According to the answers here and here, the term person means "an individual substance of a rational nature".
According to trinitarian theology, the triune god exists as three consubstantial persons consisting of one substance.
Does this mean the triune god exists as three individual substances of a rational nature, consisting of one substance of an irrational nature, in which neither nature is divided so that each are fully rational and irrational?
The Divine Substance (a.k.a. God) is supremely rational, as long as we understand the term rational to mean “capable of intellectual knowledge.”
As I mentioned in my answer to the O.P.’s question regarding person and substance, the Divine Substance (God) is perfectly identical with each one of the Persons.
The Persons are really distinct from one another, because they consist of the very relations of origin that subsist in the Godhead. However, there is no real distinction between Person and Substance. (In God, individual and universal substance coincide perfectly.)
(It is important to recall that the notions of person and substance—which strictly speaking apply only to creatures—are applied very improperly to God. That is why theses affirmation sound strange to us.)
As I mentioned in my previous answer, substance can have two slightly different connotations. For an Aristotelian like me, “substance” primarily means individual, concrete beings, like trees, flowers, stones, and human beings. Secondarily it can mean the common or universal kind of thing something is: for instance, tree-kind, flower-kind, stone-kind, and mankind. (For a Platonist, the priority is reversed—but for our purposes the fundamental idea is the same.) Also, “substance,” “essence,” and “nature” are basically synonyms for the ancient Greeks.
Now, Boethius’ definition for person, as the O.P. correctly mentions, is “individual substance with a rational nature.” We are, therefore, dealing with the two meanings of substance that I just mentioned above. Boethius is affirming, in effect, that a person is a concrete being that belongs to a particular genus or kind of being: namely those with the capacity for intellectual (rational) knowledge.
Boethius’ definition, as I mentioned, applies properly to creatures. God is beyond creaturely genera and species. However, He does possess to an infinite degree the capacity for intellectual knowledge (i.e., He is omniscient).
Therefore, both the Divine Nature or Substance (a.k.a. God) and the Divine Persons (who are, as I have mentioned, each perfectly identical to the Divine Nature) are not only rational but supremely rational.*
Hence, irrationality has no part whatsoever in God. Moreover, God is one Substance: the Persons do not divide the one Substance in any way whatsoever. In no way can He be considered three Substances.
* Note that in classical Scholastic lingo the term “rational” can be taken in a strict sense and in a broad sense. In the strict sense, “reason” and “rationality” refer to the particular kind of intellectual knowledge that we human beings possess. Our intellects are discursive; that is, we have to use concepts, enunciations, and argumentations to arrive at knowledge. Not all intellects are like that, however. For instance, angelic intellects are intuitive. In the broad sense, however, “rational” is a synonym for “intellectual.” That is what Boethius means by it.
