Dear Bob,I do not consider myself a real master of the subject, still I would like make some remarks.
Robert Boyer wrote:
Why you think that first order logic is dubious (PRA is a first order theory, isn't it?).possibly inconsistent set theory, or even on a possibly dubious first order logic, whose semantics are generally based on set theory.
At least we know that first order logic is consistent.It is true that the proof of consistency may be based on set theory (just consider formulae valid in one element model) but one can do without set theory. Just add to the theory a constant X and an axiom
for x holds x = X
then eliminate quantifiers and the whole problem is reduced to the
propositional calculus.
Actually I do not know what 'semantics` means, if it is not based on set theory.
All the best, Andrzej