It is not very fair to make a general criticism about the weakness of logic based on propositional logic, since that is the weakest of the various families of l

It is not very fair to make a general criticism about the weakness of logic based on propositional logic, since that is the weakest of the various families of logic - the other two main kinds of logic being first-order logic (FOL) and term logic. First-order logic is powerful enough that it has become the yardstick of adequacy for mathematical proofs: a mathematical proof about a mathematical domain is only adequate if (i) the mathematical domain accepts a first-order axiomatization, and (ii) we believe the proof can be rendered as a proof in FOL. So the broader point is that we have a lot of freedom in choosing axiomatisations. But in fact we do not, at least in mathematics: there is one accepted axiomatisation (up to equivalence of inessential variations) of Peano arithmetic, one of ring theory, one of vector spaces, one of C* algebras, etc.