Abstract
We present new sequent calculi for Lewis’ logics of counterfactuals. The calculi are based on Lewis’ connective of comparative plausibility and modularly capture almost all logics of Lewis’ family. Our calculi are standard, in the sense that each connective is handled by a finite number of rules with a fixed and finite number of premises; internal, meaning that a sequent denotes a formula in the language, and analytical. We present two equivalent versions of the calculi: in the first one, the calculi comprise simple rules; we show that for the basic case of logic \(\mathbb {V}\), the calculus allows for syntactic cut-elimination, a fundamental proof-theoretical property. In the second version, the calculi comprise invertible rules, they allow for terminating proof search and semantical completeness. We finally show that our calculi can simulate the only internal (non-standard) sequent calculi previously known for these logics.
B. Lellmann—Funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 660047.
G.L. Pozzato—Partially supported by the project “ExceptionOWL”, Università di Torino and Compagnia di San Paolo, call 2014 “Excellent (young) PI”.
M. Girlando—Partially supported by the LabEx Archimède, AMU.
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Notes
- 1.
It is worth noticing that in turn the connective \(\preccurlyeq \) can be defined in terms of
. - 2.
Employing this notation, satisfiability of a \( \preccurlyeq \)-formula in a model becomes the following: \(x \Vdash A\preccurlyeq B \) iff for all
. \( \alpha \Vdash ^{\forall } \lnot B \) or \( \alpha \Vdash ^{\exists } A \). - 3.
Lewis’ original presentation in [14] is slightly different: he did not assume the general condition on sphere models that for every
: \(\alpha \ne \emptyset \), and formulated normality as 
and weak centering as normality plus 
, if \(\alpha \ne \emptyset \) then \(w \in \alpha \). Furthermore, note that absoluteness can be equally stated as local absoluteness: 
. - 4.
Actually, the rules \(\mathsf {Con}_S\) and \(\mathsf {Con}_B\) are not needed for completeness (refer to Sect. 6); we have included them in our official formulation of the calculi for technical convenience.
- 5.
A quick argument: once all non redundant \( \mathsf {com}^{\mathsf {c}} \) have been applied, it holds that either \( \varSigma _{i} \subseteq \varSigma _{j} \) or \( \varSigma _{j} \subseteq \varSigma _{i} \); we then order the blocks: \( \varSigma _{1} \subseteq \varSigma _{2} \subseteq ... \subseteq \varSigma _{n} \).
- 6.
The proof uses in an essential way the fact that a backwards application of \( \mathsf {jump}\) reduces the modal degree of a sequent. Although rule \( \mathsf {A}^\mathsf {i}\) plays a similar role as \( \mathsf {jump}\), it does not reduce the modal degree when applied backwards. Thus we need another argument for handling logics including \(\mathbb {A}\); this is object of further investigation.
.
. 
: 
and weak centering as normality plus 
, if 
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Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L. (2016). Standard Sequent Calculi for Lewis’ Logics of Counterfactuals. In: Michael, L., Kakas, A. (eds) Logics in Artificial Intelligence. JELIA 2016. Lecture Notes in Computer Science(), vol 10021. Springer, Cham. https://doi.org/10.1007/978-3-319-48758-8_18
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