A theory of granular parthood based on qualitative cardinality and size measures

We present a theory of granular parthood based on qualitative cardinality and size measures. Using standard mereological relations and qualitative, context-dependent relations such as roughly the same size, we define a granular parthood relation and distinguish different ways in which a collection of smaller objects may sum to a larger object. At one extreme, an object x may be a mereological sum of a large collection p where the members of p are all negligible in size with respect to x (e.g., x is a human body and p is the collection of its molecules). At the other extreme, x may be a mereological sum of a collection q none of whose members are negligible in size with respect to x (e.g., x is again a human body and p is the collection consisting of its head, neck, torso, and limbs). We cannot give precise quantitative definitions for relations such as roughly the same size or negligible in size with respect to since these are, even within a fixed context, vague relations. The primary focus in the formal theory presented in this paper is on the context-independent logical properties of these qualitative cardinality and size relations and their interaction with mereological relations. In developing our formal theory, we draw upon work on order of magnitude reasoning.