Structural proof theory

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

Analytic proof

Main article: analytic proof

The notion of analytic proof was introduced into proof theory by Gerhard Gentzen for the sequent calculus; 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—the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting.

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 called 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 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.

Cut-elimination in the sequent calculus

Main article: Cut-elimination

Natural deduction and the formulae-as-types correspondence

Main article: Natural deduction

Logical duality and harmony

Main article: Logical harmony

Calculus of structures

Main article: Calculus of structures

References

• Sara Negri; Jan Von Plato (2001). Structural proof theory. Cambridge University Press. ISBN 978-0-521-79307-0.
• Anne Sjerp Troelstra; Helmut Schwichtenberg (2000). Basic proof theory (2nd ed.). Cambridge University Press. ISBN 978-0-521-77911-1.