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A logic for reasoning about coherent conditional probability: A modal fuzzy logic approach. (English) Zbl 1111.68683
Alferes, José Júlio (ed.) et al., Logics in artificial intelligence. 9th European conference, JELIA 2004, Lisbon, Portugal, September 27–30, 2004. Proceedings. Berlin: Springer (ISBN 3-540-23242-7/pbk). Lecture Notes in Computer Science 3229. Lecture Notes in Artificial Intelligence, 213-225 (2004).
Summary: In this paper we define a logic to reason about coherent conditional probability, in the sense of de Finetti. Under this view, a conditional probability $$\mu(\cdot\mid\cdot)$$ is a primitive notion that applies over conditional events of the form “$$\varphi$$ given $$\psi$$”, where $$\psi$$ is not the impossible event. Our approach exploits an idea already used by Hájek and colleagues to define a logic for (unconditional) probability in the frame of fuzzy logics. Namely, in our logic for each pair of classical propositions $$\varphi$$ and $$\psi$$, we take the probability of the conditional event “$$\varphi$$ given $$\psi$$”, $$\varphi | \psi$$ for short, as the truth-value of the (fuzzy) modal proposition $$P (\varphi|\psi)$$, read as “$$\varphi|\psi$$ is probable”. Based on this idea we define a fuzzy modal logic FCP(Ł$$\Pi$$), built up over the many-valued logic Ł$$\Pi\frac12$$ (a logic which combines the well known Lukasiewicz and Product fuzzy logics), which is shown to be complete with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Finally, we show that checking coherence of a probability assessment to an arbitrary family of conditional events is tantamount to checking consistency of a suitably defined theory over the logic FCP(Ł$$\Pi$$).
For the entire collection see [Zbl 1056.68002].

##### MSC:
 68T27 Logic in artificial intelligence 68T37 Reasoning under uncertainty in the context of artificial intelligence 03B52 Fuzzy logic; logic of vagueness
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