**(2017) A strong reflection principle. ***T**he 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).

**(2018) Modal structuralism and reflection.** *The Review of Symbolic Logic. *Online first view. 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.

**(2019) Classless.** *Analysis. *Online first. 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.

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