Why isn’t the soul (ψυχή) considered to be a “person” (ὑπόστασις)? From my understanding, the unity of spirit, soul, and body is considered to be a ὑπόστασις, or “person.” But, what argument is there against the soul itself (apart from the body) being a ὑπόστασις? It seems the soul possesses reason. It exists by itself. So, why is it not considered to be a ὑπόστασις? Or, if it is considered to be a ὑπόστασις, where and by whom is such an argument affirmed? I am particularly interested in an answer in accordance with Catholic beliefs.

NOTE: To clarify, I am using «ὑπόστασις» as an equivalent to the Greek πρόσωπον and Latin persona as understood in Trinitarian creeds and theology.
Humans are living souls. Soul (psyche) means the breathing existence. The body (soma) is the outward appearance and the physique of the human. 
It all means the same human being. Soul more in the sense of living and ability to feel. Body more in the sense of physical existence.

The spirit, pneuma, is what moves the person, what his thoughts are and his emotions.

It all is the human as one whole living being.

The difference is regarding what of a person one is looking at. It is about aspects to the human, not about entities or compound parts. 

In certain instances the soul is considered the person, but in a more impersonal way to say. (Name, onoma, would be the more personal expression.)

Person (prosopon) is used for the human encounter, the facial and bodily appearance towards one another.