The Formulae-as-Types Notion of Construction.¶
Howard, W. A. (1969/1980). The Formulae-as-Types Notion of Construction. To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Formal System
- Type theory re-exported formal-system structure back to philosophy through the Curry–Howard correspondence linking proofs and programs.
This sourceThe Curry–Howard correspondence linking proofs and programs, re-exporting formal-system structure between logic and computation.
- Type theory re-exported formal-system structure back to philosophy through the Curry–Howard correspondence linking proofs and programs.
- Isomorphism
- Howard's 1969 note (originally a manuscript circulated in 1969 and formally published in 1980) is the canonical statement of the formula-as-types correspondence, building on Curry's earlier observation and Lambek's categorical extension.
This source(Canonical statement of the propositions-as-types and proofs-as-programs correspondence, building on Curry's earlier observation and subsequently extended by Lambek to a categorical correspondence with Cartesian closed categories; the correspondence underwrites the design of dependently-typed programming languages and proof assistants.)
- Howard's 1969 note (originally a manuscript circulated in 1969 and formally published in 1980) is the canonical statement of the formula-as-types correspondence, building on Curry's earlier observation and Lambek's categorical extension.
Domain-specific¶
- Conjunction Introduction
- Type System
- The Curry-Howard correspondence reveals that this hierarchy has a unified logical reading: a type is a proposition, a well-typed program is a proof of that proposition, type checking is proof verification, and the richness of the type language is exactly the expressiveness of the logic it encodes
This sourceThe original source of the formulae-as-types correspondence — propositions as types, proofs as terms — for intuitionistic logic; the reading of type checking as machine proof verification, and the calibration of type-language richness against logical strength, are later developments.
Supported in partVerified against the source
- The Curry-Howard correspondence reveals that this hierarchy has a unified logical reading: a type is a proposition, a well-typed program is a proof of that proposition, type checking is proof verification, and the richness of the type language is exactly the expressiveness of the logic it encodes
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