The Axiom of Weak Supplementation (WS) says that if y is a proper part of x, then there is a part of x that doesn't overlap y. Standard arguments against WS adduce possible counterexamples. But I want to take a different tack. Proper parthood seems to be a primitive relation or a case of a primitive relation (the proper metaphysical component relation seems a good candidate; cf. here). Moreover, this relation does not involve any entities besides the two relata--it's not like the relationship of siblinghood, which holds between people who have a parent in common.
But if R is a primitive binary relation that does not involve any entities besides the two relata, then it is unlikely that the obtaining of R between two entities should non-trivially entail the existence of a third entity. (By "non-trivially", I want to rule out cases like this: everything trivially entails the existence of any necessary being; if mereological universalism is true, then the existence of any two entities trivially entails the existence of their sum.) But if WS is true, then existence of two entities in a proper parthood relationship non-trivially entails the existence of another part. Hence, WS is unlikely to be true.
Doesn't WS follow from the definition of "proper parthood"? It is my understanding that x and y cannot be equal in a case of proper parthood, and that it must be the case that x is a part of y but not the other way around. In such a case, there must be MORE to y than merely x. That "more" should likewise constitute a part (or be constituted of several parts) which are not x.
ReplyDeleteIs there some argument for the rule that a primitive binary relation not involving any entities besides the two relata should not non-trivially entail the existence of a third entity? It doesn't seem obvious to me.... For example, that my mother's relation to me is a binary one, but it's obtaining entails the existence of my father, no?
I have a friend working on something to do with metaphysical composition, who pointed the following passage out to me. I share it because the denial of WS reminds me of it, and not because I have a very real grasp of what denying WS really means.
ReplyDelete“Therefore, to the first it must be said that sometimes a third thing results from those which are joined together; as the humanity by which a man is a man is constituted from soul and body so a man is composed of soul and body. Sometimes, however, a third thing does not result from those which are joined together but a kind of composite intelligible notion results, as when the notion ‘man’ and ‘white’ go to make up the intelligible notion ‘white man’. And in such things something is composed of itself and another, just as a white thing is composed of that which is white and whiteness.” (Aquinas, Quodl. 2, Q. 2, A.1)
Call the fundamental relation behind parthood 'R'. In the post I said that R is proper parthood. But maybe the very interesting Aquinas quote shows that that was too quick. Perhaps in some cases an object stands in R to itself and in others it doesn't. (Compare: If there are Platonic forms, some self-exemplify and others do not; abstraction is abstract but concreteness isn't concrete.) In that case, R is neither proper nor improper parthood, but something in between.
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