Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics

Marianna Girlando (1) , Sara Negri (2) , Nicola Olivetti (3)

1 PARTOUT - Automatisation et ReprésenTation: fOndation du calcUl et de la déducTion
2 Helsingin yliopisto = Helsingfors universitet = University of Helsinki
3 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre

The preferential conditional logic PCL, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalise Lewis' sphere models for counterfactual logics, is proposed. Soundness and completeness of PCL and its extensions with respect to this class of models are proved directly. Labelled sequent calculi for all logics of the family are then introduced. The cal-culi are modular and have standard proof-theoretical properties, the most important of which is admissibility of cut, that entails a syntactic proof of completeness of the calculi. By adopting a general strategy, root-first proof search terminates, thereby providing a decision procedure for PCL and its extensions. Finally, the semantic completeness of the calculi is established: from a finite branch in a failed proof attempt it is possible to extract a finite countermodel of the root sequent. The latter result gives a constructive proof of the finite model property of all the logics considered.

Domaines: Logique en informatique [cs.LO]

Marianna Girlando, Sara Negri, Nicola Olivetti. Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics. 2019. ⟨hal-02330319v1⟩