We have rejected both the idea that all things would be nothing else than collections or sums of component parts and the idea that all things would be nothing else than collections or sums of attributes (leaving aside the relations they may have). Insofar as we have chosen attributes as the ultimate factors in a structure of sets interpreted in the line of the sober ontology of a nominalist calculus of individuals, we are much closer to the latter view than to the former. We do recognize some sets of attributes as entities, whereas we do not recognize any pure set of 'parts' as an entity. Nevertheless, sets of parts do play a role as the extensionalities of wholes, and thus it may be worthwhile to have a look at theories which deal with wholes in a purely extensional way.

A constructional theory which solely recognizes parts is mereology. It treats a physical aggregate as a 'sum' or 'fusion' of all elements of the class of parts of the object in question, so that all elements of the class of parts of the object are part of the fusion and no proper part of the fusion is disjoint from all parts of the object. Others have already argued against mereology that an object like a jug cannot be a mere collection, or even a mere 'fusion', of its proper parts. The reason is for us that such an object must have properties as well, in particular those bearing on its own integrity or function. Only because of those properties is the object under consideration a jug. None of its component parts is a jug on its own.

Suppose that merely a heap of fragments is left of a certain jug; and suppose that these pieces are subsequently put together again to make a certain kind of pot. The pot is then made of the same collection of pieces as the jug was, but it has taken on quite a different shape and function (as we assume). Hence, the object is another one, tho the mereological sum has remained the same. To solve this analytical identity problem it has been suggested that is would be used in two different senses in this sort of context, namely in a sense pertaining to the object's constitution and in a sense pertaining to its identity. In the first sense the jug and the pot are, then, the same (made of, or constituted of, the same bits of clay, for instance); in the second sense they are not the same (not identical). This rather drastic cleavage in the meaning of the monosyllabic is provides too easy a way out tho and is not required in our system. We must, to make it work, not apply the mereological conception to the whole things themselves but only to their extensionalities.

The jug character of a thing belongs to the attributive predicament of the whole and it is this which disappears or changes when a jug is dropped or badly damaged. The extensionality of the jug remains, if we interpret it not just as a set (as we have done hitherto) but rather as a 'fusion', so that it is a collection of all component parts of the whole however specified, thus including the collections of parts and the parts of parts. In this mereological sense the extensionality of the pot is the same as that of the original jug, and in this respect they are the same or constituted of the same matter. It should be kept in mind tho that the extensionality is not an ontic set, that is, not a real thing itself.

From the point of view of strict identity, that is, sameness in all respects, the jug and the pot are different, albeit merely because of their disparate predicaments (attributive but also relational). This does not mean, of course, that the two objects could not be the same in some other than extensional respect as well. Furthermore, it is worth noting that one does not (necessarily) damage the (proper) parts of a jug, or affect them as severely, when the jug itself is damaged -- as has been confusedly suggested. Just as somebody who paints a little figure on the jug does not (necessarily) paint this figure on every part of the jug, so somebody who damages the jug does not (necessarily) damage every part of it; if so, then this is an additional, contingent fact. Damaging the jug is affecting the predicament and extensionality of the jug; damaging a part of the jug is affecting the predicament and extensionality of that part.

It is obvious that not only the attributes of the whole and those of its parts are to be distinguished, but also that the relationships with the whole may be quite different from those with the parts of that whole. Altogether a thing may have a relation with a whole, with one of its parts and with one of its attributes. And as regards a thing which has one or more parts itself, one of its parts may have a relation with another whole, with a part of another whole or with an attribute of such a whole. These different types of relations are shown in figure I.1.5.2.1.

In the diagram of this figure one does not find any truly reflexive relation, that is, a relation between 'two' things which are identical in the strict sense. There is no problem with accepting reflexivity with respect to (nonpropositional) things in a loose sense, but the strict interpretation of the notion remains obscure. To prove that things can have (nonpropositional, nonultimate) relations which literally turn back upon themselves, one must either give plausible examples of such things which are not wholes, or which are wholes, but which definitely do not have a relation with one, or between two, of their own (proper) parts instead. The so-called 'reflexivity' does not seldom concern such a relation between a whole and one of its parts or (in an even looser sense) between two different parts of the same whole.