Structural rule

For the type of rule used in linguistics, see phrase structure 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. ^{[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]

• Weakening, where the hypotheses or conclusion of a sequence may be extended with additional members. In symbolic form weakening rules can be written as ${\displaystyle {\frac {\Gamma \vdash \Sigma }{\Gamma ,A\vdash \Sigma }}}$ on the left of the turnstile, and ${\displaystyle {\frac {\Gamma \vdash \Sigma }{\Gamma \vdash \Sigma ,A}}}$ on the right. Known as monotonicity of entailment in classical logic.
• Contraction, where two equal (or unifiable) members on the same side of a sequent may be replaced by a single member (or common instance). Symbolically: ${\displaystyle {\frac {\Gamma ,A,A\vdash \Sigma }{\Gamma ,A\vdash \Sigma }}}$ and ${\displaystyle {\frac {\Gamma \vdash A,A,\Sigma }{\Gamma \vdash A,\Sigma }}}$. 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: ${\displaystyle {\frac {\Gamma _{1},A,\Gamma _{2},B,\Gamma _{3}\vdash \Sigma }{\Gamma _{1},B,\Gamma _{2},A,\Gamma _{3}\vdash \Sigma }}}$ and ${\displaystyle {\frac {\Gamma \vdash \Sigma _{1},A,\Sigma _{2},B,\Sigma _{3}}{\Gamma \vdash \Sigma _{1},B,\Sigma _{2},A,\Sigma _{3}}}}$. (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 . ^[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

• Affine logic  – substructural logic whose proof theory rejects the structural rule of contractionPages displaying wikidata descriptions as a fallback
• Linear logic  – System of resource-aware logic
• Ordered logic (linear logic)
• Relevance logic  – mathematical logic system that imposes certain restrictions on implicationPages displaying wikidata descriptions as a fallback
• Separation logic

References

1. 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.