In positive set theory, the axiom scheme of generalized positive comprehension in $GPK^+_\infty$ [of Olivier Esser] is stated in a manner as to forbid the symbol of the asserted set to occur in the defining formula.
To quote from the above source:
Positive Comprehension: For any positive formula $\phi$ in which $A$ does not appear, $(\exists A.(\forall x.x \in A \leftrightarrow \phi)).$
Where a positive formula is defined as:
Deﬁnition: Let the class of (bounded) positive formulas be the smallest class containing the formula $x \neq x$ (useful because uniformly false), all atomic formulas, and closed under conjunction, disjunction, bounded universal quantiﬁcation $(\forall x \in A.\phi)$ and existential quantiﬁcation $(∃x ∈ A.φ)$.
Question 1: Is there a clear inconsistency involved with permitting symbol $A$ to occur in $\phi$, in the statement of positive comprehension?
Question 2: Suppose that there is no clear inconsistency, then would that allowance results in increment in consistency strength over the original system?