constructive proof

For programs to make sense as proofs, they need to be terminating (cannot run forever), since otherwise you can write infinite loops that have any type.  Under Curry-Howard this means any formula is provable, which is one way to define inconsistency.  (And logics have to be consistent to be useful.)

If you have dependent types, classical reasoning, and the Curry-Howard isomorphism, you can write programs that look like they are invoking oracles for undecidable problems -- but they are not, and this is confusing.

CH can be applied to classical logic, too.  The seminal paper is <a href="https://www.cl.cam.ac.uk/~tgg22/publications/popl90.pdf">A Formulae-as-Types Notion of Control</a> by Timothy Griffin.  I discuss how backtracking implements the law of excluded middle.

I follow up on some comments I made about Curry-Howard for first-order quantifiers in the previous episode.  Sheard's Omega language also mentioned (see links on <a href = "http://web.cecs.pdx.edu/~sheard/">his web page</a>).  First-order quantifications turn into indexed types where the indices are not program expressions but come from another syntactic domain.

Discussion of the Curry-Howard isomorphism for the connectives of propositional logic (AND, OR, NOT, FALSE, IMPLIES).  Initial consideration of Curry-Howard for first-order and higher-order logic.  Dependent types.

In which I discover why the domino analogy for explaining induction always bothered me!

We consider the basic idea of the Curry-Howard isomorphism, that constructive proofs are essentially programs, and vice versa.  Several simple examples.  Why the law of excluded middle is not a legal constructive proof.

The basic idea of the Curry-Howard isomorphism, and its connection to the contents of Chapters 1 and 2.  Constructive proof.  A famous nonconstructive proof.