MEREOLOGICAL COMPOSITION AND PLURAL QUANTIFIER SEMANTICS [Note: This is a preprint version of Lechthaler, M. & Lightfield, C. (forthcoming): Mereological Composition and Plural Quantifier Semantics. Philosophia. https://doi.org/10.1007/s11406-018-9961-1. The official version is freely viewable here.] Abstract Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one’s chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity to a debate in which opponents frequently talk past one another. With the two views stated precisely, our investigation reveals the dependence of the mereologists’ ontological commitments on the semantics of plural quantifiers. Though we discuss the debate with respect to a simplified and idealized model, the insights provided will make more complex debates on composition more productive and deflationist criticisms of the debate less substantial. Keywords: Composition, Mereology, Plural Logic, Ontology 1 Introduction The Special Composition Question (SCQ) asks “When is it true that the s compose ?” (van Inwagen 1990, 30), or in other words SCQ: Under what conditions do some objects compose an object? Two positions dominate the literature on the SCQ: According to mereological nihilism, the answer to the SCQ is Never, i.e. no objects compose. According to mereological universalism, the correct answer is Always, i.e. any objects compose. This disagreement about the right way to answer the SCQ leads to radically different ontologies, in the sense 1 of Quine (1951), nihilism and universalism1 disagree about what objects exist: Nihilists claim that there are no objects with parts and the only things that exist are atoms, partless subatomic objects, see (Dorr 2005), (Hossack 2000), (Sider 2013). Universalists assume that besides atoms – if there are any2 – there are not only ordinary objects, like animals, artifacts, and planets, but also such seemingly extra-ordinary objects as a “trout-turkey”,3 or “an object whose parts are [Michael Rea’s] left tennis shoe, W. V. Quine and the Taj Mahal” (Rea 1998, 348), or “the object composed of the moon and … six pennies” (van Cleve 2008, 321). The aim of this paper is not to settle the dispute between nihilists and universalists, but to present an analysis of the two positions which gives a better understanding of the role plural quantifier semantics has in the debate. In order to do this, we will present the two theories within a framework that allows us to spell out their differences in precise logical terms. This framework is a model which does not allow for any alternative view besides nihilism and universalism. It is important to note that this model is only used as a tool: It allows us to keep out certain discussions, for instance, about the existence of trout- turkeys, and helps us to focus on what is at stake in the discussion. In other words, we are isolating variables so as to bring clarity to the debate. We take many of the ongoing disputes in mereology to be the result of the debating parties talking past one another, so to speak. The model will help us eliminate the “noise” which surrounds the discussion on composition. 1 We drop ‘mereological’ from here on, for the sake of better readability. 2 Universalism is, contrary to nihilism, not committed to the existence of atoms since it is consistent with the assumption that there are gunky objects, i.e. objects whose parts all have themselves proper parts. 3 A trout-turkey “… is the front half of a trout plus the back half of a turkey, which is neither fish nor fowl [but] part fish and part fowl” (Lewis 1991, 7–8) 2 2 Framing the Analysis Arguments against nihilism are often taken to be arguments in favor of universalism, and vice versa. However, they are not contradictories but contraries. By recalling Aristotle’s square of opposition, this becomes clear: The central claim of nihilism has the form of a universal negative sentence, No are . Universalism’s central claim is a universal affirmative sentence, Every is a . A universal negative sentence and the according universal affirmative sentence are contraries of each other, i.e. they cannot both be true, but both may be false. The negation of nihilism is the claim Some objects compose and the negation of universalism is Some objects do not compose. The conjunction of these two particular statements, i.e. the claim that some objects compose and some objects do not compose, is the feature shared by the so-called “[m]oderate answers” (van Inwagen 1990, 61) to the SCQ. Although some have suggested a moderate answer to the SCQ, e.g. (Carmichael 2015), (van Inwagen 1990), and (Merricks 2001), the predominant view in the literature on the SCQ is that it is an “all or nothing”-question. We will follow this view here since it allows us to see the key differences between the two theories in a much clearer way. We will analyze nihilism and universalism in the context of a model that does not allow any alternative with respect to the SCQ besides those two. This model will be created on the basis of a classical first-order logic with identity and an Atomistic Extensionality Mereology. Before we spell out the specifications of the model, we present the line of thought that underlies it informally. 2.1 From Contraries to Contradictories The purpose of the model we present is to exclude the possibility of an alternative answer to the SCQ besides nihilism and universalism, i.e. the model will not be consistent with 3 the claim that some objects compose and some objects do not.4 It follows from the model that nihilism holds iff universalism does not hold. Our strategy to get to this conclusion is to specify the model in such a way that the model validates the following claims: (1) There are at least many objects and at most many objects (2) Nihilism holds in the model iff there are exactly many objects (3) Universalism holds in the model iff there are exactly many objects (4) There are at most many objects iff it is not the case that there are at least many objects From these claims it follows that nihilism holds in the model iff universalism does not hold in the model, or in other words: Nihilism and universalism are contradictory positions within the model. How can we get to a model that validates these claims? Obviously, such a model has to be relatively small, similar to the one discussed in (Black 1952). As a first step, we stipulate for the model that there are exactly two atoms, i.e. two objects which do not have any parts.5 Furthermore, we have to exclude gunky objects, i.e. objects whose parts all have themselves parts, from the model. Finally, we stipulate that if an object with at least one part6 shares all its parts with an object, then they are identical. Thereby, we exclude that the model contains distinct objects with parts, which share all their proper parts, i.e. we stipulate that our model does not resemble the model of a non-extensional universe depicted in figure 1 on the next page, if we interpret lines 4 Although Lewis (1986, 211-3) and Sider (2001, 120-32) use the “Argument from Vagueness”, which concludes that composition is unrestricted, it has – if it is sound – the effect as our model has: ruling out any alternatives besides nihilism and universalism. 5 We use the term ‘part’ here in the sense of proper part in contrast to improper part. See the next section for a formal definition of ‘improper part’. 6 This condition is needed in order to avoid that all atoms turn out identical, due to the fact that any two atoms have the same parts, none. 4 going upwards as depicting the parthood relation. [fig.1] A model of a non-extensional universe. This is already enough to give us the claims we need as we will see in the next section. Before we move on to the formal presentation of the model and the proof for our claim that nihilism and universalism are contradictories in our model, we wish to highlight the importance of the claim (1): It guarantees that both nihilism and universalism can each hold on their own in the model and avoids begging the question against one of the two theories. 2.2 The Model Our formal framework consists of classical first-order logic with identity and Atomistic Extensional Mereology ( ). We take proper parthood as our primitive relation, and define improper parthood, being an atom and overlapping as follows: (D1) (D2) (D3) AEM7 is based on three axioms: the transitivity and asymmetry of proper parthood, as 7 Our presentation of AEM is similar to that in (Casati and Varzi 1999). Like Casati and Varzi, we will drop the initial universal quantifier. All formulas are to be understood as universally closed. 5 well as an atomistic supplementation principle: (A1) (A2) (A3) The axiom (A3) allows us to derive the principles called “Atomicity”, stating that any object has an atom as improper part, and “Strong Supplementation”, which in turn gives us “Extensionality”.8 Hence, it guarantees that our model does not contain any gunky objects and that objects with at least one part are identical iff they have all the same parts.