Over the centuries there have been several attempts to show from general revelation that God is triune: e.g. Richard St Victor, Edwards, and, more recently, Swinburne. In modern times, with the increasing interest in AI and Information Theory, it is not surprising that there have been suggestions that these domains have something to contribute to the question of God’s multi-personality.
A major contribution here was from Donald M. MacKay. In the 1950’s he hypothesised that no single person could have complete knowledge of themselves. In fact, a complete specification of such a person (whether finite or infinite) does not exist for that person. This finding was proved formally (again in the context of AI and information systems) more recently by David Wolpert.
MacKay also noted that the relevant specification was, in principle, knowable by a second person, if and only if they are not in a reciprocal communication relation (because in such a case the two persons form a coupled system, and neither one can have full knowledge of the complete system). From this MacKay argued that for the incarnation when God enters His creation, He must be multi-personal, otherwise He could not have complete knowledge of the system formed thereby consisting of the incarnated God and the world. So there must be at least two persons; and here MacKay drew the distinction between God-as-creator (Father) and God-in-dialogue (Son, God incarnate).
This does not, however, say anything about the ontological/immanent aspects of the Godhead, and MacKay did not deal with that in this context. Nonetheless there are implications that suggest that God-self must be multi-personal. From the above arguments of MacKay and Wolpert, it follows that a single person cannot be absolutely omniscient, only “omniscient-after-their-kind”. Multiple persons could between them have knowledge of all things. Here the knowledge remains distributed, and each person remains omniscient-after-their kind. This gives a clear individuation between the persons, since they share all knowledge in common (and have a single will) except for those propositions which are unique to each person. Once the persons have been individuated in this way, it is not possible to identify the Godhead as another person because being “multi-personal” is inconsistent with being a person! That is, the “Third Man” argument would be ineffective here (as in fact it was ineffective in its original context). This then points to God-self as an absolutely omniscient multi-personal being.
These things do not prove the Threeness of the Godhead (but given the difficulty of doing that from the OT alone, this is not surprising). However, given the Threeness of the Godhead they suggest a model whereby the three persons are united in a unique manner as a tripersonal being who is absolutely omniscient. This model favours classical trinitarianism over societal trinitarianism, (or at the very least, is in the scope of Crisp’s chastened theism) with three persons in the Godhead, the same in substance, equal in power and glory (and knowledge).