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Completeness and Correspondence in Chellas–Segerberg Semantics


We investigate a lattice of conditional logics described by a Kripke type semantics, which was suggested by Chellas and Segerberg – Chellas–Segerberg (CS) semantics – plus 30 further principles. We (i) present a non-trivial frame-based completeness result, (ii) a translation procedure which gives one corresponding trivial frame conditions for arbitrary formula schemata, and (iii) non-trivial frame conditions in CS semantics which correspond to the 30 principles.

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  1. 1.

    Adams E. W.: The logic of conditionals. Inquiry 8, 166–197 (1965)

    Article  Google Scholar 

  2. 2.

    Adams E. W.: On the logic of high probability. Journal of Philosophical Logic 15, 255–279 (1986)

    Article  Google Scholar 

  3. 3.

    Arló-Costa H.: Bayesian epistemology and epistemic conditionals.On the status of the export–import laws. The Journal of Philosophy 98, 555–593 (2001)

    Article  Google Scholar 

  4. 4.

    Blackburn, P., M. de Rijke., and Y. Venema, Modal logic, Cambridge University Press, Cambridge, 2001.

  5. 5.

    Chellas B. F.: Basic conditional logic. Journal of Philosophical Logic 4, 133–153 (1975)

    Article  Google Scholar 

  6. 6.

    Fine, K., and G. Schurz, Transfer theorems for multimodal logics, in B. J. Copeland (ed.), Logic and reality. Essays on the legacy of Arthur Prior, Clarendon Press, Oxford, 1996, pp. 169–213.

  7. 7.

    Gabbay, D. M., A. Kurucz, F. Wolter., and M. Zakharyaschev, Many-dimensional modal logics. Theory and applications, Elsevier, Amsterdam, 2003.

  8. 8.

    Hawthorne, J., A primer on rational consequence relations, Popper functions, and their ranked structures, Studia Logica 102, 2014, this issue.

  9. 9.

    Hughes, G. E., and M. J. Cresswell, A companion to modal logic, Methuen, London, 1984.

  10. 10.

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

    Article  Google Scholar 

  11. 11.

    Leahy, B., E. Rafetseder., and J. Perner, Basic conditional reasoning: how children mimic counterfactual reasoning, Studia Logica 102, 2014, this issue.

  12. 12.

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

    Google Scholar 

  13. 13.

    Lewis, D., Counterfactuals, Blackwell, Oxford, 1973.

  14. 14.

    McGee V.: Conditional probabilities and compounds of conditionals. The Philosophical Review 98, 485–541 (1989)

    Article  Google Scholar 

  15. 15.

    Nute, D., and C. B. Cross, Conditional logic, in D. M. Gabbay, and F. Guenthner (eds.), Handbook of philosophical logic, vol. 4, 2nd ed., Kluwer, Dordrecht, 2001, pp. 1–98.

  16. 16.

    Pfeifer N., Kleiter G. D.: Coherence and nonmonotonicity in human reasoning. Synthese 146, 93–109 (2005)

    Article  Google Scholar 

  17. 17.

    Priest, G., An Introduction to non-classical logic, 2nd ed., Cambridge University Press, Cambridge, 2008.

  18. 18.

    Schurz G.: Probabilistic semantics for Delgrande’s conditional logic and a counterexample to his default logic. Artificial Intelligence 102, 81–95 (1998)

    Article  Google Scholar 

  19. 19.

    Schurz G.: Non-monotonic reasoning from an evolution-theoretic perspective. Ontic, logical and cognitive foundations. Synthese 146, 37–51 (2005)

    Article  Google Scholar 

  20. 20.

    Schurz G., Thorn P. D.: Reward versus risk in uncertain inference. Theorems and simulations. The Review of Symbolic Logic 5, 574–612 (2012)

    Article  Google Scholar 

  21. 21.

    Segerberg K.: Notes on conditional logic. Studia Logica 48, 157–168 (1989)

    Article  Google Scholar 

  22. 22.

    Stalnaker, R. C., A theory of conditionals, in N. Rescher (ed.), Studies in Logical Theory, Basil Blackwell, Oxford, 1968, pp. 98–112.

  23. 23.

    Thorn, P. D., and G. Schurz, Logico-probabilistic inference in an indifferent environment, Studia Logica 102, 2014, this issue.

  24. 24.

    Unterhuber, M., Possible worlds semantics for indicative and counterfactual conditionals? A formal-philosophical inquiry into Chellas–Segerberg semantics, Ontos Verlag (Logos Series), Frankfurt am Main, 2013.

  25. 25.

    van Benthem, J., Correspondence theory, in D. M. Gabbay, and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. 3, 2nd ed., Kluwer, Dordrecht, 2001, pp. 325–408.

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Correspondence to Matthias Unterhuber.

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Unterhuber, M., Schurz, G. Completeness and Correspondence in Chellas–Segerberg Semantics. Stud Logica 102, 891–911 (2014).

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  • Chellas–Segerberg Semantics
  • Standard Segerberg Frame Completeness
  • Correspondence
  • Non-trivial Frame Condition
  • Conditional Logic