## Notre Dame Journal of Formal Logic

### An Analytic Calculus for the Intuitionistic Logic of Proofs

#### Abstract

The goal of this article is to take a step toward the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp, the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus $\mathbf{Gilp}^{*}$. We show that $\mathbf{Gilp}^{*}$ is a conservative extension of Gilp, and that $\mathbf{Gilp}^{*}$ satisfies the subformula property.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 353-393.

Dates
Accepted: 17 May 2017
First available in Project Euclid: 11 July 2019

https://projecteuclid.org/euclid.ndjfl/1562810593

Digital Object Identifier
doi:10.1215/00294527-2019-0008

Mathematical Reviews number (MathSciNet)
MR3985617

#### Citation

Hill, Brian; Poggiolesi, Francesca. An Analytic Calculus for the Intuitionistic Logic of Proofs. Notre Dame J. Formal Logic 60 (2019), no. 3, 353--393. doi:10.1215/00294527-2019-0008. https://projecteuclid.org/euclid.ndjfl/1562810593

#### References

• [1] Artemov, S. N., “Explicit provability and constructive semantics,” Bulletin of Symbolic Logic, vol. 7 (2001), pp. 1–36.
• [2] Artemov, S. N., “Unified semantics for modality and $\lambda$-terms via proof polynomials,” pp. 1–35 in Algebras, Diagrams, and Decisions in Language, Logic, and Computation, edited by K. Vermeulen and A. Copestake, vol. 144 of CSLI Lecture Notes, CSLI, Stanford, 2001.
• [3] Avron, A., “The method of hypersequents in the proof theory of propositional non-classical logics,” pp. 1–32 in Logic: From Foundations to Applications, edited by W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, Oxford University Press, Oxford, 1996.
• [4] Barendregt, H., “Lambda calculi with types,” pp. 120–48 in Handbook of Logic in Computer Science, vol. 2 of Oxford Science Publications, Oxford University Press, New York, 1992.
• [5] Mkrtychev, A., “Models for the logic of proofs,” pp. 266-75 in Logical Foundations of Computer Science (Yaroslavl, 1997), edited by S. Adian and A. Nerode, vol. 1234 of Lecture Notes in Computer Science, Springer, Berlin, 1997.
• [6] Paoli, F., Substructural Logics: A Primer, vol. 13 of Trends in Logic—Studia Logica Library, Kluwer Academic, Dordrecht, 2002.
• [7] Poggiolesi, F., Gentzen Calculi for Modal Propositional Logic, vol. 32 of Trends in Logic—Studia Logica Library, Springer, Dordrecht, 2011.
• [8] Poggiolesi, F., “Towards a satisfying proof analysis of the logic of proofs,” pp. 371–87 in Proceedings of the Second ILCLI International Workshop on Logic and Philosophy of Knowledge, Communication, and Action, edited by X. Arrazola and M. Ponte, University of the Basque Country Press, San Sebastian, 2010.
• [9] Poggiolesi, F., “A Pragmatic Argument in Support of Analyticity” in Proceedings of the Third Workshop on Philosophy of Information, Flemish Academy, Brussels, 2012.
• [10] Poggiolesi, F., “On the importance of being analytic: The paradigmatic case of the logic of proofs,” Logique et Analyse (Nouvelle Série), vol. 55 (2012), pp. 443–61.
• [11] Renne, B., “Evidence elimination in multi-agent justification logic,” pp. 227–36 in Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge (Stanford, 2009), ACM International Conference Proceedings Series, New York, 2009
• [12] Savateev, Y., “Proof internalization in generalized Frege systems for classical logic,” Annals of Pure and Applied Logic, vol. 165 (2014), pp. 340–56.
• [13] Sørensen, M. H., and P. Urzyczyn, Lectures on the Curry-Howard Isomorphism, vol. 149 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 2006.
• [14] Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, vol. 43 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 1996.
• [15] Wansing, H., Displaying Modal Logic, vol. 3 of Trends in Logic–Studia Logica Library, Kluwer Academic, Dordrecht, 1998.
• [16] Wansing, H., “Sequent systems for modal logics,” pp. 61–145 in Handbook of Philosophical Logic, Vol. 8, edited by D. M. Gabbay and F. Guenthner, Springer, Dordrecht, 2002.