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105Finitude and Hume’s PrincipleJournal of Philosophical Logic 26 (6): 589-617. 1997.The paper formulates and proves a strengthening of ‘Frege’s Theorem’, which states that axioms for second-order arithmetic are derivable in second-order logic from Hume’s Principle, which itself says that the number of Fs is the same as the number ofGs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. ‘Finite Hume’s Principle’ al…Read more
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175Communication and Knowledge: Rejoinder to Byrne and ThauMind 105 (417). 1996.A reply to Byrne and Thau's criticisms of "The Sense of Communiction".
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1018The Existence (and Non-existence) of Abstract ObjectsIn Frege's Theorem, Oxford University Press. 2011.This paper is concerned with neo-Fregean accounts of reference to abstract objects. It develops an objection to the most familiar such accounts, due to Bob Hale and Crispin Wright, based upon what I call the 'proliferation problem': Hale and Wright's account makes reference to abstract objects seem too easy, as is shown by the fact that any equivalence relation seems as good as any other. The paper then develops a response to this objection, and offers an account of what it is for abstracta to e…Read more
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454Sir Michael Anthony Eardley Dummett, 1925-2011Philosophia Mathematica 21 (1): 1-8. 2013.A remembrance of Dummett's work on philosophy of mathematcis.
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1380Frege on Identity and Identity-Statements: A Reply to Thau and CaplanCanadian Journal of Philosophy 33 (1): 83-102. 2003.The paper argues, as against Thau and Caplan, that the traditional interpretation that Frege abandoned his earlier views about identity and identity--statements is correct
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840In "Counting and Indeterminate Identity", N. Ángel Pinillos develops an argument that there can be no cases of `Split Indeterminate Identity'. Such a case would be one in which it was indeterminate whether a=b and indeterminate whether a=c, but determinately true that b≠c. The interest of the argument lies, in part, in the fact that it appears to appeal to none of the controversial claims to which similar arguments due to Gareth Evans and Nathan Salmon appeal. I argue for two counter-claims. Fir…Read more
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746Do demonstratives have senses?Philosophers' Imprint 2 1-33. 2002.Frege held that referring expressions in general, and demonstratives and indexicals in particular, contribute more than just their reference to what is expressed by utterances of sentences containing them. Heck first attempts to get clear about what the essence of the Fregean view is, arguing that it rests upon a certain conception of linguistic communication that is ultimately indefensible. On the other hand, however, he argues that understanding a demonstrative (or indexical) utterance require…Read more
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361Julius Caesar and Basic Law VDialectica 59 (2). 2005.This paper dates from about 1994: I rediscovered it on my hard drive in the spring of 2002. It represents an early attempt to explore the connections between the Julius Caesar problem and Frege's attitude towards Basic Law V. Most of the issues discussed here are ones treated rather differently in my more recent papers "The Julius Caesar Objection" and "Grundgesetze der Arithmetik I 10". But the treatment here is more accessible, in many ways, providing more context and a better sense of how thi…Read more
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291This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of axiomatic theories of truth. These results are then used to address…Read more
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2331The Julius Caesar objectionIn Richard G. Heck (ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett, Oxford University Press. pp. 273--308. 1997.This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us.
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732Reason and LanguageIn Cynthia Macdonald & Graham Macdonald (eds.), McDowell and His Critics, Blackwell. pp. 22--45. 2006.John McDowell has often emphasized the fact that the use of langauge is a rational enterprise. In this paper, I explore the sense in which this is so, arguing that our use of language depends upon our consciously knowing what our words mean. I call this a 'cognitive conception of semantic competence'. The paper also contains a close analysis of the phenomenon of implicature and some suggestions about how it should and should not be understood.
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474Predicative Frege Arithmetic and ‘Everyday’ MathematicsPhilosophia Mathematica 22 (3): 279-307. 2014.The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results
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475Intuition and the Substitution ArgumentAnalytic Philosophy 55 (1): 1-30. 2014.The 'substitution argument' purports to demonstrate the falsity of Russellian accounts of belief-ascription by observing that, e.g., these two sentences: (LC) Lois believes that Clark can fly. (LS) Lois believes that Superman can fly. could have different truth-values. But what is the basis for that claim? It seems widely to be supposed, especially by Russellians, that it is simply an 'intuition', one that could then be 'explained away'. And this supposition plays an especially important role…Read more
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152The development of arithmetic in Frege's Grundgesetze der ArithmetikJournal of Symbolic Logic 58 (2): 579-601. 1993.Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Mor…Read more
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752A Liar ParadoxThought: A Journal of Philosophy 1 (1): 36-40. 2012.The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬…Read more
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367Cardinality, Counting, and EquinumerosityNotre Dame Journal of Formal Logic 41 (3): 187-209. 2000.Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, howev…Read more
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957The Finite and the Infinite in Frege's Grundgesetze der ArithmetikIn Matthias Schirn (ed.), Philosophy of Mathematics Today, Oxford University Press. 1998.Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.
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837Truth and disquotationSynthese 142 (3): 317--352. 2005.Hartry Field has suggested that we should adopt at least a methodological deflationism: [W]e should assume full-fledged deflationism as a working hypothesis. That way, if full-fledged deflationism should turn out to be inadequate, we will at least have a clearer sense than we now have of just where it is that inflationist assumptions ... are needed. I argue here that we do not need to be methodological deflationists. More pre-cisely, I argue that we have no need for a disquotational truth-predic…Read more
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261Frege's PrincipleIn J. Hintikka (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, Kluwer Academic Publishers. 1995.This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V.
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243Language, thought, and logic: essays in honour of Michael Dummett (edited book)Oxford University Press. 1997.In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought.…Read more
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662Frege and semanticsGrazer Philosophische Studien 75 (1): 27-63. 2007.In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. I argue here Frege did make very serious use of semantical concepts. I argue, first, that Frege had reason to be interested in the question how the axioms and rules of his formal theory might be justified and, second, that he explicitly commits himself to offering a justification that appeals to the notion of reference. I then discuss the…Read more
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705MacFarlane on relative truthPhilosophical Issues 16 (1). 2006.John MacFarlane has made relativism popular again. Focusing just on his original discussion, I argue that the data he uses to motivate the position do not, in fact, motivatie it at all. Many of the points made here have since been made, independently, by Hermann Cappelen and John Hawthorne, in their book Relativism and Monadic Truth.
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1711That There Might Be Vague Objects (So Far as Concerns Logic)The Monist 81 (1): 277-99. 1998.Gareth Evans has argued that the existence of vague objects is logically precluded: The assumption that it is indeterminate whether some object a is identical to some object b leads to contradiction. I argue in reply that, although this is true—I thus defend Evans's argument, as he presents it—the existence of vague objects is not thereby precluded. An 'Indefinitist' need only hold that it is not logically required that every identity statement must have a determinate truth-value, not that some …Read more
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721A Logic for Frege's TheoremIn Richard G. Heck (ed.), Frege’s Theorem: An Introduction, The Harvard Review of Philosophy. 1999.It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, Delta-3-1 comprehension axioms are not logical …Read more
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611Semantics and Context-Dependence: Towards a Strawsonian AccountIn Brett Sherman & Alexis Burgess (eds.), Metasemantics: New Essays on the Foundations of Meaning, Oxford University Press. pp. 327-364. 2014.This paper considers a now familiar argument that the ubiquity of context -dependence threatens the project of natural language semantics, at least as that project has usually been conceived: as concerning itself with `what is said' by an utterance of a given sentence. I argue in response that the `anti-semantic' argument equivocates at a crucial point and, therefore, that we need not choose between semantic minimalism, truth-conditional pragmatism, and the like. Rather, we must abandon the idea…Read more
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741Ramified Frege ArithmeticJournal of Philosophical Logic 40 (6): 715-735. 2011.Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic
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56Critical Notice of Michael Dummett, Frege: Philosophy of Mathematics (review)Philosophical Quarterly 43 223-33. 1993.
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452In Defense of Formal RelationismThought: A Journal of Philosophy 3 (3): 243-250. 2014.In his paper “Flaws of Formal Relationism”, Mahrad Almotahari argues against the sort of response to Frege's Puzzle I have defended elsewhere, which he dubs ‘Formal Relationism’. Almotahari argues that, because of its specifically formal character, this view is vulnerable to objections that cannot be raised against the otherwise similar Semantic Relationism due to Kit Fine. I argue in response that Formal Relationism has neither of the flaws Almotahari claims to identify
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677The Sense of CommunicationMind 104 (413). 1995.Many philosophers nowadays believe Frege was right about belief, but wrong about language: The contents of beliefs need to be individuated more finely than in terms of Russellian propositions, but the contents of utterances do not. I argue that this 'hybrid view' cannot offer no reasonable account of how communication transfers knowledge from one speaker to another and that, to do so, we must insist that understanding depends upon more than just getting the references of terms right.
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1510Are there different kinds of content?In Brian P. McLaughlin & Jonathan D. Cohen (eds.), Contemporary Debates in Philosophy of Mind, Blackwell. pp. 117-138. 2007.In an earlier paper, "Non-conceptual Content and the 'Space of Reasons'", I distinguished two forms of the view that perceptual content is non-conceptual, which I called the 'state view' and the 'content view'. On the latter, but not the former, perceptual states have a different kind of content than do cognitive states. Many have found it puzzling why anyone would want to make this claim and, indeed, what it might mean. This paper attempts to address these questions
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