Abstract
This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bencivenga, E., Free logics, in D. Gabbay, and F. Guenther, (eds.), Handbook of Philosophical Logic. Volume III: Alternatives to Classical Logic, Springer, Dortrecht, 1986, pp. 373–426.
Bostock, D., Intermediate Logic, Clarendon Press, Oxford, 1997.
Czermak, J., A logical calculus with definite descriptions, Journal of Philosophical Logic 3(3): 211–228, 1974.
Dummett, M., Frege. Philosophy of Language, 2 ed., Duckworth, London, 1981.
Fitting, M., and R. L. Mendelsohn, First-Order Modal Logic. Kluwer, Dordrecht, Boston, London, 1998.
Garson, J. W., Modal Logic for Philosophers, 2 ed., Cambridge University Press, 2013.
Gratzl, N., Incomplete symbols – definite descriptions revisited, Journal of Philosophical Logic 44(5): 489–506, 2015.
Hintikka, J., Towards a theory of definite descriptions, Analysis 19(4): 79–85, 1959.
Indrzejczak, A., Cut-free modal theory of definite descriptions, in G.M.G. Bezhanishvili, G. D’Agostino, and T. Studer, (eds.), Advances in Modal Logic, Vol. 12, College Publications, London, 2018a, pp. 359–378.
Indrzejczak, A., Fregean description theory in proof-theoretical setting, Logic and Logical Philosophy 28(1): 137–155, 2018b.
Indrzejczak, A., Existence, definedness and definite descriptions in hybrid modal logic, in N. Olivetti, R. Verbrugge, S. Negri, and G. Sandu, (eds.), Advances in Modal Logic 13, College Publications, Rickmansworth, 2020a, pp. 349–368.
Indrzejczak, A., Free definite description theory - sequent calculi and cut elimination, Logic and Logical Philosophy 29(4): 505–539, 2020b.
Indrzejczak, A., Free logics are cut free, Studia Logica 109(4): 859–886, 2021.
Kürbis, N., A binary quantifier for definite descriptions in intuitionist negative free logic: Natural deduction and normalisation, Bulletin of the Section of Logic 48(4): 81–97, 2019a.
Kürbis, N., Two treatments of definite descriptions in intuitionist negative free logic, Bulletin of the Section of Logic 48(4): 299–318, 2019b.
Kürbis, N., Definite descriptions in intuitionist positive free logic, Logic and Logical Philosophy 30(2): 327–358, 2021.
Lambert, K., Notes on “E!”: II, Philosophical Studies 12(1/2): 1–5, 1961.
Lambert, K., Notes on “E!” III, A theory of descriptions, Philosophical Studies 13(4): 51–59, 1962.
Lambert, K., Notes on “E!” IV: A reduction in free quantification theory with identity and descriptions, Philosophical Studies 15(5): 85–88, 1964.
Lambert, K., Free logic and definite descriptions, in E. Morscher, and A. Hieke, (eds.), New Essays in Free Logic in Honour of Karel Lambert, Kluwer, Dordrecht, 2001, pp. 37–47.
Morscher, E., and P. Simons, Free logic: A fifty-year past and an open future, in E. Morscher, and A. Hieke, (eds.), New Essays in Free Logic in Honour of Karel Lambert, Kluwer, Dordrecht, 2001, pp. 1–34.
Neale, S., Descriptions, MIT Press, Cambridge, Mass, 1990.
Ramsey, F. P., Philosophy, in H. Mellor, (ed.), Philosophical Papers, Cambridge University Press, 1990, pp. 1–7.
Russell, B., and A. N. Whitehead, Principia Mathematica, Vol. 1, Cambridge University Press, 1910.
Troestra, A., and H. Schwichtenberg, Basic Proof Theory, 2 ed., Cambridge University Press, 2000.
van Fraassen, B. C., On (the x) (x = Lambert), in W. Spohn, B.C. van Fraassen, and B. Skyrms, (eds.), Existence and Explanation. Essays presented in Honor of Karel Lambert, Kluwer, Dordrecht, Boston, London, 1991, pp. 1–18.
Acknowledgements
I would like to thank Andrzej Indrzejczak for his comments on this paper and a referee for Studia Logica, who also made helpful suggestions for improvement. This paper was written while I was an Alexander von Humboldt Fellow at the University of Bochum. To both institutions many thanks are due.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Jacek Malinowski
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, 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 licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence 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. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kürbis, N. A Binary Quantifier for Definite Descriptions for Cut Free Free Logics. Stud Logica 110, 219–239 (2022). https://doi.org/10.1007/s11225-021-09958-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-021-09958-x