An intuitionistic logic that proves Markov’s principle
“An intuitionistic logic that proves Markov’s principle”, Hugo Herbelin, LICS 2010.
We design an intuitionistic predicate logic that supports a limited amount of classical reasoning, just enough to prove a variant of Markov’s principle suited for predicate logic.
At the computational level, the extraction of an existential witness out of a proof of its double negation is done by using a form of statically-bound exception mechanism, what can be seen as a direct-style variant of Friedman’s A-translation.
Markov’s principle is an axiom of “Russian constructivism”. It says that if P is a decidable predicate (i.e., the formula ∀x. P(x) or ¬P(x) is constructively provable), then if we know that P is not always false (ie ¬(∀x. ¬P(x))), then there exists an x such that P holds (ie ∃x. P(x)).
One operational way of understanding this axiom is that it says is that if we know P is not always false (on, say, the natural numbers), we can find an element for which it is true with a while-loop: check to see if 1 is true, and then check to see if 2 is true, and so on, stopping when you hit a number for which it holds. This means that Markov’s principle is a principle of unbounded search, and as a result it has controversial status in intuitionistic mathematics — should we regard a program with a non-constructive termination proof as a constructively terminating program? (The answer is, as usual, “it depends”: in different applications you can want it or not.)
However, just chucking an axiom like ¬(∀x. ¬P(x)) → ∃x. P(x) into your proof rules is not very nice from a proof-theoretic perspective. It mixes up a bunch of different logical connectives, and adding it as an axiom will break things like the cut-elimination theorem for sequent calculus.
In this paper, Herbelin exploits (a) the fact that Markov’s principle is a (mildly) classical principle of reasoning, and (b) the fact that classical logic is connected with control operators like continuations, to give a logic which is proof-theoretically well-behaved, but which supports Markov’s principle. The trick is to add first-class continuations, but only for value types (ie, you can throw products and sums, but not functions).
What I find surprising is that this same class of restriction also arises in another paper on control operators at LICS — Noam Zeilberger’s paper “Polarity and the Logic of Delimited Continuations” uses it too. (This paper deserves a post here too.)