Absolute Generality and Singularization

Is it possible to assert something of absolutely everything there is? While such absolute generality appears possible, it faces serious theoretical challenges. This chapter examines one such challenge, based on the possibility of one-to-one mappings from pluralities to objects. The challenge gives rise to a trilemma. First, generality relativists deny the possibility of absolute generality, at the cost of being unable to express various important insights. Second, traditional generality absolutists deny the possibility of the mentioned mappings, but are pushed up a hierarchy of logics of higher and higher order, resulting in an expressibility deficit akin to that of the relativists. This motivates taking a closer look at the third option, which restricts traditional plural logic.

Restrictionism and Expansionism

If her view is to diffuse charges of mystical censorship, the relativist needs a well-motivated account of what prevents our quantifying over an absolutely comprehensive domain. But relativists may seek to meet this challenge in different ways. One option is to draw on more familiar cases of quantifier domain restriction in order to motivate the thesis that a quantifier’s domain is always subject to restriction. An alternative is to permit unrestricted quantifiers but maintain that even these fail to attain absolute generality on the grounds that the universe of discourse is always open to expansion. This chapter outlines restrictionist and expansionist variants of relativism and argues that the importance of the distinction comes out in two influential objections that have been levelled against relativism.

Absolutism and Relativism

Absolutism about quantifiers maintains, with a good deal of prima facie plausibility, that quantifiers like ‘everything’ sometimes range over an absolutely comprehensive domain. This view has been challenged on various grounds: some deny the availability of a universal nominal like ‘thing’ on the grounds that it lacks a non-trivial criterion of identity; others contend that absolutism is committed to objectionable views in metaontology. But the most compelling reason to support relativism about quantifiers as opposed to absolutism is bound up with the set-theoretic paradoxes. This introductory chapter offers an overview of the absolute generality debate, and sets the scene for the defence of relativism that follows in the rest of the book.

Russell Reductio Redux

By far and away the strongest argument against there being an absolutely comprehensive domain of quantification comes from the set-theoretic paradoxes. The argument from indefinite extensibility can be rigorously regimented with the help of schematic or modal resources. After dispensing with the charge that the argument relies on an incoherent conception of set, this chapter offers a defence of its premisses. Advocates of the orthodox absolutist means to defend absolute generality have yet to give a non-ad-hoc response to the paradoxes. A heterodox absolutist view, which seeks to give an absolutist-friendly account of indefinite extensibility, leads to severe problems with impure set theory. The chapter closes by considering a revenge problem for hybrid relativists, who take modalized quantifiers to achieve absolute generality.

Modal Operators

Notwithstanding her rejection of quantification over an absolutely comprehensive domain, a relativist about quantifiers may still be tempted to seek other means to generalize. This chapter concerns relativist-friendly modal operators. By modalizing her quantifiers, the relativist has a systematic way to attain absolute generality, which permits her to regiment her view with a single modal formula, and to frame an attractive modal axiomatization of the iterative conception of set. In addition to the immediate cost of admitting the relevant modality into her ideology, however, this approach leads to a hybrid version of relativism, which has some significant commonalities with absolutism about quantifiers.

Russell, Zermelo, and Dummett

Concerns about generality in the context of set theory are not new. Russell seeks to resolve the set-theoretic antinomies by maintaining that we cannot legitimately speak of ‘all classes’. Zermelo attempts to avoid the paradoxes without ‘constriction and mutilation’ by adopting an open-ended conception of the cumulative hierarchy of sets. Dummett takes the indefinite extensibility of concepts such as set and ordinal to impugn absolutism about quantifiers. But not every paradox-inspired argument is an argument for relativism about quantifiers. This chapter aims to fill in the logical and philosophical background to the contemporary absolute generality debate, with an eye to disentangling my favoured indefinite-extensibility-based argument from others in its vicinity.

Everything, more or less

Almost no systematic theorizing is generality-free. Scientists test general hypotheses; set theorists prove theorems about every set; metaphysicians espouse theses about all things regardless of their kind. But how general can we be? Do we ever succeed in theorizing about ABSOLUTELY EVERYTHING in some interestingly final, all-caps-worthy sense of ‘absolutely everything’? Not according to generality relativism. In its most promising form, this kind of relativism maintains that what ‘everything’ and other quantifiers encompass is always open to expansion: no matter how broadly we may generalize, a more inclusive ‘everything’ is always available. The importance of the issue comes out, in part, in relation to the foundations of mathematics. Generality relativism opens the way to avoid Russell’s paradox without imposing ad hoc limitations on which pluralities of items may be encoded as a set. On the other hand, generality relativism faces numerous challenges: What are we to make of seemingly absolutely general theories? What prevents our achieving absolute generality simply by using ‘everything’ unrestrictedly? How are we to characterize relativism without making use of exactly the kind of generality this view foreswears? This book offers a sustained defence of generality relativism that seeks to answer these challenges. Along the way, the contemporary absolute generality debate is traced through diverse issues in metaphysics, logic, and the philosophy of language; some of the key works that lie behind the debate are reassessed; an accessible introduction is given to the relevant mathematics; and a relativist-friendly motivation for Zermelo–Fraenkel set theory is developed.

Dynamic Abstraction

Any abstractionist approach to thin objects faces the threat of paradox, as illustrated by Frege’s inconsistent Basic Law V. The neo-Fregeans Hale and Wright respond by severely restricting the class of acceptable abstraction principles. Their approach is static in the sense that they hold the domain fixed. This approach to abstraction is criticized, and an alternative approach is developed which permits abstraction on a vast class of equivalence relations. This alternative approach is dynamic in the sense that abstraction on an extensionally specified domain (i.e. a domain specified by means of a plurality of objects) may result in a larger such domain. A form of absolute generality is nevertheless possible, provided that the associated domain is understood in an intensional sense (i.e. it cannot be specified by means of a plurality).

Dadaism: Restrictivism as militant quietism

Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some authors have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who claims to hold a positive position about ‘absolute generality’.Published in Proceedings of the Aristotelian Society 110.3: 387–98.