Structures and Connectives
The term structure in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators which we call structural operators: in, the commas to the left of the turnstile are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication. However, it is important to note that there is a fundamental difference in behaviour between these operators and the logical connectives they are interpreted by in the sequent calculus: the structural operators are used in every rule of the calculus, and are not considered when asking whether the subformula property applies. Furthermore, the logical rules go one way only: logical structure is introduced by logical rules, and cannot be eliminated once created, while structural operators can be introduced and eliminated in the course of a derivation.
The idea of looking at the syntactic features of sequents as special, non-logical operators is not old, and was forced by innovations in proof theory: when the structural operators are as simple as in Getzen's original sequent calculus there is little need to analyse them, but proof calculi of deep inference such as display logic support structural operators as complex as the logical connectives, and demand sophisticated treatment.
Read more about this topic: Structural Proof Theory
Famous quotes containing the word structures:
“The American who has been confined, in his own country, to the sight of buildings designed after foreign models, is surprised on entering York Minster or St. Peter’s at Rome, by the feeling that these structures are imitations also,—faint copies of an invisible archetype.”
—Ralph Waldo Emerson (1803–1882)