Talk: Nils Kürbis (Bochum/Lodz/University College London)

Title:

120 Year of Russell's 'On Denoting’

Abstract:

October 2025 sees the 120th anniversary of the publication of Bertrand Russell's 'On Denoting’ [1]. It is undeniably one of the most influential papers in logic and analytic philosophy. It is most famous for the theory of definite descriptions. The heart of this theory is that definite descriptions, expressions of the form 'the $F$', are not singular terms and only have meaning in the context of complete sentences 'The F is G'. The purpose of this talk is to celebrate Russell's paper by presenting new work on the proof theory of definite descriptions in natural deduction and sequent calculus. First, I present a formalisation of Neale's reconstruction of Russell's theory [2], in which definite descriptions are characterised as quantificational phrases, by rules of inference for a binary quantifier I to be added to classical logic. Ix(Fx, Gx) formalises 'The F is G' and is equivalent to 'There is exactly one F and it is G'. Secondly and thirdly, I show how the same or slightly modified rules can be added to positive or negative free logic. In the former, the requirement of existence is given up, in the latter, it is retained. Deductions in all systems of natural deduction normalise, cut is admissible in sequent systems. Russell came to regard as a defect of its logic that theorems are provable in Principia Mathematica that imply that at least one particular exists [3]. He would have been sympathetic to negative free logic.

[1] Bertrand Russell, On Denoting, Mind, vol.14 (1905), no.56, pp.479-493
[2] Stephen Neale, Descriptions, Harvard: MIT Press 1993
[3] Bertrand Russell, Introduction to Mathematical Philosophy, George Allen & Unwin 1919 (Reprinted: Routledge 1993)