14 June 2022, The Utrecht Logic in Progress Series (TULIPS), Nils Kürbis

Abstract: According to Russell, a definite description is an expression of the form `The F’. They are incomplete symbols that mean nothing in isolation, and only in the context of complete sentences of the form `The F is G’, from which they disappear upon analysis. `The F is G’ is equivalent to `There is exactly one F and it is G’. For Russell, if there is no F, then any sentence of this form is false. Free logicians argue against Russell and that sometimes such sentences can be true. In free logic, definite descriptions are generally treated as genuine singular terms. But Russell surely had a point that we can draw distinctions of scope, for instance between `The F is not G’ and `It is not the case that the F is G’. This would require a means of introducing scope distinctions into the logic, which in free logic is generally not done. In my talk I will present a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier I. I forms a formula from two formulas. Ix[F, G] means `The F is G'. This approach has the advantage of incorporating scope distinctions directly into the notation. Cut elimination is proved for a system of classical positive free logic with I and it is shown to be sound and complete for the semantics. The system has a number of novel features and is briefly compared to the usual approach of formalising `the F' by a term forming operator in free logic. It does not coincide with Hintikka's and Lambert's preferred theories, but the divergence is well-motivated and attractive. Viewed from the perspective of proof theoretic semantics, the systems lets us decide various questions concerning the logic of definite descriptions in a principled way.