CEFISES Seminar (extra): Nils Kürbis, « Some Systems for Formalising Sentences Containing Definite Descriptions by a Binary Quantifier and some Thoughts on Modal Logic »

Livestream https://youtu.be/Hq5irGh_1X4

Series: Logic&Philosophy

Speaker: Nils Kürbis (RUB Bochum)

Title: « Some Systems for Formalising Sentences Containing Definite Descriptions by a Binary Quantifier and some Thoughts on Modal Logic. »

Place: [not in the usual Salle Ladrière] Class room Socrate 25, Bâtiment SOCR, Place Cardinal Mercier, Louvain-la-Neuve

Date and time: Tuesday 4/7 from 14:00 to 16:00.

Abstract:

A definite description is an expression of the form ‘the so-and-so’. On the face of it, definite descriptions are singular terms: they purport to refer to the sole so-and-so. Since Peano it is customary to formalise them by a term forming operator: 𝜾 takes an open formula F and forms the singular term 𝜾xF, binding the variable x. This accords with the grammar of ordinary English, where the definite article ‘the’ forms a complex expression that can be the subject or object of a sentence, e.g. `the present King of France’, from a noun phrase, in this case `present King of France’. It is probably the most common way of treating definite descriptions in formal logic.

There is, however, an alternative method that builds on a distinctly Russellian point. According to Russell, a definite description has no meaning in itself, but only in the context of a complete sentence. This is because upon Russell’s celebrated analysis of sentences of the form ‘The F is G’ the definite description ‘the F’ disappears: ‘The F is G’ means ‘There is exactly one F and it is G’. Two things are worth noting. (1) Russell, too, utilises the term forming operator, but its use is defined in the context of formulas only. [𝜾xF] G(𝜾xF) is contextually defined as ∃x(∀y(Fy↔︎x=y) ∧ Gx). (2) the need for scope distinctions in the contextual definition.

The alternative method consists instead of formalising complete sentences containing definite descriptions with a binary quantifier I: I takes two open formulas and forms a formula out of them, binding a variable. Ix[F, G] formalises ‘The F is G’.

I will present a number of options for rules of inference governing I in natural deduction for positive and negative free logic. Two systems follow an established account by Lambert, adjusted to natural deduction by Tennant, of the formalisation of definite descriptions with a term forming operator quite closely, with one crucial difference: the binary quantifier permits the marking of scope distinction, while the established account avoids these. The systems are therefore not directly comparable, but there is considerable overlap. The rules for I are suitable for negative free logic. They are not, however, in the spirit of positive free logic. Thus I will also present a set of rather complicated rules suitable for positive free logic. The resulting theory is new, and thus of some interest, but it must be admitted that a much simpler theory is possible. It results by ignoring the existence assumptions in the rules for I for negative free logic. I will present this theory at the end of the talk and briefly consider the effect of adding the rules to modal logic.