 
Summary: Review of Artemov's
Explicit provability and constructive semantics
Jeremy Avigad
June 5, 2002
The meaning of the intuitionistic connectives is often explained, informally, in
terms of the BrouwerHeytingKolmogorov interpretation: a proof of A B
consists of a proof of A paired with a proof of B, a proof of A B consists of
a procedure for transforming a proof of A into a proof of B, and so on. The
simplicity and intuitive appeal of this explanation suggests that there should
be a formal semantics lurking underneath. But, after surveying the existing
semantics for intuitionistic logic (including Kripke and Beth models, algebraic
and topological semantics, realizability, and various syntactic interpretations),
Artemov concludes that none of them fit the bill. He then undertakes the
challenge of providing one that does.
Artemov's solution involves interpreting intuitionistic propositional logic in
a logic LP of propositions and proofs. Consideration of similar interpretations
will provide some useful context. Many researchers in constructive logic take the
CurryHoward (or "propositions as types") isomorphism, formally represented
by deductive type theories like MartinL¨of's, to offer a formal explication of the
BHK interpretation. But Artemov objects that this does not go far enough:
