A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences


In recent years, the effort to formalize erotetic inferences—i.e., inferences to and from questions—has become a central concern for those working in erotetic logic. However, few have sought to formulate a proof theory for these inferences. To fill this lacuna, we construct a calculus for (classes of) sequents that are sound and complete for two species of erotetic inferences studied by Inferential Erotetic Logic (IEL): erotetic evocation and erotetic implication. While an effort has been made to axiomatize the former in a sequent system, there is currently no proof theory for the latter. Moreover, the extant axiomatization of erotetic evocation fails to capture its defeasible character and provides no rules for introducing or eliminating question-forming operators. In contrast, our calculus encodes defeasibility conditions on sequents and provides rules governing the introduction and elimination of erotetic formulas. We demonstrate that an elimination theorem holds for a version of the cut rule that applies to both declarative and erotetic formulas and that the rules for the axiomatic account of question evocation in IEL are admissible in our system.

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  1. 1.

    Belnap, N., The Logic of Questions and Answers, New Haven, Yale University Press, 1976.

    Google Scholar 

  2. 2.

    Bimbo, K., Proof Theory: Sequent Calculi and Related Formalisms, CRC Press, Boca Raton, 2014.

    Book  Google Scholar 

  3. 3.

    Brandom, R., Making It Explicit: Reasoning, Representing, and Discursive Commitment, Harvard University Press, Cambridge, 1994.

    Google Scholar 

  4. 4.

    Ciardelli, I., J. Groenendijk, and F. Roelofsen, On the semantics and logic of declaratives and interrogatives, Synthese 192(6): 1689–1728, 2015.

    Article  Google Scholar 

  5. 5.

    Groenendijk, J., and M. Stokhof, Studies on the Semantics of Questions and the Pragmatics of Answers, Dissertation, University of Amsterdam, 1984.

  6. 6.

    Hamblin, C. L., Questions, Australasian Journal of Philosophy 36(3): 159–168, 1958.

    Article  Google Scholar 

  7. 7.

    Harrah, D., A logic of questions and answers, Philosophy of Science 28(1): 40–46, 1961.

    Article  Google Scholar 

  8. 8.

    Hintikka, J., The semantics of questions and the questions of semantics, Acta Philosophica Fennica 28: 4, 1976.

    Google Scholar 

  9. 9.

    Hintikka, J., Socratic Epistemology: Explorations of Knowledge-Seeking by Questioning, Cambridge, Cambridge University Press, 2007.

    Book  Google Scholar 

  10. 10.

    Karttunen, L., Syntax and semantics of questions, Linguistics and Philosophy 1(1): 3–44, 1977.

    Article  Google Scholar 

  11. 11.

    Ketonen, O., Untersuchungen zum Prädikatenkalkül, Annales Academiae Scientiarum Fennicae, 1944, Series A 1.

  12. 12.

    Kukla, R., and M. Lance, ‘Yo!’ and ‘Lo!’ : the Pragmatic Topography of the Space of Reasons, Harvard University Press, Cambridge, 2009.

    Google Scholar 

  13. 13.

    Leszczyńska-Jasion, D., M. Urbański, and A. Wiśniewski, Socratic trees, Studia Logica 101(5): 959–986, 2013.

    Article  Google Scholar 

  14. 14.

    Meheus, J., Adaptive logics for question evocation, Logique Et Analyse 173(175): 135–164, 2001.

    Google Scholar 

  15. 15.

    Millson, J., How to Ask a Question in the Space of Reasons, PhD Thesis, Emory University, 2014.

  16. 16.

    Millson, J., Queries and Assertions in Minimally Discursive Practice, Proceedings of the Society for the Study of Artificial Intelligence and the Simulation of Behavior AISB’50, 2014.

  17. 17.

    Peliš, M., Inferences with Ignorance: Logics of Questions:, Chicago, University of Chicago Press, 2017.

    Google Scholar 

  18. 18.

    Piazza, M., and G. Pulcini, Uniqueness of axiomatic extensions of cut-free classical propositional logic, Logic Journal of IGPL 24(5): 708–718, 2016.

    Article  Google Scholar 

  19. 19.

    Piazza, M., and G. Pulcini, Unifying logics via context-sensitiveness, Journal of Logic and Computation 27(1): 21–40, 2017.

    Article  Google Scholar 

  20. 20.

    Poggiolesi, F., Gentzen Calculi for Modal Propositional Logic, Springer Netherlands, 2016.

    Google Scholar 

  21. 21.

    Wiśniewski, A., The Posing of Questions: Logical Foundations of Erotetic Inferences, Kluwer Academic Publishers, Dordrecht/Boston/London, 1995.

    Book  Google Scholar 

  22. 22.

    Wiśniewski, A., Socratic proofs, Journal of Philosophical Logic 33(3): 299–326, 2004.

    Article  Google Scholar 

  23. 23.

    Wiśniewski, A., Questions, Inferences, and Scenarios, Studies in Logic, College Publications, London, 2013.

  24. 24.

    Wiśniewski, A., An axiomatic account of question evocation: the propositional case, Axioms 5(4): 1–14, 2016.

    Google Scholar 

  25. 25.

    Wiśniewski, A., G. Vanackere, and D. Leszczyńska, Socratic proofs and paraconsistency: a case study, Studia Logica 80(2-3): 431–466, 2005.

    Article  Google Scholar 

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This work has benefited from the comments and suggestions of Andrzej Wiśniewski and two anonymous referees.

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Correspondence to Jared Millson.

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Presented by Andrzej Indrzejczak

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Millson, J. A Cut-Free Sequent Calculus for Defeasible Erotetic Inferences. Stud Logica 107, 1279–1312 (2019). https://doi.org/10.1007/s11225-018-9839-z

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  • Erotetic logic
  • Proof theory
  • Sequent calculus
  • Defeasible reasoning