Does Trinity Entail Quaternity?

Christianity, like the other two Abrahamic religions, is monotheistic. But unlike Judaism and Islam, Christianity holds to a trinitarian conception of God. The idea, spelled out in the Athanasian Creed, is that there is one God in three divine Persons, the Father, the Son, and the Holy Ghost. Each person is God, and yet there is exactly one God, despite the fact that the Persons are distinct from one another. How is this possible? How can Christians convince Jews and Muslims that their position is logically tenable and does not collapse into tritheism, and thus into polytheism to the detriment of the divine unity and transcendence?

Here is one problem. God is said to be tripersonal: the one God somehow includes three numerically distinct Persons. But none of these Persons is tripersonal. The Father is not tripersonal. The Son is not tripersonal. The Holy Ghost is not tripersonal. Now if two things differ in a property, then they cannot be identical. (This is the irreproachable principle of the Indiscernibility of Identicals expressed in its contrapositive form.) Now God and each of the Persons differs in point of tripersonality: God is tripersonal while no Person is tripersonal.  It follows that God is not identical to the Father, nor to the Son, nor to the Holy Ghost. Therefore, God is not identical to any of the Persons, whence it follows that God is distinct from each of the three Persons. And if God is distinct from each of the Persons, then he is a Fourth. 

Is God a divine person? If you say yes, then we are on our way to the Quaternity, the doctrine that there is one God in four divine Persons. For if God is not identical to the Father, or to the Son, or or to the Holy Ghost, each of which is a person, and God is a person, then there are four – count ‘em – four Persons.

Some doctrines in philosophy threaten to collapse into others. Thus mind-brain identity theory threatens to collapse into eliminativism about the mind. Other doctrines seem to want to expand. How do we keep the Trinity from expanding into the Quaternity? The attentive reader will have noticed that the argument can be iterated. If the Three-in-One becomes a Four-in-One, how avoid a Five-in-One, ad infinitum?