Three Whos and one What

This article, by Frank Sheed, does a good job of explaining the orthodox understanding of the Trinity, and it is an example of one of the most common ways of arguing that the Trinity doctrine is not self-contradictory; the 'three whos and one what' argument. This argument fails, however, because if possession of the one Divine Nature means that each of the Persons is God, and if there is only one God, then it follows that each of the Persons literally is the same thing. To say that each of the Persons 'personifies' the one God, or even that each of them 'subjectifies' the one God, would not entail any contradiction but to say that each of them is (ie. is identical to) God implies that there can be no distinction whatsoever between them because each of them is the same thing, a thing that there is only one of.
In other words, by insisting that the 'who' is the 'what' (in other words, that both of those terms signify the same thing), the 'three who's and one what' argument defeats itself.