1.) Reason can prove the Trinity free of logical contradiction.
2.) Whatever is proved free of contradiction is really possible.
3.) To prove that a necessary being is really possible proves it exists.
4.) The Trinity is a necessary being.
5.) The existence of the Trinity is a mystery that cannot be proven by reason.
Regardless of whether the Trinity exists or not #4 is true by definition and so I take it as given.
A.) Take the unqualified affirmation of (1) as a truth of faith. While reason might be able to establish that it has proved the trinity free of all logical contradictions suggested so far, it cannot be sure it hasn’t missed one.
I reject this possibility since it destroys the very thing it tries to defend, sc. the reasonableness of apologetics. A reason that has to assume it’s missed some contradictions in something is one that cannot do its due diligence. True, “apologetics” is etymologically the defense of the faith, and in this sense is only a response, but the term clearly means more than this in actual practice.
B.) Take the logical space of metaphysically necessary beings as underdefined. While it is true that necessary truths must be either impossible or actual, the difference between mathematics and metaphysics is that the former domain is defined enough that we can specify the relevant possibilities while the latter is not. Ruling out contradictions can get us somewhere in mathematics while it can’t get us the same results in metaphysics. Reason must take the possibility of a Quaternity or Binity or Infinity as just as possible as a Trinity.
This account is false and possibly incoherent. How is the logical space of divine persons undefined? It’s one or not one, divided or undivided, etc. Either transcendental multiplicity is logically possible or it isn’t.
C.) Take freedom from contradiction as establishing agnosticism. To establish that something is not impossible leaves the mind undetermined as to its real existence. “Possibility” in this sense means “for all I can tell, it could be or not be so”. The contradictory of can’t be is might be, not can be.
While Aristotle argues this, it’s hard to avoid the conviction that the previous sentence is obviously false. (N.B. The most persuasive critiques of the modal Ontological Argument confuse “can be” with “might be”: viz “saying “a necessary being is possible” and then “a necessary being can not be”, from which it’s non-existence is tehn rigorously proven, is an example of this).
While the principle that a little mistake in the beginning, etc. gets overused, it is very applicable here. One should decide at the beginning of his logic whether the contradictory of “can’t” is “might” or “can” or both.