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Calculi of Epistemic Grounding Based on Prawitz’s Theory of Grounds


We define a class of formal systems inspired by Prawitz’s theory of grounds. The latter is a semantics that aims at accounting for epistemic grounding, namely, at explaining why and how deductively valid inferences have the power to epistemically compel to accept the conclusion. Validity is defined in terms of typed objects, called grounds, that reify evidence for given judgments. An inference is valid when a function exists from grounds for the premises to grounds for the conclusion. Grounds are described by formal terms, either directly when the terms are in canonical form, or indirectly when they are in non-canonical form. Non-canonical terms must reduce to canonical form, and two terms may be said to be equal when they converge towards equivalent grounds. In our systems these properties can be proved through rules distinguished according to whether they concern types or logic. Type rules involve type introduction and elimination, equality for application of operational symbols, and re-writing equations for non-canonical terms. The logic amounts to a sort of intuitionistic system in a Gentzen format. To conclude, we show that each system of our class enjoys a normalization property.

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  1. Cellucci, C., Teoria delle dimostrazione. Normalizzazioni e assegnazioni di numeri ordinali, Bollati Boringhieri, Torino, 1978.

    Google Scholar 

  2. Cozzo, C., Meaning and argument. A theory of meaning centred on immediate argumental role, Almqvist & Wiksell, Uppsala, 1994.

  3. d’Aragona, A. P., A partial calculus for Dag Prawitz’s theory of grounds and a decidability issue, in A. Christian, D. Hommen, N. Retzlaff, and G. Schurz, (eds), Philosophy of Science, vol. 9 of European Studies in Philosophy of Science, Springer, Berlin Heidelberg, New York, 2018, pp. 223–244.

  4. d’Aragona, A. P., Dag Prawitz on proofs, operations and grounding, Topoi 38(3):531–550, 2019.

  5. d’Aragona, A. P., Dag Prawitz’s theory of grounds, Ph.D. thesis, Aix-Marseille University, “Sapienza” University of Rome, 2019,

  6. d’Aragona, A. P., Denotational semantics for languages of epistemic grounding based on Prawitz’s theory of grounds, Studia Logica, 2021, online first.

  7. d’Aragona, A. P., Proofs, grounds and empty functions: epistemic compulsion in Prawitz’s semantics, Journal of Philosophical Logic, 2021, online first.

  8. Dummett, M., The logical basis of metaphysics, Harvard University Press, Cambridge, 1991.

  9. Francez, N., Proof-theoretic semantics, College Publications, London, 2015.

  10. Gentzen, G., Untersuchungen über das logische Schließen I, Matematische Zeitschrift, 39:176–210, 1935.

  11. Heyting, A., Intuitionism. An introduction, North-Holland Publishing Company, Amsterdam, 1956.

  12. Martin-Löf, P., Intuitionistic type theory, Bibliopolis, Napoli, 1984.

  13. Piecha, T., W. de Campos Sanz, and P. Schroeder-Heister, Failure of completeness in proof-theoretic semantics, Journal of Philosophical Logic 44(3):321–335, 2015.

  14. Piecha, T., and P. Schroeder-Heister, Incompleteness of intuitionistic propositional logic with respect to proof-theoretic semantics, Studia Logica 107(1):233–246, 2019.

  15. Prawitz, D., Ideas and results in proof theory, in J. E. Fenstad, (ed.), Proceedings of the Second Scandinavian Logic Symposium, vol. 63 of Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1971, pp. 235–308.

  16. Prawitz, D., Towards a foundation of a general proof-theory, in P. Suppes, L. Henkin, A. Joja, and G. C. Moisil, (eds.) Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science, Bucharest, 1971, vol. 74 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973, pp. 225–250.

  17. Prawitz, D., Meaning and proofs: on the conflict between classical and intuitionistic logic, Theoria 43(1):2–40, 1977.

  18. Prawitz, D., Natural deduction. A proof-theoretical study, Dover, New York, 2006.

  19. Prawitz, D., Explaining deductive inference, in H. Wansing, (ed.), Dag Prawitz on proofs and meaning, vol.7 of Outstanding Contributions to Logic, Springer, Berlin, Heidelberg New York, 2015, pp. 65–100.

  20. Prawitz, D., The seeming interdependence between the concepts of valid inference and proof, Topoi 38(3):493–503, 2019.

  21. Prawitz, D., The validity of inference and argument, forthcoming.

  22. Schroeder-Heister, P., A natural extension for natural deduction, Journal of Symbolic Logic 49(4):1284–1300, 1984.

  23. Schroeder-Heister, P., Generalized rules for quantifiers and the completeness of the intuitionistic operators \(\wedge \), \(\vee \), \(\rightarrow \), \(\forall \), \(\exists \), in M. M. Richter, E. Börger, W. Oberschelp, B. Schinzel, and W. Thomas, (eds.), Computation and proof theory. Proceedings of the Logic Colloquium held in Aachen, July 18–23, 1983, Part II, Springer, Berlin, 1984, pp. 399–426.

  24. Schroeder-Heister, P., Uniform proof-theoretic semantics for logical constants, Journal of Symbolic Logic 56:1142, 1991.

  25. Schroeder-Heister, P., Proof-theoretic semantics, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), 2018.

  26. Tranchini, L., Proof, meaning and paradox: some remarks, Topoi 38(3):1–13, 2019.

  27. Usberti, G., A notion of \(C\)-justification for empirical statements, in H. Wansing, (ed.), Dag Prawitz on proofs and meaning, vol. 7 of Oustanding Contributions to Logic, Springer, Berlin, Heidelberg, New York, 2015, pp. 415–450.

  28. Usberti, G., Inference and epistemic transparency, Topoi 38(3):517–530, 2019.

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I am grateful to Cesare Cozzo, Gabriella Crocco, Enrico Moriconi and, above all, Dag Prawitz for helpful suggestions. I am also grateful to the anonymous reviewers, whose comments helped me to improve an earlier draft of this paper.

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Correspondence to Antonio Piccolomini d’Aragona.

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d’Aragona, A.P. Calculi of Epistemic Grounding Based on Prawitz’s Theory of Grounds. Stud Logica 110, 819–877 (2022).

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  • Prawitz
  • Type
  • Grounding
  • Formal system
  • Normalization