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Inlogic, asubstructural logic is a logic lacking one of the usualstructural rules (e.g. ofclassical andintuitionistic logic), such asweakening,contraction, exchange or associativity. Two of the more significant substructural logics arerelevance logic andlinear logic.

In asequent calculus, one writes each line of a proof as

${\displaystyle \mathrm{\Gamma }⊢\mathrm{\Sigma }}$.

Here the structural rules are rules forrewriting theLHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is asconjunction: we expect to read

${\displaystyle \mathcal{A},\mathcal{B}⊢\mathcal{C}}$

as the sequent notation for

(AandB)impliesC.

Here we are taking theRHS Σ to be a single propositionC (which is theintuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of theturnstile symbol${\displaystyle ⊢}$.

Since conjunction is acommutative andassociative operation, the formal setting-up of sequent theory normally includesstructural rules for rewriting the sequent Γ accordingly—for example for deducing

${\displaystyle \mathcal{B},\mathcal{A}⊢\mathcal{C}}$

from

${\displaystyle \mathcal{A},\mathcal{B}⊢\mathcal{C}}$.

There are further structural rules corresponding to theidempotent andmonotonic properties of conjunction: from

${\displaystyle \mathrm{\Gamma },\mathcal{A},\mathcal{A},\mathrm{\Delta }⊢\mathcal{C}}$

we can deduce

${\displaystyle \mathrm{\Gamma },\mathcal{A},\mathrm{\Delta }⊢\mathcal{C}}$.

Also from

${\displaystyle \mathrm{\Gamma },\mathcal{A},\mathrm{\Delta }⊢\mathcal{C}}$

one can deduce, for anyB,

${\displaystyle \mathrm{\Gamma },\mathcal{A},\mathcal{B},\mathrm{\Delta }⊢\mathcal{C}}$.

Linear logic, in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, whilerelevant (or relevance) logics merely leaves out the latter rule, on the ground thatB is clearly irrelevant to the conclusion.

The above are basic examples of structural rules. It is not that these rules are contentious, when applied in conventional propositional calculus. They occur naturally in proof theory, and were first noticed there (before receiving a name).

There are numerous ways to compose premises (and in the multiple-conclusion case, conclusions as well). One way is to collect them into a set. But since e.g. {a,a} = {a} we have contraction for free if premises are sets. We also have associativity and permutation (or commutativity) for free as well, among other properties. In substructural logics, typically premises are not composed into sets, but rather they are composed into more fine-grained structures, such as trees or multisets (sets that distinguish multiple occurrences of elements) or sequences of formulae. For example, in linear logic, since contraction fails, the premises must be composed in something at least as fine-grained as multisets.

Substructural logics are a relatively young field. The first conference on the topic was held in October 1990 in Tübingen, as "Logics with Restricted Structural Rules". During the conference, Kosta Došen proposed the term "substructural logics", which is now in use today.

• Substructural type system
• Residuated lattice

• F. Paoli (2002),Substructural Logics: A Primer, Kluwer.
• G. Restall (2000)An Introduction to Substructural Logics, Routledge.

• Galatos, Nikolaos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono (2007),Residuated Lattices. An Algebraic Glimpse at Substructural Logics, Elsevier,ISBN 978-0-444-52141-5.

• Media related toSubstructural logic at Wikimedia Commons
• Restall, Greg."Substructural logics". InZalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy.

• Substructural logic
• Non-classical logic

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