Between math and software development, I feel I've encountered this idea at least three times (+ rediscovered it independently), under different names - "3-value logic", "3-state booleans" - without ever hearing about "intuitionism". Thanks for mentioning it, and now I wonder how it all relates (i.e. I'm about to get stuck on Wikipedia).
> interpreting "A is true" to mean "I have a proof of A", and likewise "not A is true" means "I have a proof of not A".
Makes me think of an interesting progression I've also seen when studying maths and then (classical) AI algorithms. You start with boolean logic, then figure out you need to represent lack of knowledge/certainty and get 3-state logic. Then someone notices that there are degrees to being sure, and you get fuzzy logic[0], representing logical values as real numbers. And then someone notices that this smells awful lot like probability theory, and you get Bayesian probability as applied to reasoning, with all the extra concepts like using whole probability distributions as first-class logic values.
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[0] - I'm not sure how much mathematical rigour there is to it; it's something I've learned when studying AI for videogames.