Quantum Logic, Part I of IV

So when I asked for reader requests last week, a couple of people asked for a post about quantum logic. The phrase "quantum logic" means a lot of different things these days, as indeed the word "logic" can mean a lot of different things. (See, for example, inductive "logic", computer-programming "logics" and so on.) So when many people talk about "quantum logic", they're talking about various formal or even mathematical constructions that model certain kinds of experimental results, the kind of thing for which no deeper philosophical justification is either offered nor required.

My interest in logic, however, veers towards what we can think of as 'logical metaphyics,' questions like, 'Do the inferences that we classically take to be universally truth-preserving really universally preserve truth? Are the claims we take to be logical truths really true?' (Hence my central research focus is in the semantic paradoxes, the question of whether there are any true contradictions, whether the Law of the Excluded Middle always holds and so on.) Because of that--and because I'm a Quineish confirmational holist--my interest in quantum logic is specifically on the question of whether the best explanation of the relevant physics might involve rejecting some of our current ideas about logical truth.

Historically, the most common proposal along these lines has been that, in response to quantum phenomena, we should reject Distribution, the principle that [P & (Q v R)] entails [(P & Q) v (P & R)]. Unless otherwise indicated, when I talk about "quantum logicians" or "the quantum logician", I'll be talking about the Distribution-rejecting quantum logician. In the new few posts, I'll argue that the prospects for their proposal are fairly bleak.

If we understand entailment in terms of truth-preservation (as I think we should), then, no matter how creative we get about adding in extra truth-values, there doesn't seem to be any plausible way to (a) get the result that the usual inferences about conjunction and disjunction that the quantum logician doesn't want to revise away remain valid, while (b) getting the result that Distribution is invalid. Without (a)--if, for example, it turns out that the truth of P isn't enough to guarantee the truth of (P v Q) in some truth-functional quantum logic--the 'change of meaning' charges often lobbed against quantum logic start seeming pretty hard to refute.

Of course, one could take all of this (in combination with whatever empirical case one thinks there is for quantum logic) as a good reason to reject truth-preservationism in favor of switching over to an inferentialist account of logical consequence, where primitive inference rules are taken to be "meaning-constituting" for logical connectives. Unfortunately, on closer inspection, things look even worse for the quantum logician here. It seems terribly implausible that experimental results about esoteric sub-atomic phenomena should show us that we were mistaken about the meaning of the terms "and" and "or."

At the level of description I'm giving here, it might seem like these are generic criticisms that would apply to *any* proposal to revise logic--"aren't classical logicians always accusing people with heterodox views of these sorts of things?"--but this isn't the case. Comparisons to other revisionary proposals will be instructive. After all, as we'll see, paracomplete theorists who reject instances of the Excluded Middle and Disjunctive Syllogism-rejecting dialetheists both pass the tests which (I argue) the Distribution-rejecting quantum logician fails.

Finally, though, I'll argue that even if the proposal that we reject Distribution to make sense of quantum phenomena isn't particularly plausible, that doesn't let classical orthodoxy 'off the hook.' Given the experiments that establish superposition and the rest, there are well-grounded worries that some sort of logically revisionary solution may be needed, even if rejecting Distribution doesn't fit the bill. I'll conclude with some tentative thoughts about that.

Meanwhile, though, I have to prep for teaching Philosophy of Art to some Koreans, so this post will have to remain nothing more than a preview for coming attractions. Stay tuned for Wednesday!