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126Thin objectsIn Hieke Alexander & Leitgeb Hannes (eds.), Reduction, Abstraction, Analysis, Ontos Verlag. pp. 11--227. 2009.
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1Against Limitation of SizeThe Baltic International Yearbook of Cognition, Logic and Communication 1. 2005.
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191Some Criteria for Acceptable AbstractionNotre Dame Journal of Formal Logic 52 (3): 331-338. 2011.Which abstraction principles are acceptable? A variety of criteria have been proposed, in particular irenicity, stability, conservativeness, and unboundedness. This note charts their logical relations. This answers some open questions and corrects some old answers
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4Logic and PluralsIn Kirk Ludwig & Marija Jankovic (eds.), The Routledge Handbook of Collective Intentionality, Routledge. pp. 451-463. 2018.This chapter provides an overview of the philosophical and linguistic debate about the logic of plurals. We present the most prominent singularizing analyses of plurals as well as the main criticisms that such analyses have received. We then introduce an alternative approach to plurals known as plural logic, focusing on the question whether plural logic can count as pure logic.
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355Platonism in the Philosophy of MathematicsIn Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab. 2014.Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.
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185Mending the master: John P. Burgess, fixing Frege. Princeton, N. J.: Princeton university press, 2005. ISBN 0-691-12231-8. Pp. XII + 257 (review)Philosophia Mathematica 14 (3): 338-351. 2006.Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege's writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well-informed overview provided by the first of its three chapters. Specialists will find the boo…Read more
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61What is the infinite?The Philosophers' Magazine 61 42-47. 2013.The paper discusses some different conceptions of the infinity, from Aristotle to Georg Cantor (1845-1918) and beyond. The ancient distinction between actual and potential infinity is explained, along with some arguments against the possibility of actually infinite collections. These arguments were eventually rejected by most philosophers and mathematicians as a result of Cantor’s elegant and successful theory of actually infinite collections.
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245Review of P. Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (review)Philosophy 87 (1): 133-137. 2012.
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144The individuation of the natural numbersIn Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics, Palgrave. 2009.It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal part of this water? According to Quine, you must at least …Read more
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185Entanglement and non-factorizabilityStudies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (3): 215-221. 2013.Quantum mechanics tells us that states involving indistinguishable fermions must be antisymmetrized. This is often taken to mean that indistinguishable fermions are always entangled. We consider several notions of entanglement and argue that on the best of them, indistinguishable fermions are not always entangled. We also present a simple but unconventional way of representing fermionic states that allows us to maintain a link between entanglement and non-factorizability.
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218Plurals and modalsCanadian Journal of Philosophy 46 (4-5): 654-676. 2016.Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.
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164Frege's proof of referentialityNotre Dame Journal of Formal Logic 45 (2): 73-98. 2004.I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successf…Read more
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357The potential hierarchy of setsReview of Symbolic Logic 6 (2): 205-228. 2013.Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
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271Bad company tamedSynthese 170 (3). 2009.The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting w…Read more
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267Superplurals in EnglishAnalysis 68 (3). 2008.where ‘aa’ is a plural term, and ‘F’ a plural predicate. Following George Boolos (1984) and others, many philosophers and logicians also think that plural expressions should be analysed as not introducing any new ontological commitments to some sort of ‘plural entities’, but rather as involving a new form of reference to objects to which we are already committed (for an overview and further details, see Linnebo 2004). For instance, the plural term ‘aa’ refers to Alice, Bob and Charlie simultaneo…Read more
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272On the Innocence and Determinacy of Plural QuantificationNoûs 50 (3). 2016.Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-…Read more
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