## Propositions as Types

Propositions as Types by Philip Wadler.

From the Introduction::

Powerful insights arise from linking two fields of study previously thought separate. Examples include Descartes’s coordinates, which links geometry to algebra, Planck’s Quantum Theory, which links particles to waves, and Shannon’s Information Theory, which links thermodynamics to communication. Such a synthesis is offered by the principle of Propositions as Types, which links logic to computation. At first sight it appears to be a simple coincidence—almost a pun—but it turns out to be remarkably robust, inspiring the design of automated proof assistants and programming languages, and continuing to influence the forefronts of computing.

Propositions as Types is a notion with many names and many origins. It is closely related to the BHK Interpretation, a view of logic developed by the intuitionists Brouwer, Heyting, and Kolmogorov in the 1930s. It is often referred to as the Curry-Howard Isomorphism, referring to a correspondence observed by Curry in 1934 and refined by Howard in 1969 (though not published until 1980, in a Festschrift dedicated to Curry). Others draw attention to significant contributions from de Bruijn’s Automath and Martin-Löf’s Type Theory in the 1970s. Many variant names appear in the literature, including Formulae as Types, Curry-Howard-de Bruijn Correspondence, Brouwer’s Dictum, and others.

Propositions as Types is a notion with depth. It describes a correspondence between a given logic and a given programming language, for instance, between Gentzen’s intuitionistic natural deduction (a logic) and Church’s simply-typed lambda calculus (which may be viewed as a programming language). At the surface, it says that for each proposition in the logic there is a corresponding type in the programming language—and vice versa…

Important work even if it is very heavy sledding!

BTW, Wadler mentions two textbook treatments of the subject:

M. H. Sørensen and P. Urzyczyn. Lectures on the Curry-Howard isomorphism. Elsevier, 2006. Amazon has it listed for \$146.33.

S. Thompson. Type Theory and Functional Programming. Addison-Wesley, 1991. Better luck here, out of print and posted online by the author: https://www.cs.kent.ac.uk/people/staff/sjt/TTFP/. Errata page was last updated October 2013.

I just glanced at 4.10 Equality and 5.1 Assumptions – 5.2 Naming and abbreviations in Thompson and it promises to be an interesting read!

Enjoy!

I first saw this in a tweet by Chris Ford.