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Conditional Beliefs: from Neighbourhood Semantics to Sequent Calculus

Croyances conditionnelles : de la Sémantique neighbourhood au calcul des sequents

Abstract : The logic of Conditional Beliefs (CDL) has been introduced by Board, Baltag and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this paper both the semantics and the proof theory for this logic are studied. First, a natural semantics for CDL is defined in terms of neighbourhood models, a multi-agent generalisation of Lewis' spheres models, and it is shown that the axiomatization of CDL is sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus for CDL is obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of the neighbourhood semantics for the analysis of epistemic/doxastic notions.