In his textbook Mathematical Logic, Shoenfield states the Hilbert-Ackermann Consistency Theorem as follows:
"Consistency Theorem (Hilbert-Ackermann): An open theory $T$ is inconsistent iff there is a quasi-tautology [i.e. a tautological consequence of instances of identity axioms and equality axioms--this from the first sentence of the paragraph (my comment)] which is a disjunction of negations of instances of nonlogical axioms of $T$."
He defines the term "open theory" as follows:
"[Pg. 48] A theory is open if all of its nonlogical axioms are open."
"[Pg. 36] A formula is open if it contains no quantifiers."
Since the usual formulation of $PRA$ (Primitive Recursive Arithmetic) is quantifier-free, it is, from a naive point of view, an 'open theory'.
Question: Does the Hilbert-Ackermann Consistency theorem hold for $PRA$? If so, can one (from the literature or otherwise) produce a theorem of the consistency of $PRA$ using the Hilbert-Ackermann Consistency Theorem? If not, could someone please explain to me why the theorem does not apply to $PRA$?