Curry–Howard Correspondence - Wikipedia - logic programming  ## Metadata - Author: **logic programming** - Full Title: Curry–Howard Correspondence - Wikipedia - Category: #articles - URL: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence#:~:text=The%20expression%20Curry%E2%80%93Howard%E2%80%93Lambek,morphisms%20as%20terms%20or%20proofs ## Highlights - Joachim Lambek showed in the early 1970s that the proofs of intuitionistic propositional logic and the combinators of typed combinatory logic share a common equational theory which is the one of cartesian closed categories. The expression Curry–Howard–Lambek correspondence is now used by some people to refer to the three way isomorphism between intuitionistic logic, typed lambda calculus and cartesian closed categories, with objects being interpreted as types or propositions and morphisms as terms or proofs. The correspondence works at the equational level and is not the expression of a syntactic identity of structures as it is the case for each of Curry's and Howard's correspondences: i.e. the structure of a well-defined morphism in a cartesian-closed category is not comparable to the structure of a proof of the corresponding judgment in either Hilbert-style logic or natural deduction. To clarify this distinction, the underlying syntactic structure of cartesian closed categories is rephrased below. - In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs.