From Collapse Theorems to Proof-Theoretic Arguments

On some views, we can be sure that parties to a dispute over the logic of `exists' are not talking past each other if they can characterise `exists' as the only monadic predicate up to logical equivalence obeying a certain set of rules of inference. Otherwise, we ought to be suspicious about the reality of their disagreement. This is what we call a proof-theoretic argument. Pace some critics, who have tried to use proof-theoretic arguments to cast doubts about the reality of disagreements about the logic of `exists', we argue that proof-theoretic arguments can be deployed to establish the reality of several such disagreements. Along the way, we will also utilise this technique to establish similar results about some disagreements over the logic of identity.