Abstract
We outline a class of term-languages for epistemic grounding inspired by Prawitz’s theory of grounds. We show how denotation functions can be defined over these languages, relating terms to proof-objects built up of constructive functions. We discuss certain properties that the languages may enjoy both individually (canonical closure and universal denotation) and with respect to their expansions (primitive/non-primitive and conservative/non-conservative expansions). Finally, we provide a ground-theoretic version of Prawitz’s completeness conjecture, and adapt to our framework a refutation of this conjecture due to Piecha and Schroeder-Heister.
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References
Cozzo, C., Meaning and argument. A theory of meaning centred on immediate argumental role, Almqvist & Wiksell, 1994.
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. European Studies in Philosophy of Science, vol 9. Springer, Berlin, Heidelberg, New York, 2018, pp. 223–244.
d’Aragona, A. P., Dag Prawitz on proofs, operations and grounding, Topoi 38(3):531–550, 2019.
d’Aragona, A. P., Dag Prawitz’s theory of grounds, Ph.D. thesis, Aix-Marseille University, “La Sapienza” University of Rome, 2019.
d’Aragona, A. P., Calculi of epistemic grounding based on Prawitz’s theory of grounds, submitted, 2021.
d’Aragona, A. P., Proofs, grounds and empty functions: epistemic compulsion in Prawitz’s semantics, Journal of Philosophical Logic, forthcoming, 2021.
Dean, W., Recursive functions, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy (Spring 2021 Edition), 2021.
Díez, G. F., Five observation concerning the intended meaning of the intuitionistic logical constants, Journal of Philosophical Logic 29(4):409–424, 2000.
Došen, K., Inferential semantics, in H. Wansing, (ed.), Dag Prawitz on proofs and meaning, Springer, Berlin, Heidelberg, New York, 2015, pp. 147–162.
Dummett, M., The logical basis of metaphysics, Harvard University Press, Cambridge, 1991.
Dummett, M., What is a theory of meaning (I), in M. Dummett, The seas of language, Oxford University Press, Oxford, 1996, pp. 1–33.
Dummett, M., What is a theory of meaning (II), in M. Dummett, The seas of language, Oxford University Press, Oxford, 1996, pp. 34–93.
Francez, N., Proof-theoretic semantics, College Publications, London, 2015.
Gentzen, G., Untersuchungen über das logische Schließen. I, Matematische Zeitschrift 39:176–210, 1935.
Heyting, A.,Intuitionism. An introduction, North-Holland Publishing Company, Amsterdam, 1956.
Howard, W., The formula-as-types notion of construction, in J. R. Hindley, and J. P. Seldin, (eds.), To H. B. Curry: essays on combinatory logic, lambda calculus and formalism, Academic Press, London, 1980, pp. 479–490.
Martin-Löf, P., Intuitionistic type theory, Bibliopolis, Napoli, 1984.
Peter, R., Rekursive Functionen, Budapest, Akademiai Kiado, 1959.
Piecha, T., Completeness in proof-theoretic semantics, in T. Piecha, and P. Schroeder-Heister, (eds.), Advances in proof-theoretic semantics, Springer, Berlin, Heidelberg, New York, 2016, pp. 231–251.
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.
Piecha, T., and P. Schroeder-Heister, Incompleteness of intuitionistic propositional logic with respect to proof-theoretic semantics, Studia Logica 107(1):233–246, 2019.
Poggiolesi, F., A critical overview of the most recent logics of grounding, in F. Boccuni, and A. Sereni, (eds.), Objectivity, realism and proof, vol. 318 of Boston Studies in the Philosophy and History of Science, Springer, 2016, pp. 291–309.
Poggiolesi, F., Logics of grounding, in M. Raven, (ed.), Routledge handbook for metaphysical grounding, Routledge, 2020, pp. 213–227.
Prawitz, D., Constructive semantics, in Proceedings of the First Scandinavian Logic Symposium, Åbo 1968, Filosofiska studier 8, Uppsala, 1970, pp. 96–114.
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, 1971, pp. 235–307.
Prawitz, D., Towards a foundation of a general proof-theory, in P. Suppes, L. Henkin, A. Joja, and Gr C. Moisil, (eds.), Proceedings of the Fourth International Congress for Logic, Methodology and Philosophy of Science, Bucharest, vol. 74 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1973, pp. 225–250.
Prawitz, D., Meaning and proofs: on the conflict between classical and intuitionistic logic, Theoria 43(1):2–40, 1977.
Prawitz, D., Proofs and the meaning and completeness of the logical constants, in J. Hintikka, I. Niiniluoto, and E. Saarinen, (eds.), Essays on mathematical and philosophical logic, Reidel, Dordrecht, 1979, pp. 25–40.
Prawitz, D., Natural deduction. A proof-theoretical study, Dover, New York, 2006.
Prawitz, D., Inference and knowledge, in M. Peliš, (ed.), The Logica Yearbook 2008, College Publications, London, 2009, pp. 175–192.
Prawitz, D., The epistemic significance of valid inference, Synthese 187(3): 887–898, 2012.
Prawitz, D., Truth and proof in intuitionism, in P. Dybier, S. Lindström, E. Palmgren, and G. Sundholm, (eds.), Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf, Springer, Berlin, Heidelberg, New York, 2012, pp. 45–67.
Prawitz, D., Validity of inferences, in M. Frauchiger, (ed.), Reference, rationality, and phenomenology: themes from Føllesdal, Dordrecht, Ontos Verlag, 2013, pp. 179–204.
Prawitz, D., An approach to general proof theory and a conjecture of a kind of completeness of intuitionistic logic revisited, in L. C. Pereira, E. H. Haeusler, and V. de Paiva, (eds.), Advances in natural deduction, vol. 39 of Trends in Logic, Springer, Dordrecht, 2014, pp. 269–279.
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.
Prawitz, D., The seeming interdependence between the concepts of valid inference and proof, Topoi 38(3):493–503, 2019.
Prawitz, D., Validity of inferences, forthcoming, 2020.
Prawitz, D., Validity of inferences reconsidered, forthcoming, 2020.
Schroeder-Heister, P., A natural extension for natural deduction, Journal of Symbolic Logic 49(4):1284–1300, 1984.
Schroeder-Heister, P., Generalized rules for quantifiers and the completeness of the intuitionistic operators&, \(\vee \), \(\supset \), \(\curlywedge \), \(\forall \), \(\exists \), in M. M. Richter, E. Börger, W. Oberschelp, and B. Schinzel & W. Thomas, (eds.), Computation and proof theory. Proceedings of the Logic Colloquium held in Aachen, July 18–23, 1983, Part II, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984, pp. 399–426.
Schroeder-Heister, P., Uniform proof-theoretic semantics for logical constants (Abstract), Journal of Symbolic Logic 56:1142, 1991.
Schroeder-Heister, P., Validity concepts in proof-theoretic semantics, Synthese 148(3):525–571, 2006.
Schroeder-Heister, P., Proof-theoretic versus model-theoretic consequence, in M. Peliš, (ed), The Logica Yearbook 2007, Filosofia, Prague, 2008, pp. 187–200.
Schroeder-Heister, P., The categorical and the hypothetical: a critique of some fundamental assumptions of standard semantics, Synthese 187(3):925–942, 2012.
Schroeder-Heister, P., Proof-theoretic semantics, in E. N. Zalta, (ed.), The Stanford Encyclopedia of Philosophy (Spring 2018 Edition), 2018.
Sundholm, G., Proofs as acts and proofs as objects, Theoria 64(2-3):187–216, 1998.
Tranchini, L., Dag Prawitz, APhEx 9, 2014.
Tranchini, L., Proof-theoretic semantics, proofs and the distinction between sense and denotation, Journal of Logic and Computation 26(2):495–512, 2016.
Tranchini, L., Proof, meaning and paradox. Some remarks, Topoi 38(3):591–603, 2019.
Troelstra, A. S., and D. Van Dalen, Constructivism in mathematics, vol. I, North-Holland Publishing Company, Amsterdam, 1988.
Usberti, G., A notion of C-justification for empirical statements, in H. Wansing, (ed.), Dag Prawitz on proofs and meaning, vol. 7 of Outstanding Contributions to Logic, Springer, Berlin, Heidelberg, New York, 2015, pp. 415–450.
Usberti, G., Inference and epistemic transparency, Topoi 38(3):517–530, 2019.
Acknowledgements
I am grateful to Cesare Cozzo, Gabriella Crocco, Ansten Klev, Dag Prawitz and Peter Schroeder-Heister 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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d’Aragona, A.P. Denotational Semantics for Languages of Epistemic Grounding Based on Prawitz’s Theory of Grounds. Stud Logica 110, 355–403 (2022). https://doi.org/10.1007/s11225-021-09969-8
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DOI: https://doi.org/10.1007/s11225-021-09969-8