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5Plural quantificationIn Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, The Metaphysics Research Lab. 2014.
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2026 Potential Infinity, Paradox, and the Mind of God: Historical SurveyIn Mirosław Szatkowski (ed.), Ontology of Divinity, De Gruyter. pp. 531-560. 2024.
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44Eklund, Maximalism, and the Problem of Incompatible Objects.
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417Abstraction and GroundingPhilosophy and Phenomenological Research. forthcoming.The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one: (HP) #xx=#yy iff xx≈yy. The principal aim of this article is to use the noti…Read more
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35No Easy Road to Impredicative DefinabilismPhilosophia Mathematica 32 (1): 21-33. 2024.Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and p…Read more
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39Early Analytic Philosophy (review)Philosophical Review 109 (1): 98-101. 2000.Analytic philosophy has traditionally been little concerned with the history of philosophy, including that of analytic philosophy itself. But in recent years the study of the early period of the analytic tradition has become an active and lively branch of Anglo-American philosophy. Early Analytic Philosophy, a collection of papers presented in honor of professor Leonard Linsky at the University of Chicago in April 1992, is an example of this. The contributors, many of them leading scholars in th…Read more
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56Potential Infinity and De Re Knowledge of Mathematical ObjectsIn Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner, Springer. pp. 79-98. 2023.Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.
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3Thin ObjectsIn Alexander Hieke & Hannes Leitgeb (eds.), Reduction, abstraction, analysis: proceedings of the 31th International Ludwig Wittgenstein-Symposium in Kirchberg, 2008, De Gruyter. pp. 227-238. 2009.
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5Føllesdal and Frege on ReferenceIn Michael Frauchiger (ed.), Reference, Rationality, and Phenomenology: Themes from Føllesdal, De Gruyter. pp. 259-280. 2013.
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84Rumfitt on the logic of set theoryInquiry: An Interdisciplinary Journal of Philosophy 62 (7): 826-841. 2019.ABSTRACTAccording to a famous argument by Dummett, the concept of set is indefinitely extensible, and the logic appropriate for reasoning about the instances of any such concept is intuitionistic, not classical. But Dummett's argument is widely regarded as obscure. This note explains how the final chapter of Rumfitt's important new book advances our understanding of Dummett's argument, but it also points out some problems and unanswered questions. Finally, Rumfitt's reconstruction of Dummett's a…Read more
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111Cardinality and Acceptable AbstractionNotre Dame Journal of Formal Logic 59 (1): 61-74. 2018.It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for acceptable abstraction.
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48RepliesTheoria 89 (3): 393-406. 2023.Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
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30PrécisTheoria 89 (3): 247-255. 2023.Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes.
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133Weyl and Two Kinds of Potential DomainsNoûs. forthcoming.According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
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Modality in mathematicsIn Otávio Bueno & Scott A. Shalkowski (eds.), The Routledge Handbook of Modality, Routledge. 2018.
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302Generality ExplainedJournal of Philosophy 119 (7): 349-379. 2022.What explains the truth of a universal generalization? Two types of explanation can be distinguished. While an ‘instance-based explanation’ proceeds via some or all instances of the generalization, a ‘generic explanation’ is independent of the instances, relying instead on completely general facts about the properties or operations involved in the generalization. This intuitive distinction is analyzed by means of a truthmaker semantics, which also sheds light on the correct logic of quantificati…Read more
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29Introduction to special issue on ‘critical views of logic’Inquiry: An Interdisciplinary Journal of Philosophy 65 (6): 631-637. 2022.Critical views of logic are presented. These are views that are critical of logic in a sense akin to the way in which Kant is critical rather than dogmatic about traditional metaphysics. Such approaches differ from the Fregean ‘logic-first’ view. In accordance with the latter, logic is often regarded as epistemologically and methodologically fundamental. Hence, all disciplines – including mathematics – are considered as answerable to logic, rather than vice versa. In critical views of logic, by …Read more
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Putnam on Mathematics as Modal LogicIn John Burgess (ed.), Hilary Putnam on Logic and Mathematics, Springer Verlag. 2018.Two uses of modal logic to explicate mathematics—due primarily to Hilary Putnam and Charles Parsons—are compared and contrasted. The approaches differ both technically and concerning ontology. Some reasons to push the former approach in the direction of the latter are articulated and discussed.
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81Predicativism as a Form of PotentialismReview of Symbolic Logic 16 (1): 1-32. 2023.In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.
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97Divergent Potentialism: A Modal Analysis With an Application to Choice SequencesPhilosophia Mathematica 30 (2): 143-172. 2022.Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripk…Read more
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33Bob Hale. Essence and Existence: Selected EssaysPhilosophia Mathematica 29 (3): 420-427. 2021.Essence and Existence: Selected Essays brings together fifteen essays by Bob Hale, mostly written between the publication of his last book, Necessary Beings, in.
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19Freges oppfatning av logikk: fra Kant til GrundgesetzeNorsk Filosofisk Tidsskrift 48 (3-4): 219-228. 2013.I first argue that Frege started out with a conception of logic that is closer to Kant’s than is generally recognized, after which I analyze Frege’s reasons for gradually rejecting this view. Although conceding that the demands posed by Frege’s logicism played some role, I argue that his increasingly vehement anti-psychologism provides a deeper and more interesting reason for rejecting his earlier view.
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233Which abstraction principles are acceptable? Some limitative resultsBritish Journal for the Philosophy of Science 60 (2): 239-252. 2009.Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these …Read more
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104Riemann’s Scale: A Puzzle About InfinityErkenntnis 88 (1): 189-191. 2020.Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well known by mathematic…Read more
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48Peacocke on magnitudes and numbersPhilosophical Studies 178 (8): 2717-2729. 2020.Peacocke’s recent The Primacy of Metaphysics covers a wide range of topics. This critical discussion focuses on the book’s novel account of extensive magnitudes and numbers. First, I further develop and defend Peacocke’s argument against nominalistic approaches to magnitudes and numbers. Then, I argue that his view is more Aristotelian than Platonist because reified magnitudes and numbers are accounted for via corresponding properties and these properties’ application conditions, and because the…Read more
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55Early analytic philosophy: Frege, Russell, Wittgenstein (review)Philosophical Review 109 (1): 98-101. 2000.
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95Critical Plural LogicPhilosophia Mathematica 28 (2): 172-203. 2020.What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.
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1137The Many and the One: A Philosophical Study of Plural LogicOxford University Press. 2021.Plural expressions found in natural languages allow us to talk about many objects simultaneously. Plural logic — a logical system that takes plurals at face value — has seen a surge of interest in recent years. This book explores its broader significance for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct? After introducing plural logic and its main applications, the book provides a systematic analysis of the relation between th…Read more
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164Review of Kit Fine, Modality and Tense: Philosophical Papers (review)Philosophical Quarterly 57 (227): 294-297. 2007.
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