Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

Bob Hale presents a broadly Fregean approach to metaphysics, according to which ontology and modality are mutually dependent upon one another. He argues that facts about what kinds of things exist depend on facts about what is possible. Modal facts are fundamental, and have their basis in the essences of things--not in meanings or concepts.

Bob Hale presents a broadly Fregean approach to metaphysics, according to which ontology and modality are mutually dependent upon one another. He argues that facts about what kinds of things exist depend on facts about what is possible. Modal facts are fundamental, and have their basis in the essences of things--not in meanings or concepts.

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...) explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)

This paper discusses some serious difficulties for what we shall call the standard account of various kinds of relative necessity, according to which any given kind of relative necessity may be defined by a strict conditional - necessarily, if C then p - where C is a suitable constant proposition, such as a conjunction of physical laws. We argue, with the help of Humberstone, that the standard account has several unpalatable consequences. We argue that Humberstone’s alternative account has certain disadvantages, (...) and offer another - considerably simpler - solution. (shrink)

This volume provides a survey of contemporary philosophy of language. As well as providing a synoptic view of the key issues, figures, concepts and debates, each essay makes new and original contributions to ongoing debate.

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant (...) or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

We may define words or concepts, and we may also, as Aristotle and others have thought, define the things for which words stand and of which concepts are concepts. Definitions of words or concepts may be explicit or implicit, and may seek to report preexisting synonymies, as Quine put it, but they may instead be wholly or partly stipulative. Definition by abstraction, of which Hume’s principle is a much discussed example, seek to define a term-forming operator, such as the number (...) operator, by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things, or real definitions, are, by contrast, typically conceived as true or false statements about the nature or essence of their definienda, and so not stipulative. There thus appears to be an obvious and head-on clash between taking Hume’s principle as an implicit and at least partly stipultative definition of the number operator and taking it as a real definition, stating the nature or essence of cardinal numbers. This paper argues that this apparent tension can be resolved, and that resolving it sheds light on part of the epistemology or essence and necessity, showing how some of our knowledge of essence and necessity can be a priori. (shrink)

Must we believe in logical necessity? I examine an argument for an affirmative answer given by Ian McFetridge in his posthumously published paper 'Logical Necessity: Some Issues', and explain why it fails, as it stands, to establish his conclusion. I contend, however, that McFetridge's argument can be effectively buttressed by drawing upon another argument aimed at establishing that we ought to believe that some propositions are logically necessary, given by Crispin Wright in his paper 'Inventing Logical necessity'. My contention is (...) that Wright's argument, whilst it likewise fails, as it stands, to establish the necessity of necessity, established enough to close off what appears to me to be the only effective-looking sceptical response to McFetridge's original argument. My paper falls into four principal parts. In the first I expound McFetridge's argument and draw attention to the possibility of a certain type of sceptical counter to it. In the second, I begin a response to this sceptical move, taking it as far as I can without reliance upon argument of the kind given by Wright. Turning, then, to Wright's argument, I explain to what extent I think it is successful and seek to rebut some objections to the argument which, were they well-taken, would show that the argument cannot enjoy even the partial success I which to claim for it. Finally, I return to my main theme and try to show, with the assistance of what I take to be solidly established by Wright's argument, that the sceptical response collapses. (shrink)

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects simpler, (...) proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

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 possible and what is necessary, the nature of modal knowledge, modal (...) logic and its relations to necessary existence and to counterfactual reasoning. The general introduction locates the individual contributions in the wider context of the contemporary discussion of the metaphysics and epistemology of modality. (shrink)

I investigate two asymmetrical approaches to knowledge of absolute possibility and of necessity--one which treats knowledge of possibility as more fundamental, the other according epistemological priority to necessity. Two necessary conditions for the success of an asymmetrical approach are proposed. I argue that a possibility-based approach seems unable to meet my second condition, but that on certain assumptions--including, pivotally, the assumption that logical and conceptual necessities, while absolute, do not exhaust the class of absolute necessities--a necessity-based approach may be able (...) to do so. (shrink)

While Frege’s own attempt to provide a purely logical foundation for arithmetic failed, Hume’s principle suffices as a foundation for elementary arithmetic. It is known that the resulting system is consistent—or at least if second-order arithmetic is. Some philosophers deny that HP can be regarded as either a truth of logic or as analytic in any reasonable sense. Others—like Crispin Wright and I—take the opposed view. Rather than defend our claim that HP is a conceptual truth about numbers, I explain (...) one way it may be possible to extend our view beyond elementary arithmetic to encompass the theory of real numbers. My approach has affinities to the leading ideas of Frege’s own treatment of the reals, although differing in one fundamental way. I attempt, like the HP approach to elementary arithmetic, to obtain the reals very directly by means of abstraction principles without any essential reliance on a theory of sets. This is the most natural way of extending the neo-Fregean position to the reals. (shrink)

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main difficulties confronting the (...) most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...) defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role. (shrink)

Michael Dummett mounts, in Frege: Philosophy of Mathematics, a concerted attack on the attempt, led by Crispin Wright, to salvage defensible versions of Frege's platonism and logicism in which Frege's criterion of numerical identity plays a leading role. I discern four main strands in this attack—that Wright's solution to the Caesar problem fails; that explaining number words contextually cannot justify treating them as enjoying robust reference; that Wright has no effective counter to ontological reductionism; and that the attempt is vitiated (...) by the unavoidable impredicativity of its leading principle—and argue that none of them succeeds. (shrink)

An early objection to Simon Blackburn’s first attempts to breathe new life into expressivism—by solving the Frege-Geach problem—was that whilst viewing compound sentences featuring moral components as expressive of attitudes towards combinations of attitudes might enable one to make out that a thinker who, to take the usual example, asserts the premisses but will not accept the conclusion of a moral modus ponens is at fault because they are involved in a “clash of attitudes”, this does no justice to the (...) data of the problem, since the failing here is logical, not just moral or more generally practical. Blackburn claims to have an effective answer to this objection—a way to see how, as he now puts it, expressivism can after all “deliver the mighty ‘musts’ of logic”. I remain unconvinced. (shrink)

How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object , property , and relation —are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical (...) form: for example, the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach—including Frege’s notorious paradox of the concept horse—are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments. (shrink)

How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories—such as object, property, and relation—are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical form: for example, (...) the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach—including Frege’s notorious paradox of the concept horse—are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments. (shrink)

On pourrait définir la nécessité absolue comme la vérité dans absolument tous les mondes possibles sans restriction. Mais nous devrions être capables de l’expliquer sans invoquer les mondes possibles. J’envisage trois definitions alternatives de : « Il est absolument nécessaire que p » et défends une définition contrefactuelle généralisée : ∀q.Je montre que la nécessité absolue satisfait le principe S5 et soutiens que la nécessité logique est absolue. Je discute ensuite des relations entre la nécessité logique et la nécessité métaphysique, (...) en expliquant comment il peut y avoir des nécessités absolues non logiques. J’esquisse également une théorie essentialiste selon laquelle la nécessité a son fondement dans la nature des choses. Je discute certaines de ses conséquences, y compris concernant l’existence nécessaire de certaines propriétés et celle de certains objets.Absolute necessity might be defined as truth at absolutely all possible worlds without restriction. But we should be able to explain it without invoking possible worlds. I consider three alternative definitions of ’it is absolutely necessary that p’ and argue for a generalized counterfactual definition: ∀q. I show that absolute necessity satisfies the S5 principle, and argue that logical necessity is absolute. I discuss the relations between logical and metaphysical necessity, explaining how there can be non-logical absolute necessities, and sketch an essentialist theory according to which necessity is grounded in the natures of things, and discuss some of its consequences, including the necessary existence of certain properties and objects. (shrink)