If two equally sovereign creator gods disagree on any point, noncontradiction is no longer a valid test of truth. Without noncontradiction, there can be no test of truth at all, and rational thought goes out the window.
But the problem is worse than that.
Since the gods are equally sovereign, neither is in control of the other one. This means that neither one can guarantee the behavior of the other one. This means that even if they agreed on every point all the time, neither one of them would ever be able to guarantee their agreement.
So why is this a problem?
This is a problem because humans come pre-programmed with the belief, the first principle, the presupposition, that contradictions are always false. But which one of these gods pre-programmed us humans to believe that all things are non-contradictory? God A, who doesn't know whether God B will contradict him? Or God B, who doesn't know if God A will contradict him? Either way, our first principle or initial assumption of universal noncontradiction is rationally unjustified, meaning there is no good reason for us to believe it, even if it is true.
If there is no good reason for us to hold to universal noncontradiction, there is no good reason for us to hold to anything. All of our conclusions are based on the premise of noncontradiction. If we hold to it irrationally, we hold to every belief irrationally.
According to Plato, Socrates saw this problem with respect to moral facts, and his conversation with Euthyphro in Plato's Trial and Death of Socrates provides the basis for the argument against a plurality of ultimate authorities.
Some may wish to use a different definition for polytheism, claiming that only one God created everything, and lesser gods are then subject to Him. I would call this a form of monotheism. Others may say that there are lots of gods, but none of them created the universe. I would call this a form of atheism and it is even easier to prove atheism is false than it is to prove that polytheism is false! Polytheism then stands apart from both monotheism and atheism as having multiple, equally sovereign creator gods.
Under Christianity, there is one God -- one Creator. His identity corresponds directly to His identity, establishing the Law of Identity (Exodus 3:14). He never contradicts Himself, meaning that if He creates something, it is created, establishing the Law of Noncontradiction (2 Timothy 2:13). Our omniscient, omnipotent God can rationally justify our pre-programmed idea of noncontradiction. He knows all of reality is non-contradictory, because He made it! The Christian God Himself makes rational thought possible for His created humans.
As Greg Bahnsen rightly said, the most basic proof of Christianity is that without the Christian God, it would be impossible to prove anything at all.
Under your scenario, if I'm not mistaken, there is no rational justification for any claim. No god can rationally guarantee the behavior of any other god. Contradictions are not guaranteed to be false in anyone's mind. Thus, there can be no reason behind denying contradictions. This scenario appears to be tantamount to a denial of rational thought, and is thus a deductive disproof of the premise of polytheism.
Trinitarians differ on many details of that doctrine, so please understand up front that I do not speak for all Trinitarians in my answer. Instead, please accept my answer as nothing more than my answer. But I believe it to be coherent, in line with the Bible, and in line with historic creeds of the Christian faith.
In the Bible, it is abundantly clear that there is a sense in which the Son is God. But it is also clear that there is a sense in which He is not. In other words, it is true both that He is God; and that it is not the case that He is God, but in different ways.
John 1:1 claims that the Son (the "logos") is both God and with God. I take this to mean exactly what I just said -- He is God in one sense, but not in another.
In 1 Corinthians 8:6, the Father, whom Trinitarians assert is definitely not the Son, is the only one God.
Trinitarian theology states that there is only one God in terms of nature or essence; and that He is revealed in three persons; and that these persons are eternally distinct. So the Father is God, the Son is God, and the Spirit is God, but the Father is not the Son, and so forth.
The Son is described in scripture as "begotten" of the Father, and the "image" of God.
Jonathan Edwards described the Son as being, effectively, the Father's self-conception.
I agree with Edwards in essence. I have used his ideas (modified slightly with insight from Heraclitus) to develop a deductive argument against unitarianism, which can be found here:
So, to directly answer your question, my understanding is that the Father is the ontologically ultimate singular person of the Godhead, and that the Son is eternally begotten (as the Nicene Creed and Westminster Confession both affirm), and thus ontologically dependent. Unlike the heresy of Arianism, I do not claim that "there was when Christ was not", because the Son is eternal, transcending time.
Ontologically, therefore, the Father is the One God in all valid senses, as scripture proclaims, and the Son is God in some of those same senses, but not all, as scripture also proclaims.
In my mind, the Father, combined with His divine nature, would be the singular unit you ask about.
Other Trinitarians would argue that the one nature is the one unit you ask about. I find that answer unsatisfying as it seems to ultimately remove all personhood from God.
Yet others would make similar arguments to the arguments I have heard from henadologists (is that the right word?) like yourself, but I find those lacking as well for the same reason I have brought up to you -- rational unjustifiability.
I hope this clarifies my thoughts on your excellent and insightful question.
I elaborate on my thoughts on Bahnsen's proof-concept here:
Trinitarians don't advocate for multiple gods, but multiple persons in the Godhead. The distinction may sound like semantics, but it is actually huge.
We understand the Son, the second person in the Godhead, to be both begotten of and the Image of the Father. There appears to be a logical primacy in the Trinitarian Godhead, Father first, that is absent in the forms of polytheism I was speaking to. This is not to say that the Son was created -- not at all -- merely that He does not have primacy. Instead, His unique personhood is begotten of and the image of the Father.
This is not just my opinion about the Biblical texts (though it is my opinion about them), but also seems to be the opinion of both Jonathan Edwards and Aquinas, and can be found in the Nicene Creed and the Westminster Confession (2:3).
If two equally sovereign senators/MPs disagree on any point, noncontradiction is no longer a valid test of legislation. Without noncontradiction, there can be no test of legislation at all, and common law goes out the window.
You know human governments can reach agreements (eg constitutions and legislation) but assume you a multitude of divine minds cannot do the same, to assume two (or more) equally great beings cannot reach a timeless agreement is unfounded.
Euthyphro's dilemma applies equally in Christianity; is some good because god wills it or does god will it because it's good?
Your argument is circular; it presupposes objective morality to justify monotheism on the basis objective morality comes from God.
Senators exist in an environment they did not create. They each relate to their outside world -- and thus to one another -- based on their initial programming regarding what the outside world is like. They can successfully coordinate things specifically because there is one God who created the environment in which they find themselves, and who also programmed them both to comprehend it. There could be no rational cooperation between senators if they did not share the same singular creator: there would be no conceivable justification for their beliefs about one another.
In the above you say that Jesus was "begotten of" the Father. Can you explain in more detail what you mean by that? What does "begotten of" mean in the context of that answer? And what does "primacy" mean?