A Sequent Calculus for Preferential Conditional Logic Based on Neighbourhood Semantics

  • Sara Negri
  • Nicola Olivetti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9323)


The basic preferential conditional logic PCL, initially proposed by Burgess, finds an interest in the formalisation of both counterfactual and plausible reasoning, since it is at the same time more general than Lewis’ systems for counterfactuals and it contains as a fragment the KLM preferential logic P for default reasoning. This logic is characterised by Kripke models equipped with a ternary relational semantics that represents a comparative similarity/normality assessment between worlds, relativised to each world. It is first shown that its semantics can be equivalently specified in terms of neighbourhood models. On the basis of this alternative semantics, a new labelled calculus is given that makes use of both world and neighbourhood labels. It is shown that the calculus enjoys syntactic cut elimination and that, by adding suitable termination conditions, it provides a decision procedure.


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  1. 1.
    Alenda, R., Olivetti, N., Pozzato, G.L.: Nested sequent calculi for normal conditional logics. J. Logic Computation (2013) (published online)Google Scholar
  2. 2.
    Burgess, J.: Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic 22, 76–84 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chellas, B.F.: Basic conditional logic. J. of Philosophical Logic 4, 133–153 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Archive for Mathematical Logic 51, 71–92 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Garg, G., Genovese, V., Negri, S.: Countermodels from sequent calculi in multi-modal logics. In: LICS 2012, pp. 315–324 (2012)Google Scholar
  6. 6.
    Giordano, L., Gliozzi, V., Olivetti, N., Schwind, C.: Tableau calculus for preference-based conditional logics: Pcl and its extensions. ACM Trans. Comput. Logic 10(3) (2009)Google Scholar
  7. 7.
    Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Analytic tableaux calculi for KLM logics of nonmonotonic reasoning. ACM Trans. Comput. Logic 10(3), 1–47 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Friedman, N., Joseph, Y., Halpern, J.: On the complexity of conditional logics. In: KR 1994, pp. 202–213 (1994)Google Scholar
  9. 9.
    Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1-2), 167–207 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lellmann, B., Pattinson, D.: Sequent systems for lewis’ conditional logics. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS, vol. 7519, pp. 320–332. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Lewis, D.: Counterfactuals. Blackwell (1973)Google Scholar
  12. 12.
    Marti, J., Pinosio, R.: Topological semantics for conditionals. In: Dančák, M., Punčochář, V. (eds.) The Logica Yearbook 2013. College Publications (2014)Google Scholar
  13. 13.
    Negri, S.: Proof analysis in modal logic. J. of Philosophical Logic 34, 507–544 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Negri, S.: Proofs and countermodels in non-classical logics. Logica Universalis 8, 25–60 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Negri, S., von Plato, J.: Proof Analysis. Cambridge University Press (2011)Google Scholar
  16. 16.
    Negri, S., Sbardolini, G.: Proof analysis for Lewis counterfactuals (submitted) (2014),
  17. 17.
    Nute, D.: Topics in Conditional Logic. Dordrecht, Reidel (1980)CrossRefzbMATHGoogle Scholar
  18. 18.
    Olivetti, N., Pozzato, G.L., Schwind, C.: A Sequent Calculus and a Theorem Prover for Standard Conditional Logics. ACM Trans. Comput. Logic 8(4), 1–51 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pattinson, D., Schröder, L.: Generic modal cut elimination applied to conditional logics. Logical Methods in Computer Science 7(1) (2011)Google Scholar
  20. 20.
    Pollock, J.: A refined theory of counterfactuals. Journal of Philosophical Logic 10, 239–266 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schröder, L., Pattinson, D., Hausmann, D.: Optimal tableaux for conditional logics with cautious monotonicity. In: ECAI 2010, pp. 707–712 (2010)Google Scholar
  22. 22.
    Stalnaker, R.: A theory of conditionals. In: Rescher, N. (ed.) Studies in Logical Theory, Oxford, pp. 98–112 (1968)Google Scholar
  23. 23.
    Stalnaker, R., Thomason, R.H.: A semantic analysis of conditional logic. Theoria 36, 23–42 (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of HelsinkiHelsinkiFinland
  2. 2.Aix Marseille University, CNRS, ENSAM, Toulon University, LSIS UMR 7296MarseilleFrance

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