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. Logics that deny one or more of the structural rules are classified as substructural logics.

Common structural rules

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.

These are not the only possible structural rules. A famous structural rule is known as cut.[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. The successful 'removal' of cut rules, known as cut elimination, is directly related to the philosophy of computation as normalization (see Curry–Howard correspondence); it often gives a good indication of the complexity of deciding a given logic.

See also


  1. ^ a b Gentzen, Gerhard (1935). "Untersuchungen über das logische Schließen. I, Mathematische Zeitschrift". Mathematische Zeitschrift (in German). 39 (1): 176–210. doi:10.1007/BF01201353. ISSN 0025-5874.
  2. ^ Szabo, M. E. (1969). Collected papers of Gerhard Gentzen. Place of publication not identified: Elsevier. ISBN 978-0-444-53419-4.
  3. ^ Jacobs, Bart (1994). "Semantics of weakening and contraction". Annals of Pure and Applied Logic. 69 (1): 73–106. doi:10.1016/0168-0072(94)90020-5.