TitleCoherent conditional probability in a fuzzy logic setting
Publication TypeJournal Article
Year of Publication2006
AuthorsGodo L, Marchioni E
JournalLogic Journal of the IGPL
Volume14
Number3
Pagination457-481
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 $\mu(\cdot \mid \cdot)$ is taken as a primitive notion defined over conditional events of the form ``$\varphi$ {\em given} ψ'', $\varphiěrtψ$ for short, where ψ is not the impossible event. The logic, called \FCP, 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 ``$\varphiěrtψ$'' as the truth-value of the (fuzzy) modal proposition $P(\varphi \mid \psi)$, read as ``$\varphiěrtψ$ is probable". The logic \FCP, which is built up over the many-valued logic \LPii (a logic which combines the well-known {Ł}ukasiewicz 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. In this paper we provide further results for the logic \FCP. 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. Then, we improve the results concerning checking consistency of suitably defined theories in \FCP to determine coherence showing parallel results w.r.t. the notion of {\em generalized coherence} when dealing with imprecise assessments. Moreover we also show and discuss compactness results for our logic. Finally, \FCP is shown to be a powerful tool for knowledge representation. Indeed, following ideas already investigated in the related literature, we show how \FCP allows the definition of suitable notions of default rules which enjoy the core properties of nonmonotonic reasoning characterizing system {\bf P} and {\bf R}.