Abstract
We present the first internal calculi for Lewis’ conditional logics characterized by uniformity and reflexivity, including non-standard internal hypersequent calculi for a number of extensions of the logic \(\mathbb {V}\mathbb {T}\mathbb {U}\). These calculi allow for syntactic proofs of cut elimination and known connections to \(\mathsf {S5}\). We then introduce standard internal hypersequent calculi for all these logics, in which sequents are enriched by additional structures to encode plausibility formulas as well as diamond formulas. These calculi provide both a decision procedure for the respective logics and constructive countermodel extraction from a failed proof search attempt.
Supported by the Project TICAMORE ANR-16-CE91-0002-01, by the EU under Marie Skłodowska-Curie Grant Agreement No. [660047], and by the project “ExceptionOWL”, Università di Torino and Compagnia di San Paolo, call 2014 “Excellent (young) PI”.
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Notes
- 1.
Using this notation we thus have: \(x \Vdash A\preccurlyeq B \) iff for all
. \( \alpha \Vdash ^{\forall } \lnot B \) or \( \alpha \Vdash ^{\exists } A \). - 2.
Observe that \( \mathbb {V}\mathbb {T}\mathbb {A}\)+weak centering collapses to S5, since in any model over a set of worlds W it must be for all \(w\in W\),
. Furthermore, \( \mathbb {V}\mathbb {T}\mathbb {A}\) + centering collapses to Classical Logic, as in any model the set of worlds must be a singleton \(\{w\}\) and 
, so that \(A\preccurlyeq B\) is equivalent to the material implication \(B \rightarrow A\). See also Proposition 16 below..
. 
. Furthermore, 
, so that References
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Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L. (2017). Hypersequent Calculi for Lewis’ Conditional Logics with Uniformity and Reflexivity. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_8
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