In the logical discipline of proof theory, a structural rule is an inference rule of a sequent calculus that does not refer to any logical connective but instead operates on the sequents directly.
[1][2] Structural rules often mimic the intended meta-theoretic properties of the logic.
Three common structural rules are:[3] A logic without any of the above structural rules would interpret the sides of a sequent as pure sequences; with exchange, they can be considered to be multisets; and with both contraction and exchange they can be considered to be sets.
[1] Considerable effort is spent by proof theorists in showing that cut rules are superfluous in various logics.
More precisely, what is shown is that cut is only (in a sense) a tool for abbreviating proofs, and does not add to the theorems that can be proved.