Virtues of Presentation of FO Logic in Kleene's Mathematical Logic I refer to Stephen Cole Kleene, Mathematical Logic (1967 - Dover reprint : 2002).
What are the "pedagogical benefits" (if any) of the presentation chosen by Kleene, mixing Natural Deduction and Hilbert-style ?
Propositional Calculus - at pag.33 he refers back to formulas 1a-10b of Th.2 pag.15, which are usual Intro- and Elimination- rules for propositional connectives "rewritten" as axiom schemata.
Predicate Calculus - at pag.107 he add two axiom schemata : $\forall x A(x) \rightarrow A(r)$  (the A-schema) and : $A(r) \rightarrow \exists x A(x)$ (the E-schema), and the two rules - the A-rule : from $C \to A(x)$  to $C \to \forall x A(x)$ and the E-rule : from $A(x) \to C$  to $\exists x A(x) \to C$ , with x not free in C.
Then (Th.21 pag.118) he proves as derived rules the four standard rules of Intro- and Elim- for quantifiers.

@Peter Smith wrote: So it is worth noting that e.g. John Corcoran can write "Three Logical Theories" as late as 1969 (Philosophy of Science, Vol. 36, No. 2 (Jun., 1969), pp. 153-177), finding it still novel and necessary to stress the distinctions between different types of logical theory. Here it is https://www.academia.edu/9855795/Three_logical_theories 
