TitleReasoning about coherent conditional prbability in the logic FCP (??)
Publication TypeConference Paper
Year of Publication2004
AuthorsGodo L, Marchioni E
EditorKern-Isberner G, Rödder W, Kulmann F.
Conference NameCII04 Workshop Conditionals, Information, and Inference. co, located with KI2004, 27th German Conference on Artificial Intelligence. September 21, 2004, Ulm, Germany

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 that applies 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 frame 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 Lukasiewicz and Product fuzzy logics), is 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 suitable defined theory over the logic \FCP. In this paper we review and provide further results for the logic \FCP. In particular, we extend the previous completeness result when we allow 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. Moreover we also show 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 for the definition of suitable notions of default rules which enjoy the core properties of nonmonotonic reasoning characterizing system {\bf P} and {\bf R}.