But what about:
P1. All men are mortal.
C. Everything is such that (it is a man ⊃ it is mortal)?
This is a logical inference. It is not of the form 'A therefore A'. It embodies a very clever logical discovery! P1 and C are not the same statement. Talk of 'translating' the former by means of the latter papers over all this.
Modern quantificational logic does not really capture the inferences captured by traditional logic, any more than it captures this link between the two. It does capture inferences which, given logical insight, can be seen to parallel those codified by traditional logic, but that is not the same thing.