
188Carnap: an Open Framework for Formal Reasoning in the BrowserElectronic Proceedings in Theoretical Computer Science 267 7088. 2018.This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnapbased applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where easeofuse is crucial for students and adaptability to different teaching strategies and…Read more

165StructuralAbstraction PrinciplesPhilosophia Mathematica. 2015.In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neologicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neologicists call the ‘bad company’ problem for structural abstractio…Read more

65Yablifying the Rosser SentenceJournal of Philosophical Logic 43 (5): 827834. 2014.In a recent paper , Urbaniak and Cieśliński describe an analogue of the Yablo Paradox, in the domain of formal provability. Just as the infinite sequence of Yablo sentences inherit the paradoxical behavior of the liar sentence, an infinite sequence of sentences can be constructed that inherit the distinctive behavior of the Gödel sentence. This phenomenon—the transfer of the properties of selfreferential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—…Read more

61What Russell Should Have Said to Burali–FortiReview of Symbolic Logic 10 (4): 682718. 2017.The paradox that appears under BuraliForti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of BuraliForti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years…Read more

37BuraliForti as a Purely Logical ParadoxJournal of Philosophical Logic 48 (5): 885908. 2019.Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all nonselfmembered sets, in pure firstorder logic—the firstorder logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all nonselfmembered sets. BuraliF…Read more

29Generalizing boolos’ theoremReview of Symbolic Logic 10 (1): 8091. 2017.It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of secondorder Peano Arithmetic in the theory ￼ HP2, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of HP2￼ in PA2 . Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of ￼PA2 to be recovered from some model of ￼HP2. So the space of possi…Read more

Coalgebra And AbstractionNotre Dame Journal of Formal Logic 62 (1): 3366. 2021.Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of secondorder ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of Falgebras and coalgebras. In particular, we show how Boolos’s construction amounts to id…Read more

Provability, Mechanism, and the Diagonal ProblemIn Leon Horsten & Philip Welch (eds.), Gödel's Disjunction: the Scope and Limits of Mathematical Knowledge. pp. 211240. 2016.
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Philosophy of Computing and Information 