Another Example of a Necessary Being Depending for its Existence on a Necessary Being

The Father and the Son are both necessary beings.  And yet the Father 'begets' the Son.  Part, though not the whole, of the notion of begetting here must be this: if x begets y, then y depends for its existence on x.  If that were not part of the meaning of 'begets'' in this context, I would have no idea what it means.  But how can a necessary being depend for its existence on a necessary being?  I gave a non-Trinitarian example yesterday, but it was still a theological example. Now I present a non-theological example.

I assume that there are mathematical (as opposed to commonsense) sets.  And I assume that numbers are necessary beings.  (There are powerful arguments for both assumptions.) Now consider the set S = {1, 3, 5} or any set, finite or infinite, the members of which are all of them necessary beings, whether numbers, propositions, whatever.  Both S and its membership are necessary beings.  If you are worried about the difference between members and membership, we can avoid that wrinkle by considering the singleton set T = {1}.

T and its sole member are both necessary beings.  And yet it seems obvious to me that one depends on the other for its existence:  the set is existentially dependent on the member, and not vice versa.  The set exists because -- though this is not an empirically-causal use of 'because' -- the members exist, and not the other way around.   Existential dependence is an asymmetrical relation.  I suppose you either share this intuition or you don't.  In a more general form, the intuition is that collections depend for their existence on the things collected, and not vice versa.  This is particularly obvious if the items collected can also exist uncollected.  Think of Maynard's stamp collection.  The stamps in the collection will continue to exist if Maynard sells them, but then they will no longer form Maynard's collection. The point is less obvious if we consider the set of stamps in Maynard's collection.  That set cannot fail to exist as long as all the stamps exist.  Still, it seems to me that the set exists because the members exist and not vice versa.

And similarly in the case of T.  {1} depends existentially on 1 despite the fact that there is no possible world in which the one exists without the other.  If, per impossibile, 1 were not to exist, then {1} would not exist either. But it strikes me as false to say: If, per impossibile, {1} were not to exist, then 1 would not exist either.  These counterfactuals could be taken to unpack the sense in which the set depends on the member, but not vice versa.

It therefore is reasonable to hold that two necessary beings can be such that one depends for its existence on the other.  And so one cannot object to the notion of the Father 'begetting'  the Son by saying that no necessary being can be existentially dependent upon a necessary being.  Of course, this is not to say that other objections cannot be raised.

Comments

[Jason Hamza van Boom]

A thoughtful argument, but it is deficient because of the possible ambiguity of the word "depends."

When we say that y "depends" on x, we imply that x is a necessary cause of y (although, of course, not necessarily a sufficient cause). For the past 200-300 years, most Westerners have assumed that "cause" has a single meaning. From the time of Aristotle to the 17th or 18th centuries, however, the common practice was to distinguish between several kinds of causality: formal, material, efficient, final, and, in some cases, exemplary.

The idea that "cause" does not have a uniform meaning, and that there are four or five types of causality, is still the norm among Catholic and Shia Muslim philosophers and theologians, so it cannot be easily dismissed. Furthermore, "classical" formulations of Christian doctrine were made assuming this manifold sense of "cause." Any discussion of them needs to take this position into account.

Your example, taken from set theory, is an instance of formal causality. For philosophers and theologians in the "classical" tradition (Catholics, Shias, certain kinds of Protestants, etc) this isn't a big revelation.

"Classical" arguments for the existence of God, however, consider "cause" as efficient, final or exemplary (Duns Scotus' "On the First Principle" argues explicitly for all three). It is in those senses of "cause" that a necessary being cannot depend on another. In fact, when Ibn Sina (Avicenna) makes the distinction between "contingent" and "necessary" existence prominent in philosophy, it is this kind of causal dependence he has in mind. In fact, he defines "existing necessarily" vs "existing contingently" as existing through one's self vs through another ("per se" vs. "per aliud", in the Western phrasing).

Thomas Aquinas would reply to your argument that "a necessary being cannot depend on another" is true in the cases of efficient, final and exemplary causality, not so in the case of formal causality.