The goal of this paper is to explore the significance of Montague’s paradox—that is, any arithmetical theory $T\supseteq Q$ over a language containing a predicate $P$ satisfying $P\rightarrow \varphi $ and $T\vdash \varphi \,\therefore\,T\vdash P$ is inconsistent—as a limitative result pertaining to the notions of formal, informal, and constructive provability, in their respective historical contexts. To this end, the paradox is reconstructed in a quantified extension $\mathcal {QLP}$ of Artemov’s logic of proofs. $\mathcal {QLP}$ contains both explicit modalities $t:\varphi $ (...) and also proof quantifiers $x:\varphi $. In this system, the basis for the rule NEC is decomposed into a number of distinct principles governing how various modes of reasoning about proofs and provability can be internalized within the system itself. A conceptually motivated resolution to the paradox is proposed in the form of an argument for rejecting the unrestricted rule NEC on the basis of its subsumption of an intuitively invalid principle pertaining to the interaction of proof quantifiers and the proof-theorem relation expressed by explicit modalities. (shrink)

In “Properties and the Interpretation of Second-Order Logic” Bob Hale develops and defends a deflationary conception of properties where a property with particular satisfaction conditions actually exists if and only if it is possible that a predicate with those same satisfaction conditions exists. He argues further that, since our languages are finitary, there are at most countably infinitely many properties and, as a result, the account fails to underwrite the standard semantics for second-order logic. Here a more lenient version of (...) the view is explored, which allows for the possibility of countably infinite predicates understood as the product of linguistic supertasks. This enriched deflationist account of properties—the Infinitary Deflationary Conception of Existence—supports the standard semantics for models with countable first-order domains, and allows one to prove the categoricity of the second-order Peano axioms. (shrink)