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.

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.

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)

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)

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)

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)

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)

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)

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)

This chapter concentrates on two discussions, both of which enlist Wittgenstein's rule‐following considerations in support of radical and highly revisionary conclusions about the objectivity of meaning ‐ conclusions which may appear to entail, and have been taken to entail, consequences for the objectivity of truth and judgment which are no less radical and revisionary. There is widespread agreement that Wittgenstein advances, in the rule‐following sections of Philosophical Investigations and Remarks on the Foundations of Mathematics, considerations that are quite destructive of (...) certain conceptions of meaning, understanding, and rule‐following into which one may easily slide when attempts a general philosophical account of them. Saul Kripke interprets central sections of Philosophical Investigations as developing a 'skeptical paradox' about meaning. Kripke himself describes the skeptical conclusion as 'insane and intolerable'. (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)

According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.

Bob Hale; XII*—Grundlagen §64, Proceedings of the Aristotelian Society, Volume 97, Issue 1, 1 June 1997, Pages 243–262, https://doi.org/10.1111/1467-9264.00015.

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)

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)

From the philosophy of mind and language, through physics and mathematics, to the philosophy of the human sciences, morality and religion, there is almost no area of philosophy to which Hilary Putnam has not made highly original and influential contributions. This wide-ranging collection of papers provides a critical assessment and exploration of Putnam's Seminal Work. Written by Philosophers themselves well known for their work in the field, each essay bears witness to the continuing influence and importance of Putnam's thought. Putnam's (...) reply constitutes an extensive new essay which clarifies, and develops further, central themes in his philosophy. This volume includes papers by Simon Blackburn, George Boolos, Michael Hallett, Michael Redhead, Thomas Ricketts, David Wiggins and Crispin Wright. (shrink)

The article argues that modal concepts should be explained in terms of the essences or nature of things: necessarilypif, and because, there is something the nature of which ensures thatp; possiblypif, and because, there is nothing whose nature rules out its being true thatp. The theory is defended against various objections and difficulties, including ones arising from attributing essences to contingent individuals.

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(q□→p).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. (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)

What is wrong with abstraction’, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their ‘Hale on Caesar’ and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are: To get the character of this objection into sharper focus; To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and (...) about the effects of impredicativity in abstraction principles—which underlie it; and To advance the debate of the issues thereby raised. Thanks for helpful comments to Roy Cook and to an anonymous referee. CiteULike Connotea Del.icio.us What's this? (shrink)

One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it (...) requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence. (shrink)

This book is a collection of essays written by Bob Hale (three co-authored), with a critical introduction from Kit Fine. They comprise Hale’s final years of work, adding to and extending beyond his landmark monograph Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them (OUP, 2013, 2nd edition 2015). The essays develop and consolidate several key themes in Hale’s work, most notably the notion of definition, especially as it extends beyond definition of a word to definition of (...) a thing more generally, and its relations to essence and existence; how the recently influential notion of truthmaking relates to and illuminates some of these issues; and ontological questions connected with Hale’s metaphysical commitments on the one hand, and his commitments in the foundations of mathematics on the other. Several of the essays engage with and respond to work by Kit Fine, and contributions to Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (OUP 2018). As such, these essays also demonstrate a rich and fruitful philosophical dialogue between Hale and his critics and friends. (shrink)

At least some of us, at least some of the time—when not in the grip of radical sceptical doubt—are inclined to believe that we know, for example, that if we infer a conclusion from two true premises, one a conditional whose consequent is that conclusion and the other the antecedent of that conditional, then our conclusion must be true, or that we know similar things about other simple patterns of inference. If we do indeed have knowledge of this sort, it (...) is what I mean by logical knowledge. Logical knowledge is, roughly speaking, knowledge about logic—such as knowledge that a certain principle of inference necessarily preserves truth, or that every proposition of a certain form must be true—and so is not the same thing as knowledge that is gained by using logic, i.e. inferential knowledge. That is not to say, of course, that logical knowledge can't be inferential. On the contrary, it is barely open to question that—if there is any logical knowledge at all—there is a lot of inferential logical knowledge. For example, if we know that the introduction and elimination principles for the conditional are truth-preserving, we can surely get to know, by inference, that the principle of hypothetical syllogism is so too, not to mention other, less obvious and more recondite, examples of putative logical knowledge. (shrink)