In my experience a lot of the problems in modern philosophy of religion come about from not taking enough care to get right the religious position the philosopher is analyzing. Part of this difficulty stems from the way terminology shifts across the centuries, so that the modern philosopher takes for granted an anachronistic understanding of key terms.
Dr Vallicella and others at Maverick Philosopher have been discussing the logical coherence of the doctrine of the Trinity, as well as giving links to contemporary philosophy of religion discussions of the topic. Here is one of the posts: http://maverickphilosopher.typepad.com/maverick_philosopher/2010/01/some-water-analogies-for-the-trinity.html
More than once Dr Vallicella points out that God's substance can't be understood as matter, which is correct. But he fails to understand what "substance" means in trinitarian doctrine. He writes: "The sense in which water is a substance is not the sense in which God is a substance. Water is a substance in the sense of a stuff; God is a substance in the sense of a hypostasis (that which stands under) or hypokeimenon (that which is placed under), or as I prefer to say, an individual."
From the standpoint of traditional, classical Trinitarian theology, this is incorrect. God is a substance neither in the sense of stuff (hyle) nor in the sense of individual (hypostasis). Here's a representative explanatory snippet from St John of Damascus, showing the universal traditional use of the terms, from "De Fide Orthodoxa" c.48: "Substantia quidem communem speciem et complectivam speciem homoiodon (id est earum quae unum sunt specie) hypostaseon (id est personarum) significat, utputa Deus, homo; hypostasis autem atomon (id est individuum) demonstrat, scilicet Patrem, Filium, Spiritum Sanctum, Petrum, Paulum."
So "substance" here means something like "essence" or "being" (in the sense of ousia) rather than hypostasis; the whole doctrine of the Trinity depends on this distinction between the one nature, substance, being, essence, etc. on the one hand and the three individual persons or hypostases on the other. In most cases where there is one existing human nature (man), there is one individual hypostasis (Peter or Paul); in the case of the Trinity there is one divine nature (God) instantiated in three hypostases (Father and Son and Holy Spirit); conversely, in the Incarnation there are two existing natures (God and man), but only one hypostasis (Christ the Incarnate Logos). It is not good Trinitarian doctrine to say that there is one individual or hypostasis (God) who is identical to three individual persons (Father, Son, and Holy Spirit).
There's no point in discussing the coherence of a doctrine if the doctrine is not first clearly understood. But the doctrine turns into unintelligible mush if these crucial distinctions are not carefully preserved.
