On the Innocence and Determinacy of Plural Quantification

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. Both assumptions are problematic, as is shown by providing a Henkin-style semantics for plural logic that does not resort to sets but takes a plural term to have plural reference. This semantics gives rise to a generalized notion of ontological commitment, which is used to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.

Much Ado About the Many

English distinguishes between singular quantifiers like "a donkey" and plural quantifiers like "some donkeys". Pluralists hold that plural quantifiers range in an unusual, irreducibly plural, way over common objects, namely individuals from first-order domains and not over set-like objects. The favoured framework of pluralism is plural first-order logic, PFO, an interpreted first-order language that is capable of expressing plural quantification. Pluralists argue for their position by claiming that the standard formal theory based on PFO is both ontologically neutral and really logic. These properties are supposed to yield many important applications concerning second-order logic and set theory that alternative theories supposedly cannot deliver. I will show that there are serious reasons for rejecting at least the claim of ontological innocence. Doubt about innocence arises on account of the fact that, when properly spelled out, the PFO-semantics for plural quantifiers is committed to set-like objects. The correctness of my worries presupposes the principle that for every plurality there is a coextensive set. Pluralists might reply that this principle leads straight to paradox. However, as I will argue, the true culprit of the paradox is the assumption that every definite condition determines a plurality.

Collective quantification and the homogeneity constraint

<p>The main theoretical claim of the paper is that a slightly revised version of the analysis of mass quantifiers proposed in Roeper 1983, Lønning 1987 and Higginbotham 1994 extends to <em>collective</em> quantifiers: such quantifiers denote <em>relations between sums of entities</em> (type e), <em>rather than relations between sets of sums</em> (type &lt;e,t&gt;). Against this background I will explain a puzzle observed by Dowty (1986) for <em>all</em> and generalized to all quantifiers by Winter 2002: plural quantification is not allowed with all the predicates that are traditionally classified as ”collective”. The Homogeneity Constraint – as well as the weaker requirement of divisiveness - will be shown to be too strong (for both collective and mass quantifiers). What is required is that the nominalization of the nuclear-scope predicate denotes a maximal sum (rather than a group). Divisiveness is a sufficient, but not a necessary condition for this to happen. Non-divisive predicates such as <em>form a circle</em>, which denote sets of ‘extensional’ groups are allowed, because extensional groups are equivalent to the maximal sum of their members. It is only intensional group predicates that block collective Qs.</p><p><strong>Keywords</strong>: collective quantification, mass quantification, homogeneous, cumulative, divisive, groups, sums, maximality operator, plural logic</p>

Atom Predicates and Set Predicates: Towards a General Theory of Plural Quantification

No abstract.

ON THE INFINITE IN MEREOLOGY WITH PLURAL QUANTIFICATION

In “Mathematics is megethology,” Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, if—as Lewis maintains—MPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of objects.

ON THE ONTOLOGICAL COMMITMENT OF MEREOLOGY

In Parts of Classes (1991) and Mathematics Is Megethology (1993) David Lewis defends both the innocence of plural quantification and of mereology. However, he himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out as a whole. In the case of plural quantification “we have many things, in no way do we mention one thing that is the many taken together”. Instead, in the mereological case: “we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many” (Lewis, 1991, p. 87). The aim of the paper is to argue that—for a certain use of mereology, weaker than Lewis’ one—an innocence thesis similar to that of plural reference is defensible. To give a precise account of plural reference, we use the idea of plural choice. We then propose a virtual theory of mereology in which the role of individuals is played by plural choices of atoms.