Hyperintensionality, Identity and Excluded Middle

Szymon Chlebowski

doi:10.12775/LLP.2025.025

Authors

Szymon Chlebowski Department of Logic and Cognitive Science, Adam Mickiewicz University in Poznań https://orcid.org/0000-0003-4787-7187

DOI:

https://doi.org/10.12775/LLP.2025.025

Keywords

intuitionistic logic, identity, Kripke semantics, hyperintensionality, natural deduction

Abstract

We introduce new semantics for the intuitionistic variant of the weakest non-Fregean logic, SCI. This semantics captures the notion of identity within a constructive framework and aligns with our intuitive understanding of it, which is provided by the Brouwer-Heyting–Kolmogorov interpretation. Moreover, we demonstrate how this approach naturally leads to interesting extensions of ISCI.

References

Artemov, Sergei, and Tudor Protopopescu, 2016, “Intuitionistic epistemic logic”, The Review of Symbolic Logic, 9(2): 266–298. DOI: CrossRef

Bloom, Stephen L., and Roman Suszko, 1972, “Investigations into the sentential calculus with identity”, Notre Dame Journal of Formal Logic, 13(3): 289–308.

Chlebowski, Szymon, and Dorota Leszczyńska-Jasion, 2019, “An investigation into intuitionistic logic with identity”, Bulletin of the Section of Logic, 48 (4): 259–283. DOI: CrossRef

Chlebowski, Szymon, Marta Gawek, and Agata Tomczyk, 2022, “Natural deduction systems for intuitionistic logic with identity”, Studia Logica, 110(6): 1381–1415. DOI: CrossRef

DiCosmo, Roberto, 1995, Isomorphisms of Types: From Lambda-Calculus to Information Retrieval and Language Design, volume 14, Springer Science & Business Media.

Dummett, Michael, 2021, “Sense and reference from a constructivist standpoint”, Bulletin of Symbolic Logic, 27(4): 485–500.

Golińska-Pilarek, Joanna, 2025, “Non-Fregean world of logics”, Journal of Philosophical Logic, 54: 575–620. DOI: CrossRef

Ishii, Tadao, 1998, “Propositional calculus with identity”, Bulletin of the Section of Logic, 27(3): 96–104.

Kleene, Stephen Cole, 1952, Introduction to Metamathematics, P. Noordhoff N.V., Groningen.

Klonowski, Mateusz, 2021, “History of relating logic. The origin and research directions”, Logic and Logical Philosophy, 30(4): 579–629. DOI: CrossRef

Kuznets, Roman, 2007, “Proof identity for classical logic: Generalizing to normality”, pages 332–348 in International Symposium on Logical Foundations of Computer Science, Springer. DOI: CrossRef

Martin-Löf, Per, 2021, “The sense/reference distinction in constructive semantics”, Bulletin of Symbolic Logic, 27(4): 501–513. DOI: CrossRef

Nowak, Marek, 2008, “The logics of analytic equivalence”, Bulletin of the Section of Logic, 37(3).

Sørensen, Morten Heine, and Pawel Urzyczyn, 2006, Lectures on the Curry-Howard Isomorphism, Elsevier.

Suszko, Roman, 1975, “Abolition of the Fregean axiom”, pages 169–239 in R. Parikh (ed.), Logic Colloquium, Berlin, Heidelberg. Springer. DOI: CrossRef

Troelstra, Anne Sjerp, and Helmut Schwichtenberg, 2000, Basic Proof Theory, Cambridge University Press.

Wansing, Heinrich, 2024, “A note on synonymy in proof-theoretic semantics”, pages 339–362 in T. Piecha and K. F. Wehmeier (eds.), Peter Schroeder-Heister on Proof-Theoretic Semantics, Outstanding Contributions to Logic, vol 29, Springer, Cham. DOI: CrossRef

Wittgenstein, Ludwig, 1956, Remarks on the Foundations of Mathematics, Macmillan, Oxford.