Philosophical distinctions; Trinitarian applicationNumerical identity concerns one and the same thing and remains transitive: if A = B and B = C, then A = C. Qualitative similarity concerns agreement in some feature. Two printed copies can contain the same words without being the same book token. One person can remain the same person while changing knowledge, clothing, or location. What we predicate or count needs a stated respect; the ordinary rules of identity do not change with that respect.
Contradiction requires incompatible claims about the same subject in the same respect. A physician can be knowledgeable about anatomy and ignorant about a patient's present condition without contradiction. Anatomy and this patient's condition are different objects of knowledge. The distinction identifies what the claims concern rather than changing the meaning of identity.
Consider the Trinity. Simplicity excludes constituents through which God's actuality is assembled or received. Catholic faith confesses three really distinct Persons, each wholly the one undivided God. Real distinction would contradict simplicity if it required constituents that together make God actual. The classical claim is that opposed relations of origin distinguish the Persons without performing that constituting work (CCC 253–255).
A relation can constitute who a Person is without being a component from which divine being is made. Paternity identifies the Father; filiation identifies the Son. Neither is an added property completing a prior divine subject. On the account of subsisting relations, their real opposition distinguishes personal subsistence within the same undivided divine actuality. If that account holds, personal distinction introduces no assembly of deity. Calling each Person wholly God must preserve this undivided deity without turning the statements into ordinary identity equations that identify the Father with the Son.