A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation

  • Nils Kürbis Department of Philosophy, University College London, London, UK
Keywords: definite descriptions, negative intuitionist free logic, natural deduction, normalization


This paper presents a way of formalising definite descriptions with a binary quantifier ℩, where ℩x[F, G] is read as `The F is G'. Introduction and elimination rules for ℩ in a system of intuitionist negative free logic are formulated. Procedures for removing maximal formulas of the form ℩x[F, G] are given, and it is shown that deductions in the system can be brought into normal form.


[1] D. Bostock, Intermediate Logic, Oxford: Clarendon Press, 1997.

[2] M. Dummett, Frege. Philosophy of Language, 2 ed., Cambridge: Harvard University Press, 1981.

[3] A. Indrzejczak, Cut-Free Modal Theory of Definite Descriptions, [in:] Advances in Modal Logic, G. Bezhanishvili, G. D'Agostino, G. Metcalfe and T. Studer (eds.), vol. 12, pp. 359–378, London: College Publications, 2018.

[4] A. Indrzejczak, Fregean Description Theory in Proof-Theoretical Setting, Logic and Logical Philosophy, vol. 28, no. 1 (2018), pp. 137–155. http://dx.doi.org/10.12775/LLP.2018.008

[5] D. Prawitz, Natural Deduction: A Proof-Theoretical Study, Stockholm, Göteborg, Uppsala: Almqvist and Wiksell, 1965.

[6] D. Scott, Identity and Existence in Intuitionistic Logic, [in:] Applications of Sheaves, Michael Fourman, Christopher Mulvery, Dana Scott (eds.), Berlin, Heidelberg, New York: Springer, 1979. https://doi.org/10.1007/BFb0061839

[7] N. Tennant, A General Theory of Abstraction Operators, The Philosophical Quarterly, vol. 54, no. 214 (2004), pp. 105–133. https://doi.org/10.1111/j.0031-8094.2004.00344.x

[8] N. Tennant, Natural Logic, Edinburgh: Edinburgh University Press, 1978.

[9] A. S. Troelstra and H. Schwichtenberg, Basic Proof Theory, Cambridge University Press, 2 ed., 2000. https://doi.org/10.1017/CBO9781139168717

How to Cite
Kürbis, N. (2019). A Binary Quantifier for Definite Descriptions in Intuitionist Negative Free Logic: Natural Deduction and Normalisation. Bulletin of the Section of Logic, 48(2), 81-97. https://doi.org/10.18778/0138-0680.48.2.01
Research Article