Counterfactual logic: labelled and internal calculi, two sides of the same coin?

Abstract : Lewis' Logic V is the fundamental logic of counterfactuals. Its proof theory is here investigated by means of two sequent calculi based on the connective of comparative plausibility. First, a labelled calculus is defined on the basis of Lewis' sphere semantics. This calculus has good structural properties and provides a decision procedure for the logic. An internal calculus, recently introduced, is then considered. In this calculus, each sequent in a derivation can be interpreted directly as a formula of V. In spite of the fundamental difference between the two calculi, a mutual correspondence between them can be established in a constructive way. In one direction, it is shown that any derivation of the internal calculus can be translated into a derivation in the labelled calculus. The opposite direction is considerably more difficult, as the labelled calculus comprises rules which cannot be encoded by purely logical rules. However, by restricting to derivations in normal form, derivations in the labelled calculus can be mapped into derivations in the internal calculus. On a general level, these results aim to contribute to the understanding of the relations between labelled and internal proof systems for logics belonging to the realm of modal logic and its extensions, a topic still relatively unexplored.
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Marianna Girlando, Sara Negri, Nicola Olivetti. Counterfactual logic: labelled and internal calculi, two sides of the same coin?. Advances in Modal Logics 2018, Aug 2018, Bern, Switzerland. pp.291-310. ⟨hal-02077043⟩

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