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.

Schemas

Relativism about quantifiers maintains that we never quantify over an absolutely comprehensive domain. Some philosophers are quick to reject this view on the grounds that it cannot be coherently formulated. But not every coherent theory can be captured in a single thesis. Sooner or later, everyone needs to make sense of theories that are infinitely axiomatized using schemas. This chapter offers a schematic regimentation of relativism. It then goes on to investigate the use of schemas to provide a relativist-friendly means to simulate absolutely general quantification. The chapter closes by discussing whether the relativist is able to frame an appropriate side-condition on the schema intended to capture her view.

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.

How Universes Expand

Advocates of an expansionist version of relativism face an important explanatory challenge: how might we cause the universe of discourse to expand? To make progress with this question, it’s helpful to begin with another: how might a quantifierless linguistic community come to quantify? On the basis of some standard semantic assumptions and a metasemantic assumption about how use determines meaning, this chapter offers an idealized answer to both questions. It is further argued that natural patterns of use may cause a linguistic community’s universe to repeatedly expand. The chapter closes with a summary of the defence of relativism given in the rest of the book.

Quantifiers

The notion of quantification is clearly central to the absolutist’s characteristic claim that we sometimes quantify over an absolutely comprehensive domain. Barwise and Cooper give a widely accepted semantics for natural language quantifiers, building on the usual model-theoretic semantics for first-order languages. But only a relativist about quantifiers can take these semantic theories at face value. An absolutist who denies that absolutely general quantifiers range over a set-domain may seek to free these semantic theories from their set-theoretic trappings by employing plural and superplural resources. More radically, he may reject the Barwise–Cooper semantics altogether. This chapter argues that neither approach is cost-free.