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POWhat are the most interesting equivalences arising from the Curry-Howard Isomorphism?
 

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  1. POWhat are the most interesting equivalences arising from the Curry-Howard Isomorphism?
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    <p>I came upon the <a href="http://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence" rel="noreferrer">Curry-Howard Isomorphism</a> relatively late in my programming life, and perhaps this contributes to my being utterly fascinated by it. It implies that for every programming concept there exists a precise analogue in formal logic, and vice versa. Here's a "basic" list of such analogies, off the top of my head:</p> <pre><code>program/definition | proof type/declaration | proposition inhabited type | theorem/lemma function | implication function argument | hypothesis/antecedent function result | conclusion/consequent function application | modus ponens recursion | induction identity function | tautology non-terminating function | absurdity/contradiction tuple | conjunction (and) disjoint union | disjunction (or) -- corrected by Antal S-Z parametric polymorphism | universal quantification </code></pre> <p>So, to my question: <strong>what are some of the more interesting/obscure implications of this isomorphism?</strong> I'm no logician so I'm sure I've only scratched the surface with this list.</p> <p>For example, here are some programming notions for which I'm unaware of pithy names in logic:</p> <pre><code>currying | "((a &amp; b) =&gt; c) iff (a =&gt; (b =&gt; c))" scope | "known theory + hypotheses" </code></pre> <p>And here are some logical concepts which I haven't quite pinned down in programming terms:</p> <pre><code>primitive type? | axiom set of valid programs? | theory </code></pre> <p>Edit:</p> <p>Here are some more equivalences collected from the responses:</p> <pre><code>function composition | syllogism -- from Apocalisp continuation-passing | double negation -- from camccann </code></pre>
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    1. COHow about "goto" ? =)
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    2. COcare to elaborate?
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    3. COclosure ~= set of axioms
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