Substructural logic

In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevant logic and linear logic.

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

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

as the sequent notation for

Here we are taking the RHS Σ to be a single proposition C (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol ${\displaystyle \vdash }$.

Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes structural rules for rewriting the sequent Γ accordingly—for example for deducing

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