Definition, Abstraction, Postulation, and Magic

In recent work, Kit Fine proposes a new approach to the philosophy of mathematics, which he calls procedural postulationism: the postulates from which a mathematical theory is derived are imperatival, rather than indicative, in character. According to procedural postulationism, what is postulated in mathematics are not propositions true in a given mathematical domain, but rather procedures for the construction of that domain. Fine claims some very significant advantages for procedural postulationism over other approaches. This chapter raises some questions for the view and its promised advantages. One crucial set of questions concerns how exactly the commands of procedural postulationism are to be understood. And in particular, how literally are we to take talk of construction?

S5 as the Logic of Metaphysical Modality: Two Arguments for and Two Arguments against

Two arguments for S5 being the logic of metaphysical modality are favourably discussed: one from the logic of absolute necessity, one from Timothy Williamson. Two arguments against S5 being the logic of metaphysical modality are discussed and rebuffed: one from Nathan Salmon against S4, and thereby S5, being the logical of metaphysical modality; and one from Michael Dummett against the B principle for metaphysical modality. In the Appendix, some comments are offered on the logics of ‘true in virtue of the nature of’, and its relation to logical necessity. It is argued that the logic both of ‘true in virtue of the nature of x’ and of essentialist logical necessity is S5.

Essence and Definition by Abstraction

We may define words. We may also define the things for which words stand. Definitions of words may be explicit or implicit, and may seek to report pre-existing synonymies, but they may instead be wholly or partly stipulative. Definition by abstraction seeks to define a term-forming 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 (real definitions) are typically conceived as statements about the essence of their definienda, and so not stipulative. There thus appears to be a clash between taking Hume's principle as an implicit, at least partly stipulative definition of the number operator and as a real definition of cardinal numbers. This chapter argues that this apparent tension can be resolved, and that resolving it shows how some modal knowledge can be a priori.

The Problem of Mathematical Objects

If fundamental mathematical theories such as arithmetic and analysis are taken at face value, any attempt to provide an epistemological foundation—roughly, an account which explains how we can know standard mathematical theories to be true, or at least justifiably believe them—must confront the problem of mathematical objects—the problem of explaining how a belief in the existence of an infinity of natural numbers, an uncountable infinity of real numbers, etc., is to be justified. One small but fundamental part of the problem is discussed: whether we can be justified in believing that there is a denumerable infinity of natural numbers, or, more generally, an infinity of objects of any kind. The chapter considers two broad approaches to this problem—what are called object-based and property-based approaches.

Second-order Logic: Properties, Semantics, and Existential Commitments

Quine’s charge against second-order logic is that it carries massive existential commitments. This chapter argues that if we interpret second-order variables as ranging over properties construed in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This need not preclude the use of model-theoretic semantics for second-order languages; but it precludes the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach of this chapter has revisionary implications; it is, however, compatible with the different special case in which second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, important metalogical results obtainable under the standard semantics may still be obtained. Finally, the chapter discusses the relations between second-order logic, interpreted as recommended, and a strong version of schematic ancestral logic promoted in recent work by Richard Kimberly Heck.

Relative Necessity Reformulated with Jessica Leech

This chapter discusses some serious difficulties for what it calls 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. It is argued, with the help of Humberstone (1981), that the standard account has several unpalatable consequences. It is argued that Humberstone’s alternative account has certain disadvantages, and another—considerably simpler—solution is offered. The proposed alternative takes seriously the idea that the standard account omits crucial information which, if suitably replaced, allows the problems to be solved.

The Problem of De Re Modality

Quine has two arguments against the intelligibility of de re modality: a ‘logical’ argument and a ‘metaphysical’ argument. That the ‘logical’ argument is central to Quine’s attack is surely indisputable. This chapter claims that this ‘logical argument’ is his basic argument. However, Kit Fine disagrees. It is conceded that Fine is correct that there are some significant differences between the two arguments. However, the most important question for the purposes of this chapter is whether Fine is right to claim that the two arguments have force independently of one another; that the metaphysical argument raises a separate and independent objection to the intelligibility of quantifying into modal contents. This chapter suggests not.

Essence and Existence

The essentialist theory faces two problems concerning contingent beings. First, it apparently leads to the conclusion, unpalatable to believers in contingently existing individuals, that Aristotle is a necessary being. Second, if, as is reasonable to suppose, some natures exist contingently, then they will, it seems, be unable to ground necessities. In this chapter, it is attempted to explain how these problems are best solved. The heart of the first problem concerns how essence interacts with existence. In short: statements of essence—including statements of individual essence—are not existence-entailing with respect to the entities whose essences they purport to state. The key to solving the second problem is a distinction between a proposition being true in and true of a possible situation. The essences of actually yet contingently existing entities ground necessary truths, and in particular, the truth of propositions about those entities of, but not in, all possible situations.

Bolzano’s Definition of Analytic Propositions with Crispin Wright

The chapter begins by drawing attention to some drawbacks of the Frege-Quine definition of analytic truth. With this the definition of analytic propositions given by Bolzano in his Wissenschaftslehre is contrasted. If Bolzano’s definition is viewed, as Bolzano himself almost certainly did not view it, as attempting to capture the notion of analyticity as truth-in-virtue-of-meaning, which occupied centre stage during the first half of the last century and which, Quine’s influential assault on it notwithstanding, continues to attract philosophical attention, it runs into some very serious problems. It is argued that Bolzano’s central idea can, nevertheless, be used as the basis of a new definition which avoids these problems and possesses definite advantages over the Frege-Quine approach.

Ordinals by Abstraction

The neo-Fregean programme in the philosophy of mathematics seeks to provide foundations for fundamental mathematical theories in abstraction principles. Ian Rumfitt (2018) proposes to introduce ordinal numbers by means of an abstraction principle, (ORD), which says, roughly, that ‘the ordinal number attaching to one well-ordered series is identical with that attaching to another if, and only if, the two series are isomorphic’. Rumfitt’s proposal poses a sharp and serious challenge to those seeking to advance the neo-Fregean programme, for Rumfitt proposes to save (ORD) from threatening paradox by avoiding dependence on an impredicative comprehension principle. However, such a principle is usually taken to be required by the neo-Fregean account of the cardinal numbers. Thus if neo-Fregean foundations for elementary arithmetic are to be saved, we must explain how we can avoid paradox for (ORD) in another way. In this chapter, the prospects for doing so are explored.