In a
sequent calculus, one writes each line of a proof as :\Gamma\vdash\Sigma. Here the structural rules are rules for
rewriting the
LHS of the sequent, denoted Γ, initially conceived of as a finite
string (sequence) of propositions. The standard interpretation of this string is as
conjunction: we expect to read :\mathcal A,\mathcal B \vdash\mathcal C as the sequent notation for :(
A and B)
implies C. 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 \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 :\mathcal B,\mathcal A\vdash\mathcal C from :\mathcal A,\mathcal B\vdash\mathcal C. There are further structural rules corresponding to the
idempotent and
monotonic properties of conjunction: from : \Gamma,\mathcal A,\mathcal A,\Delta\vdash\mathcal C we can deduce : \Gamma,\mathcal A,\Delta\vdash\mathcal C. Also from : \Gamma,\mathcal A,\Delta\vdash\mathcal C one can deduce, for any
B, : \Gamma,\mathcal A,\mathcal B,\Delta\vdash\mathcal C.
Linear logic, in which duplicated hypotheses 'count' differently from single occurrences, leaves out both of these rules, while
relevant (or relevance) logics 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). == Premise composition ==