Hypersequent calculus for the logic of conditional belief: preliminary results Marianna Girlando 1 , Bjoern Lellmann 2 , Nicola Olivetti 1 ∗ 1 Aix Marseille Univ, Universit´ e de Toulon, CNRS, LIS, Marseille, France; 2 Techniche Universit¨ at Wien, Austria 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. Our aim is to develop standard internal calculi for this logic. As a preliminary result we propose an internal hypersequent calculus for the logic in the single agent case. 1 Introduction Knowledge and belief are the most important propositional attitudes to reason about epistemic interaction among agents. Conditional Doxastic Logic (CDL) has been pro- posed by Board [4] and Baltag and Smets [2, 1, 3] for modelling both belief and knowl- edge in a multi-agent setting (refer to [9] for a survey). Differently from knowledge, the essential feature of beliefs is that they are revisable whenever the agent learns new information. To capture the revisable nature of beliefs, CDL contains the conditional belief operator Bel (C |B), the meaning of which is that agent i would believe C in case she learnt B. Both unconditional beliefs and knowledge can be defined in CDL : Bel B (agent i believes B) as Bel (B|⊤), K i B (agent i knows B) as Bel (⊥|¬B), the latter meaning that i considers impossible (inconsistent) to learn ¬B. The logic of conditional belief has been significantly employed in game theory [10], and it has been used as the basic formalism to study further dynamic extensions of epistemic logics, determined by several kinds of epistemic/doxastic actions. Not sur- prisingly, the axiomatization of the operator Bel in CDL internalises the well-known AGM postulates of belief revision. The original semantics of this logic is defined in terms of the so-called Plausibility Models: these are standard epistemic models, where each agent is further equipped by a “comparative plausibility” relation between worlds needed to evaluate her own (condi- tional) beliefs. However, as shown in [6], it is possible to give an alternative semantics of this logic in terms of multi-agent neighbourhood models, which are essentially a multi- agent version of Lewis’ sphere models for counterfactual logics. In particular, it turns out that the semantics of CDL coincides with a multi-agent version of Lewis’ logic VTA. From a proof-theoretical point of view the logic CDL has not been studied very much, the only existing calculus for it being the one given in [6]: a labelled sequent calculus based on the neighbourhood semantics mentioned above. Our aim here is to design an internal calculus for this logic. By an internal calculus we mean a calculus where the syntactic structures (sequents) can be directly interpreted as formulas of CDL. An internal calculus for CDL would be particularly significant, * Supported by the ANR Project TICAMORE ANR-16-CE91-0002-01 1