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110If A then B: How the World Discovered LogicHistory and Philosophy of Logic 35 (3): 301-303. 2014.If A then B: How the World Discovered Logic is a historically oriented introduction to the basic notions of logic. In particular, and in the words of the authors, it is focused on the idea that ‘lo...
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109Patricia A. Blanchette. Frege's Conception of Logic. Oxford University Press, 2012. ISBN 978-0-19-926925-9 (hbk). Pp. xv + 256 (review)Philosophia Mathematica (1). 2013.
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107The No-No Paradox Is a ParadoxAustralasian Journal of Philosophy 89 (3): 467-482. 2011.The No-No Paradox consists of a pair of statements, each of which ?says? the other is false. Roy Sorensen claims that the No-No Paradox provides an example of a true statement that has no truthmaker: Given the relevant instances of the T-schema, one of the two statements comprising the ?paradox? must be true (and the other false), but symmetry constraints prevent us from determining which, and thus prevent there being a truthmaker grounding the relevant assignment of truth values. Sorensen's vie…Read more
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100Philosophy of Mathematics: An Introduction to the World of Proofs and PicturesMind 113 (449): 154-157. 2004.
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99Possible predicates and actual propertiesSynthese 196 (7): 2555-2582. 2019.In “Properties and the Interpretation of Second-Order Logic” Bob Hale develops and defends a deflationary conception of properties where a property with particular satisfaction conditions actually exists if and only if it is possible that a predicate with those same satisfaction conditions exists. He argues further that, since our languages are finitary, there are at most countably infinitely many properties and, as a result, the account fails to underwrite the standard semantics for second-orde…Read more
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96Conservativeness, Stability, and AbstractionBritish Journal for the Philosophy of Science 63 (3): 673-696. 2012.One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principl…Read more
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94Hintikka's Revolution: The Priciples of Mathematics Revisited (review)British Journal for the Philosophy of Science 49 (2): 309-316. 1998.
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93Frege's Cardinals and Neo-LogicismPhilosophia Mathematica 24 (1): 60-90. 2016.Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternativ…Read more
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90Abstraction and Four Kinds of InvariancePhilosophia Mathematica 25 (1). 2017.Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this par…Read more
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89Charles E. Rickart. Structuralism and Structures: A Mathematical Perspective. Singapore: World Scientific Publishing, 1995. pp. xiii + 219. ISBN 981-02-1860-5 (review)Philosophia Mathematica 6 (2): 227-231. 1998.
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79Frege's RecipeJournal of Philosophy 113 (7): 309-345. 2016.In this paper, we present a formal recipe that Frege followed in his magnum opus “Grundgesetze der Arithmetik” when formulating his definitions. This recipe is not explicitly mentioned as such by Frege, but we will offer strong reasons to believe that Frege applied it in developing the formal material of Grundgesetze. We then show that a version of Basic Law V plays a fundamental role in Frege’s recipe and, in what follows, we will explicate what exactly this role is and explain how it differs f…Read more
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76Aristotelian logic, axioms, and abstractionPhilosophia Mathematica 11 (2): 195-202. 2003.Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to…Read more
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67The Yablo Paradox: An Essay on CircularityOxford University Press. 2012.Roy T Cook examines the Yablo paradox--a paradoxical, infinite sequence of sentences, each of which entails the falsity of all others that follow it. He focuses on questions of characterization, circularity, and generalizability, and pays special attention to the idea that it provides us with a semantic paradox that involves no circularity
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66RICHARD G. HECK, Jr. Frege's Theorem. Oxford: Clarendon Press, 2011. ISBN 978-0-19-969564-5. Pp. xiv + 307Philosophia Mathematica 20 (3): 346-359. 2012.
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59Necessity, Necessitism, and NumbersPhilosophical Forum 47 (3-4): 385-414. 2016.Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to a…Read more
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58Logic: A Very Short Introduction (review)History and Philosophy of Logic 40 (2): 204-205. 2019.Volume 40, Issue 2, May 2019, Page 204-205.
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56An Intensional Theory of Truth: An Informal ReportPhilosophical Forum 51 (2): 115-126. 2020.Saul Kripke’s theory of truth suffers from expressive limitations – in particular, there are no extensional operators within that framework that allow one to characterize those sentences that fail to receive a truth value within the framework. Especially worrisome is the fact that there is no operator that outputs true on exactly the paradoxical sentences. In this paper I extend Kripke’s approach via the addition of extensional operators, which allows us to characterize many (but not all) such s…Read more
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55Canonicity and Normativity in Massive, Serialized, Collaborative FictionJournal of Aesthetics and Art Criticism 71 (3): 271-276. 2013.
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52Embracing the technicalities: Expressive completeness and revengeReview of Symbolic Logic 9 (2): 325-358. 2016.The Revenge Problem threatens every approach to the semantic paradoxes that proceeds by introducing nonclassical semantic values. Given any such collection Δ of additional semantic values, one can construct a Revenge sentence:This sentence is either false or has a value in Δ.TheEmbracing Revengeview, developed independently by Roy T. Cook and Phlippe Schlenker, addresses this problem by suggesting that the class of nonclassical semantic values is indefinitely extensible, with each successive Rev…Read more
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52Inverted space: Minimal verificationism, propositional attitudes, and compositionalityPhilosophia 32 (1-4): 73-92. 2005.
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51Iteration one more timeNotre Dame Journal of Formal Logic 44 (2): 63--92. 2003.A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that ther…Read more
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48B. Jack Copeland, Carl J. Posy, and Oron Shagrir, eds, Computability: Turing, Gödel, Church, and Beyond. Cambridge, Mass.: MIT Press, 2013. ISBN 978-0-262-01899-9. Pp. x + 362 (review)Philosophia Mathematica 22 (3): 412-413. 2014.
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48Mathematics, Models, and ModalityHistory and Philosophy of Logic 31 (3): 287-289. 2010.John P. Burgess, Mathematics, Models, and Modality: Selected Philosophical Essays. Cambridge: Cambridge University Press, 2008. xiii + 301 pp. $90.00, £50.00. ISBN 978-0-521-88034-3. Adobe eBook, $...
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46Revising Benardete’s ZenoJournal of Philosophical Logic 48 (1): 37-56. 2019.The majority of disucssions of Benardete’s Paradox conclude that the traveller approaching the infinite series of gods will be mysteriously halted despite none of the gods erecting any barriers. Using a revision-theoretic analysis of Benardete’s puzzle, four distinct possible outcomes that might occur given Benardete’s set-up are distinguished. This analysis provides additional insight into the puzzle at hand, via identifying heretofore unnoticed possible outcomes, but it also serves as an examp…Read more
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45Robert Lorne Victor Hale FRSE May 4, 1945 – December 12, 2017Philosophia Mathematica 26 (2): 266-274. 2018.
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38Yablo Paradox. 2015.The Yablo Paradox The Yablo Paradox implies there is no way to coherently assign a truth value to any of the sentences in the countably infinite sequence of sentences, each of the form, “All of the subsequent sentences are false.” Specifically, the Yablo Paradox arises when we consider the following infinite sequence of sentences: The … Continue reading Yablo Paradox →.
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University of St. Andrews3- Year Post-doctoral Fellow
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University of MinnesotaTenured
Ohio State University
PhD, 2000
St Andrews, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
Science, Logic, and Mathematics |
PhilPapers Editorships
Theories of Mathematics |