Peter Richard Forrest

This chapter argues that we have inconsistent intuitions about the structure of space (or spacetime, or some stuff, the aether, that fills space or spacetime). A contradiction is obtained from eleven premises, each of which is either directly intuitive or supported by intuitions. The use of so many premises results from the desire to exhibit as clearly as possible the places where one might decide some intuition is to be undermined. Nonetheless the argument for inconsistency is straightforward: our intuitions support the existence of a _supersponge_, namely a region of less than the total volume but not disjoint from any connected part of space of positive diameter. But the complement of a supersponge is an intuitively impossible region. Yet, intuitively a region of less than maximal volume must have a complement. So some intuition must be abandoned.

Typical structural universals are not just the mereological sum of their constituents. Hence, there is the Structure Problem of explaining this non-mereological structure. The Instantiation Problem is that the predicate "U is instantiated by x, y, etc., in that order" is ill-suited to be a primitive, unanalyzed predicate. The proposed solution to these problems is based on the observation that if universal U is said to supervene upon universals V, W, etc., then it is the instantiation of U that supervenes on the instantiation of V, W, etc. Assuming there are few subvenient universals, which is admittedly a speculation, their instantiation may be explained in ways that would be uneconomic if there were too many.

Consider the things that exist—the entities—and let us suppose they are mereologically structured, that is, some are parts of others. The project of ontology within the bounds of bare mereology use this structure to say which of these entities belong to various ontological kinds, such as properties and particulars. My purpose in this paper is to defend the most radical section of the project, the mereological theory of the exemplification of universals. Along the way I help myself to several hypotheses: the existence of merely possible worlds; that particulars have thisnesses; and that mereology is far from classical. Moreover, the way I characterize instantiation might be judged too complicated to be plausible. At the end of the paper, I reply to these objections based on complexity.