papers

published

(2024) Truth and finite conjunction (with Leon Horsten and Guanglong Lou). Mind, 133 (532), pp. 1121–1135.

This note is a critical response to Kentaro Fujimoto’s new conservativeness argument about truth, which centres on the notion of finite conjunction. We argue that Fujimoto’s arguments turn on a specific way of formalizing the notions of finite collection and finite conjunction in first-order logic. In particular, by instead formalizing these concepts in a natural way in set theory or in second-order logic, Fujimoto’s new conservativeness argument can be resisted.

(2024) No easy road to impredicative definabilism (with Øystein Linnebo). Philosophia Mathematica, 32 (1), pp. 21-33.

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that it fails to characterize the domain uniquely. Thus, we conclude, there is no easy road to impredicative definabilism.

(2023) Hume’s principle, bad company, and the axiom of choice (with Stewart Shapiro). The Review of Symbolic Logic, 16 (4), pp. 1158-1176. Penultimate draft: PDF.

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject.

(2022) Pluralities as nothing over and above. Journal of Philosophy, 119 (8), pp. 405–424. Penultimate draft: PDF.

This paper develops an account of pluralities based on the following simple claim: some things are nothing over and above the individual things they comprise. For some, this may seem like a mysterious statement, perhaps even meaningless; for others, like a truism, trivial and inferentially inert. I show that neither reaction is correct: the claim is both tractable and has important consequences for a number of debates in philosophy.

(2020) Classless. Analysis, 80(1), pp. 76-83. Penultimate draft: PDF.

This note proves a new conservativity result for class theories. It tells us that as long as our set theory T contains an independently well-motivated reflection principle, anything provable about the sets in any reasonable class theory extending T is already provable in T itself. 

(2019) Modal structuralism and reflection. The Review of Symbolic Logic, 12(4), pp. 823-860. Penultimate draft: PDF.

This paper investigates the assumptions underlying modal structuralism, and looks at the prospects for supplementing them with a reflection principle. It shows that the viability of modal structuralism about set theory turns on a non-trivial assumption — the Stability principle — about the behaviour of structures across modal space. Once this assumption is accepted, however, I show that the modal structuralist can make sense of a significant fragment of set theory. The axiom schema of Replacement requires further assumptions, though, and I show that a recent proposal to use reflection principles to obtain it fails.

(2017) A strong reflection principle. The Review of Symbolic Logic, 10(4), pp. 651-662. Penultimate draft: PDF.

This paper introduces a new reflection principle. It says that whatever is true in all entities of some kind is also true in a small collection of them. When applied to sets and classes, it turns out to be remarkably strong (implying that there are so-called 1-extendible cardinals).

drafts

Ultimate V.

Potentialism is the view that the universe of sets is inherently potential. It comes in two main flavours: height-potentialism and width-potentialism. It is natural to think that height and width potentialism are just aspects of a broader phenomenon of potentialism, that they might both be true. The main result of this paper is that this is mistaken: height and width potentialism are jointly inconsistent. Indeed, I argue that height potentialism is independently committed to an ultimate background universe of sets, an ultimate V, up to its height.

Possibilities for sets.

As central as the method of forcing is within set theory, it has yet to be incorporated into the philosopher’s toolbox. That strikes me as a shame, since it may well have important applications within philosophy. One barrier is that typical presentations of forcing are overly dry and technical and make it seem inherently bound up with its applications within set theory. The purpose of this note is to try to rectify this.  In particular, I will explain how the method of forcing can be seen as a way of constructing a certain kind of intensional model that philosophers are already interested in: namely, possibility models.

Sets as structures.

Structuralism is the view that mathematics is about structures. According to the orthodoxy, mathematical objects like natural numbers and sets are places in structures. In this paper, I propose a new idea: namely, that mathematical objects like natural numbers and sets are structures. I will focus almost exclusively on the case of sets. So, the proposal is that: sets are structures. More precisely: the proposal is that we should think of sets as well-founded extensional structures with a top element.

slides

A Potential Hierarchy of Properties.

I develop a potentialist view of untyped properties based on the idea that all that’s required for the possible existence of a property is the possible availability of its definition. I provide a formal theory for the view and show that it delivers a rich universe of potential properties. In particular, I show that it delivers enough potential properties to satisfy the full second-order comprehension schema.