Thomas Aquinas, Compendium of Theology

That there cannot be several souls in one body is proved as follows. The soul is evidently the substantial form of any being possessing a soul, because a living being is constituted in genus and species by its soul. But the same thing cannot have several substantial forms. A substantial form differs from an accidental form in this, that a substantial form causes a particular thing simply to be, whereas an accidental form is added to a particular being already constituted as such, and determines its quality or quantity or its mode of being. Hence, if several substantial forms belong to one and the same thing, either the first of them causes it to be this particular thing or it does not. If it does not, the form is not substantial; if it does, then all the subsequent forms accrue to what is already this particular thing. Therefore none of the subsequent forms will be the substantial form, but only some accidental form.

Clearly, therefore, one and the same thing cannot have several substantial forms; and so one and the same person cannot have several souls.

Furthermore, it is evident that a man is said to be living because he has a vegetative soul, that he is called an animal because he has a sensitive soul, and that he is a man because he has an intellectual soul. Consequently, if there were three souls in man, namely, vegetative, sensitive, and rational, man would be placed in a genus because of one of his souls, and in a species because of another. But this is impossible. For thus genus and specific difference would constitute, not what is simply one, but what is one per accidens, or a sort of conglomeration, such as musical and white; but such is not a being that is simply one. Accordingly a man can have only one soul.