I've also explained elsewhere why Clark's theory of persons would imply the unbiblical view that God is metaphysically dependent on creation (here).
Others have pointed out problems with Clark's theory (for example, see here).
I'm going to note a few more problems. But firstly, it isn't clear whether Clark consistently held the same metaphysical view of persons throughout his life. For instance:
Aristotle admitted that individuals cannot be known. Hegel’s fault, or one of them, was to make the concept rather than the propositions the object of knowledge. But a concept is as unknowable as an individual. “Pen” is neither true nor false. Only a proposition can be true. “The pen belongs to Herr Krug” may be true; it may be false; but a concept in isolation is not an object of knowledge. Truth always comes in propositions.
Two quotations from Leibniz enforced the application of this principle to persons. In fact the citations will do double work. They will show that knowledge of a person is propositions (and thus they bear on what several of my critics consider paradoxical, to wit, persons are propositions), and at the same time they will bring home the lesson from Plotinus that knowledge of oneself is no easy, off-hand, immediate experience, but of all things immensely difficult...
Far from my making it impossible for God to know human beings, it is rather Professor Nash who does so. His view of the self is that of some Ich-an-sich. Leibniz suggests that the ego is a complex definition, including the life history of the person, and no doubt his state in a future world as well. This definition is not unknowable in essence, and God knows it because he determined what it should be. On the other hand, it is something that the person himself does not know, at least in this life. (Clark and His Critics, 2009, pgs 148-149)
On this view, persons are just propositions. Clark is here silent as to whether or not they are propositions they think, as he argued later in his life:
Accordingly the proposal is that a man is a congeries, a system, sometimes an agglomeration of miscellany, but at any rate a collection of thoughts. A man is what he thinks: and no two men are precisely the same combination.
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.
Several romantically inclined students, and a few professors as well, have complained that “this makes your wife merely a set of propositions.” Well, so it does. This suits me, for I am a set of propositions too. And those who complain are as they think. (The Trinity, 2010, pg. 129)
Let's forget about that for now. Let's say we are what we think - specifically, the propositions we think. But sometimes, we think falsely as well as truly. Clark admits as much himself, including the false propositions we think in our individual, personal definitions as well as the true ones:
Therefore, since God is Truth, we shall define person, not as a composite of sensory impressions, as Hume did, but rejecting with him the meaningless term substance, we shall define person as a composite of truths. A bit more exactly, since all men make mistakes and believe some falsehoods, the definition must be a composite of propositions. As a man thinketh in his (figurative) heart, so is he. A man is what he thinks... Whether the propositions be true or false, a person is the propositions he thinks (The Incarnation, 1988, pgs. 54-55).
However, given Clark's statements that “No one more than I insists on the necessity of a single self-consistent worldview” (Today’s Evangelism: Counterfeit or Genuine? 1990, pg. 111), doesn't that mean that, metaphysically speaking, we are contradictions? If there is a single, self-consistent worldview, any false thought we have must be contradictory to any true thought we have. The result is that either God doesn't know us or God is a dialetheist, which is about as far removed from Clark's "consistency" theory of truth (Clark and His Critics, 2009, pgs. 142-145, 290-291, etc.) as one could get.
Of course, if one bites the bullet and argues God doesn't have to know us, then Clark's whole motivation for persons metaphysically being propositions in the first place is gone. God either doesn't need to know us or, as I think, we don't have to metaphysically just be propositions in order for God to know us, for what we metaphysically are was determined by God to correspond to some truth which God knows.
Now instead, let's say we are just a set of propositions and disregard what it is that we think. We are a complex definition that God has determined, per the above exposition of Leibniz. In that case, mustn't whatever set of propositions God knows us to be, whatever set of propositions we are, be changeless - which would make us eternal - on pain of making God's knowledge change? As propositions, we must be the objects of God's thoughts; if we change, God's thoughts and knowledge must change. Most Clarkians don't believe God's knowledge can change, but the resultant implication goes much farther than this or even a corollary to a B-series theory of time called eternal creation; in this case, we ourselves would cease to be temporal. We wouldn't change. This is opposed to Clark's own beliefs, and, at any rate, clearly unbiblical.
But suppose we allow that the set of propositions we are changes, and so God's knowledge changes. I've argued elsewhere (without endorsing the view) that God could be eternally omniscient and yet have determined that His knowledge will change in accordance with changes in time. In fact, per the above quote, it seems Clark unwittingly admits this to be the case in the incarnation (“I or my thoughts will assume or have assumed a human nature”). [Lest anyone think Clark's change of view on the incarnation may have affected this, he says on pg. 55 of The Incarnation (1988): “Neither the complex of truths we call the Father nor those we call the Spirit, has the proposition, “I was incarnated.” This proposition occurs only in the Son’s complex.”]
It would take someone extremely committed to Clark's metaphysic of personhood to goes so far as to admit God is temporal just to save it, for he would have to give up Clark's motivations for necessitarianism and divine eternality. Worse, however, I think this view leads to a kind of process theology or divine becoming. For if persons are propositions, the persons of the Trinity must be propositions. And if "the Father is a knower of [person] x as [a set of propositions] y" is true at one time and false at another (corresponding to the time[s] at which He decreed we change as persons), does this not imply metaphysical change on the part of the Father?
One would have to state that this proposition ("the Father is a knower of [person] x as [a set of propositions] y") isn't essential to or found in the complex definition of the Father at any time (and likewise the Son and Spirit). But then, this implies "the Father is omniscient" isn't to be found in the definition of the Father either, for the truth of this latter proposition hinges on the truth[s] of the former. And then by parity of reasoning, all the other divine attributes appear equally unessential, and thus one couldn't even say that "the Father (or Son or Spirit) is God (or divine)" is essential to their personhood. Clearly this has been ad hoc reasoning for more than a while now, so the view that persons metaphysically just are propositions is problematic too.