> It's hard to see what the obstacle to mechanisation could be but logical complexity.
There is no impenetrable obstacle if humans could do it (for some definition of "could"), but the difficulty is handling infinite state spaces. Model checkers can do it in some situations, but the could do it in more. Neither model checkers nor humans can handle infinite state spaces -- or even very large ones -- in all situations (halting problem and its generalizations).
> Model theory is not some magical deduction-free paradise.
I just pointed out that some of the most successful automated methods, like model checkers (hence their name, BTW) and SAT solvers (which are really special kinds of model checkers), work in the model, not the proof theory.
> The reason we care about model theory in logic is because we want to have evidence that the deductive system at hand is consistent.
The reason we care about models is that we want to know that the "symbols game" of formal logic corresponds to mathematical objects; i.e. that it is sound -- a stronger property than consistency. The model existed first. Proof theory in deductive systems -- and formal logic in general as we know it -- is a 20th-century invention (well, late 19th but only accepted by mathematicians in the 20th).
There is no impenetrable obstacle if humans could do it (for some definition of "could"), but the difficulty is handling infinite state spaces. Model checkers can do it in some situations, but the could do it in more. Neither model checkers nor humans can handle infinite state spaces -- or even very large ones -- in all situations (halting problem and its generalizations).
> Model theory is not some magical deduction-free paradise.
I just pointed out that some of the most successful automated methods, like model checkers (hence their name, BTW) and SAT solvers (which are really special kinds of model checkers), work in the model, not the proof theory.
> The reason we care about model theory in logic is because we want to have evidence that the deductive system at hand is consistent.
The reason we care about models is that we want to know that the "symbols game" of formal logic corresponds to mathematical objects; i.e. that it is sound -- a stronger property than consistency. The model existed first. Proof theory in deductive systems -- and formal logic in general as we know it -- is a 20th-century invention (well, late 19th but only accepted by mathematicians in the 20th).