 One can raise a number of questions about whether the modern textbook procedure does really capture a sensible notion of logical consequence. For example in Boole’s case the set-theoretic consequences that he relies on are all easily provable by formal proofs in first-order logic, not even using any set-theoretic axioms; and by the completeness theorem (see the entry on  classical logic)  the same is true for first-order logic. But for some other logics it is certainly not true. For instance the model-theoretic consequence relation for some logics of time presupposes some facts about the physical structure of time. Also, as Boole himself pointed out, his translation from an English argument to its set-theoretic form requires us to believe that for every property used in the argument, there is a corresponding class of all the things that have the property. This comes dangerously close to Frege’s inconsistent comprehension axiom!