Substructural logic
In
mathematical logic, in particular in connection with
proof theory, a number of
substructural logics have been introduced, as systems of
propositional calculus that are weaker than the conventional one. They differ in having fewer
structural rules available: the concept of structural rule is based on the
sequent presentation, rather than the
natural deduction formulation. Two of the more significant substructural logics are
relevant logic and linear logic.
In discussing the sequent calculus, one writes each line of a proof as
- .
Here the structural rules are rules for
rewriting the
LHS Γ of the sequent, initially conceived of as a
string of propositions. The standard interpretation of this string is as
conjunction: we expect to read
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.
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
from
- .
There are further structural rules corresponding to the
idempotent and
monotonic properties of conjunction: from
we can deduce
- .
Also from
one can deduce, for any
B,
- .
The former of these rules is left out of systems of
linear logic, in which duplicated hypotheses 'count' differently from single occurrences; the latter is left out of
relevant (
relevance) logics, on the ground that
B is clearly irrelevant to the conclusion.
These 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).