English contains singular terms, quantifiers and predicates (e.g. ‘it’, ‘something ’ and ‘... is an elephant’). But it also contains plural terms, quantifiers and predicates (e.g. ‘they’, ‘some things ’ and ‘... are scattered on the floor’). 1 Philosophers have become increasingly interested in plurals over the past couple of decades. The purpose of this paper is to explain why plurals might be thought to have philosophical importance, and why they have led to philosophical debate. 1

. There are conflicting intuitions concerning the status of a boundary separating two adjacent entities (or two parts of the same entity). The boundary cannot belong to both things, for adjacency excludes overlap; and it cannot belong to neither, for nothing lies between two adjacent things. Yet how can the dilemma be avoided without assigning the boundary to one or the other thing at random? Some philosophers regard this as a reductio of the very notion of a boundary, which should accordingly be treated a mere faon de parler. In this paper I resist this temptation and examine some ways of taking the puzzle at face value within a realist perspective---treating boundaries as ontologically on a par with (albeit parasitic upon) voluminous parts. 1. Introduction The world of everyday experience is mostly a world of physical things separated in various ways from their environment: things with surfaces, skins, crusts, boundaries of some sort. These may not always be sharply defined, or so...

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and investigates how much of Frege’s Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension. 1

this article has been submitted for publication in: J. Glasgow, H. Narayanan, B. Chandrasekaran (Eds.), Diagrammatic Reasoning: Computational and Cognitive Perspectives, AAAI/MIT Press. The research described here was supported in part by the Doctoral Program in Cognitive Science at the University of Hamburg (funded by the German Science Foundation; Ha 1237/3) and the Center for Interdisciplinary Research (ZIF) at the University of Bielefeld. We wish to thank the members of the research group Mental Models in Discourse Processing at ZIF and the participants of the workshop on "Principles of Hybrid Systems and Reasoning " at IJCAI '93 for discussions; and Stephanie Kelter, Carola Eschenbach and two anonymous reviewers for helpful comments on the manuscript.

The paper starts with an examination and critique of Tarski's wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L1;1 . Also characterized similarly is a natural generalization of Tarski's thesis, due to Sher, in terms of bijections between domains. My main objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation is introduced to accomplish the latter. The main result is that an operation is definable in the first-order predicate calculus without equality just in case it is lambda-definable from homomorphism invariant mona...

Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical

We often speak of an object being composed of various other objects. We say that the deck is composed of the cards, that a road is the sum total of its sections, that a house is composed of its walls, ceilings, floors, doors, etc. Suppose we have some material objects. Here is a philosophical question: what conditions must obtain for those objects to compose something? In his recent book Material Beings, Peter van Inwagen addresses this question, which he calls the ‘special composition question’; his answer is: 1 (1) For any material objects X, the X s compose something iff the activity of the X s constitutes a life, or there is only one of the X s. Additionally, he accepts a simpler thesis that follows from (1): 2 (2) Every material object is either a mereological atom or a living thing, where a mereological atom is an object lacking proper parts. (2) may seem radical. If it is true then there are no tables, chairs, planets,

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Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,

Consider a statue made of a piece of clay. Call the statue “Statue ” and the piece of clay “Clay. ” Clay materially constitutes Statue. What is this relation? A standard way to ask this question is to ask whether Clay is strictly identical to Statue. Or is Clay numerically distinct from Statue? The more general way to ask the question is to ask what it means for an object to materially constitute another. Is constitution simply identity? If not, what are the features of this relation? Some contemporary metaphysicians argue that material constitution is simply identity. In other words, when Statue is materially constituted by Clay, Statue just is Clay. 1 Fine 2003 calls this view monism. Others follow the lead of what seem to be intuitive judgments about the natures, essences and sorts of objects and defend pluralism about material constitution, denying that constitution is identity. When Clay materially constitutes Statue, Clay is not identical to Statue. 2 After all, it seems as though Statue and Clay differ in their properties. Statue is essentially statue-shaped, but Clay is not. Clay could be shaped into a vase, but Statue would not persist through this change. Statue represents beauty and grace incarnate. Clay does not. Clay has its molecules essentially, while Statue could persist even if a few fragments were chipped off. If Statue and Clay do not share all of their properties they cannot be identical, for if a and b are the same object, a has exactly the same properties as b. 3 Why be monist? Given the difference in properties, isn’t it just obvious that Statue cannot be identical to Clay? No, for monists can deny there are any real differences in essence or other properties. According to the monist, the seeming differences in essential and other properties are just differences in description. Statue is just Clay called by a