Editing Sequent calculus (section)

===Structural rules===

The structural rules deserve some additional discussion.

Weakening (W) allows the addition of arbitrary elements to a sequence.  Intuitively, this is allowed in the antecedent because we can always restrict the scope of our proof (if all cars have wheels, then it's safe to say that all black cars have wheels); and in the succedent because we can always allow for alternative conclusions (if all cars have wheels, then it's safe to say that all cars have either wheels or wings).

Contraction (C) and Permutation (P) assure that neither the order (P) nor the multiplicity of occurrences (C) of elements of the sequences matters. Thus, one could instead of [[sequence]]s also consider [[Set (mathematics)|sets]].

The extra effort of using sequences, however, is justified since part or all of the structural rules may be omitted. Doing so, one obtains the so-called [[substructural logic]]s.