A Game Semantics for System P

Abstract

In this paper we introduce a game semantics for System P, one of the most studied axiomatic systems for non-monotonic reasoning, conditional logic and belief revision. We prove soundness and completeness of the game semantics with respect to the rules of System P, and show that an inference is valid with respect to the game semantics if and only if it is valid with respect to the standard order semantics of System P. Combining these two results leads to a new completeness proof for System P with respect to its order semantics. Our approach allows us to construct for every inference either a concrete proof of the inference from the rules in System P or a countermodel in the order semantics. Our results rely on the notion of a witnessing set for an inference, whose existence is a concise, necessary and sufficient condition for validity of an inferences in System P. We also introduce an infinitary variant of System P and use the game semantics to show its completeness for the restricted class of well-founded orders.

References

  1. 1.

    Baltag A., Smets S.: Conditional doxastic models: a qualitative approach to dynamic belief revision. Electronic Notes in Theoretical Computer Science 165, 5–21 (2006)

    Article  Google Scholar 

  2. 2.

    Boutilier C.: Conditional logics of normality: a modal approach. Artificial Intelligence 68(1), 87–154 (1994)

    Article  Google Scholar 

  3. 3.

    Burgess J.: Quick completeness proofs for some logics of conditionals. Notre Dame Journal of Formal Logic 22(1), 76–84 (1981)

    Article  Google Scholar 

  4. 4.

    Friedman N., and J. Y. Halpern, On the complexity of conditional logics, in Principles of Knowledge Representation and Reasoning: Proceedings of the Fourth International Conference (KR’94), Morgan Kaufmann, Burlington, 1994, pp. 202–213.

  5. 5.

    Grove A.: Two modellings for theory change. Journal of Philosophical Logic 17(2), 157–170 (1988)

    Article  Google Scholar 

  6. 6.

    Hintikka, J., and G. Sandu, Game-theoretical semantics, in J. van Benthem and A. ter Meulen (ed.), Handbook of Logic and Language, 2nd edn. Elsevier, New York, 2010, pp. 415–465.

  7. 7.

    Korte B., Lovász L., Schrader R.: Greedoids. Springer, New York (1991)

    Book  Google Scholar 

  8. 8.

    Kraus S., Lehmann D., Magidor M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1–2), 167–207 (1990)

    Article  Google Scholar 

  9. 9.

    Lehmann D., Magidor M.: What does a conditional knowledge base entail?. Artificial Intelligence 55(1), 1–60 (1992)

    Article  Google Scholar 

  10. 10.

    Lewis D.: Counterfactuals. Blackwell Publishers, Oxford (1973)

    Google Scholar 

  11. 11.

    Lorenzen, P., and K. Lorenz, Dialogische Logik, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978.

  12. 12.

    Pozzato, G. L., Conditional and Preferential Logics: Proof Methods and Theorem Proving. Vol. 208. Frontiers in Artificial Intelligence and Applications. IOS Press, Amsterdam, 2010.

  13. 13.

    Rahman, S., and T. Tulenheimo, From games to dialogues and back, in O. Majer, A.-V. Pietarinen, and T. Tulenheimo (ed.), Games: Unifying Logic, Language, and Philosophy, Springer, New York, 2009, pp. 153–208.

  14. 14.

    Väänänen, J., Models and Games, Cambridge University Press, Cambridge, 2011.

  15. 15.

    Veltman, F., Logics for conditionals, PhD thesis, University of Amsterdam, Amsterdam, 1985.

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Correspondence to J. Marti.

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Marti, J., Pinosio, R. A Game Semantics for System P. Stud Logica 104, 1119–1144 (2016). https://doi.org/10.1007/s11225-016-9669-9

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Keywords

  • Non-monotonic consequence relations
  • Conditional logic
  • Belief revision
  • Game semantics
  • Dialogical logic