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60John P. Burgess. Rigor and Structure. Oxford: Oxford University Press, 2015. ISBN: 978-0-19-872222-9 ; 978-0-19-103360-5 . Pp. xii + 215 (review)Philosophia Mathematica 24 (1): 129-136. 2016.
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75Accuracy and the belief-credence connectionPhilosophers' Imprint 15 1-20. 2015.Probabilism says an agent is rational only if her credences are probabilistic. This paper is concerned with the so-called Accuracy Dominance Argument for Probabilism. This argument begins with the claim that the sole fundamental source of epistemic value for a credence is its accuracy. It then shows that, however we measure accuracy, any non-probabilistic credences are accuracy-dominated: that is, there are alternative credences that are guaranteed to be more accurate than them. It follows that …Read more
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1506Transformative Experience and Decision TheoryPhilosophy and Phenomenological Research 91 (3): 766-774. 2015.This paper is part of a book symposium for L. A. Paul (2014) Transformative Experience (OUP).
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177Aristotle on the subject matter of geometryPhronesis 54 (3): 239-260. 2009.I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics , I explain how such objects exist in the sensibles in a special way. I conclude by considering the passages …Read more
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192The Population Ethics of Belief: In Search of an Epistemic Theory XNoûs 52 (2): 336-372. 2018.Consider Phoebe and Daphne. Phoebe has credences in 1 million propositions. Daphne, on the other hand, has credences in all of these propositions, but she's also got credences in 999 million other propositions. Phoebe's credences are all very accurate. Each of Daphne's credences, in contrast, are not very accurate at all; each is a little more accurate than it is inaccurate, but not by much. Whose doxastic state is better, Phoebe's or Daphne's? It is clear that this question is analogous to a qu…Read more
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157Beliefs come in different strengths. What are the norms that govern these strengths of belief? Let an agent's belief function at a particular time be the function that assigns, to each of the propositions about which she has an opinion, the strength of her belief in that proposition at that time. Traditionally, philosophers have claimed that an agent's belief function at any time ought to be a probability function, and that she ought to update her belief function upon obtaining new evidence by c…Read more
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368An Objective Justification of Bayesianism II: The Consequences of Minimizing InaccuracyPhilosophy of Science 77 (2): 236-272. 2010.One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its prequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In the prequel, we made this norm mathematically precise; in this paper, we derive its consequences. We show that the two core tenets of Bayesianism follo…Read more
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187On interpretations of bounded arithmetic and bounded set theoryNotre Dame Journal of Formal Logic 50 (2): 141-152. 2009.In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.
THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.
In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforw…Read more -
141Accuracy and the Laws of CredenceOxford University Press UK. 2016.Richard Pettigrew offers an extended investigation into a particular way of justifying the rational principles that govern our credences. The main principles that he justifies are the central tenets of Bayesian epistemology, though many other related principles are discussed along the way. Pettigrew looks to decision theory in order to ground his argument. He treats an agent's credences as if they were a choice she makes between different options, gives an account of the purely epistemic utility…Read more
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42Précis and replies to contributors for book symposium on accuracy and the laws of credenceEpisteme 14 (1): 1-30. 2017.ABSTRACTThis book symposium onAccuracy and the Laws of Credenceconsists of an overview of the book’s argument by the author, Richard Pettigrew, together with four commentaries on different aspects of that argument. Ben Levinstein challenges the characterisation of the legitimate measures of inaccuracy that plays a central role in the arguments of the book. Julia Staffel asks whether the arguments of the book are compatible with an ontology of doxastic states that includes full beliefs as well as…Read more
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223Deference Done RightPhilosophers' Imprint 14 1-19. 2014.There are many kinds of epistemic experts to which we might wish to defer in setting our credences. These include: highly rational agents, objective chances, our own future credences, our own current credences, and evidential probabilities. But exactly what constraint does a deference requirement place on an agent's credences? In this paper we consider three answers, inspired by three principles that have been proposed for deference to objective chances. We consider how these options fare when a…Read more
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231The foundations of arithmetic in finite bounded Zermelo set theoryCahiers du Centre de Logique 17 99-118. 2010.In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<…Read more
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130II—Pluralism about Belief StatesAristotelian Society Supplementary Volume 89 (1): 187-204. 2015.With his Humean thesis on belief, Leitgeb seeks to say how beliefs and credences ought to interact with one another. To argue for this thesis, he enumerates the roles beliefs must play and the properties they must have if they are to play them, together with norms that beliefs and credences intuitively must satisfy. He then argues that beliefs can play these roles and satisfy these norms if, and only if, they are related to credences in the way set out in the Humean thesis. I begin by raising qu…Read more
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640Category theory as an autonomous foundationPhilosophia Mathematica 19 (3): 227-254. 2011.Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in …Read more
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188Platonism and aristotelianism in mathematicsPhilosophia Mathematica 16 (3): 310-332. 2008.Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that…Read more
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417Accuracy, Chance, and the Principal PrinciplePhilosophical Review 121 (2): 241-275. 2012.In ‘A Non-Pragmatic Vindication of Probabilism’, Jim Joyce attempts to ‘depragmatize’ de Finetti’s prevision argument for the claim that our partial beliefs ought to satisfy the axioms of probability calculus. In this paper, I adapt Joyce’s argument to give a non-pragmatic vindication of various versions of David Lewis’ Principal Principle, such as the version based on Isaac Levi's account of admissibility, Michael Thau and Ned Hall's New Principle, and Jenann Ismael's Generalized Principal Prin…Read more
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28Reviewed Work(s): An introduction to the philosophy of mathematics by Mark ColyvanAssociation for Symbolic Logic: The Bulletin of Symbolic Logic 19 (3): 396-397. 2013.Review by: Richard Pettigrew The Bulletin of Symbolic Logic, Volume 19, Issue 3, Page 396-397, September 2013
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163Epistemic Utility and Norms for CredencesPhilosophy Compass 8 (10): 897-908. 2013.Beliefs come in different strengths. An agent's credence in a proposition is a measure of the strength of her belief in that proposition. Various norms for credences have been proposed. Traditionally, philosophers have tried to argue for these norms by showing that any agent who violates them will be lead by her credences to make bad decisions. In this article, we survey a new strategy for justifying these norms. The strategy begins by identifying an epistemic utility function and a decision-the…Read more
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341What Chance‐Credence Norms Should Not BeNoûs 47 (3): 177-196. 2013.A chance-credence norm states how an agent's credences in propositions concerning objective chances ought to relate to her credences in other propositions. The most famous such norm is the Principal Principle (PP), due to David Lewis. However, Lewis noticed that PP is too strong when combined with many accounts of chance that attempt to reduce chance facts to non-modal facts. Those who defend such accounts of chance have offered two alternative chance-credence norms: the first is Hall's and Thau…Read more
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279Indispensability arguments and instrumental nominalismReview of Symbolic Logic 5 (4): 687-709. 2012.In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that abstract mathematical objects exist. I wish to defend a particular objection to such arguments that has become increasingly popular recently. It is called instrumental nominalism. I consider the recent versions of this view and conclude that it has yet to be given an adequate formulation. I provide such a formulation and show that it can be used to answer the indispensability argumen…Read more
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481An Objective Justification of Bayesianism I: Measuring InaccuracyPhilosophy of Science 77 (2): 201-235. 2010.One of the fundamental problems of epistemology is to say when the evidence in an agent’s possession justifies the beliefs she holds. In this paper and its sequel, we defend the Bayesian solution to this problem by appealing to the following fundamental norm: Accuracy An epistemic agent ought to minimize the inaccuracy of her partial beliefs. In this paper, we make this norm mathematically precise in various ways. We describe three epistemic dilemmas that an agent might face if she attempts to f…Read more
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