DEPARTMENT OF PHILOSOPHY

Published articles

• Explicit Abstract Objects in Predicative Settings. Journal of Philosophical Logic 53 (5): 1347-1382. 2024. With Francesca Boccuni.
  Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 1…Read more

  Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society 87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula $$\phi (F)$$, with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA $$^2$$.
• What is the aim of (contradictory) Christology? In Jonathan C. Rutledge (ed.), Paradox and Contradiction in Theology, Routledge Academic. pp. 33-51. 2023.
  How good a theory is depends on how well it meets the goals of its inquiry.  Thus, for example, theories in the natural sciences are better if in addition to stating truths, they also impart a kind of understanding.  Recent proposals—such as Jc Beall’s Contradictory Christology—to set Christian theology within non-classical logic  should be judged in a like manner:  according to how well they meet…Read more

  How good a theory is depends on how well it meets the goals of its inquiry.  Thus, for example, theories in the natural sciences are better if in addition to stating truths, they also impart a kind of understanding.  Recent proposals—such as Jc Beall’s Contradictory Christology—to set Christian theology within non-classical logic  should be judged in a like manner:  according to how well they meet the goals of Christology.  This paper examines some of the effects of changing the logic of Christology on meeting the proposed goals of that inquiry.  The paraconsistent logic Beall favors has (FDE) features that inhibit increasing our understanding of the Incarnation.  Of course, there is good reason to think understanding is not the goal of Christological inquiry, that its goals are (for lack of a better word) devotional.  But FDE’s features don’t foster these goals either—raising questions about how much of a theological advance proposals like Beall’s could be. [NOTE: The published version has some formatting errors, specifically with respect to block quotes. Please cite the published version, but if you are confused, consult the correctly formatted version on PhilArchive]
• On Number-Set Identity: A Study. Philosophia Mathematica 30 (2): 223-244. 2022.
  Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspec…Read more

  Benacerraf’s 1965 multiple-reductions argument depends on what I call ‘deferential logicism’: his necessary condition for number-set identity is most plausible against a background Quineanism that allows autonomy of the natural number concept. Steinhart’s ‘folkist’ sufficient condition on number-set identity, by contrast, puts that autonomy at the center — but fails for not taking the folk perspective seriously enough. Learning from both sides, we explore new conditions on number-set identity, elaborating a suggestion from Wright.
• Deductive Cardinality Results and Nuisance-Like Principles. Review of Symbolic Logic 14 (3): 592-623. 2021.
  The injective version of Cantor’s theorem appears in full second-order 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 neo-Fregean philosophers of mathematics. This paper u…Read more

  The injective version of Cantor’s theorem appears in full second-order 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 neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.
• Identifying finite cardinal abstracts. Philosophical Studies 178 (5): 1603-1630. 2020.
  Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists 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 determ…Read more

  Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists 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 subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2.
• Abstraction Principles and the Classification of Second-Order Equivalence Relations. Notre Dame Journal of Formal Logic 60 (1): 77-117. 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 well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—thos…Read more

  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 well-behaved second-order 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 three profiles. The improvements to Fine’s theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem.
• The Nuisance Principle in Infinite Settings. Thought: A Journal of Philosophy 4 (4): 263-268. 2015.
  Neo-Fregeans 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
• Relative categoricity and abstraction principles. Review of Symbolic Logic 8 (3): 572-606. 2015. With Sean Walsh.
  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 traditio…Read more

  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, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work.