A Theory of Granular Partitions

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Imagine that you are standing on a bridge above a highway checking off the makes and models of the cars that are passing underneath. Or that you are a postal clerk dividing envelopes into bundles; or a laboratory technician sorting samples of bacteria into species and subspecies. Or imagine that you are making a list of the fossils in your museum, or of the guests in your hotel on a certain night. In each of these cases you are employing a certain grid of labeled cells, and you are recognizing certain objects as being located in those cells. Such a grid of labeled cells is an example of what we shall call a granular partition. We shall argue that granular partitions are involved in all naming, listing, sorting, counting, cataloguing and mapping activities. Division into units, counting and parceling out, mapping, listing, sorting, pigeonholing, cataloguing are activities performed by human beings in their traffic with the world. Partitions are the cognitive devices designed and built by human beings to fulfill these various listing, mapping and classifying purposes. In almost all current work in areas such as common-sense reasoning and natural language semantics it is the naive portion of set theory that is used as basic framework. The theory of granular partitions as it is developed in this paper is intended to serve as an alternative to set theory both as a tool of formal ontology and as a framework for the representation of human cognition. Kinds, sorts, species and genera are standardly treated as sets of their instances; subkinds as subsets of these sets. Set theory nicely does justice to the granularity that is involved in our sorting and classification of reality by giving us a means of treating objects as elements of sets, i.e. as single whole units within which further parts are not recognized. But set theory also has its problems, not the least of which is that it supports no distinction between natural totalities (such as the species cat) and such ad hoc totalities as, for example, {the moon, Napoleon, justice}. Set theory has problems, too, when it comes to dealing with time, and with the fact that biological species and similar entities may remain the same even when there is a turnover in their instances. For sets are identical if and only if they have the same members. If we model the species cat as the set of its instances, then this means that cats form a different species every time a cat is born or dies. If, similarly, we identify an organism as the set of