Conceiving of premises as collected into sets or multisets, instead of sequences, may lead to triviality for classical and intuitionistic logic in general proof theory, where we investigate identity of deductions. Any two deductions with the same premises and the same conclusions become equal. In terms of categorial proof theory, this is a consequence of a simple fact concerning adjunction with...

We present a sequent calculus for intuitionistic non-commutative linear logic (INCLL), show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cut-free derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for...