On Partitioning Reality Barry Smith and Thomas Bittner Department of Philosophy, Center for Cognitive Science and NCGIA State University of New York, Buffalo, phismith@buffalo.edu Department of Computer Science, Northwestern University, bittner@cs.nwu.edu Abstract: In Smith and Brogaard (2000, 2001) the notion of partition is introduced as a generalization of David Lewis’s (1991) conception of a class as the mereological sum of its singletons. A partition is a mereological sum of labeled cells. Partitions thus conceived are involved in all human mapping, classifying, listing and theorizing activity. We here provide a more detailed formal characterization of partitions. We define notions of well- formedness and truth for partitions and we classify partitions along three axes: (a) degree of correspondence between partition cells and objects in reality; (b) degree of how well a partition represents the mereological structure of the domain it is projected onto; and (c) degree of completeness and exhaustiveness. 1. Introduction 1.1 Partitions as cognitive devices Sorting, slicing, counting and parceling out, dividing and gathering into units and portions, listing, pigeonholing, cataloguing and checking off – all of these are activities performed by human beings in their everyday and scientific traffic with the world. Partitions are the cognitive devices designed and built by human beings to fulfill these various purposes. When making lists or classifying objects you are in every case, we shall suggest, employing a certain grid of cells and you are recognizing certain objects as being located in those cells. A partition, as we shall conceive it, is just such a grid of cells. Some partitions are flat: they amount to nothing more than a mere list. Other partitions are hierarchical: they consist of cells and subcells in a hierarchical array. Some partitions are built in order to reflect independently existing divisions on the side of objects in the world. Other partitions – for example the partitions created by nightclub doormen or electoral redistricting commissions – are themselves such as to create the necessary divisions on the side of their objects. Quite different sorts of partitions, having cells of different resolutions and effecting slicings and unifyings of different types, can be applied simultaneously to the same domain of objects. The people in your building can be divided according to gender, zip code, social class, tax bracket, blood type, current location, golf handicap or (to bring this list to an end) according to Erdös number or blood cesium level. As will be clear from what follows, the notion of partition that is hereby implied is only distantly related to the more familiar notion defined in terms of equivalence classes. The latter notion is indeed parasitic on the one presented here. This is because it carries with it the presupposition that the domain to which an equivalence relation is applied has already been divided up into units (the elements of the set with which we begin), and it is this very notion of division into units which our present theory is designed to illuminate. 1.2 Better than Sets In Smith and Brogaard (2000, 2001) the notion of partition is introduced as a generalization of David Lewis’s (1991) conception of classes as the mereological sums of their singletons. Partitions, as we conceive them, are similarly the mereological sums of their constituent cells. The cells within a partition may however manifest a range of properties which singletons lack. This is because, where a set is defined in terms of its members, a partition is a device that is, as it were, seeking out objects which might fall