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structural rule wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 common structural rules 2 see also 3 references toggle the table of contents structural rule 2 languages português 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia rule of mathematical logic for the use in linguistics see phrase structure rules 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 edit 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 γ σ γ a σ displaystyle frac gamma vdash sigma gamma a vdash sigma on the left of the turnstile and γ σ γ σ a 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 γ a a σ γ a σ displaystyle frac gamma a a vdash sigma gamma a vdash sigma and γ a a σ γ a σ 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 γ 1 a γ 2 b γ 3 σ γ 1 b γ 2 a γ 3 σ displaystyle frac gamma _ 1 a gamma _ 2 b gamma _ 3 vdash sigma gamma _ 1 b gamma _ 2 a gamma _ 3 vdash sigma and γ σ 1 a σ 2 b σ 3 γ σ 1 b σ 2 a σ 3 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 edit affine logic resource sensitive logic allowing each assumption to be used at most once linear logic system of resource aware logic ordered logic linear logic extension of linear logic pages displaying short descriptions of redirect targets relevance logic kind of non classical logic separation logic concept in computer science references edit 1 2 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 szabo m e 1969 collected papers of gerhard gentzen place of publication not identified elsevier isbn 978 0 444 53419 4 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 v t e non classical logic intuitionistic intuitionistic logic constructive analysis heyting arithmetic intuitionistic type theory constructive set theory fuzzy degree of truth fuzzy rule fuzzy set fuzzy finite element fuzzy set operations substructural structural rule relevance logic linear logic paraconsistent dialetheism description ontology information science ontology language many valued three valued four valued łukasiewicz digital logic three state logic tri state buffer four valued verilog ieee 1164 vhdl others deviant logic dynamic semantics inquisitive logic intermediate logic non monotonic logic retrieved from https en wikipedia org w index php title structural_rule oldid 1343095628 categories proof theory rules of inference hidden categories articles with short description short description is different from wikidata cs1 german language sources de pages displaying short descriptions of redirect targets via module annotated link this page was last edited on 12 march 2026 at 06 14 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents structural rule 2 languages add topic