
111Relative categoricity and abstraction principlesReview of Symbolic Logic 8 (3): 572606. 2015.Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, wh…Read more

62The Nuisance Principle in Infinite SettingsThought: A Journal of Philosophy 4 (4): 263268. 2015.NeoFregeans have been troubled by the Nuisance Principle, an abstraction principle that is consistent but not jointly satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP

25Identifying finite cardinal abstractsPhilosophical Studies 178 (5): 16031630. 2021.Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap.ionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subjectmatter of arithmetic—there are too many numbers. That is, …Read more

16Abstraction Principles and the Classification of SecondOrder Equivalence RelationsNotre Dame Journal of Formal Logic 60 (1): 77117. 2019.This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a wellbehaved secondorder logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only …Read more

7Deductive cardinality results and nuisancelike principlesReview of Symbolic Logic 133. forthcoming.The injective version of Cantor’s theorem appears in full secondorder logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neoFregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, witho…Read more
Evanston, Illinois, United States of America
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
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