The Trinity and the Incarnation are two central Christian doctrines. Moreover, these are interrelated. One particular member of the Trinity becomes Incarnate.
Unitarians attack both doctrines. There are different kinds of unitarians. Muslims and Jews. They attack these Christian doctrines from a different religious framework. You also have heretics who present unitarianism as the true Christian position. And you have religious pluralists like John Hick who attack the Incarnation because it's "intolerant."
The Incarnation, if true, is a contingent truth. It happened by the will of God. In principle, God might have refrained from willing the Incarnation.
The Trinity, if true, is a necessary truth. It figures in the essential nature of God.
At least, that's the case if you regard God as a necessary being–or espouse classical theism. If, on the other hand, you espouse process theology, then the question of whether God has an essential nature is disputable.
Christianity is a historical religion. By that I mean, it's an actual religion, practiced by billions of adherents past and present. It's not a thought-experiment or philosophical idea, like monadology or the brain-in-the-vat.
As such, if you (the unitarian) see fit to attack the Incarnation or the Trinity, then it seems natural to attack the specifically Christian versions. There are different approaches. Jews and Muslims contend that the NT is false. Heretics content that the NT has been misinterpreted.
However, unitarians often raises supposedly logical or metaphysical objections to the Incarnation and the Trinity. But in that event, are their objections to the Christian doctrines in particular, or to the possibility of an Incarnation, and the possibility of the Trinity?
Suppose, for the sake of argument, that unitarianism is true. That rules out the Incarnation of the Son, inasmuch as that requires a Trinitarian presupposition. That, however, doesn't ipso facto rule out the possibility of a divine Incarnation. Even if you deny that God is Trinitarian, it doesn't follow that the Incarnation of God is impossible. Even if you deny the reality of a divine Incarnation, it doesn't follow that the very idea of a divine Incarnation is logically or metaphysically impossible.
To contend that a divine Incarnation is not even possible is a very philosophically demanding claim. Yet unless unitarians can show that an Incarnation is impossible in general or in principle, they can't show that it's impossible in particular (i.e. Christology).
If they were intellectually honest, they'd recast the debate. Rather than beginning with the Christian doctrine of the Incarnation as their target, they'd try to prove that a divine Incarnation, even in the abstract, is essentially incoherent. But that's a tall order. It turns on different models of a divine incarnation.
What about the Trinity? Suppose we didn't know that much about what God was like. What if we were left to speculate, not knowing one way or the other. Are unitarians entitled to claim, a priori, that God could not be Trinitarian? Unitarians like Dale Tuggy invoke " the law of identity." Here are some examples of the law:
The Identity of Indiscernibles (hereafter called the Principle) is usually formulated as follows: if, for every property F, object x has F if and only if object y has F, then x is identical to y.
The Indiscernibility of Identicals (Roughly, if a = b, then whatever is true of a is true of b, and vice versa.)
The principle of the identity of indiscernibles, which states that any entities which are indiscernible with respect to their properties are identical. Leibniz is fond of using leaves as an example. Two leaves often look absolutely identical. But, Leibniz argues, if "two" things are alike in every respect, then they are the same object, and not two things at all. So, it must be the case that no two leaves are ever exactly alike.
Problem is, most philosophers think personal identity is consistent with diachronic identity and/or counterfactual identity. But they didn't get that from applying the "law of identity" to diachronic identity or counterfactual identity. Rather, they make allowance for diachronic identity and counterfactual identity despite the "law of identity." For instance:
There may seem to be an obvious objection to the employment of transworld identity to interpret or paraphrase statements such as ‘Bertrand Russell might have been a playwright’. A fundamental principle about (numerical) identity is Leibniz's Law: the principle that if A is identical with B, then any property of A is a property of B, and vice versa. In other words, according to Leibniz's Law, identity requires the sharing of all properties; thus any difference between the properties of A and B is sufficient to show that A and B are numerically distinct. (The principle here referred to as ‘Leibniz's Law’ is also known as the Indiscernibility of Identicals. It must be distinguished from another (more controversial) Leibnizian principle, the Identity of Indiscernibles, which says that if A and B share all their properties then A is identical with B.) However, the whole point of asserting a transworld identity is to represent the fact that an individual could have had somewhat different properties from its actual properties. Yet does not (for example) the claim that a philosopher in the actual world is identical with a non-philosopher in some other possible world conflict with Leibniz's Law?
It is generally agreed that this objection can be answered, and the appearance of conflict with Leibniz's Law eliminated. We can note that the objection, if sound, would apparently prove too much, since a parallel objection would imply that there can be no such thing as genuine (numerical) identity through change of properties over time. But it is generally accepted that no correct interpretation of Leibniz's Law should rule this out.
i) To say it "proves too much" because that would apply perforce to diachronic identity begs the question.
ii) Likewise, to say it "proves too much" doesn't entail that the interpretation of the law is wrong. The law itself may be too stringent.
The law of identity, as commonly formulated (see above) doesn't have any give. It doesn't permit change or identity across possible worlds. To make allowance for diachronic identity or transworld identity is to reduce "identity" to a matter of degree.
This is a case of conflicting intuitions. On the one hand, there's an intuition about numerical identity. On the other hand, there's an intuition about personal identity. Since these are in tension, philosophers propose ad hoc modifications of the "law of identity" to make room for common sense exceptions. Sometimes the tradeoff is made explicit. For instance:
Endurantism seems able to accommodate our prephilosophical belief that Henry persists through the change only at the expense of rejecting the Indiscernibility of Identicals. M. Loux, Metaphysics: A Contemporary Introduction (Routledge, 3rd ed. 2006), 244.
Now, I don't object to the common sense exceptions. But if you must weaken or relax the "law of identity," then there's a problem with how numerical identity is formulated in the first place. If you can only accommodate personal identity in spite of how the law is formulated, then that needs to be revised. They didn't begin with the law of identity, then use that as a criterion to determine the nature of personal identity. Rather, they begin with their experience of personal identity, including their hypothetical deliberations, and adjust the law of identity accordingly, as necessary, to conform to experience or common sense. But the whole point of having a law of identity is to distinguish identity from nonidentity. If, however, identity is consistent with difference, then what demarcates identity from nonidentity?
Is it a particular kind (or kinds) of difference that's incompatible with identity? Perhaps. But that's not how the law of identity is formulated. And if it were reformulated to take that into account, it would be less rigorous. The point of a law is to make a general or preferably universal statement. Everything on one side is identity, everything on the other side is nonidentity.