Basic Concepts of Formal Ontology

The term 'formal ontology' was first used by the philosopher Edmund Husserl in his Logical investigations to signify the study of those formal structures and relations above all relations of part and whole which are exemplified in the subject-matters of the different material sciences. We follow Husserl in presenting the basic concepts of formal ontology as falling into three groups: the theory of part and whole, the theory of dependence, and the theory of boundary, continuity and contact. These basic concepts are presented in relation to the problem of providing an account of the formal ontology of the mesoscopic realm of everyday experience, and specifically of providing an account of the concept of individual substance. 1 . Basic History of Formal Ontology We owe the idea of a f01mal ontology to the philosopher Edmund Husserl, whose Logical Investigations [1] draws a distinction between formal logic, on the one hand, and formal ontology, on the other. Formal logic deals with the interconnections of truths (or of propositional meanings in general) with inference relations, consistency, proof and validity. Formal ontology deals with the interconnections of things, with objects and properties, parts and wholes, relations and collectives. As formal logic deals with properties of inferences which are formal in the sense that they apply to inferences in virtue of their form alone, so formal ontology deals with properties of objects which are formal in the sense that they can be exemplified, in principle, by objects in all material spheres or domains of reality [2]. · Husserl's formal ontology is based on mereology, on the theory of dependence, and on topology. The title of his third Logical Investigation is "On the Theory of Wholes and Parts" and it divides into two chapters: "The Difference between Independent and Dependent Objects" and "Thoughts Towards a Theory of the Pure Forms of Wholes and Parts". Unlike more familiar 'extensional' theories of wholes and parts, such as those propounded by Lesniewski (whom Husserl influenced), and by Leonard and Goodman (see [3]), Husserl's theory does not concern itself merely with what we might think of as the vertical relations between parts and the wholes which comprehend them on successive levels of comprehensiveness. Rather, his theory is concerned also with the horizontal relations between co-existing parts, relations which serve to give unity or integrity to the wholes in question. To put the matter simply: some parts of a whole t;xist merely side by side, they can be destroyed or removed from the whole without detriment to the residue. A whole, all of whose parts manifest exclusively such side-by-sideness relations with each other, is called a heap or aggregate or, more technically, a purely summative whole. In many wholes, however, and one might say in all wholes manifesting any kind of unity, certain parts stand to each other in formal relations of what Husserl called necessary dependence (which is sometimes, but not always, necessary interdependence). Such parts, for example the individual instances of hue, saturation and brightness involved in a given instance of colour, cannot, as a matter of necessity, exist, except in association with their complementary parts in a whole of the given type. There is a huge variety of such lateral dependence relations, giving rise to a correspondingly huge variety of different types of whole which the more standard approaches of extensional mereology are unable to distinguish [ 4]. The topological background of Husserl's work makes itself felt already in his theory of dependence[5] . It comes most clearly to the fore, however, in his treatment of the notion of fusion: the relation which holds between two adjacent parts of an extended totality when there is no qualitative discontinuity between the two [6]. Adjacent squares on a chess-board array are not fused together in this sense; but if we 19 20 B. Smith I Basic Concepts of Formal Ontology imagine a band of colour that is subject to a gradual transition from red through orange to yellow, then each region of this band is fused with its immediately adjacent regions. In the field of what is experienced perceptually, then, we can draw a distinction, between intuitively separated contents, contents set in relief from or separated off from adjoining contents, on the one hand, and contents which are fused with adjoining contents, or which flow over into them without separation, on the other ([l], Investigation III,§ 8, p. 449). That Husserl was at least implicitly aware of the topological aspect of his ideas, even if not under this name, is unsurprising given that he was a student of the mathematician Weierstrass in Berlin, and that it was Cantor, Husserl's friend and colleague in Halle during the period when the Logical Investigations were being written, who first defined the fundamental topological notions of open, closed, dense, perfect set, boundary of a set, accumulation point, and so on. Husserl consciously employed Cantor's topological ideas, not least in his writings on the general theory of (extensive and intensive) magnitudes which make up one preliminary stage on the road to the formal ontology of the Logical Investigations. (See [8], pp. 83f, 95, 413; [l], "Prolegomena",§§ 22 and 70.) In what follows we shall outline the basic concepts of formal ontology as Husserl conceived it, concentrating our remarks on the specific case of the world of everyday human experience. We shall thus cover some of the same ground that is covered by Patrick Hayes and others working in the territory of 'na'ive physics' [9]. Most recently, formal ontology has been used as a tool of knowledge representation [10], in ways which draw on the insight that the categories of formal ontology, because they are fundamental to a wide variety of different domains, can be fruitfully employed in providing frameworks for translating between knowledge systems constructed on divergent bases. 2 . Mereology vs Set Theory When modern-day philosophers and those working in the artificial intelligence field turn their attentions to ontology, they standardly begin not with mereology or topology but with set-theoretic tools of the sort that are employed in standard model-theoretic semantics. The rationale for insisting on a mereological rather than a set-theoretic foundation for the purposes of formal ontology can be stated as follows. The world of everyday human experience is made up of objects extended in space, objects which thus manifest a certain sort of boundary-continuum structure. The standard set-theoretic account of the continuum, initiated by Cantor and Dedekind and contained in all standard textbooks of the theory of sets, will be inadequate as a theory of this everyday 'qualitative' continuum for at least the following reasons: 1. The application of set theory to a subject-matter presupposes the isolation of some basic level of urelements in such a way as to make possible a simulation of all structures appearing on higher levels by means of sets of successively higher types. If, however, as holds in the case of investigations of the ontology of the experienced world, we are dealing with mesoscopic entities and with their mesoscopic constituents (the latter the products of more or less arbitrary real or imagined divisions along a variety of distinct axes), then there are no urelements fit to serve as our starting-point [ 11 ]. 2. Experienced continua are in every case concrete, changing phenomena, phenomena existing in time; they are wholes which can gain and lose parts. Sets, in contrast, are abstract entities, existing outside time, which are defined entirely, and once and for all, via the specification of their members. Now certainly, in order to do justice to the changing relations between parts and wholes in the realm of mesoscopic objects, we might conceive of a theory analogous to the theory of sets which would be constmcted on the basis of a tensed or time-indexed membership relation. To this end, I