Abstract
The logic of conditional belief, called Conditional Doxastic Logic (\(\mathsf {CDL}\)), was proposed by Board, Baltag and Smets to model revisable belief and knowledge in a multi-agent setting. We present a proof system for \(\mathsf {CDL}\) in the form of a nested sequent calculus. To the best of our knowledge, ours is the first internal and standard calculus for this logic. We take as primitive a multi-agent version of the “comparative plausibility operator”, as in Lewis’ counterfactual logic. The calculus is analytic and provides a decision procedure for \(\mathsf {CDL}\). As a by-product we also obtain a nested sequent calculus for multi-agent modal logic \(\mathsf {S5}_i\).
This work was partially supported by the Project TICAMORE ANR-16-CE91-0002-01 and by WWTF project MA 16-28.
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Notes
- 1.
- 2.
- 3.
Refer to next section on \( \mathsf {S5}_i\).
- 4.
Evaluating KA at a world x corresponds to evaluating \( \bot \preccurlyeq \lnot A \) in the outer neighbourhood of N(x) . For this reason, Lewis calls \( \mathsf {S5}\) the outer modal logic of \( \mathbb {V}\mathbb {T}\mathbb {A}\).
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Girlando, M., Lellmann, B., Olivetti, N. (2019). Nested Sequents for the Logic of Conditional Belief. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_46
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