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Intuitionistic Conditional Logics

Abstract

Building on recent work by Yale Weiss, we study conditional logics in the intuitionistic setting. We consider a number of semantic conditions which give rise, among others, to intuitionistic counterparts of Lewis’s logic VC and Stalnaker’s C2. We show how to obtain a sound and complete axiomatization of each logic arising from a combination of these conditions. On the way, we remark how, in the intuitionistic setting, certain classically equivalent principles of conditional logic come apart, and how certain logical connections between different principles no longer hold.

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References

  1. 1.

    Adams, E. (1965). The logic of conditionals. Inquiry, 8(1-4), 166–197.

    Article  Google Scholar 

  2. 2.

    Alechina, N., Mendler, M., De Paiva, V., Ritter, E. (2001). Categorical and Kripke semantics for constructive S4 modal logic. In Fribourg, L. (Ed.) Computer Science Logic (pp. 292–307). Berlin: Springer.

  3. 3.

    Arlo-Costa, H., Egré, P., Rott, H. (2019). The logic of conditionals. In Zalta, E.N. (Ed.) The stanford encyclopedia of philosophy. Metaphysics Research Lab, Stanford University, summer 2019 edition.

  4. 4.

    Bellin, G., De Paiva, V., Ritter, E. (2001). Extended curry-howard correspondence for a basic constructive modal logic. In Proceedings of methods for modalities, Vol. 2.

  5. 5.

    Benton, N., Bierman, G., De Paiva, V. (1998). Computational types from a logical perspective. Journal of Functional Programming, 8, 177–193.

    Article  Google Scholar 

  6. 6.

    Bezhanishvili, N., & de Jongh, D. (2006). Intuitionistic logic. Lecture Notes. Institute for Logic Language and Computation (ILLC), University of Amsterdam.

  7. 7.

    Bierman, G.M., & de Paiva, V. (2000). On an intuitionistic modal logic. Studia Logica, 65(3), 383–416.

    Article  Google Scholar 

  8. 8.

    Chellas, B. (1975). Basic conditional logic. Journal of Philosophical Logic, 4 (2), 133–153.

    Article  Google Scholar 

  9. 9.

    Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40(1), 55–94.

    Article  Google Scholar 

  10. 10.

    Ciardelli, I., Zhang, L., Champollion, L. (2018). Two switches in the theory of counterfactuals. Linguistics and Philosophy, 41(6), 577–621.

    Article  Google Scholar 

  11. 11.

    Genovese, V., Giordano, L., Gliozzi, V., Pozzato, G.L. (2014). Logics in access control: a conditional approach. Journal of Logic and Computation, 24(4), 705–762.

    Article  Google Scholar 

  12. 12.

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

    Article  Google Scholar 

  13. 13.

    Lewis, D. (1973). Counterfactuals. Blackwell.

  14. 14.

    Nute, D. (1980). Conversational scorekeeping and conditionals. Journal of Philosophical Logic, 9(2), 153–166.

    Article  Google Scholar 

  15. 15.

    Nute, D. (1984). Conditional logic. In Handbook of philosophical logic (pp. 387–439): Springer.

  16. 16.

    de Paiva, V., & Ritter, E. (2011). Basic constructive modality. Logic without Frontiers: Festschrift for Walter Alexandre Carnielli on the occasion of his 60th Birthday, pp. 411–428.

  17. 17.

    Segerberg, K. (1989). Notes on conditional logic. Studia Logica, 48(2), 157–168.

    Article  Google Scholar 

  18. 18.

    Servi, G.F. (1981). Semantics for a class of intuitionistic modal calculi. In Dalla Chiara, M.L. (Ed.) Italian Studies in the Philosophy of Science. Studies in the Philosophy of Science, Vol. 47. Dordrecht: Springer.

  19. 19.

    Simpson, A. (1994). The proof theory and semantics of intuitionistic modal logic. Ph.D. thesis, University of Edinburgh.

  20. 20.

    Stalnaker, R. (1968). A theory of conditionals. In Rescher, N. (Ed.) Studies in Logical Theory. Blackwell, Oxford.

  21. 21.

    Stalnaker, R.C., & Thomason, R.H. (1970). A semantic analysis of conditional logic. Theoria, 36(1), 23–42.

    Article  Google Scholar 

  22. 22.

    Veltman, F. (1996). Defaults in update semantics. Journal of Philosophical Logic, 25(3), 221–261.

    Article  Google Scholar 

  23. 23.

    Weiss, Y. (2019a). Basic intuitionistic conditional logic. Journal of Philosophical Logic, 48(3), 447–469.

    Article  Google Scholar 

  24. 24.

    Weiss, Y. (2019b). Frontiers of conditional logic. Ph.D. thesis, CUNY.

  25. 25.

    Wolter, F., & Zakharyaschev, M. (1999). Intuitionistic modal logic. In Cantini, A., Casari, E., Minari, P. (Eds.) Logic and foundations of mathematics. Synthese library (Studies in epistemology, logic, methodology, and philosophy of science), Vol. 280. Dordrecht: Springer.

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Acknowledgements

The authors would like to thank two anonymous reviewers for precious comments on a previous version of this paper, as well as the audience at the LoRI -VII conference in Chongqing for inspiring discussions.

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Correspondence to Ivano Ciardelli or Xinghan Liu.

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Ciardelli, I., Liu, X. Intuitionistic Conditional Logics. J Philos Logic 49, 807–832 (2020). https://doi.org/10.1007/s10992-019-09538-4

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Keywords

  • Conditional logic
  • Intuitionistic logic
  • Variably strict conditionals
  • Modal logic
  • Non-monotonic reasoning