The problem is in your logical relationships.
Jesus ~=~ God
The Father ~=~ God
Jesus =/= God
The ~=~ operator denotes a function which is only fully understood by God. It’s relationship with equality is also impossible for mortals such as us to understand.
I think you hit the nail squarely on the head, good buddy. The idea of a “mystery” as being necessary, desirable, and the final answer to questions is central to this religious concept.
It’s plausible that the idea that the trinity is an incomprehensible mystery functions the way you’re saying. But there are many (orthodox and even conservative) theologians who do not affirm that it’s an incomprehensible mystery. So I don’t think that what you’re talking about applies to the concept of the trinity per se.
I have very little exposure to theology to work with, but I haven’t noticed any theologians or theologies which don’t rely on those old ‘mysterious ways’ to tie up the loose ends when confronted with formal logic.
I seem to recall both Barth and Tillich give accounts of the Trinity which they take to be complete expositions that aren’t contradictory. (It’s been years since I read them though.) I’ve got Tillich’s Systematic Theology on my shelf so I’ll try to take a look some time.
At the Stanford Encyclopedia article, you can see a lot of philosophers and theologians giving accounts of the Trinity which (according to the article anyway) purport not to be contradictory. In fact only one view is recounted there (out of quite a number) which explicitly embraces contradiction–and even there he insists that it’s only an apparent contradiction caused by an unarticulated equivocation.
I don’t think that “the most natural sense of ‘is’” is identical with “the standard interpretation of the identity symbol.” I have to admit, I don’t know what the standard interpretation of the identity symbol actually is, but I think it’s clear that identity is a somewhat problematic concept philosophically, and I don’t think there is a unitary “most natural sense of ‘is.’”
As a possible analogy, the person I am today is Alan Smithee, and the person I was ten years ago is Alan Smithee, but the person I am today is not the person I was ten years ago. Each of those clauses is true in a way that everyone reading this understands and agrees is meaningful, but I don’t know how a philosopher would represent those clauses symbolically, though I believe all three statements use ‘is’ in its most natural sense.
Not long ago I for some reason looked up the Trinity on Wikipedia and ran across the “Shield of the Trinity,” which may be relevant to the OP’s question.
Recommend thread merger.
I’ve merged the threads started by virtually yours and Frylock since they’re about the same general topic.
One OP seemed to be asking for information (presumably neutral) about the doctrine of the Trinity and the other OP was looking for a debate (with people taking sides) on the topic. So combining the two threads might cause some unfortunate crossover.
Just waxing anecdotal here, but I can recall conversations with Protestant ministers where their only explanation for some incomprehensible things was “You can’t explain it, Son, it’s a mystery. A wonderful mystery.” And there it ended.
A mystery, to some, is something to be celebrated, a beautiful example of the wonderfulness and divine incomprehensibility of the unknowable deity.
Remember, “Faith means not wanting to know what is true.”
I just thought of a light-hearted example. From the lyrics of Carly Simon’s The Night Before Christmas:
Part of the fun of a mystery, is to try and solve it.
Is the Trinity related to the “Triune Brain”? The poet Robert Bly suggested that the human brain is three distinct brains and these three brains only communicate with each other through rare moments of Deja Vu or through ritual.
My problem with the idea of a unknowable mystery is you have to believe something you can’t understand. And if you don’t understand it, how can you know if you should believe it? Suppose Buddhism and Christianity both have an unknowable mystery at the heart of their faith - you can’t choose which is the true faith by learning about them. Apparently all you can do is flip a metaphorical coin and hope you pick the right faith.
Is God going to make decisions regarding the afterlife on this basis? He’s set up a situation where it’s impossible to know the answer but he’s going to grade people on whether they guess right or wrong?
Why is it so hard to accept some mysteries? People haven’t a clue about another great mystery-like the one why people destroy themselves with drugs (for example). They have ample information, they know the consequences of becoming addicted, yet they do it anyway. To me, that is a greater mystery than that posed by the Trinity.
How is this a mystery? Some people looking for short term pleasures don’t consider long term costs. Mystery solved.
I should apologize too, and in fact do; the tendency toward branding all Christians/theists/believers generally with the moral turpitude that characterizes many of those “leading the religion business” has left me with a sour taste. I believe because I consider I have reasonable personal subjective grounds to believe; I’ve done my best to combat any tendency to self-delusion, to believing what one wants to believe; and I am certainly not out to defraud anyone nor to adduce reasons to abridge their rights based on my reading of a large old collection of ancient literature.
This is not an excuse for rudeness; there is none. It’s an explanation of why I jumped to a conclusion that had me returning rudeness for perceived rudeness.
One of the frequent times I have trouble sleeping I’ll try reading one of those 
I won’t be surprised to see a claim of a logical explanation, but I doubt it will hold up. It’s easy to claim logical consistency when preaching to the choir, quite different to address it in formal terms. But I see it as many others here do, it’s theology, not logic. To me logic dictates that there is no God, if there is one, one must have faith.
On the subject of mystery, the idea to me is one of those which fits logically with theory but does not correspond to common sense, like the wavicle nature of photons or how quantum mechanics works. To have a single TV set that sometimes shows the face and speaks with the voice of Gary Sinise and sometimes with those of Connie Chung, is not peculiar; to have a human being which did so, is so far beyond human experience that it “flies in the face of common sense.” But the former is technically sound. I think the problem lies in presuming the identity function to convey all meanings of “is” – and as the great philosopher W.J. Clinton demonstrated, it all depends! Cats are carnivores and dogs are carnivores, but cats are not dogs. McCain is Senator from Arizona and Kyl is Senator from Arizona, but McCain is not Kyl. So what’s so hard about “The Father is God and the Son is God, but the Father is not the Son.”
There are even simpler things to consider. I am known to some people as my father’s son, and to others as my son’s father. There are a lot of relationships that extend the concept of ‘is’ beyond simple equivalence.
Right, if they’re all true, it can’t be the is of (necessary) identity (and I still don’t think TATG’s attempt to use contingent identity to interpret the doctrine will work, but I haven’t replied to that yet).
So that’s as much as to say that the statement Jesus == God is false. (Because ‘==’ refers to identity).
But that in turn means that Jesus =/= God. In plain English, Jesus is not God. And that seems to be something trinitarians avoid ever saying. Do they avoid it because it’s false, or because they’re afraid of being misunderstood?