Editing Proof-theoretic semantics (section)

==Overview==

[[Gerhard Gentzen]] is the founder of proof-theoretic semantics, providing the formal basis for it in his account of [[cut-elimination]] for the [[sequent calculus]], and some provocative philosophical remarks about locating the meaning of logical connectives in their [[Natural_deduction#Introduction_and_elimination|introduction rule]]s within [[natural deduction]]. The history of proof-theoretic semantics since then has been devoted to exploring the consequences of these ideas.{{Citation needed|date=December 2014}}

[[Dag Prawitz]] extended Gentzen's notion of [[analytic proof]] to [[natural deduction]], and suggested that the value of a proof in natural deduction may be understood as its normal form.{{Citation needed|date=March 2019}}  This idea lies at the basis of the [[Curry&ndash;Howard isomorphism]], and of [[intuitionistic type theory]].  His [[inversion principle]] lies at the heart of most modern accounts of proof-theoretic semantics.

[[Michael Dummett]] introduced the very fundamental idea of [[logical harmony]], building on a suggestion of [[Nuel Belnap]].  In brief, a [[formal language|language]], which is understood to be associated with certain patterns of inference, has logical harmony if it is always possible to recover analytic proofs from arbitrary demonstrations, as can be shown for the sequent calculus by means of cut-elimination theorems and for natural deduction by means of normalisation theorems.  A language that lacks logical harmony will suffer from the existence of incoherent forms of inference: it will likely be inconsistent.