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We present hypersequent calculi for the strongest logics in Lewis’ family of conditional systems, characterized by uniformity and total reflexivity. We first present a non-standard hypersequent calculus, which allows a syntactic proof of cut elimination. We then introduce standard hypersequent calculi, in which sequents are enriched by additional structures to encode plausibility formulas and diamond formulas. Proof search using these calculi is terminating, and the completeness proof shows how a countermodel can be constructed from a branch of a failed proof search. We then describe  tuCLEVER , a theorem prover that implements the standard hypersequent calculi. The prover provides a decision procedure for the logics, and it produces a countermodel in case of proof search failure. The prover  tuCLEVER  is inspired by the methodology of  lean  T A P  and it is implemented in Prolog. Preliminary experimental results show that the performances of  tuCLEVER  are promising.  1

Calculi, countermodel generation and theorem prover for strong logics of counterfactual reasoning