Consider the following propositions:
"I am me" "I am my Father's son"
In both these cases, the predicate is the same as the subject by definition of the very subject and predicate.
Is there a special name for these kinds of propositions?
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In technical logic, a predicate is an entire statement. In your usage, with respect to grammar, the predicate is the verb and object (or other parts) which apply to the subject; the predicate is usually some relation about the subject. I think the latter concept of 'predicate' is what you are referring to.
In your statements, the subject is 'I' and the predicate is 'am me' or ' am my father's son'. The predicate relation, in both instances, is an equivalence relation ('is', equals'). The object of that particular predicate is proposed as equivalent to the subject. As such a proposition is then referred to as an equivalence.
The first one, because the pronouns refer to the same thing, is a tautology, because x=x is already an axiom of equivalence relations. The second is not, because you could be female.