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Natural Deduction Systems for Intuitionistic Logic with Identity


The aim of the paper is to present two natural deduction systems for Intuitionistic Sentential Calculus with Identity (ISCI); a syntactically motivated \(\mathsf {ND}^1_{\mathsf {ISCI}}\) and a semantically motivated \(\mathsf {ND}^2_{\mathsf {ISCI}}\). The formulation of \(\mathsf {ND}^1_{\mathsf {ISCI}}\) is based on the axiomatic formulation of ISCI. Its rules cannot be straightforwardly classified as introduction or elimination rules; ISCI-specific rules are based on axioms characterizing the identity connective. The system does not enjoy the standard subformula property, but due to the normalization procedure non-subformulas can label only leaves of proofs. In \(\mathsf {ND}^2_{\mathsf {ISCI}}\), we propose only two general identity-related rules, in reference to the treatment of the identity connective in First-Order Logic.

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  1. Bloom, S. L., and R. Suszko, Investigations into the Sentential Calculus with Identity, Notre Dame Journal of Formal Logic 13(3):289–308, 1972.

    Article  Google Scholar 

  2. Chlebowski, S., Sequent Calculi for SCI, Studia Logica 106(3):541–563, 2018.

    Article  Google Scholar 

  3. Chlebowski, S., and D. Leszczyńska-Jasion, An Investigation into Intuitionistic Logic with Identity, Bulletin of the Section of Logic 48(4):259–283, 2019.

    Article  Google Scholar 

  4. Di Cosmo, R., Isomorphisms of Types: from\(\lambda \)-calculus to Information Retrieval and Language Design, Springer Science & Business Media, 2012.

  5. Frege, F. L. G., Sens i znaczenie, in F. L. G. Frege, Pisma semantyczne, PanstwoweWydawnictwo Naukowe, 2014, pp. 60–88.

    Google Scholar 

  6. Golińska-Pilarek, J., Rasiowa-Sikorski Proof System for the Non-Fregean Sentential Logic SCI, Journal of Applied Non-Classical Logics 17:511–519, 2007.

    Article  Google Scholar 

  7. Indrzejczak, A., Natural deduction, hybrid systems and modal logics, vol. 30 of Trends in Logic, Springer Science & Business Media, 2010.

  8. Łukowski, P., Intuitionistic Sentential Calculus with Identity, Bulletin of the Section of Logic 19(3):92–99, 1990.

    Google Scholar 

  9. Michaels, A., A Uniform Proof Procedure for SCI Tautologies, Studia Logica 33(3):299–310, 1974.

    Article  Google Scholar 

  10. Negri, S., and J. von Plato, Proof Analysis: A Contribution to Hilbert’s Last Problem, Cambridge University Press, 2011.

  11. Orłowska, E., and J. Golińska-Pilarek, Dual Tableaux: Foundations, Methodology, Case Studies, vol. 33 of Trends in Logic, Springer Science & Business Media, 2011.

  12. Suszko, R., Abolition of the Fregean Axiom, in R. Parikh, (ed.), Logic Colloquium. Symposium on Logic held at Boston, 1972–73, vol. 453 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1975, pp. 169–239.

  13. Tomczyk, A., M. Gawek, and S. Chlebowski, A Proof-Theoretical Analysis of Intuitionistic Logic with Identity. Retrieved from, access: 22.02.2021.

  14. Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory. Second Edition, vol. 43 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2000.

    Book  Google Scholar 

  15. Wittgenstein, L., Tractatus Logico-Philosophicus. PWN, 2000.

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Correspondence to Szymon Chlebowski.

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Chlebowski, S., Gawek, M. & Tomczyk, A. Natural Deduction Systems for Intuitionistic Logic with Identity. Stud Logica 110, 1381–1415 (2022).

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  • Natural deduction
  • Non-Fregean logics
  • Propositional identity
  • Intuitionistic logic