Eigenvariables, bracketing and the decidability of positive minimal predicate logic

We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic is decidable. The idea of the proof is to replace the eigenvariable condition of sequent calculus by an appropriate scoping mechanism. The algorithm given by this proof seems to be more practical than that given by the original proof. A naive implementation is given at the end of the paper. Another contribution is to show that this result extends to a large class of theories, including simple type theory (higher-order logic) and second-order propositional logic. We obtain this way a new proof of the decidability of the inhabitation problem for positive types in system F.