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188Logic in the deep endAnalysis 84 (2): 282-291. 2024.Weak enough relevant logics are often closed under depth substitutions. To determine the breadth of logics with this feature, we show there is a largest sublogic of R closed under depth substitutions and that this logic can be recursively axiomatized.
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388Hyperdoctrine Semantics: An InvitationIn Shay Logan & Graham Leach-Krouse (eds.), The Logica Yearbook, 2021, College Publications. pp. 115-134. 2022.Categorial logic, as its name suggests, applies the techniques and machinery of category theory to topics traditionally classified as part of logic. We claim that these tools deserve attention from a greater range of philosophers than just the mathematical logicians. We support this claim with an example. In this paper we show how one particular tool from categorial logic---hyperdoctrines---suggests interesting metaphysics. Hyperdoctrines can provide semantics for quantified languages, but this …Read more
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325On Not Saying What We Shouldn't Have to SayAustralasian Journal of Logic 18 (5): 524-568. 2021.In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.
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1Coalgebra And AbstractionNotre Dame Journal of Formal Logic 62 (1): 33-66. 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 second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to id…Read more
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Provability, Mechanism, and the Diagonal ProblemIn Leon Horsten & Philip Welch (eds.), Godel's Disjunction: The Scope and Limits of Mathematical Knowledge, Oxford University Press Uk. pp. 211-240. 2016.
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613Carnap: an Open Framework for Formal Reasoning in the BrowserElectronic Proceedings in Theoretical Computer Science 267 70-88. 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. Carnap-based 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 ease-of-use is crucial for students and adaptability to different teaching strategies and…Read more
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56Generalizing boolos’ theoremReview of Symbolic Logic 10 (1): 80-91. 2017.It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order 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
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211Structural-Abstraction 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 neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstractio…Read more
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128What Russell Should Have Said to Burali–FortiReview of Symbolic Logic 10 (4): 682-718. 2017.The paradox that appears under Burali-Forti’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 Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years…Read more
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73Burali-Forti as a Purely Logical ParadoxJournal of Philosophical Logic 48 (5): 885-908. 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 non-self-membered sets, in pure first-order logic—the first-order 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 non-self-membered sets. Burali-F…Read more
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97Yablifying the Rosser SentenceJournal of Philosophical Logic 43 (5): 827-834. 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 self-referential sentences of formal mathematics to their “unwindings” into infinite sequences of sentences—…Read more
Areas of Specialization
Logic and Philosophy of Logic |
Philosophy of Mathematics |
Philosophy of Computing and Information |