Abstract
The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold.
Chapter PDF
Similar content being viewed by others
References
Avron, A. and I. Lev (2001). Canonical propositional gentzen-type systems. In: Proceedings of IJCAR’01. Vol. 2083. LNCS, 529–543.
Baaz, M. and A. Leitsch (2011). Methods of Cut-Elimination. Springer.
Bell, J. L. and M. Machover (1977). A Course in Mathematical Logic. Amsterdam: North-Holland.
Binder, D. and T. Piecha (2021). Popper on quantification and identity. In: Karl Popper’s Science and Philosophy. Ed. by Z. Parusnikova and D. Merrit. To appear. Springer.
Church, A. (1956). Introduction to Mathematical Logic. Vol. I. Princeton: Princeton University Press.
Degtyarev, A. and A. Voronkov (2001). Equality reasoning in sequent-based calculi. In: Handbook of Automated Reasoning vol I. Ed. by A. Robinson and A. Voronkov. Elsevier, 611–706.
Došen, K. (1985). Sequent-systems for modal logic. Journal of Symbolic Logic 50, 149–159.
Došen, K. (1989). Logical constants as punctuation marks. Notre Dame Journal of Formal Logic 30, 362–381.
Gallier J., H. (1986). Logic for Computer Science. New York: Harper and Row.
Gazzari, R. (2019). The calculus of natural calculation. In: Proof-Theoretic Semantics: Assesment and Future Perspectives. Proceedings of the Third Tübingen Conference of Proof-Theoretic Semantics. Ed. by T. Piecha and P. Schroeder-Heister. Tübingen, 123–139.
Gratzl, N. and E. Orlandelli (2017). Double-line harmony in sequent calculi. In: The Logica Yearbook 2016. Ed. by P. Arazim and T. Lavicka. College Publications, 157–171.
Griffiths, O. (2014). Harmonious rules for identity. The Review of Symbolic Logic 7, 499–510.
Hacking, I. (1979). What is logic? Journal of Philosophy 76, 285–319.
Hertz, P. (1929). Über axiomensysteme für beliebige satzsysteme. Mathematische Annalen 101, 457–514.
Hintikka, J. (1956). Identity, variables and impredicative definitions. Journal of Symbolic Logic 21, 225–245.
Indrzejczak, A. (2017). Tautology elimination, cut elimination and s5. Logic and Logical Philosophy 26, 461–471.
Indrzejczak, A. (2018a). Cut-free modal theory of definite descriptions. In: Advances in Modal Logic 12. Ed. by G. B. et al. College Publications, 387–406.
Indrzejczak, A. (2018b). Rule-generation theorem and its applications. The Bulletin of the Section of Logic 47, 265–281.
Indrzejczak, A. (2019). Fregean description theory in proof-theoretical setting. Logic and Logical Philosophy 28, 137–155.
Indrzejczak, A. (2021). A novel approach to equality. Synthese 199, 4749–4774.
Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica 1, 5–32.
Kahle, R. (2016). Towards a proof-theoretic semantics of equalities. In: Advances in Proof-theoretic Semantics. Ed. by T. Piecha and P. Schroeder-Heister. Springer, 153–160.
Kalish, D. and R. Montague (1957). Remarks on descriptions and natural deduction. Archiv. für Mathematische Logik und Grundlagen Forschung 3, 65–73.
Kalish, D. and R. Montague (1964). Logicical Techniques of Formal Reasoning. New York.
Kanger, S. (1957). Provability in Logic. Stockholm: Almqvist & Wiksell.
Klev, A. (2019). The harmony of identity. Journal of Philosophical Logic 48, 867–884.
Koslow, A. (1992). A Structuralist Theory of Logic. Cambridge: Cambridge University Press.
Kürbis, N. (2019). Proof and Falsity. A Logical Investigation. Cambridge: Cambridge University Press.
Lemmon, E. J. (1965). Beginning Logic. London: Nelson.
Manzano, M. (1999). Model Theory. Oxford: Oxford University Press.
Manzano, M. (2005). Extensions of First-Order Logic. Cambridge: Cambridge University Press.
Manzano, M. and M. C. Moreno (2017). Identity, equality, nameability and completeness. The Bulletin of the Section of Logic 46, 169–196.
Martin-Löf, P. (1971). Hauptsatz for the intuitionistic theory of iterated inductive definitions. In: Proceedings of the Second Scandinavian Logic Symposium. Ed. By J. E. Fenstad. North-Holland.
Maruyama, Y. (2016). Categorical harmony and paradoxes in proof-theoretic semantics. In: Advances in Proof-Theoretic Semantics. Ed. by T. Piecha and P. Schroeder-Heister. Springer, 95–114.
Mates, B. (1965). Elementary Logic. Oxford: Oxford University Press.
Mates, B. (1986). The Philosophy of Leibniz. Metaphysics and Language. Oxford: Oxford University Press.
Mints, G. (1968). Some calculi of modal logic. Trudy Mat. Inst. Steklov 98. in Russian, 88–111.
Nagashima, T. (1966). An extension of the craig-schütte interpolation theorem. Annals of the Japan Association for the Philosophy of Science 3, 12–18.
Negri, S. and J. von Plato (2001). Structural Proof Theory. Cambridge: Cambridge University Press.
Negri, S. and J. von Plato (2011). Proof Analysis. A Contribution to Hilbert’s Last Problem. Cambridge: Cambridge University Press.
Parlamento, F. and F. Previale (2019). The elimination of atomic cuts and the semishortening property for Gentzen’s sequent calculus with equality. on-line first. The Review of Symbolic Logic.
Poggiolesi, F. (2011). Gentzen Calculi for Modal Propositional Logic. Springer.
Popper, K. (1947a). Logic without assumptions. Proceedings of the Aristotelian Society 47, 251–292.
Popper, K. (1947b). New foundations for logic. Mind 56. Quine, W. V. (1966). Set Theory and its Logic. Harvard University Press.
Popper, K. (1970). Philosophy of Logic. Prentice Hall.
Read, S. (2004). Identity and harmony. Analysis 64, 113–119.
Reeves, S. V. (1987). Adding equality to semantic tableau. Journal of Automated Reasoning 3, 225–246.
Restall, G. (2019). Generality and existence 1: quantification and free logic. The Review of Symbolic Logic 12, 354–378.
Restall, G. (2020). Assertions, denials, questions, answers and the common ground. url: http://consequently.org/presentation.
Sambin, G., G. Battilotti, and C. Faggian (2000). Basic logic: reflection, symmetry, visibility. Journal of Symbolic Logic 65, 979–1013.
Schroeder-Heister, P. (1984). Popper’s theory of deductive inference and the concept of a logical constant. History and Philosophy of Logic 5, 79–110.
Schroeder-Heister, P. (1994). Definitional reflection and the completion. In: Extensions of Logic Programming. Fourth International Workshop, St. Andrews, Scotland, April 1993, Proceedings. Ed. by R. Dyckhoff. Vol. 798. Berlin/Heidelberg/New York: Springer Lecture Notes in Artificial Intelligence, 333–347.
Schroeder-Heister, P. (2006). Popper’s structuralist theory of logic. In: Karl Popper: A Centenary Assesment, vol III: Science. Ed. by I. Jarvie, K. Milford, and D. Miller. Ashgate Publishing: Aldershot, 17–36.
Schroeder-Heister, P. (2012). Proof-theoretic semantics. In: Stanford Encyclopedia of Philosophy. url: https://plato.stanford.edu/entries/proof-theoretic-semantics/.
Schroeder-Heister, P. (2016). Open problems in proof-theoretic semantics. In: Advances in Proof theoretic Semantics. Ed. by T. Piecha and P. Schroeder-Heister. Springer, 253–283.
Scott, D. (1974). Rules and derived rules. In: Logical Theory and Semantical Analysis. Ed. by S. Stenlund, 147–161.
Seligman, J. (2001). Internalization: the case of hybrid logics. Journal of Logic and Computation 11, 671–689.
Suppes, P. (1957). Introduction to Logic. Princeton: Van Nostrand.
Takeuti, G. (1987). Proof Theory. Amsterdam: North-Holland.
Tarski, A. (1941). Introduction to Logic. Oxford University Press.
Tennant, N. (2010). Harmony in a sequent setting. Analysis 70, 462–468.
Textor, M. (2017). Towards a neo-brentanian theory of existence. Philosophers’ Imprint 17, 1–20.
Troelstra, A. S. and H. Schwichtenberg (1996). Basic Proof Theory. Oxford: Oxford University Press.
Wang, H. (1960). Toward mechanical mathematics. IBM Journal of Research and Development 4, 2–22.
Wansing, H. (1999). Displaying Modal Logics. Dordrecht: Kluwer Academic Publishers.
Wehmeier, K. (2014). How to live without identity – and why. Australasian Journal of Philosophy 90, 761–777.
Więckowski, B. (2011). Rules for subatomic derivations. Review of Symbolic Logic 4, 219–236.
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Brace and Co., New York: Harcourt.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License (http://creativecommons.org/licenses/by-sa/4.0/), which permits use, sharing, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. If you remix, transform, or build upon this chapter or a part thereof, you must distribute your contributions under the same license as the original.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2024 The Author(s)
About this chapter
Cite this chapter
Indrzejczak, A. (2024). The Logicality of Equality. In: Piecha, T., Wehmeier, K.F. (eds) Peter Schroeder-Heister on Proof-Theoretic Semantics. Outstanding Contributions to Logic, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-031-50981-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-031-50981-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-50980-3
Online ISBN: 978-3-031-50981-0
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)