It is sometimes asked if God could include one less or more persons in the Godhead. The obvious answer is in the negative but interesting metaphysical reasons may be behind why.

Sufficient meditation on the above observations yields the conclusion that God cannot be any more or any less than three divine persons, without being reduced to a common class with those finite “impersonally-contained” deities mentioned earlier. Any other number of divine persons would create a disparity between the personal contexts and the personal relationships; between the “one” and the “many” of the Godhead. For example, if God were bi-personal— perhaps as Father and Son but excluding the Holy Spirit— one of two equally problematic conclusions would follow. Either, (a) the Father and the Son would be reliant on an impersonal context as that which could facilitate their mutual relationship and definition; or, (b) one of the two persons would have to function as an unrelated and thus undefined context, in whom the other resides as an individual monad (see fig. 26). In the former case, both divine persons are at the mercy of an impersonal universe, in the latter case, the two persons represent an indefinable context and an indefinable individual, respectively. One encounters the same sort of difficulties when the number of divine persons is haphazardly increased. For example, a quadrinity in whom the Father, Son, and Spirit are married to some forth person, “x,” must subordinate the individual persons to abstract and impersonal “groups,” and/ or render the persons divisible into finite impersonal parts (see fig. 27). If the relationship between the Spirit and person “x” were facilitated by the Father and the Son together, then the actual mediator of that relationship, and that thing which comprehends the Trinity, would be an abstract and impersonal “group” formed by the Father and Son, rather than a concrete person. Neither the Father nor the Son, but that “something” between them, would be the true facilitator of the relationship between the Spirit and person “x.” But the very profundity of the Trinity lies in the fact that that which unifies all things is not an unknown something, but a personal Authority Who we might come to know and trust because He is also a concrete individual. It is equally unacceptable that the Father alone might function as the context of the Spirit-“ x” relationship, for then (a) the Son would be indifferent and uninvolved in some activity of the godhead, and so fail to fully express/ comprehend the entire divine being in himself; and (b) the Father would cease to be an absolute person since only the relationship between Spirit-“ x,” and not the relationship between Himself and His Son, would fall within the sphere of His own self-activity (see fig. 27). Both the Father and the Son would be mere parts of a generic divine nature that comprehends them both. Finally, it is also to no avail to suppose that the seven possible pairs of relationships between any two persons (F-S, F-X, F-H, S-X, S-H, and X-H) could be mediated by a single third person. Such a scenario would create a disparity between the divine persons, as some would have to function as the contexts of multiple pairs of relationships, and others of only one. Again, those persons whose natures were fully expressed as the context of only one personal relationship would not comprehend the entirety of the divine nature in themselves (as several relations would fall outside of themselves). And, that one person who did comprehend all of the others and their relationships would represent an abstract unity, who depends on an irrationalist principle of individuation, since he fails to behold or to differentiate himself with respect to other co-equal persons who in turn comprehend him. Hence, to add or subtract from the number of three divine persons is to compromise an ultimately personalist view of reality, by making God identical with, or subordinate to, an impersonal context. Such a God could not speak with authority, and the servants of such a deity would lack any ground for placing their trust in Him, because He would not “know” himself solely with reference to His own person.

Bosserman, B. A.. The Trinity and the Vindication of Christian Paradox: An Interpretation and Refinement of the Theological Apologetic of Cornelius Van Til (pp. 180-181). Pickwick Publications, an Imprint of Wipf and Stock Publishers. Kindle Edition.