Structural proof theory

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In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.

The notion of analytic proof was introduced by Gerhard Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as was shown by Dag Prawitz; the definition is slightly more complex — we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting.

1 Structures and connectives
2 Cut-elimination in the sequent calculus
3 Natural deduction and the formulae-as-types correspondence
4 Logical duality and harmony
5 Display logic
6 Calculus of structures

[edit] 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 $A_1, \dots, A_m \vdash B_1, \dots, B_n$ , 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 are free to be made and unmade 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.