we improve the results concerning checking consistency of suitably defined theories in FCP(Ł∏) to determine coherence by showing parallel results w.r.t. the notion of generalized coherence when dealing with imprecise assessments. Moreover we also show and discuss compactness results for our logic. Finally, FCP(Ł∏) is shown to be a powerful tool for knowledge representation. Indeed, following ideas already investigated in the related literature, we show how FCP(Ł∏) allows the definition of suitable notions of default rules which enjoy the core properties of nonmonotonic reasoning characterizing system P and R. © 2006 Oxford University Press.
Coherence, Compactness, Conditional probability, Default reasoning, Fuzzy Logics, Generalized coherence
1367-0751
457-481
Godo, Lluís
0518f088-84ec-4816-ad43-4e240ae66dc1
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Godo, Lluís
0518f088-84ec-4816-ad43-4e240ae66dc1
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96

Godo, Lluís and Marchioni, Enrico (2006) Coherent conditional probability in a fuzzy logic setting. Journal of IGPL, 14 (3), 457-481. (doi:10.1093/jigpal/jzl019).

Record type: Article

Abstract

Very recently, a (fuzzy modal) logic to reason about coherent conditional probability, in the sense of de Finetti, has been introduced by the authors. Under this approach, a conditional probability μ (· ·) is taken as a primitive notion defined over conditional events of the form "phi; given Ψ", φ Ψ for short, where Ψ is not the impossible event. The logic, called FCP(Ł∏), exploits an idea already used by Hájek and colleagues to define a logic for (unconditional) probability in the framework of fuzzy logics. Namely, we take the probability of the conditional event "φ Ψ" as the truth-value of the (fuzzy) modal proposition P (φ Ψ), read as "φ Ψ is probable". The logic FCP(Ł∏), which is built up over the many-valued logic Ł∏1/2 (a logic which combines the well-known Łukasiewicz and Product fuzzy logics), was shown to be complete for modal theories with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Indeed, checking coherence of a (generalized) probability assessment to an arbitrary family of conditional events becomes tantamount to checking consistency of a suitably defined theory over the logic FCP(Ł∏). In this paper we provide further results for the logic FCP(Ł∏). In particular, we extend the previous completeness result by allowing the presence of non-modal formulas in the theories, which are used to describe logical relationships among events. This increases the knowledge modelling power of FCP(Ł∏). Then, we improve the results concerning checking consistency of suitably defined theories in FCP(Ł∏) to determine coherence by showing parallel results w.r.t. the notion of generalized coherence when dealing with imprecise assessments. Moreover we also show and discuss compactness results for our logic. Finally, FCP(Ł∏) is shown to be a powerful tool for knowledge representation. Indeed, following ideas already investigated in the related literature, we show how FCP(Ł∏) allows the definition of suitable notions of default rules which enjoy the core properties of nonmonotonic reasoning characterizing system P and R. © 2006 Oxford University Press.

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More information

Published date: 1 June 2006
Keywords: Coherence, Compactness, Conditional probability, Default reasoning, Fuzzy Logics, Generalized coherence

Identifiers

Local EPrints ID: 417386
URI: http://eprints.soton.ac.uk/id/eprint/417386
ISSN: 1367-0751
PURE UUID: 669f598a-7495-4122-8fe6-8afc850c7d78

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Date deposited: 30 Jan 2018 17:31
Last modified: 13 Mar 2019 19:04

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