Quantum logicians* claim that there are cases where the behavior of true statements about the properties of subatomic particles fails to conform to the distribution of conjunction over disjunction in classical logic (i.e. the rule that lets us go from P & (Q v R) to (P & Q) v (P & R)). Now, for there to be a counter-example to this law, we'd need a case where P was true, (Q v R) was true, but both Q and R had some status other than "true."**
After all (holding the truth of P constant in all of these cases), if Q was true and R wasn't, then the premise of the relevant instance of Distribution would be true, but so would the conclusion. The same would be true if R was true and Q wasn't. And, of course, it would still be true if P and Q were both true. If, on the other hand, Q and R were both false, then once again, we wouldn't have a counter-example to Distribution, because the premise would be false.
Fair enough, you might think, but that just shows you that the old bivalent conception of truth is wrong, and that's exactly the sort of thing we should expect to be shown once we've really absorbed the quantum revolution, really exposed the ancient dogmas encoded in classical logic to the searing light of empirical revision.
OK. Maybe. But postulating a third truth-value, by itself, doesn't clarify much here. What third truth-value would get the job done? A natural first thought is that what we're talking about here is a truth-value gaps--i.e. the joint absence of the two classical values--but that's not going to get it done. If Q is neither true nor false, and R is neither true nor false, then why should (Q v R) be true rather than neither true nor false itself? Now, if we think of third truth-value not as a gap but as a glut--the joint presence of the two classical values--the situation might seem to be a little bit better. Now, after all, the premise of the instances of Distribution where Q and R are both oddly-valued comes out true (whatever else it might be). The problem, of course, is that the conclusion also comes out true.
One might say that the third value is not a matter of being definitely neither or definitely both but being in some sense vague or ambiguous or indeterminate between the two. Fine. But why, then, wouldn't both the premise and the conclusion come out as vague or ambiguous or indeterminate or whatever? If it's ambiguous whether or not Q is true, and ambiguous whether or not R is true, but P is unambiguously true, shouldn't it be ambiguous whether (Q v R) is true, and also ambiguous whether (P & Q) is true, whether (P & R) is true, and whether ((P & Q) v (P & R)) is true?
One might set up the truth tables differently here, but its hard to see how one could do so, in a principled way and without opening oneself up to some "change of meaning" charges. Of course, some people routinely level those charges against all heterodox proposals about the behavior of logical connectives, but it would be much harder to answer them here. To see why, think of it likes this:
In classical logic, "either P or Q is true" and "at least one of the following things is true: P, Q" are different ways of saying the same thing. Now, if a heterodox logician comes along and says "sometimes it's ambiguous whether P is true, and it's equally ambiguous whether Q is true," and then concludes that in those cases it's ambiguous whether at least of the two is true, then the "change of meaning" charge seems unfair. It seems more natural to say that they mean the same thing by "or" as the classical logician, but that they admit possibilities that the classical logician rejects. If, by contrast, they say that it's ambiguous whether P is true, and ambiguous whether Q is true (and not epistemically ambiguous, but in terms of its objective truth-status), but that "either P or Q" is unambiguously true, it really does start to seem like they're using "or" in a new way.
...or maybe not. A more radical move yet would be to simply reject truth-functionality entirely here. Just as "for any collection of numbers, there is a sum of those numbers" is true (and might even seem so obvious as to follow from the meaning of "number" or "collection") so long as we restrict our focus to finite (and countably infinite) collections of numbers, but it breaks down when we get to uncountably infinite collections (like the collection of all real numbers), to which addition simply doesn't apply, one could argue that "the truth-value of disjunctions is a function of the truth-values of their disjuncts" holds when we restrict our attention to normal situations, but that it breaks down when we turn our attention to the outer edges of logical possibility that are physically actualized by quantum weirdness.
OK, fair enough, but if they do choose to take that line, it's surely incumbent on the quantum logician to give us a clear account of exactly what exactly the distinction is between normal and non-normal situations. If the distinction is simply a matter of shifting truth-values, then this isn't a proposal about a break-down of truth-functionality, it's simply about non-standard truth-functionality, and given the failure of the third truth-value to transfer from the disjuncts to the disjunction, the change of meaning question looms large. If the distinction between normal and non-normal situation is about something other than truth-values--e.g. we have a situation where Q and R are both false but where (Q v R) somehow manages to be true, or where P, Q and R are all true but (P & Q) v (P & R) somehow fails to be true--then they really owe us a very clear explanation of how the inclusive "or" of formal logic can retain its customary meaning at the same time as two false disjuncts somehow jointly yield a true disjunction, or two true disjuncts can fail to yield a true disjunction, and exactly what the difference is between situations where logical connectives behave in this strange way and the situations in which they don't, and exactly how to distinguish between which situations are which.
Now, from an orthodox perspective, it's tempting to conclude from the whole mess that the proposal that Distribution fails in quantum contexts is just deeply confused, and that might even be the right answer here, but I'd be far more interested in hearing attempts to resolve it and explain just how the trick can be turned--e.g. exactly how we can conceptualize a third truth-value that would plausibly behave in the right way, or how to make sense of the idea that the standard truth-values would in the relevant situations stop combining in the standard ways.
*For our purposes here, the phrase "quantum logicians" refers to full-on, 1970s-Putnam-style, realist, monist quantum logicians, not the namby-pamby kind that just take quantum logic to be an interesting mathematical representation of certain experimental results and leave it at that. The latter might be far easier to plausibly argue for, but it's also far more boring.
**Note that, for the sake of simplicity, in everything that follows I'm assuming that conjunction behaves in the standard way. If anyone wants to get into that in the comments, and provide a quantum-logical motivation for questioning that, that's fine too.