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Logical and mathematical truths differ from the empirical sciences in being necessary; they can be discovered a priori and in general can be proved (contra Quine). How is this possible? This problem is partly met by recognizing that the rational sciences are sciences of the possible. Only the mathematical structures that are instantiated in space‐time are existents. Furthermore, using the Entailment Principle, it is seen that only the logico‐mathematical axioms require truthmakers. We should recognize laws in these sciences, but laws that are necessary. Such laws will be truthmakers for truths about uninstantiated structures, for instance large infinite numbers. What is the source of these necessary laws? Perhaps it is a necessity in the nature of things.

An account is given of the Forrest‐Armstrong theory of number (Peter Forrest). Natural numbers are argued to be relations holding between a certain property and a certain mereological whole (black swan on the lake now, and the whole that these swans make). With the rational numbers and the real numbers the relation becomes one of proportion, they are the units that measure the proportion. It is pointed out, however, that this view is largely to be found in Isaac Newton, and is even anticipated in Aristotle. What it is for a mathematical entity such as a number to be ‘instantiated’ is considered.

David Lewis' major contribution to the ontology of classes is to perceive that many‐membered classes are mereological sums of the singletons (unit‐classes of their members: {a, b, c, d, . . .} is identical with {a + {b} + }c} + {d} . . . This raises the question of what singletons are ontologically. My proposal that they are all states of affairs was refuted by Gideon Rosen. But a more limited claim is tenable. Only some singletons are states of affairs. Because set theory is a mathematical system it will produce structures that may not be instantiated, and therefore are no more than possibilities. Nelson Goodman's protest at using set theory to do metaphysics is endorsed. The null class is rejected as an existent.

This book has shown that mereology was dominated by a single theory: classical extensional mereology (CEM), present in two logical guises — the Calculus of Individuals and Mereology — each in a number of variants. CEM is algebraically neat: only a complete Boolean algebra is neater. It is also strong. CEM is tenseless, non-modal, upholds extensionality of parts, and upholds the conditioned existence of general sums. The first two characteristics are privative, while the last two are positive. Among approaches at variance with CEM, most retain the first three characteristics and drop the last in favour of some weaker conditional existence principle. In the face of apparent temporal and modal variation, two major strategies have been followed. The first ignores modality and attempts to retain the third characteristic by recourse to an ontology of four-dimensional objects. The second strategy, that of Roderick Chisholm, takes both time and modality seriously, but preserves the third characteristic by putting forward an ontology, opposed to common sense, of modally and temporally invariable objects.

This book provides a connected account of the various kinds of mereology, or formal theory of part, whole, and related concepts, which exist in the literature. It also exposes the philosophical defects of most of this tradition, and suggests why, where, and how it should be put right. The standardly accepted formal theory of part-whole is classical extensional mereology, which is known in two logical guises, the Calculus of Individuals of Henry Leonard and Nelson Goodman, and the Mereology of Stanislaw Leśniewski. Despite the discrepancies between the underlying logics of these two approaches, there is a precise sense in which both say the same things about parts and wholes. The book also considers the mereology of continuants and brings modality and mereology together as they are found in the work of Edmund Husserl at the beginning of the century and later in that of Roderick Chisholm.

This chapter is devoted to an exposition of the main concepts and principles of standard mereology, the system originally developed by Stanislaw Leśniewski and introduced into the English-speaking world in the guise of Henry Leonard and Nelson Goodman's ‘Calculus of Individuals’. The main source for this chapter is Peter Simons', Parts: A Study in Ontology, in particular his instructive gradual development of standard mereology, which shows how stronger and stronger principles may be added gradually to a minimal core, until we arrive at the full-strength theory of standard mereology. Despite standard mereology's merits as a formal theory, however, it remains to be seen whether it is of any use to the metaphysician in characterizing ordinary mereological concepts, as they apply to our scientifically informed, common-sense ontology.

This chapter returns to the question, postponed in Chapter 2: why, from a three-dimensionalist perspective, the properties of ordinary material objects are not already adequately accounted for by Thomson's modified and weakened version of standard mereology. To this end, the work of Kit Fine is examined, who, in a series of papers, has provided powerful reasons for abandoning the standard conception of composition. In its stead, Fine proposes an alternative, neo-Aristotelian model, which is in some respects close to the positive framework advocated later in this book. The chapter begins by examining Fine's reasons for parting ways with the standard conception; these reasons are found to be utterly persuasive and fatal to the standard conception. The later sections of the chapter provide a detailed discussion of Fine's own positive proposal and indicate in what respects it might be found to be unsatisfactory.

This chapter demonstrates that a structure-based neo-Aristotelian mereology for ordinary material objects can be defended utilizing a single relation of parthood with relatively straightforward formal properties. A substantive restriction on composition can be derived from a comparatively minimal and metaphysically neutral independently motivated ontology of kinds. The thoroughly mereological conception of composition defended here recommends itself based on Leibniz's Law and the Weak Supplementation Principle. Among its benefits are the following: it yields the Uniqueness of Composition as a derived principle; it contributes to the Problem of Constitution and, possibly, the Grounding Problem; and it clarifies the relation between a whole and both its material and its structural components. Finally, it is demonstrated how, on the approach defended here, wholes can be thought of as both ontologically committing and genuinely unified, despite the apparent Aristotelian regress caused by a never-ending demand for further principles of unity.

Although Stanislaw Leśniewski's Mereology was the first extensional part—whole theory to be rigorously developed, it is expressed in the formal language of the general logic Leśniewski called ‘Ontology’, and which is sometimes called his calculus of names. This language is, by comparison with those of predicate logic and set theory, still relatively little known, so mereological concepts and principles are first introduced. Leśniewski's notation can then be introduced without any great difficulty in its intended meaning. As far as the narrower issues of extensional mereology are concerned, there is no important difference between Leśniewski and other writers. This chapter discusses meteorological principles, classical extensional meteorology, and the questions of atomism.

This chapter has two aims. The first aim is to present an overview of Bolzano's universe from the point of view of his metaphysics and its relationship to logic, relying fundamentally on Bolzano's Wissenschaftslehre. The author's preferred reading of Bolzano is one according to which he is a ‘platonistic nominalist’: a platonist about propositions and a nominalist about properties. Bolzano's nominalistic tendencies are particularly conspicuous in his mereological analyses, which play a major role in every aspect of his philosophy. The author's second aim is to answer the open question of whether in Bolzano there is any ‘ontology of truth’. This chapter argues for a negative answer. Bolzano does not make room for any special object which plays the role of the counterpart of a truth-bearer, or at least his propositions do not connect semantically in a direct way to anything that would count as such a special object.

This chapter begins with an attempt to analyze the relation of being in perfect contact. Five proposals are evaluated and the final proposal endorsed. Along the way, puzzles invoking zeno-series and mereology yield refutations of intuitive views about contact and surprising results about receptacles and topology. The topics of boundaries and borders furnish a number of metaphysical puzzles, and some of these are explored with emphasis on the proper analysis of ‘boundary’, boundary ownership, vague boundaries, and concealed boundaries. A delightful puzzle about contact and boundaries is presented without final solution, yet together with a number of suggested solutions each of which seems genuinely terrible. Finally, these reflections on contact and boundaries make their way into the History of Zenopia, a brief story in which the celebrated four-color conjecture is subjected to counterexample (or at least in which two of its most famous three formulations are subjected to counterexample).

A substantivalist dualist who believes in spacetime and material objects is committed to a relation of occupation or location. This chapter explores different candidate descriptions of this relation and investigates some of the philosophical difficulties which arise for the resulting conceptions. One reason to focus on the metaphysics of occupation relations is that some of them seem to make room for the possibility of extended mereological simples. The chapter critically evaluates several philosophical attempts to promote the cause of extended simples. Four problems are then advanced to yield the conclusion that extended mereological simples are conceptually possible yet metaphysically impossible: the problem of spatial intrinsics, the problem of shapes, the problem of parsimony, and the problem of diachoric identity.