Richard Barcellos has a pretty good post up on the trinity that can be found here.  The quotation taken from the 1689 Baptist Confession that is analyzed is: “In this divine and infinite Being there are three subsistences, the Father, the Word or Son, and Holy Spirit, of one substance, power, and eternity, each having the whole divine essence, yet the essence undivided.”

He rightfully points out that it is a dangerous contradiction, “even an absurdity,” to confuse the sense in which we believe that God is one and the sense in which we believe that God is three.  The questions should be: “one what?” and “three what?”  Barcellos writes the following:

That God is one is very clearly stated in the Bible (Deut. 6:4; 1 Cor. 8:4, 6). However, historic Christianity claims that in one sense God is also three. How can God be one and three at the same time? It all depends on what is meant by “one” and “three.” One in what sense? And three in what sense? Is God one in every sense? No. Is God three in every sense? No. If the answer were yes then we would have an obvious contradiction. In fact, many throughout history and in our own day have claimed that the doctrine of the Trinity is a contradiction, even an absurdity. However, historic Christian orthodoxy has always been extremely careful to qualify and define what it means by one and three as they relate to the doctrine of the Trinity.

It is a terrible thing that even recent theologians in Reformed circles (such as Cornelius Van Til) claimed that God was one and three in the same sense.  This reduces our understanding of God to a logical absurdity.  Gordon Clark, in his book on the Trinity interacts with Van Til when he comments:

“Note the situation. When opponents have objected that the doctrine of the Trinity is logically self- contradictory because it makes three equal to one, Christians have usually replied that there are many examples of situations that are three in one sense and one in a different sense. Hence there is no contradiction. Here Van Til rejects this defense of the Trinity and asserts that the Trinity is both one and three in the same sense: not one substance and three Persons, but one Person and three Persons. This is indeed contradictory and utterly irrational. Look at his words again: “We do assert that God, that is the whole Godhead, is one person.” He defends this irrationalism on the ground that “each attribute is co-extensive with the Being of God.” Now, some attributes apply equally to all three Persons; for example, omnipotence and omniscience. But the attribute of Fatherhood and Sonship are not “co-extensive with the Being of God.” Sonship is not attributable to the Father, nor to the Spirit.”

Later, while commenting on the distinctions between the three persons of the Trinity, Barcellos quotes A.A. Hodge who wrote that the

three Divine Persons are distinguished from one another by certain Personal properties, and
are revealed in a certain order of subsistence and of operation.

This is interesting.  It is common to point out that the persons are distinguished by “properties” but what, exactly, are these properties? The Confession itself says that the three persons are “distinguished by several peculiar relative properties.” Surely they aren’t physical.  Barcellos writes: “Though they share in the fact of personhood, each person is distinguished and separate from the other as a subsistence of the one divine essence.”  What does it mean for a person of the Godhead, which is completely spiritual, to be “separate?” We can’t be referring to a separation of time and space, for the persons of the Trinity are not physical (even Christ, who existed before the incarnation).  One solution is Gordon Clark’s who wrote that “[w]hat modern Christian academics needs therefore, among other things, is a theory of individuation.”  His suggestion is as follows:

“This is true of the Trinity also, for although each of the three Persons is omniscient, one thinks “I or my collection of thoughts is the Father,” and the second thinks, “I or my thoughts will assume or have assumed a human nature.” The Father does not think this second thought, nor does the Son think the first. This is the qualitative theory of individuation, as opposed to the space-time theory: No two leaves in the forest are exactly alike, and Leibniz’ Alexander the Great is defined by his history. Even if trees could be individuated by space and time, the persons of the Trinity, as said above, could not; nor could human souls or other spirits.”

Every person then, whether a member of the Godhead or the personal souls of human beings created by the Godhead in His image, is “separated” or “individuated” or “distinct” from all other persons based on the propositions that he thinks. No mind is the same and therefore no person is the same either.  It is by this understanding of the nature of the person –which includes you and I — that Clark saw the three persons as distinct from each other within the united and one God.  I recommend both Barcellos’ post and Clark’s book.