Online Papers

Electronic versions of the papers listed below are available for download. Papers are orgranized alphabetically by title within the following categories:

Comments, criticisms, and suggestions are, of course, most welcome.

What kinds of links are provided below depends upon who holds copyright to the paper. If I do, then a link will be provided that should make it available to everyone, free of charge. If someone else holds copyriht, then links are provided to journal websites or other repositories, such as jstor.org, which may not be acessible to everyone. If you are unable to access these repositories, then, in many cases, you can request a copy from me by clicking on the "Request Copy" link. There are also, in most cases, "pre-publication" versions of the paper available.


Work in Progress

Please do not cite these papers, or quote from them, without contacting me for permission first.

Disquotationalism and the Compositional Principles

rgheck.frege.org

What Bar-On and Simmons call 'Conceptual Deflationism' is the thesis that truth is a 'thin' concept in the sense that it is not suited to play any explanatory role in our scientific theorizing. One obvious place it might play such a role is in semantics, so disquotationalists have been widely concerned to argued that 'compositional principles', such as

(C) A conjunction is true iff its conjuncts are true

are ultimately quite trivial and, more generally, that semantic theorists have misconceived the relation between truth, meaning, and logic. This paper argues, to the contrary, that even such simple compositional principles as (C) have substantial content that cannot be captured by deflationist 'proofs' of them. The key thought is that (C) is supposed, among other things, to affirm the truth-functionality of conjunction and that disquotationalists cannot, ultimately, make sense of truth-functionality.

This paper is something of a companion to "The Logical Strength of Compositional Principles".

Show Abstract

Forthcoming Papers

The Basic Laws of Cardinal Number

Forthcoming in P. Ebert and M. Rossberg, eds., A Companion to Frege's Grundgesetze

rgheck.frege.org

An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.
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Formal Arithmetic Before Grundgesetze

Forthcoming in P. Ebert and M. Rossberg, eds., A Companion to Frege's Grundgesetze

rgheck.frege.org

A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism.
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The Frontloading Argument

Forthcoming in Philosophical Studies

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Maybe the most important argument in David Chalmers's monumental book Constructing the World is the one he calls the 'Frontloading Argument', which is used in Chapter 4 to argue for the book's central thesis, A Priori Scrutability. And, at first blush, the Frontloading Argument looks very strong. I argue here, however, that it is incapable of securing the conclusion it is meant to establish. My interest is not in the conclusion for which Chalmers is arguing. As it happens, I am skeptical about A Priori Scrutability. Indeed, my views about the a priori are closer to Quine's than to Chalmers's. But my goal here is not to argue for any substantive conclusion but just for a dialectical one: Despite its initial appeal, the Frontloading Argument fails as an argument for A Priori Scrutability.

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The Logical Strength of Compositional Principles

Forthcoming in the Notre Dame Journal of Formal Logic

rgheck.frege.org

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call `compositional principles, such as 'A conjunction is true iff its conjuncts are true', have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.

This paper is something of a companion to "Disquotationalism and the Compositional Principles".

Show Abstract

Logicism, Ontology, and the Epistemology of Second-Order Logic

In Ivette Fred and Jessica Leech, eds, Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (Oxford: Oxford University Press)

rgheck.frege.org

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms.

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Speaker's Reference, Semantic Reference, and Intuition

Forthcoming in Review of Philosophy and Psychology

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Some years ago, Machery, Mallon, Nichols, and Stich reported the results of experiments that reveal, they claim, cross-cultural differences in speakers' `intuitions' about Kripke's famous Gödel-Schmidt case. Several authors have suggested, however, that the question they asked they subjects is ambiguous between speaker's reference and semantic reference. Machery and colleagues have since made a number of replies. It is argued here that these are ineffective. The larger lesson, however, concerns the role that first-order philosophy should, and more importantly should not, play in the design of such experiments and in the evaluation of their results.

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Truth In Frege (with Robert May)

Forthcoming in M. Glanzberg, ed., The Oxford Handbook of Truth

rgheck.frege.org

A general survey of Frege's views on truth, the paper explores the problems in response to which Frege's distinctive view that sentences refer to truth-values develops. It also discusses his view that truth-values are objects and the so-called regress argument for the indefinability of truth. Finally, we consider, very briefly, the question whether Frege was a deflationist.
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Unpublished Papers

These are papers I wrote but never published, and which I do not now have plans to publish. Sometimes the material has been absorbed into other papers; sometimes they're just kind of out of date. They still seem to me worth reading, though.

Is Indeterminate Identity Incoherent?

rgheck.frege.org

In "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. First, the formal argument fails to establish its conclusion, for essentially the same reason Evans's and Salmon's arguments fail to establish their conclusions. Second, the phenomena in which Pinillos is interested, which concern the cardinalities of sets of vague objects, manifest the existence of what Kit Fine called `penumbral connections', phenomena that the logics Pinillos considers are already known not to handle well.
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The Strength of Truth-Theories

rgheck.frege.org

This 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 a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism.

This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now.

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What Is a Singular Term?

rgheck.frege.org

This paper discusses the question whether it is possible to explain the notion of a singular term without invoking the notion of an object or other ontological notions. The framework here is that of Michael Dummett's discussion in Frege: Philosophy of Language. I offer an emended version of Dummett's conditions, accepting but modifying some suggestions made by Bob Hale, and defend the emended conditions against some objections due to Crispin Wright.

This paper dates from about 1989. It originally formed part of a very early draft of what became my Ph.D. dissertation. I rediscovered it and began scanning it, when I had nothing better to do, in Fall 2001, making some minor editing changes along the way. Suffice it to say that it no longer represents my current views.

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Published Papers

Alphabetical by title. If you want to see a list organized by date of publication, please see the publications page.

Are There Different Kinds of Content?

In J. Cohen and B. McLaughlin, eds, Contemporary Debates in the Philosophy of Mind (Oxford: Blackwells, 2007), pp. 117-38

philpapers.org, rgheck.frege.org

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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Cardinality, Counting, and Equinumerosity

Notre Dame Journal of Formal Logic 41 (2000), pp. 187-209
Reprinted in Frege's Theorem, pp. 156-79

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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. I argue, however, that Frege was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many.
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Cognitive Hunger: Remarks on Imogen Dickie's Fixing Reference

Philosophy and Phenomenological Research 95 (2017), pp. 738-44

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The main focus of my comments is the role played in Dickie's view by the idea that "the mind has a need to represent things outside itself". But there are also some remarks about her (very interesting) suggestion that descriptive names can sometimes fail to refer to the object that satisfies the associated description.

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The Composition of Thoughts (with Robert May)

Noûs 45 (2011), pp. 126-66

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Are Fregean thoughts compositionally complex and composed of senses? We argue that, in Begriffsschrift, Frege took 'conceptual contents' to be unstructured, but that he quickly moved away from this position, holding just two years later that conceptual contents divide of themselves into 'function' and 'argument'. This second position is shown to be unstable, however, by Frege's famous substitution puzzle. For Frege, the crucial question the puzzle raises is why "The Morning Star is a planet" and "The Evening Star is a planet" have different contents, but his second position predicts that they should have the same content. Frege's response to this antinomy is of course to distinguish sense from reference, but what has not previously been noticed is that this response also requires thoughts to be compositionally complex, composed of senses. That, however, raises the question just how thoughts are composed from senses. We reconstruct a Fregean answer, one that turns on an insistence that this question must be understood as semantic rather than metaphysical. It is not a question about the intrinsic nature of residents of the third realm but a question about how thoughts are expressed by sentences.
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Consistency and the Theory of Truth

Review of Symbolic Logic 8 (2015), pp. 424-66

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This 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? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or less equivalent to a kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories.

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The Consistency of Predicative Fragments of Frege's Grundgesetze der Artithmetik

History and Philosophy of Logic 17 (1996), pp. 209-20

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As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell's Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege's Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is relatively interpretable in the simple predicative fragment. The philosophical significance of the result is discussed.
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The Development of Arithmetic in Frege's Grundgesetze der Arithmetik

Journal of Symbolic Logic 58 (1993), pp. 579-601
Reprinted, with a postscript, in W. Demopoulos, ed., Frege's Philosophy of Mathematics (Cambridge MA: Harvard University Press, 1995), pp. 257-94
Reprinted in M. Beaney and E. H. Reck, eds., Gottlob Frege: Critical Assessments of Leading Philosophers, vol. III (New York: Routledge, 2005), pp. 323-48
Reprinted, with a new Postscript, in Frege's Theorem, pp. 40-68
Reprinted, more or less, as Chapter 6 of Reading Frege's Grundgesetze

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Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik had 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 the Basic Law of the system, Basic Law 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. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them.
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Die Grundlagen der Arithmetik §§82-83 (with George Boolos)

In M. Schirn, ed., Philosophy of Mathematics Today (Oxford: Oxford University Press, 1998), pp. 407-28
Reprinted in George Boolos, Logic, Logic, and Logic (Cambridge MA: Harvard University Press, 1998), pp. 315-38
Reprinted, with a Postscript, in Frege's Theorem, pp. 69-89

philpapers.org, rgheck.frege.org

This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in Grundgesetze der Arithmetik, but fidelity to what Frege wrote in Die Grundlagen and in Grundgesetze requires us to reject the charitable suggestion that it was this (beautiful) proof that he had in mind in §§82-83.
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Do Demonstratives Have Senses?

Philosophers' Imprint 2 (2002)
Reprinted in The Philosopher's Annual 25 (2002)

www.philosophersimprint.org, philpapers.org

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 sense 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 requires one to think of the object denoted in an appropriate way. This fact makes it difficult to reconcile the view that referring expressions are 'directly referential' with any view that seeks (as Grice's does) to ground meaning in facts about communication.
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The Existence (and Non-existence) of Abstract Objects

In P. Ebert and M. Rossberg, eds., Abstractionism: Essays in Philosophy of Mathematics (Oxford: Oxford Univerity Press, 2016), pp. 50-78
Also in Frege's Theorem, pp. 200-26

philpapers.org, rgheck.frege.org

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 exist that is Fregean in spirit but more robust than familiar views.
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The Finite and the Infinite in Frege's Grundgesetze der Arithmetik

In M. Schirn, ed., The Philosophy of Mathematics Today (Oxford: Clarendon Press, 1998), pp. 429-66
Reprinted, more or less, as Chapter 8 of Reading Frege's Grundgesetze

philpapers.org, rgheck.frege.org

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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Finitude and Hume's Principle

Journal of Philosophical Logic 26 (1997), pp. 589-617
Reprinted in R. T. Cook, ed., The Arché Papers on the Mathematics of Abstraction (Dordrecht: Springer, 2007), pp. 62-84
Reprinted, with a Postscript, in Frege's Theorem, pp. 237-66

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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 of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite 'srsquos Principle' also suffices for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege's definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed.
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Frege and Semantics

Grazer Philosophische Studien 75 (2007), pp. 27-63
Reprinted in The Cambridge Companion to Frege, ed. by T. Ricketts and M. Potter (Cambridge: Cambridge University Press, 2010), pp. 342-78
Reprinted, more or less, as Chapter 2 of Reading Frege's Grundgesetze

www.ingentaconnect.com, philpapers.org, rgheck.frege.org

This paper discusses the question to what extent Frege made serious use of semantical notions such as reference and truth. It focuses on his apparent uses of these notions in his apparently semantical discussions of his formal system in Grundgesetze der Arithmetik and defends the view that they are to be taken at face value. This paper is in some ways a companion to "Grundgesetze der Arithmetik I §§29-32", in which there is an extended, but mostly technical, discussion of Frege's attempt to prove that every well-formed expression in his formal language denotes: This paper contains more in the way of a discussion of the wider, interpretive significance of the technical interpretation given there.
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Frege on Identity and Identity-Statements: A Reply to Thau and Caplan

Canadian Journal of Philosophy 33 (2003), pp. 83-102 (Awarded the Canadian Journal of Philosophy's 2002 Essay Prize)

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In "What's Puzzling Gottlob Frege?", Michael Thau and Ben Kaplan argue that, contrary to the common wisdom, Frege never abandoned the view of identity-statments he had held in Begriffsschrift. I argue (a) that the textual evidence Thau and Caplan present does not support their view and (b) that there is overwhelming textual evidence in favor of the orthodox reading of "On Sense and Reference".
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Frege's Contribution to Philosophy of Language (with Robert May)

In E. Lepore and B. Smith, eds., The Oxford Handbook of Philosophy of Language (Oxford: Oxford University Press, 2006), pp. 3-39

philpapers.org, rgheck.frege.org

An investigation of Frege's various contributions to the study of language, focusing on three of his most famous doctrines: that concepts are unsaturated, that sentences refer to truth-values, and that sense must be distinguished from reference.
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Frege's Principle

In J.Hintikka, ed., From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics (Dordrecht: Kluwer, 1995), pp. 119-42
Reprinted, with a Postscript, in Frege's Thoerem, pp. 90-110

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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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The Function is Unsaturated (with Robert May)

In M. Beaney, ed., The Oxford Handbook of the History of Analytic Philosophy (Oxford: Oxford University Press, 2013), pp. 825-50

philpapers.org, rgheck.frege.org

An investigation of what Frege means by his doctrine that functions (and so concepts) are 'unsaturated'. We argue that this doctrine is far less peculiar than it is usually taken to be. What makes it hard to understand, oddly enough, is the fact that it is so deeply embedded in our contemporary understanding of logic and language. To see this, we look at how it emerges out of Frege's confrontation with the Booleans and how it expresses a fundamental difference between Frege's approach to logic and theirs.
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Grundgesetze der Arithmetik I §§29-32

Notre Dame Journal of Formal Logic 38 (1998), pp. 437-74
Reprinted, more or less, as Chapter 3 of Reading Frege's Grundgesetze, though some material is contained in Chapter 5

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Frege's intention in section 31 of Grundgesetze is to show that every well-formed expression in his formal system denotes. But it has been obscure why he wants to do this and how he intends to do it. It is argued here that, in large part, Frege's purpose is to show that the smooth breathing, from which names of value-ranges are formed, denotes; that his proof that his other primitive expressions denote is sound and anticipates Tarski's theory of truth; and that the proof that the smooth breathing denotes, while flawed, rests upon an idea now familiar from the completeness proof for first-order logic. The main work of the paper consists in defending a new understanding of the semantics Frege offers for the quantifiers: one which is objectual, but which does not make use of the notion of an assignment to a free variable.
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Grundgesetze der Arithmetik I §10

Philosophia Mathematica 7 (1999), pp. 258-92
Reprinted, more or less, as Chapter 4 of Reading Frege's Grundgesetze

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In section 10 of Grundgesetze, Frege confronts an indeterminacy left by his stipulations regarding his 'smooth breathing', from which names of valueranges are formed. Though there has been much discussion of his arguments, it remains unclear what this indeterminacy is; why it bothers Frege; and how he proposes to respond to it. The present paper attempts to answer these questions by reading section 10 as preparatory for the (fallacious) proof, given in section 31, that every expression of Frege's formal language denotes.
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Idiolects

In J. J. Thomson and A. Byrne, eds., Content and Modality: Themes from the Philosophy of Robert Stalnaker (Oxford: Oxford University Press, 2006), pp. 61-92

philpapers.org, rgheck.frege.org

Defends the view that the study of language should concern itself, primarily, with idiolects. The main objections considered are forms of the normativity objection.
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In Defense of Formal Relationism

Thought 3 (2014), pp. 243-50

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In his paper "Flaws of Formal Relationism", Mahrad Almotahari argues against the sort of response to Frege's Puzzle I defended in "Solving Frege's Puzzle". Almotahari argues that, because of its specifically formal character, Formal Relationism 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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Intuition and the Substitution Argument

Analytic Philosophy 55 (2014), pp. 1-30

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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 in Jennifer Saul's defense of Russellianism, based upon the existence of an allegedly similar contrast between these two sentences:
(PC) Superman is more popular than Clark.
(PS) Superman is more popular than Superman.
The latter contrast looks pragmatic. But then, Saul asks, why shouldn't we then say the same about the former?

The answer to this question is that the two cases simply are not similar. In the case of (PC) and (PS), we have only the facts that these strike us differently, and that people will sometimes say things like (PC), whereas they will never say things like (PS). By contrast, there is an argument to be given that (LS) can be true even if (LC) is false, and this argument does not appeal to anyone's 'intuitions'.

The main goal of the paper is to present such a version of the substitution argument, building upon the treatment of the Fregan argument against Russellian accounts of belief itself in "Solving Frege's Puzzle". A subsidiary goal is to contribute to the growing literature arguing that 'intuitions' simply do not play the sort of role in philosophical inquiry that so-called 'experimental philosophers' have supposed they do.

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Is Compositionality a Trivial Principle?

Frontiers of Philosophy in China 8 (2103), pp. 140-55

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Primarily a response to Paul Horwich's "Composition of Meanings", the paper attempts to refute his claim that compositionality—roughly, the idea that the meaning of a sentence is determined by the meanings of its parts and how they are there combined—imposes no substantial constraints on semantic theory or on our conception of the meanings of words or sentences.
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Is Frege's Definition of the Ancestral Adequate?

Philosophia Mathematica 24 (2016), pp. 91-116

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Why should one think that Frege's definition of the ancestral is correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that fact might seem to undermine Frege's claim to have justified induction in purely logical terms—a worry that goes back to Bruno Kerry and Henri Poincaré. In this paper, I discuss such circularity objections and then offer a new definition of the ancestral, one that is intended to be intensionally correct; its extensional correctness then follows without proof. It can then be proven to be equivalent to Frege's definition, without any use of arithmetical induction. This constitutes a proof that Frege's definition is extensionally correct that does not make any use of arithmetical induction, thus answering the circularity objections.

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Julius Caesar and Basic Law V

Dialectica 59 (2005), pp. 161-78
Reprinted in Frege's Theorem, pp. 111-26

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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 this issue relates to broader issues in Frege's philosophy.
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The Julius Caesar Objection

In Language, Thought, and Logic: Essays in Honour of Michael Dummett, pp. 273-308
Reprinted in Frege's Theorem, pp. 127-55

philpapers.org, rgheck.frege.org

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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A Liar Paradox

Thought 1 (2012), pp. 36-40

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The purpose of this note is to present some strong forms of the liar paradox. They are strong because the logical resources needed to generate them paradox are weak. The only logical resources used are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p & p), infer ¬p. We then get a paradox if we assume either (a) the `transparency' of truth and the law of non-contradiction or (b) the schemata: ¬(S & T(¬S); and ¬(¬S & ¬T(¬S).

The lesson I would like to draw is: There can be no consistent solution to the Liar paradox that does not involve abandoning truth-theoretic principles that should be every bit as dear to our hearts as the T-scheme. So we shall have to learn to live with the Liar, one way or another.

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A Logic for Frege's Theorem

In Frege's Theorem, pp. 267-96
Also to appear in a A. Miller, ed., Essays for Crispin Wright: Logic, Language and Mathematics

philpapers.org, rgheck.frege.org

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 truths. What I suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic".
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MacFarlane on Relative Truth

Philosophical Issues 16 (2006), pp. 88-100

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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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Meaning and Truth-conditions

In D. Griemann and G. Siegwart, eds., Truth and Speech Acts: Studies in the Philosophy of Language (New York: Routledge, 2007), pp. 349-76

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Develops a conception of how knowledge of meaning is put to use in communication. Along the way, it defends the view that understanding can be identified with knowledge of T-sentences against the classical criticisms of Foster and Soames.
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Meaning and Truth-conditions: A Reply to Kemp

Philosophical Quarterly 52 (2002), pp. 82-87

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In his "Meaning and Truth-Conditions", Gary Kemp offers a reconstruction of Frege's infamous 'regress argument', which purports to rely only upon the premises that the meaning of a sentence is its truth-condition and that each sentence expresses a unique proposition. If cogent, the argument would show that only someone who accepts a form of semantic holism can use the notion of truth to explain that of meaning. I respond that Kemp relies heavily upon what he himself styles "a literal, rather wooden" understanding of truth-conditions. I explore alternatives, and say a few words about how Frege's regress argument might best be understood.
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Non-conceptual Content and the 'Space of Reasons'

Philosophical Review 109 (2000), pp. 483-523

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In Mind and World, John McDowell argues against the view that perceptual representation is non-conceptual. The central worry is that this view cannot offer any reasonable account of how perception bears rationally upon belief. I argue that this worry, though sensible, can be met, if we are clear that perceptual representation is, though non-conceptual, still in some sense 'assertoric': Perception, like belief, represents things as being thus and so.
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A Note on the Logic of Higher-order Vagueness

Analysis 53 (1993), pp. 201-8
Reprinted in D. Graff and T. Williamson, eds., Vagueness (Dartmouth: Ashgate, 2002), pp. 315-22

analysis.oxfordjournals.org, philpapers.org, rgheck.frege.org

A discussion of Crispin Wright's 'paradox of higher-order vagueness', I suggest that the paradox may be resolved by careful attention to the logical principles used in its formulation. In particular, I focus attention on the rule of inference that allows for the inference from A to 'Definitely A', and argue that this rule, though valid, may not be used in subordinate deductions, e.g., in the course of a conditional proof. Wright's paradox uses the rule (or its equivalent) in this way.
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On the Consistency of Second-order Contextual Definitions

Noûs 26 (1992), pp. 491-4
Reprinted, with a Postscript, in Frege's Theorem, pp. 227-36

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One of the earliest discussions of the so-called 'bad company' objection to Neo-Fregeanism, I show that the consistency of an arbitrary second-order 'contextual definition' (nowadays known as an 'abstraction principle' is recursively undecidable. I go on to suggest that an acceptable such principle should satisfy a condition nowadays known as 'stablity'.
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Predicative Frege Arithmetic and "Everyday Mathematics"

Philosophia Mathematica 22 (2014), pp. 279-307

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This paper shows that predicative Frege arithmetic naturally interprets some weak but non-trivial arithmetical theories. The weak theories in question are all relational versions of Tarski, Mostowski, and Robinson's R and Q, i.e., they are formulated using predicates Pxy, Axyz, and Mxyz in place of the usual function symbols Sx, x+y, and x×y. We lose the existence and uniqueness of successor, sum, and product, as generalizations, but retain these in each particular case (much as we lose the recursion clauses for addition in R, but retain them in each particular case). In saying that the interpretation is "natural", I mean that it relies only upon "definitions" of arithmetical notions that are themselves "natural", that is, that have some claim to be "definitions" in something other than a purely formal sense.
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Ramified Frege Arithmetic

Journal of Philosophical Logic 40 (2011), pp. 715-35

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Øystein Linnebo has shown that the existence of successors cannot be proven in predicative Frege arithmetic, that is, predicative second-order logic plus "Hume's Principle" and Frege's definitions of zero, predecessor, and natural number. It is shown in the present paper that the existence of successors can be proven if the logic is strengthened to ramified predicative second-order logic. It then follows from work by John Burgess and Allen Hazen that Robinson arithmetic, Q, can be interpreted in ramified Frege arithmetic.
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Reason and Language

In C. Macdonald and G. Macdonald, eds., McDowell and His Critics (Oxford: Blackwell Publishing, 2006), pp. 22-45

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John McDowell has often emphasized the fact that the use of language 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 meana. 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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Reply to Hintikka and Sandu: Frege and Second-order Logic (with Jason Stanley)

Journal of Philosophy 90 (1993), pp. 416-24

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Hintikka and Sandu had argued that 'Frege's failure to grasp the idea of the standard interpretation of higher-order logic turns his entire foundational project into a hopeless daydream' and that he is 'inextricably committed to a non-standard interpretation' of higher-order logic. We disagree.
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Self-reference and the Languages of Arithmetic

Philosophia Mathematica 15 (2007), pp. 1-29

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It is often said that diagonalization allows one to construct sentences that are self-referential. This paper investigates the sense in which that is true. I argue first that, in the standard language of arithmetic, in which we have only the symbols 0, S, +, and ×, truly self-referential sentences cannot be constructed. This problem can be resolved by expanding the language to include function-symbols for all primitive recursive functions. It can also be resolved by proving a stronger form of the diagonal lemma that I call the "structural" diagonal lemma. At the end of the paper, it is argued, however, that there are some contexts in which the latter method is insufficient.
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Semantic Accounts of Vagueness

In J.C. Beall, ed., Liars and Heaps (Oxford: Oxford University Press, 2003), pp. 106-27

philpapers.org, rgheck.frege.org

Read as a comment on Crispin Wright's "Vagueness: A Fifth Column Approach", this paper defends a form of supervaluationism against Wright's criticisms. Along the way, however, it takes up the question what is really wrong with Epistemicism, how the appeal of the Sorities ought properly to be understood, and why Contextualist accounts of vagueness won't do.
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Semantics and Context-Dependence: Towards a Strawsonian Account

In A. Burgess and B. Sherman, eds., Metasemantics: New Essays on the Foundations of Meaning (Oxford: Oxford University Press, 2014), pp. 327-64

rgheck.frege.org

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, familiar from Kaplan and others, that utterances express propositions `relative to contexts' and replace it with the Strawonian idea that speakers express propositions by making utterances in contexts. The argument for this claim consists in a detailed investigation of the particular case of demonstratives, which I argue demand such a Strawsonian treatment. I then respond to several objections, the most important of which allege that the Strawsonian account somehow undermines the project of natural language semantics, or threatens the semantics-pragmatics distinction.

Please note that the paper posted here is an extended version of what was published.

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The Sense of Communication

Mind 104 (1995), pp. 79-106

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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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Solving Frege's Puzzle

Journal of Philosophy 109 (2012), pp. 132-74

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So-called 'Frege cases' pose a challenge for anyone who would hope to treat the contents of beliefs (and similar mental states) as Russellian propositions: It is then impossible to explain people's behavior in Frege cases without invoking non-intentional features of their mental states, and doing that seems to undermine the intentionality of psychological explanation. In the present paper, I develop this sort of objection in what seems to me to be its strongest form, but then offer a response to it. I grant that psychological explanation must invoke non-intentional features of mental states, but it is of crucial importance which such features must be referenced.

It emerges from a careful reading of Frege's own view that we need only invoke what I call 'formal' relations between mental states. I then claim that referencing such 'formal' relations within psychological explanation does not undermine its intentionality in the way that invoking, say, neurological features would. The central worry about this view is that either (a) 'formal' relations bring narrow content in through back door or (b) 'formal' relations end up doing all the explanatory work. Various forms of each worry are discussed. The crucial point, ultimately, is that the present strategy for responding to Frege cases is not available either to the 'psycho-Fregean', who would identify the content of a belief with its truth-value, nor even to someone who would identify the content of a belief with a set of possible worlds. It requires the sort of rich semantic structure that is distinctive of Russellian propositions. There is therefore no reason to suppose that the invocation of 'formal' relations threatens to deprive content of any work to do.

Note: The pre-publication version is longer than what was published.

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Syntactic Reductionism

Philosophia Mathematica 8 (2000), pp. 124-49
Reprinted in Frege's Theorem, pp. 180-99

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Syntactic Reductionism, as understood here, is the view that the 'logical forms' of sentences in which reference to abstract objects appears to be made are misleading so that, on analysis, we can see that no expressions which even purport to refer to abstract objects are present in such sentences. After exploring the motivation for such a view, and arguing that no previous argument against it succeeds, sentences involving generalized quantifiers, such as 'most', are examined. It is then argued, on this basis, that Syntactic Reductionism is untenable.
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Tarski, Truth, and Semantics

Philosophical Review 106 (1997), pp. 533-54

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John Etchemendy has argued that it is but "a fortuitous accident" that Tarski's work on truth has any signifance at all for semantics. I argue, in response, that Etchemendy and others, such as Scott Soames and Hilary Putnam, have been misled by Tarski's emphasis on definitions of truth rather than theories of truth and that, once we appreciate how Tarski understood the relation between these, we can answer Etchemendy's implicit and explicit criticisms of neo-Davidsonian semantics.
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That There Might Be Vague Objects (So Far as Concerns Logic)

The Monist 81 (1998), pp. 277-99

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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 such statements might actually fail to have a determinate truth-value. That makes Indefinitism a cousin of mathematical Intuitionism.
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Truth and Disquotation

Synthese 142 (2004), pp. 317-52

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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 precisely, I argue that we have no need for a disquotational truth-predicate; that the word 'true', in ordinary language, is not a disquotational truth-predicate; and that it is not at all clear that it is even possible to introduce a disquotational truth-predicate into ordinary language. If so, then we have no clear sense how it is even possible to be a methodological deflationist. My goal here is not to convince a committed deflationist to abandon his or her position. My goal, rather, is to argue, contrary to what many seem to think, that reflection on the apparently trivial character of T-sentences should not incline us to deflationism.
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Use and Meaning

In R. E. Auxier and L. E. Hahn, eds., The Philosophy of Michael Dummett (Chicago: Open Court, 2007), pp. 531-57

philpapers.org, rgheck.frege.org

Many philosophers have been attracted to the idea that meaning is, in some way or other, determined by use—chief among them, perhaps, Michael Dummett. But John McDowell has argued that Dummett, and anyone else who would seek to draw serious philosophical conclusions from this claim, must face a dilemma: Either the use of a sentence is characterized in terms of what it can be used to say, in which case profound philosophical consequences can hardly follow, or it will be impossible to make out the sense in which the use of language is a rational activity. The paper evaluates McDowell's arguments and, in so doing, attempts to offer an initial sketch of how the notion of use might be so understood that the claim that use determines meaning is a substantive one. (I do not take any stand here on whether one should accept that claim.)
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