Standard Sequent Calculi for Lewis’ Logics of Counterfactuals

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10021)


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.


Sequent Calculus Terminating Proof Search Comparative Plausibility Semantic Completeness Proof Lewis Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Alenda, R., Olivetti, N., Pozzato, G.L.: Nested sequent calculi for normal conditional logics. J. Log. Comput. 26(1), 7–50 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baltag, A., Smets, S.: The logic of conditional doxastic actions. Texts Logic Games 4, 9–31 (2008). Special Issue on New Perspectives on Games and InteractionMathSciNetGoogle Scholar
  3. 3.
    Board, O.: Dynamic interactive epistemology. Games Econ. Behav. 49(1), 49–80 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ciabattoni, A., Metcalfe, G., Montagna, F.: Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. Fuzzy Sets Syst. 161, 369–389 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delgrande, J.P.: On first-order conditional logics. Artif. Intell. 105(1), 105–137 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Friedman, N., Halpern, J.Y.: On the complexity of conditional logics. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) KR 1994, pp. 202–213. Morgan Kaufmann (1994)Google Scholar
  7. 7.
    Gent, I.P.: A sequent or tableaux-style system for Lewis’s counterfactual logic \(\mathbb{VC}\). Notre Dame J. Formal Logic 33(3), 369–382 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ginsberg, M.L.: Counterfactuals. Artif. Intell. 30(1), 35–79 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Giordano, L., Gliozzi, V., Olivetti, N., Schwind, C.: Tableau calculus for preference-based conditional logics: PCL and its extensions. ACM Trans. Comput. Logic (TOCL) 10(3), 21 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grahne, G.: Updates and counterfactuals. J. Logic Comput. 8(1), 87–117 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44(1–2), 167–207 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lellmann, B.: Sequent calculi with context restrictions and applications to conditional logic. Ph.D. thesis, Imperial College London.
  13. 13.
    Lellmann, B., Pattinson, D.: Sequent Systems for Lewis’ Conditional Logics. In: Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS (LNAI), vol. 7519, pp. 320–332. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33353-8_25 CrossRefGoogle Scholar
  14. 14.
    Lewis, D.: Counterfactuals. Blackwell, Oxford (1973)zbMATHGoogle Scholar
  15. 15.
    Negri, S., Olivetti, N.: A sequent calculus for preferential conditional logic based on neighbourhood semantics. In: Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 115–134. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-24312-2_9 CrossRefGoogle Scholar
  16. 16.
    Negri, S., Sbardolini, G.: Proof analysis for Lewis counterfactuals. Rev. Symbolic Logic 9(1), 44–75 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Olivetti, N., Pozzato, G.L.: A standard internal calculus for Lewis’ counterfactual logics. In: Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 270–286. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-24312-2_19 CrossRefGoogle Scholar
  18. 18.
    Pattinson, D., Schröder, L.: Generic modal cut elimination applied to conditional logics. Log. Methods Comput. Sci. 7(1: 4), 1–28 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Poggiolesi, F.: Natural deduction calculi and sequent calculi for counterfactual logics. Stud. Logica 104, 1003–1036 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    de Swart, H.C.M.: A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics \(\mathbb{VC}\) and \(\mathbb{VCS}\). J. Symbolic Logic 48(1), 1–20 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Aix Marseille Univ, CNRS, ENSAM, Université de Toulon, LSIS UMR 7296MarseilleFrance
  2. 2.Technische Universität WienViennaAustria
  3. 3.Dipartimento di InformaticaUniversitá di TorinoTurinItaly

Personalised recommendations