The question is whether the doctrine of Trinity meets the principle of Rational Anticipation?
We'll quickly look at two arguments to check out the same.
1. The Argument from the Possibility of Knowledge
a. If God exists, He must be an intelligent being (or else, intelligence is an accident and truth is impossible- but, to say truth is impossible is to contradict self; therefore, truth exists and has its eternal ground in God).
b. Intelligence involves Knowledge and Knowledge involves a Subject-Object distinction.
c. Eternal intelligence must involve eternally a Subject-Object distinction.
d. This distinction must be internal and eternal (since, nothing can be infinite and eternal outside the Godhead - God is by nature infinite, and there cannot be more than one infinite).
e. Complete distinction requires at least three persons (I, You, He/She/They).
f. Therefore, possibility of knowledge rationally anticipates the Three Persons in a Subject-Object relationship.
2. The Argument from Morality
a. If God exists, He must be a moral being (or else, morality is a temporal category and ultimately and eternally meaningless).
b. Morality involves community (Without community, morality is meaningless; for where there is only one person there is no moral obligation to anyone because there is no other person).
c. A community involves persons who are morally responsible to each other.
d. Responsibility involves a witness (which in turn requires the community to be composed of at least three persons, necessarily speaking: beyond that is not-necessary).
e. Therefore, the existence of morality rationally anticipates the Three Persons in an eternal Community relationship.