Crosslinguistic influence

Yet this is not the whole truth of the matter. We do assert that God, that is, the whole Godhead, is one person…. In other words, we are bound to maintain the identity of the attributes of God with the being of God in order to avoid the specter of brute fact.” 

…Over against all other beings, that is over against created beings, we must therefore hold that God’s being presents an absolute numerical identity. And even within the the ontological Trinity we must maintain that God is numerically one. He is one person. We we say that we believe in a personal God we do not merely mean that we believe in a God to whom the adjective “personality” may be attached. God is not an essence that has personality; He is absolute personality. Yet, within the being of the one person we are permitted and compelled by Scripture to make the distinction between a specific or generic type of being, and three personal subsistences.

—Cornelius Van Til, An Introduction to Systematic Theology (1971).

i) This wasn't Van Til's finest hour. It's a bad way of making a good point. The basic point, I take it, is that the Triune God isn't three persons stuck onto an impersonal essence. Rather, God is personal through-and-through. 

However, Van Til's formulation, as it stands, is contradictory and unorthodox. 

ii) So what was Van Til thinking? What did he put it that way? And is it possible to gives his statement a coherent, orthodox sense?

This may be a case of crosslinguistic influence. To begin with, English was not Van Til's first language. 

An example of crosslinguistic influence is the fact that the sense of some NT words is based, not on secular Greek, but on OT Hebrew filtered through the LXX. What David Hill calls Greek words with Hebrew meanings.

By the same token, it's possible that Van Til is using "person" in the sense of "hypostasis. That's a standard term in Trinitarian usage. And it's a flexible term, because it can mean both "person" and "substance." 

Suppose we were to substitute "hypostasis" for "person" in Van Til's statement. Suppose Van Til said "God is one hypostasis and three hypostases."

That could mean "God is one substance and three persons." That's both coherent and orthodox. 

If he's using the same word with the same intended sense throughout, then the statement is contradictory and unorthodox. If, however, he's using the same word with alternating senses, then the statement could be logically consistent and theologically orthodox.

It doesn't really work in idiomatic English. It only succeeds on the assumption that he's using "person" as a synonym for "hypostasis."