
140From the Knowability Paradox to the existence of proofsSynthese 176 (2). 2010.The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verificationtranscendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or MartinLöf using the sort of propositional moda…Read more

47The Paradox of the Knower revisitedAnnals of Pure and Applied Logic 165 (1): 199224. 2014.The Paradox of the Knower was originally presented by Kaplan and Montague [26] as a puzzle about the everyday notion of knowledge in the face of selfreference. The paradox shows that any theory extending Robinson arithmetic with a predicate K satisfying the factivity axiom K → A as well as a few other epistemically plausible principles is inconsistent. After surveying the background of the paradox, we will focus on a recent debate about the role of epistemic closure principles in the Knower. We…Read more

51Montague’s Paradox, Informal Provability, and Explicit Modal LogicNotre Dame Journal of Formal Logic 55 (2): 157196. 2014.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…Read more

30Incompleteness via paradox and completenessReview of Symbolic Logic 13 (3): 541592. 2020.This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of secondorder arithmetic and set theory in which various “paradoxical notions” for firstorder languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays and also how it was later adapted by Kreisel and Wan…Read more

11It is commonly held that the natural numbers sequence 0, 1, 2,... possesses a unique structure. Yet by a well known model theoretic argument, there exist nonstandard models of the formal theory which is generally taken to axiomatize all of our practices and intentions pertaining to use of the term “natural number.” Despite the structural similarity of this argument to the influential set theoretic indeterminacy argument based on the downward L ̈owenheimSkolem theorem, most theorists agree th…Read more

35Computational Complexity Theory and the Philosophy of Mathematics†Philosophia Mathematica 27 (3): 381439. 2019.Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the noti…Read more

11Bernays and the Completeness TheoremAnnals of the Japan Association for Philosophy of Science 25 4555. 2017.

83Arithmetical Reflection and the Provability of SoundnessPhilosophia Mathematica 23 (1): 3164. 2015.Prooftheoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truththeo…Read more