Higher-order free logic and the Prior-Kaplan paradox

Abstract

1 See Russell [22], Ramsey [21], Kripke [12], and Kaplan [11], Tucker and Thomason [26]. 2 Note that there are many different ways of implementing the ramificationist approach (it's very natural to only allow types (τ1, ..., τn)/k where k is greater than the levels of τ1, ..., τn). In what follows we shall be following Church [4] in considering only cumulative ramified type theories: informally a relation that accepts arguments of a given level, accepts arguments of any lower level. For the connection between this and the various versions of Russell's original theory see footnote 2 of [4]. 3 We assume that applying the logical connectives does not take us to a higher level. 4 Provided B contains no free occurrences of p, and no free variables that would become bound if substituted for p in A; in what follows we will take this condition to be tacitly assumed. 5 Without loss of generality we give the proof of one direction (up to a change of variables):1. (∀pA → A(q/p)) (instance of UI for ∀.)2. Πq(∀pA → A(q/p)) (by universal generalisation for Π.) 3. ∀pA → ΠqA(q/p) (by the principle Πp(A → B) → (A → ΠpB) when p is not free in A.) 6 See also McGee [27] for a discussion of the second-order variant. 7 It's arguable that the rejection of UI is also the key part to a Russell-style treatment of many other paradoxes, but we shall not defend the broader claim here. 8 Suppose that p ≡ p. Then by EG, we can infer ∃q p ≡ q. We then have ∃q p ≡ q by necessitation, ∀p ∃q p ≡ q by universal generalisation, and finally ∀p ∃q p ≡ q by necessitation again.

To Be F Is To Be G

“…See Bacon, Hawthorne and Uzquiano , §7 for a survey of some of the forms that ramification might take, including an approach that (unlike that of Whitehead and Russell) keeps the syntax of the language intact and merely replaces each of our quantifiers with a hierarchy of “restricted” quantifiers. For considerations against ramification, see Ramsey and Prior , ch.…”

Paradoxes and Restricted Quantification

“…Andrew Bacon, John Hawthorne, and Gabriel Uzquiano (Bacon, Hawthorne, and Uzquiano ) have recently considered a number of ways of rejecting or restricting Universal Instantiation (UI) and argued that they are ultimately not promising approaches to resolving a family of intensional paradoxes due to Arthur Prior (Prior ). I present a novel approach to the paradoxes by describing models that validate a restricted form of UI and avoid their concerns.…”

Radical Anti‐disquotationalism

“…For a recent approach to the liar that relaxes classical propositional logic, see Priest [], Brady [], Field []. For a discussion of approaches that relax the classical rules for the propositional quantifiers see Bacon, Hawthorne and Uzquiano [].…”