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A novel approach to equality

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Abstract

A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance with Dos̆en’s criteria for logical constants. The latter is a standard Gentzen’s sequent calculus and satisfies Hacking’s criteria for logicality, including cut elimination. It is also shown that provided rules are harmonious in the sense advocated by Gratzl and Orlandelli.

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Notes

  1. It should be noted however that it is possible to develop Wittgenstein’s view on identity in a formally satisfactory way as witnessed by Hintikka (1956) or Wehmeier (2014).

  2. See e.g. Troelstra and Schwichtenberg (1996) or Negri and von Plato (2001).

  3. See e.g. Schroeder-Heister (2012), Poggiolesi (2011), Francez (2015) or Kürbis (2019).

  4. For example, Negri and von Plato (2001) applied \((2=)\), Manzano (1999) used \((3=)\), Reeves (1987) \((4=)\), Indrzejczak (2019) \((5=)\) and Indrzejczak (2018a) \((8=)\), Nagashima (1966) \((7=)\).

  5. For example by Gallier (1986), Negri and von Plato (2001) and Nagashima (1966).

  6. See e.g. Ciabattoni et al. (2010), Indrzejczak (2015) and Kurokawa (2014).

  7. See e.g. Indrzejczak (2016, 2018a).

  8. Strictly speaking in Reeves’ case it is n conclusions since he worked with tableux being upside-down versions of SC.

  9. The terminology is borrowed from Troelstra and van Dalen (1988); in the context of modal logic the former is called an indistinguishability by Goldblatt (2011), and the latter, term equality by Fitting and Mendelsohn (1998).

References

  • Avron, A., & Lev, I. (2001). Canonical propositional gentzen-type systems. In Proceedings of IJCAR’01. LNCS (Vol. 2083, pp. 529–543).

  • Carlström, J. (2005). Interpreting descriptions in intensional type theory. The Journal of Symbolic Logic, 70(2), 488–514.

    Google Scholar 

  • Ciabattoni, A., Metcalfe, G., & Montagna, F. (2010). Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. Fuzzy Sets and Systems, 161(3), 369–389.

    Google Scholar 

  • Dos̆en, K. (1985). Sequent-systems for modal logic. Journal of Symbolic Logic, 50, 149–159.

    Google Scholar 

  • Dos̆en, K. (1989). Logical constants as punctuation marks. Notre Dame Journal of Formal Logic, 30, 362–381.

    Google Scholar 

  • Fitting, M., & Mendelsohn, R. L. (1998). First-order modal logic. Dordrecht: Kluwer.

    Google Scholar 

  • Francez, N. (2015). Proof-theoretic semantics. London: College Publications.

    Google Scholar 

  • Frege, G. (1892). Über Sinn und Bedeutung. Zeitschrift für Philosophie und Philosophische Kritik, 100, 25–50.

    Google Scholar 

  • Gallier, J. H. (1986). Logic for computer science. New York: Harper and Row.

    Google Scholar 

  • Gazzari, R. (2021). The calculus of natural calculation. Studia Logica (to appear).

  • Goldblatt, R. (2011). Quantifiers, propositions and identity. Cambridge: Cambridge University Press.

    Google Scholar 

  • Gratzl, N., & Orlandelli, E. (2017). Double-line harmony in sequent calculi. In P. Arazim & T. Lavicka (Eds.), The logica yearbook 2016 (pp. 157–171). Rickmansworth: College Publications.

    Google Scholar 

  • Hacking, I. (1979). What is logic? The Journal of Philosophy, 76, 285–319.

    Google Scholar 

  • Hintikka, J. (1956). Identity, variables and impredicative definitions. The Journal of Symbolic Logic, 21(3), 225–245.

    Google Scholar 

  • Indrzejczak, A. (2015). Eliminability of cut in hypersequent calculi for some modal logics of linear frames. Information Processing Letters, 115(2), 75–81.

    Google Scholar 

  • Indrzejczak, A. (2016). Simple cut elimination proof for hybrid logic. Logic and Logical Philosophy, 25(2), 129–141.

    Google Scholar 

  • Indrzejczak, A. (2018). Cut-free modal theory of definite descriptions. In G. Bezhanishvili, et al. (Eds.), Advances in modal logic (Vol. 12, pp. 387–406). Rickmansworth: College Publications.

    Google Scholar 

  • Indrzejczak, A. (2018). Rule-generation theorem and its applications. The Bulletin of the Section of Logic, 47(4), 265–281.

    Google Scholar 

  • Indrzejczak, A. (2019). Fregean description theory in proof-theoretical setting. Logic and Logical Philosophy, 28(1), 137–155.

    Google Scholar 

  • Indrzejczak, A. (2021). Logicality of equality. In P. Schroeder-Heister (Ed.), Proof-theoretic semantics. Outstanding contributions to logic. Berlin: Springer. (to appear).

    Google Scholar 

  • Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 5–32.

    Google Scholar 

  • Kanger, S. (1957). Provability in logic. Stockholm: Almqvist & Wiksell.

    Google Scholar 

  • Kurokawa, H. (2014). Hypersequent calculi for modal logics extending S4. In New frontiers in artificial intelligence (pp. 51–68). Springer.

  • Kürbis, N. (2019). Proof and falsity: A logical investigation. Cambridge: Cambridge University Press.

    Google Scholar 

  • Manzano, M. (1999). Model theory. Oxford: Oxford University Press.

    Google Scholar 

  • Manzano, M. (2005). Extensions of first-order logic. Cambridge: Cambridge University Press.

    Google Scholar 

  • Manzano, M., & Moreno, M. C. (2017). Identity, equality, nameability and completeness. The Bulletin of the Section of Logic, 46(3–4), 169–196.

    Google Scholar 

  • Maruyama, Y. (2016). Categorical harmony and paradoxes in proof-theoretic semantics. In T. Piecha & P. Schroeder-Heister (Eds.), Advances in proof-theoretic semantics (pp. 95–114). Berlin: Springer.

    Google Scholar 

  • Metcalfe, G., Olivetti, N., & Gabbay, D. (2008). Proof theory for fuzzy logics. Berlin: Springer.

    Google Scholar 

  • Nagashima, T. (1966). An extension of the Craig-Schütte interpolation theorem. Annals of the Japan Association for the Philosophy of Science, 3, 12–18.

    Google Scholar 

  • Negri, S., & von Plato, J. (2001). Structural proof theory. Cambridge: Cambridge University Press.

    Google Scholar 

  • Parlamento, F., & Previale, F. (2019). The elimination of atomic cuts and the semishortening property for Gentzen’s sequent calculus with equality. The Review of Symbolic Logic,. https://doi.org/10.1017/S175502031900039X. (on-line first).

    Article  Google Scholar 

  • Poggiolesi, F. (2011). Gentzen calculi for modal propositional logic. Berlin: Springer.

    Google Scholar 

  • Priest, G. (2010). Non-transitive identity. In R. Dietz & S. Moruzzi (Eds.), Cuts and clouds: Vagueness, its nature and its logic (pp. 406–416). Oxford: Oxford University Press.

    Google Scholar 

  • Quine, W. V. (1970). Philosophy of logic. Upper Saddle River: Prentice Hall.

    Google Scholar 

  • Read, S. (2004). Identity and harmony. Analysis, 64(2), 113–119.

    Google Scholar 

  • Reeves, S. V. (1987). Adding equality to semantic tableau. Journal of Automated Reasoning, 3, 225–246.

    Google Scholar 

  • Restall, G. (2019a). Generality and existence 1: Quantification and free logic. The Review of Symbolic Logic, 12(1), 354–378.

  • Restall, G. (2019b). Assertions, denials, questions, answers and the common ground. Retrieved from http://consequently.org/writing. Accessed 20 Oct 2019.

  • Sambin, G., Battilotti, G., & Faggian, C. (2000). Basic logic: Reflection, symmetry, visibility. Journal of Symbolic Logic, 65, 979–1013.

    Google Scholar 

  • Schroeder-Heister, P. (2012). Proof-theoretic semantics. In Stanford encyclopedia of philosophy. Retrieved from https://plato.stanford.edu/entries/proof-theoretic-semantics/. Accessed 12 Nov 2018.

  • Schroeder-Heister, P. (2016). Open problems in proof-theoretic semantics. In T. Piecha & P. Schroeder-Heister (Eds.), Advances in proof-theoretic semantics (pp. 253–283). Berlin: Springer.

    Google Scholar 

  • Stenlund, S. (1973). The logic of description and existence. Filosofiska Studier no 18, Uppsala.

  • Stenlund, S. (1975). Descriptions in intuitionistic logic. In S. Kanger (Ed.), Proceedings of the third Scandinavian logic symposium (pp. 197–212).

  • Tennant, N. (2010). Harmony in a sequent setting. Analysis, 70, 462–468.

    Google Scholar 

  • Textor, M. (2017). Towards a Neo-Brentanian theory of existence. Philosophers’ Imprint, 17(6), 1–20.

    Google Scholar 

  • Troelstra, A. S., & van Dalen, D. (1988). Constructivism in mathematics. Amsterdam: North-Holland.

    Google Scholar 

  • Troelstra, A. S., & Schwichtenberg, H. (1996). Basic proof theory. Oxford: Oxford University Press.

    Google Scholar 

  • Wansing, H. (1999). Displaying modal logics. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Wansing, H., & Skurt, D. (2019). On non-transitive “identity”. In C. Baskent & T. M. Fergusson (Eds.), Graham priest on dialetheism and paraconsistency. Outstanding contributions to logic (Vol. 18, pp. 535–553). Berlin: Springer.

    Google Scholar 

  • Wehmeier, K. (2014). How to live without identity- and why. Australasian Journal of Philosophy, 90, 761–777.

    Google Scholar 

  • Wittgenstein, L. (1922). Tractatus logico-philosophicus. New York: Harcourt Brace and Co.

    Google Scholar 

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Acknowledgements

I am greatly indebted to Nils Kürbis, René Gazzari, and to all reviewers for insightful remarks which helped to improve the final version of this text. The results reported in this paper are supported by the National Science Centre, Poland (Grant Number: DEC-2017/25/B/HS1/01268).

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  1. Department of Logic, University of Łódź, Lindleya 3/5, 90-131, Łódź, Poland

    Andrzej Indrzejczak

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Indrzejczak, A. A novel approach to equality. Synthese 199, 4749–4774 (2021). https://doi.org/10.1007/s11229-020-03000-8

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