That's right: a classical Aristotelian is defending a version of the so-called "transcendental argument" for God's existence. First, what does it mean for something to be transcendental? In a few quick words, the term refers to a thing's necessary preconditions. In order for water to exist, it must be composed of hydrogen and oxygen. H2O is, therefore, a transcendental of water.
More seriously, we might ask: how do we know X is true? Before determining whether X corresponds to reality in fact, we must determine whether X is logically coherent. Unless something is consistent with the laws of logic, then, it cannot exist. I realize I'm dumbing it down a bit here, but believe it or not, there are actually people who reject this line of thought. I won't pursue such a radical perspectivism at this juncture, though. In any case, the laws of logic are a necessary precondition of rational inquiry. Let's start our argument like this:
Axiom 1. One cannot be rational while rejecting rational inquiry.
Axiom 2. One cannot be rational while undermining the necessary preconditions of rational inquiry.
Here, now, is the first part of the proof, via reductio ad absurdum:
Prove A: The laws of logic are universal.
Assume ~A: The laws of logic are not universal.
~A --> B: If the laws logic are not universal, X can be ~X.
~B: X cannot be ~X.
~~A: by modus tollens.
Therefore, A: by negation, the laws of logic are universal.
Most fundamental to the laws of logic are the laws of identity, non-contradiction, and excluded middle. I would also add the transitive axiom (if A = B, and B = C, then A = C) to that list. In order to engage in rational inquiry, one must be consistent with at least these four laws of logic. They are transcendentals of rational inquiry.
Now comes the next part: from universality to existence. Does the universality of logic entail that it exists? One reason to think it does is that if X possesses the attribute of universality (I've also mentioned indispensability in the past), then X must exist. For, non-existent entities cannot possess any attributes whatsoever. The positive attribute of universality provides us with solid ground to accept the laws of logic as having an ontological instantiation throughout reality.
However, the laws of logic are abstract objects. They don't stand in any causal relations to other entities. For example, the number 5 cannot mow my lawn or pick up my laundry. So in what sense do these abstract objects, especially the laws of logic, exist?
We have already ruled out a strong form of nominalism. We are left Platonism and conceptualism. The argument may go something like this:
1. Abstract objects are either a) non-existent, b) mind-independent entities, or c) mental concepts.
2. Abstract objects exist. (contrary to 1a)
3. Abstract objects are not mind-independent. (contrary to 1b)
4. Therefore, abstract objects are mental concepts. (From 1 - 3)
Now, why think premises (2) and (3) are correct? We have already given a defense the ontological instantiation of abstract objects, based on their having the attributes of universality and indispensability. As I stated above, the real meat of the debate is between the Platonist and the conceptualist. I concede that I find the arguments for the existence of abstract objects so compelling that if, hypothetically, I were to abandon conceptualism, I would gladly embrace Platonism.
Nevertheless, I think there are good arguments in favor of conceptualism, as well as good arguments against Platonism. Assuming that abstract objects are mind-independent entities that also lack any causal relations, how can we possibly have knowledge of them? There is a causal relationship between the mind knowing and the object being known by the mind. For example, as I look into my laptop's monitor, my eyes act as part of a bridge in the causal relationship between my mind and the monitor.
But on Platonism, there just is no causal relationship to appeal to because abstract object are causally effete. It is for this reason alone that I believe Platonism should be abandoned in favor of conceptualism.
The really interesting part here is that abstract objects cannot be grounded in just any mind. For there are contingent minds, much as myself, who are incapable of grounding any necessary truths. Instead, we ought to conclude that there exists a necessary mind that grounds abstract objects. But not only is this necessary mind the ground of abstract objects. Still further, this necessary mind must know all true propositions. The union of all true propositions is itself an abstract object. Since only an omniscient mind could know all true propositions, it follows that a necessary, eternal, omniscient mind exists.