Interesting thread so far. I haven't much to add yet, but I read this article (originally in New Scientist) recently which some here might find of interest:
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CHRIS ISHAM has a problem with truth. And he suspects his fellow physicists do too. It is not their honesty he doubts, but their approach to understanding the nature of the universe, the laws that govern it and reality itself. Together with a small band of allies, Isham is wrestling with questions that lie at the very core of physics. Indeed they run even deeper, to such basic concepts as logic, existence and truth. What do they mean? Are they immutable? What lies beyond them?
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But there is a price to pay, and it is precisely what the Kochen-Specker theorem warned of: the demise of simple truth and falsity. For all its drawbacks, Boolean algebra does at least allow every statement about our universe to be either true or false. Yet this turns out to be the exception among all the different types of algebras -- including the one underpinning quantum theory. The logic associated with quantum topoi encompasses true, false and many shades of grey in between.
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Source:
Impossible Things for Breakfast