Editing Substructural logic (section)

==Examples==
In a [[sequent calculus]], one writes each line of a proof as

:<math>\Gamma\vdash\Sigma</math>.

Here the structural rules are rules for [[rewriting]] the [[Sides of an equation|LHS]] of the sequent, denoted Γ, initially conceived of as a string (sequence) <!-- CS link for string was wrong --> of propositions. The standard interpretation of this string is as [[Logical conjunction|conjunction]]: we expect to read

:<math>\mathcal A,\mathcal B \vdash\mathcal C</math>

as the sequent notation for

:(''A'' '''and''' ''B'') '''implies''' ''C''.

Here we are taking the [[Sides of an equation|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)|turnstile symbol]] <math>\vdash</math>.

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

:<math>\mathcal B,\mathcal A\vdash\mathcal C</math>

from

:<math>\mathcal A,\mathcal B\vdash\mathcal C</math>.

There are further structural rules corresponding to the ''[[idempotent]]'' and ''[[Monotonicity of entailment|monotonic]]'' properties of conjunction: from

:<math> \Gamma,\mathcal A,\mathcal A,\Delta\vdash\mathcal C</math>

we can deduce

:<math> \Gamma,\mathcal A,\Delta\vdash\mathcal C</math>.

Also from

:<math> \Gamma,\mathcal A,\Delta\vdash\mathcal C</math>

one can deduce, for any ''B'',

:<math> \Gamma,\mathcal A,\mathcal B,\Delta\vdash\mathcal C</math>.

[[Linear logic]], in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, while [[relevant logic|relevant (or relevance) logic]]s merely leaves out the latter rule, on the ground that ''B'' 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).