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56Logic and Existence [Corrected Portion of an Article appearing in Proceedings of the Aristotelian Society Supplementary Volumes, Vol. 73 (1999)] (review)Proceedings of the Aristotelian Society 100. 2000.The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward L?wenheim-Skolem theorem to …Read more
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16Generalized Quantification in an Axiomatic Truth TheoryAustralasian Journal of Philosophy. forthcoming.Bruno Whittle (2019) has recently extended Kripke’s semantical theory of truth to languages containing generalized quantifiers. There are reasons for axiomatizing semantical theories, and for regarding Halbach and Horsten’s PKF as a good axiomatization of Kripke’s. PKF is a theory in Partial Logic. The present paper complements Whittle’s by showing how Partial Logic, and then PKF, may be extended to cover binary quantifiers meaning ‘every’, ‘some’, and ‘most’.
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11Against HarmonyIn Bob Hale, Crispin Wright & Alexander Miller (eds.), A Companion to the Philosophy of Language, Wiley-blackwell. 1997.This chapter concerns that harmony is a particular relationship between the introduction rule and the elimination rule for a given connective. The Harmony Thesis says that a connective is defective unless its associated introduction and elimination rules are in harmony. It also says that a connective is defective if the logical principles which regulate its use go beyond a pair of harmonious introduction and elimination rules. The chapter scrutinizes the most influential arguments which have bee…Read more
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55Reply to Øystein Linnebo and Stewart ShapiroInquiry: An Interdisciplinary Journal of Philosophy 62 (7): 842-858. 2019.ABSTRACTIn reply to Linnebo, I defend my analysis of Tait's argument against the use of classical logic in set theory, and make some preliminary comments on Linnebo's new argument for the same conclusion. I then turn to Shapiro's discussion of intuitionistic analysis and of Smooth Infinitesimal Analysis. I contend that we can make sense of intuitionistic analysis, but only by attaching deviant meanings to the connectives. Whether anyone can make sense of SIA is open to doubt: doing so would invo…Read more
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30Logic and ExistenceAristotelian Society Supplementary Volume 73 151-203. 1999.[Ian Rumfitt] Frege's logicism in the philosophy of arithmetic consisted, au fond, in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the ari…Read more
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53In defence of PKFSynthese 201 (2): 1-21. 2023.I advance arguments in favour of PKF as an articulation of a central sense of the predicate ‘true’, and show how it illuminates the relationship between that sense and the ‘external’ notion of truth found in such claims as ‘An utterance of the Liar Sentence does not say anything, and so is not true’.
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Objects of ThoughtIn Gary Ostertag (ed.), Meanings and Other Things: Themes From the Work of Stephen Schiffer, Oxford University Press. 2016.
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8Infinitesimals, Nations, and PersonsPhilosophy 94 (4): 513-528. 2019.I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation as Englan…Read more
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33Truth, Marks of Truth, and ConditionalsPhilosophy 97 (3): 295-320. 2022.This essay assesses the account of truth presented in Wiggins's 2002 paper ‘An indefinibilist cum normative view of truth and the marks of truth'. I agree with Wiggins that we should seek, not to define truth, but to elucidate it by unfolding its connections with other basic notions. However, I give reasons for preferring an elucidation based on Ramsey's account of truth to Wiggins's Tarski-inspired approach. I also cast doubt on Wiggins's thesis that convergence is a mark of truth, arguing inst…Read more
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Neo-Fregeanism and the Burali-Forti ParadoxIn Ivette Fred Rivera & Jessica Leech (eds.), Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale, Oxford University Press. pp. 188-223. 2018.Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a predicative …Read more
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29What is Logic?In Zsolt Novák & András Simonyi (eds.), Truth, reference, and realism, Central European University Press. pp. 125-176. 2010.
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61When I was a student in the mid-1980s, Donald Davidson loomed larger over the philosophical scene than any other living thinker. His writings figured prominentl.
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796Intuitionism and the Modal Logic of VaguenessJournal of Philosophical Logic 49 (2): 221-248. 2020.Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advant…Read more
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19Prospects for JustificationismIn Michael Frauchiger (ed.), Truth, Meaning, Justification, and Reality: Themes From Dummett, De Gruyter. pp. 123-152. 2017.
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21Dummett LaudatioIn Michael Frauchiger (ed.), Truth, Meaning, Justification, and Reality: Themes From Dummett, De Gruyter. pp. 13-24. 2017.
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38Brouwer Wittgenstein on the Infinite and the Law of Excluded MiddleGrazer Philosophische Studien 89 (1): 93-108. 2014.
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81I—Ian Rumfitt: Truth and MeaningAristotelian Society Supplementary Volume 88 (1): 21-55. 2014.Should we explicate truth in terms of meaning, or meaning in terms of truth? Ramsey, Prior and Strawson all favoured the former approach: a statement is true if and only if things are as the speaker, in making the statement, states them to be; similarly, a belief is true if and only if things are as a thinker with that belief thereby believes them to be. I defend this explication of truth against a range of objections.Ramsey formalized this account of truth as follows: B is true =df ∃P; in §i, I…Read more
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113The vagaries of paraphrase: A reply to Holton on the counting problemAnalysis 56 (4). 1996.In his 'paratactic' analysis of indirect speech reports, Davidson took the occurrence of 'that' in 'Galileo said that the Earth moves' to be a demonstrative expression which refers to the reporter's subsequent utterance of 'the Earth moves'. Ian McFetridge used his 'counting problem' to argue that we get a better version of the paratactic theory if we take the demonstrative 'that' to refer to the proposition expressed by the reporter's utterance, rather than to the utterance itself. In this note…Read more
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137Co-ordination principles: A replyMind 117 (468): 1059-1063. 2008.I explain why Fernando Ferreira's interesting formal result does not threaten the bilateralist account of the sense of the connectives.
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105Singular terms and arithmetical logicismPhilosophical Books 44 (3): 193--219. 2003.This article is a critical notice of Bob Hale and Crispin Wright's *The Reason's Proper Study* (OUP). It focuses particularly on their attempts (crucial to their neo-logicist project) to say what a singular term is. I identify problems for their account but include some constructive suggestions about how it might be improved.
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611Objects of ThoughtIn Gary Ostertag (ed.), Meanings and Other Things: Themes From the Work of Stephen Schiffer, Oxford University Press. 2016.In his book The Things We Mean, Stephen Schiffer advances a subtle defence of what he calls the ‘face-value’ analysis of attributions of belief and reports of speech. Under this analysis, ‘Harold believes that there is life on Venus’ expresses a relation between Harold and a certain abstract object, the proposition that there is life on Venus. The present essay first proposes an improvement to Schiffer’s ‘pleonastic’ theory of propositions. It then challenges the face-value analysis. There w…Read more
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3Plural terms: another variety of referring expression?In José Luis Bermúdez (ed.), Thought, reference, and experience: themes from the philosophy of Gareth Evans, Clarendon Press. 2005.
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159Logical NecessityIn Bob Hale & Aviv Hoffmann (eds.), Modality: metaphysics, logic, and epistemology, Oxford University Press. 2010.Book synopsis: The philosophy of modality investigates necessity and possibility, and related notions--are they objective features of mind-independent reality? If so, are they irreducible, or can modal facts be explained in other terms? This volume presents new work on modality by established leaders in the field and by up-and-coming philosophers. Between them, the papers address fundamental questions concerning realism and anti-realism about modality, the nature and basis of facts about what is…Read more
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46Concepts and CountingProceedings of the Aristotelian Society 102 (1): 41-68. 2002.Frege's analysis of Zahlangaben is expounded and evaluated.
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128The categoricity problem and truth-value gapsAnalysis 57 (4). 1997.In his article 'Rejection' (1996), Timothy Smiley had shown how a logical system allowing rules of rejection could provide a categorical axiomatization of the classical propositional calculus. This paper shows how rules of rejection, when placed in a multiple conclusion setting, can also provide categorical axiomatizations of a range of non-classical calculi which permit truth-value gaps, among them the calculus in Smiley's own 'Sense without denotation' (1960).
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