Structural rule

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. 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:

, where the hypotheses or conclusion of a sequence may be extended with additional members. In symbolic form weakening rules can be written as <math>\frac{\Gamma \vdash \Sigma}{\Gamma, A \vdash \Sigma}</math> on the left of the turnstile, and <math>\frac{\Gamma \vdash \Sigma}{\Gamma \vdash \Sigma, A}</math> on the right. Known as monotonicity of entailment in classical logic.

, where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). Symbolically: <math>\frac{\Gamma, A, A \vdash \Sigma}{\Gamma, A \vdash \Sigma}</math> and <math>\frac{\Gamma \vdash A, A, \Sigma}{\Gamma \vdash A, \Sigma}</math>. Also known as factoring in automated theorem proving systems using resolution. Known as idempotency of entailment in classical logic.
Exchange, where two members on the same side of a sequent may be swapped. Symbolically: <math>\frac{\Gamma_1, A, \Gamma_2, B, \Gamma_3 \vdash \Sigma}{\Gamma_1, B, \Gamma_2, A, \Gamma_3 \vdash \Sigma}</math> and <math>\frac{\Gamma \vdash \Sigma_1, A, \Sigma_2, B, \Sigma_3}{\Gamma \vdash \Sigma_1, B, \Sigma_2, A, \Sigma_3}</math>. (This is also known as the permutation rule.)

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.