Now I have come to appreciate a form of Social Trinitarianism, which basically means there are three distinct centers of self-consciousness (i.e., persons), each with their own proper intellect and will who are related to each other in such a way as to form one single entity (i.e., God). Now some, especially non-Christians, would charge that Social Trinitarianism is incoherent and falls into Tritheism. Is it incoherent though? There are a few forms of Social Trinitarianism, but the one I will dig into here is Trinity Monotheism. This position holds that while the persons of the Trinity are each divine, the Trinity as a whole is properly God. However, the question then arises that if each person is not a full instance of God, in what way can each person be individually divine? William Lane Craig and J.P. Moreland provide the following analogy in response:
However, doesn't this part-whole relation view downgrade the divinity of persons? To this question Craig and Moreland continue their response:One way of being feline is to instantiate the nature of a cat. But there are other ways to be feline as well. A cat's DNA or skeleton is feline, even if neither is a cat. Nor is this a sort of downgraded or attenuated felinity: A cat's skeleton is fully and unambiguously feline. Indeed, a cat just is a feline animal, as a cat's skeleton is a feline skeleton. Now if a cat is feline in virtue of being an instance of the cat nature, in virtue of what is a cat's DNA or skeleton feline? One plausible answer is that they are parts of a cat. This suggests that we could think of the persons of the Trinity as divine because they are parts of the Trinity, that is, part of God. Now obviously, the persons are not parts of God in the sense in which a skeleton is part of a cat; but given that the Father, for example, is not the whole Godhead, it seems undeniable that there is some sort of part-whole relation obtaining between the persons of the Trinity and the entire Godhead.
Now the Trinitarian Monotheism explanation presents a coherent doctrine of the Trinity. It avoids the contradictory position Jehovah's Witnesses charge Trinitarian Christians with, which is that we believe there is only one God and three Gods. For what is actually being said is that there are three persons who make up the one and only God. Muslims (and others) would no doubt still charge Trinitarian Monotheists (TMs) with believing in three Gods, seeing as TMs believe each person within the Godhead has their own consciousness. Craig and Moreland recognise this and respond by way of an analogy to the question, "What is the salient difference between three divine persons who are each a being [tritheism] and three divine persons who are together one being? [Trinitarian Monotheism]?":Far from downgrading the divinity of the persons, such an account can be very illuminating of their contribution to the divine nature. For parts can possess properties which the whole does not, and the whole can have a property because some part has it. Thus, when we ascribe omniscience and omnipotence to God..., God has these properties because the persons do. Divine attributes like omniscience, omnipotence and goodness are grounded in the persons' possessing these properties, while divine attributes like necessity, aseity and eternity are not so grounded. With respect to the latter, the persons have these properties because God as a whole has them. For parts can have some properties in virtue of the whole of which they are parts. The point is that if we think of divinity of the persons in terms of a part-whole relation to the Trinity that God is, then, their deity seems in no way diminished because they are not instances of the divine nature.
The church fathers rejected analogies like Cerberus, although such does seem to represent an image of the Trinity amongst creatures. Now if Rover, Bowser and Spike each have a rationality and self-consciousness, Cerberus is still a single being inspite of having multiple minds because all share the one physical body. However, suppose Cerberus was killed and his minds survived physical death. How would each still be one being? Such a question is relevant to the Trinitarian concept of God, for God does not possess a physical body. As Craig and Moreland question, "Since the divine persons are, prior to the Incarnation, three unembodied minds, in virtue of what are they one being rather than three individual beings?"In Greco-Roman mythology there is said to stand guarding the gates of Hades a three-headed dog name Cerberus. We may suppose that Cerberus has three brains and therefore three distinct states of consciousness of whatever it is like to be a dog. Therefore, Cerberus, while a sentient being, does not have a unified consciousness. He has three consciousnesses. We could even assign proper names to each of them: Rover, Bowser and Spike... Despite the diversity of his mental states, Cerberus is clearly one dog. He is a single biological organism having a canine nature. Rover, Bowser and Spike may be said to be canine, too, though they are not three dogs, but parts of the one dog Cerberus. If Hercules were attempting to enter Hades and Spike snarled at him or bit his leg, he might well report, "Cerberus snarled at me" or "Cerberus attacked me."
One response to this question hinges upon how one understands the soul. I'm going to quote Craig and Moreland again who provide a more concise answer than I could produce, and one which also briefly covers the relevant position of understanding the soul (something which I can also expand upon further personally if desired):
Kurieuo.... souls are immaterial substances and ... it is plausible that animals have souls... Souls come in a spectrum of varying capacities and faculties. Higher animals such as chimpanzees and dolphins possess souls more richly endowed with powers than those of iguanas and turtles. What makes the human soul a person is that the human soul is equipped with rational faculties of intellect and volition that enable it to be a self-reflective agent capable of self-determination. Now God is very much like an unembodied soul; indeed, as a mental substance God just seems to be a soul. We naturally equate a rational soul with a person, since the human souls with which we are acquainted are persons. Suppose, then, that God is a soul which is endowed with three complete sets of rational cognitive faculties, each sufficient for personhood. Then God, though one soul, would not be one person but three, for God would have three centers of self-consciousness, intentionality and volition, as social trinitarians maintain. God would clearly not be three discrete souls because the cognitive faculties in question are all faculties belonging to just one soul, one immaterial substance. God would therefore be one being that supports three persons, just as our own individual beings each support one person. Such a model of Trinity monotheism seems to give a clear sense to the classical formula "three persons in one substance."
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Quotes taken from: J. P. Moreland and William Lane Craig, Philosophical Foundations for a Christian Worldview. (Downers Grove, IL: InterVarsity Press, 2003), 591-594.
