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582How Many Angels Can Dance on the Point of a Needle? Transcendental Theology Meets Modal MetaphysicsMind 120 (477): 53-81. 2011.We argue that certain modal questions raise serious problems for a modal metaphysics on which we are permitted to quantify unrestrictedly over all possibilia. In particular, we argue that, on reasonable assumptions, both David Lewis's modal realism and Timothy Williamson's necessitism are saddled with the remarkable conclusion that there is some cardinal number of the form ℵα such that there could not be more than ℵα-many angels in existence. In the last section, we make use of similar ideas to …Read more
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2Unrestricted Unrestricted Quantification: the cardinal problem of absolute generalityIn Agustín Rayo & Gabriel Uzquiano (eds.), Absolute generality, Oxford University Press. pp. 305--32. 2006.
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293Plural Quantification and ModalityProceedings of the Aristotelian Society 111 (2pt2): 219-250. 2011.Identity is a modally inflexible relation: two objects are necessarily identical or necessarily distinct. However, identity is not alone in this respect. We will look at the relation that one object bears to some objects if and only if it is one of them. In particular, we will consider the credentials of the thesis that no matter what some objects are, an object is necessarily one of them or necessarily not one of them
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38Hale Bob and Wright Crispin. The reason's proper study: Essays toward a neo-Fregean philosophy of mathematics. Oxford University Press, New York. 2001, 472 pp (review)Bulletin of Symbolic Logic 12 (2): 291-294. 2006.
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100IntroductionIn Agustín Rayo & Gabriel Uzquiano (eds.), Absolute generality, Oxford University Press. 2006.Whether or not we achieve absolute generality in philosophical inquiry, most philosophers would agree that ordinary inquiry is rarely, if ever, absolutely general. Even if the quantifiers involved in an ordinary assertion are not explicitly restricted, we generally take the assertion’s domain of discourse to be implicitly restricted by context.1 Suppose someone asserts (2) while waiting for a plane to take off.
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615Mereology and modalityIn Shieva Kleinschmidt (ed.), Mereology and Location, Oxford University Press. pp. 33-56. 2014.Do mereological fusions have their parts necessarily? None of the axioms of non-modal formulations of classical mereology appear to speak directly to this question. And yet a great many philosophers who take the part-whole relation to be governed by classical mereology seem to assume that they do. In addition to this, many philosophers who make allowance for the part-whole relation to obtain merely contingently between a part and a mereological fusion tend to depart from non-modal formulations o…Read more
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Ontology and the Foundations of MathematicsDissertation, Massachusetts Institute of Technology. 1999."Ontology and the Foundations of Mathematics" consists of three papers concerned with ontological issues in the foundations of mathematics. Chapter 1, "Numbers and Persons," confronts the problem of the inscrutability of numerical reference and argues that, even if inscrutable, the reference of the numerals, as we ordinarily use them, is determined much more precisely than up to isomorphism. We argue that the truth conditions of a variety of numerical modal and counterfactual sentences place ser…Read more
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123Before Effect Without Zeno CausalityNoûs 46 (2): 259-264. 2012.We argue that not all cases of before-effect involve causation and ask how to demarcate cases of before-effect in which the events that follow exert causal influence over the before-effect from cases in which they do not
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145Well- and non-well-founded Fregean extensionsJournal of Philosophical Logic 33 (5): 437-465. 2004.George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation …Read more
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1652Higher-order free logic and the Prior-Kaplan paradoxCanadian Journal of Philosophy 46 (4-5): 493-541. 2016.The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramifica…Read more
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161ReceptaclesPhilosophical Perspectives 20 (1). 2006.This paper looks at the question of what regions of space are possibly exactly occupied by a material object.
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400How to solve the hardest logic puzzle ever in two questionsAnalysis 70 (1): 39-44. 2010.Rabern and Rabern (2008) have noted the need to modify `the hardest logic puzzle ever’ as presented in Boolos 1996 in order to avoid trivialization. Their paper ends with a two-question solution to the original puzzle, which does not carry over to the amended puzzle. The purpose of this note is to offer a two-question solution to the latter puzzle, which is, after all, the one with a claim to being the hardest logic puzzle ever.
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188Toward a Theory of Second-Order ConsequenceNotre Dame Journal of Formal Logic 40 (3): 315-325. 1999.There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.
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230Recombination and ParadoxPhilosophers' Imprint 15. 2015.The doctrine that whatever could exist does exist leads to a proliferation of possibly concrete objects given certain principles of recombination. If, for example, there could have been a large infinite number of concrete objects, then there is at least the same number of possibly concrete objects in existence. And further cardinality considerations point to a tension between the preceding doctrine and the Cantorian conception of the absolutely infinite. This paper develops a parallel problem fo…Read more
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229Plurals and SimplesThe Monist 87 (3): 429-451. 2004.I would like to discuss the claim that the resources of plural reference and plural quantification are sufficient for the purpose of paraphrasing all ordinary statements apparently concerned with composite material objects into plural statements concerned exclusively with simples.
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