Bridging levels of "granularity" and "scale" are frequently cited as key problems for biomedical informatics. However, detailed accounts of what is meant by these terms are sparse in the literature. We argue for distinguishing two notions: "size range," which deals with physical size, and "collectivity," which deals with aggregations of individuals into collections, which have emergent properties and effects. We further distinguish these notions from "specialisation," "degree of detail," "density," and "connectivity." We argue that the notion of "collectivity"--molecules in water, cells in tissues, people in crowds, stars in galaxies--has been neglected but is a key to representing biological notions, that it is a pervasive notion across size ranges--micro, macro, cosmological, etc.--and that it provides an account of a number of troublesome issues including the most important cases of when the biomedical notion of parthood is, or is not, best represented by a transitive relation. Although examples are taken from biomedicine, we believe these notions to have wider application.

-There are some who defend a view of vagueness according to which there are intrinsically vague objects or attributes in reality. Here, in contrast, we defend a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show that this theory provides a general framework within which we can understand the relation between vague terms and concepts on the one hand and correlated portions of reality on the other. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches.

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Mereological relations such as part-of and its inverse has-part are fundamental to the description of the structure of living organisms. Whereas classical mereology focuses on individual entities, mereological relations in biomedical ontologies are generally asserted between classes of individuals. In general, this practice leaves some basic issues unanswered: type constraints of mereological relations, e.g., concerning artifacts and biological entities, the relation between parthood and time, inferred parts and wholes as well as a delimitation of parthood against spatial inclusion. Furthermore, mereological relations can be asserted not only between physical objects but also between biological processes and medical procedures. We analyze these ambiguities and make suggestions for a standardization of mereological relations in biomedical ontologies.

This paper presents an axiomatic formalisation of a theory of top-level relations between three categories of entities: individuals, universals, and collections. We deal with a variety of relations between entities in these categories, including the sub-universal relation among universals and the parthood relation among individuals, as well as cross-categorial relations such as instantiation and membership. We show that an adequate understanding of the formal properties of such relations -in particular their behavior with respect to time -is critical for geographic information processing.The axiomatic theory is developed using Isabelle, a computational system for implementing logical formalisms. All proofs are computerverified and the computational representation of the theory is available online.