Quantum Logic, Part III of IV

Quantum logicians argue that, when it comes to statements about the behavior of subatomic particles, Distribution--the inference from [P & (Q v R)] to [(P & Q) v (P & R)]--fails.

In Part II, I suggested that it's very hard to make sense of this claim if you take validity to be a matter of universal truth-preservation. If one extends the classical truth tables for conjunction and disjunction with a third truth value, it's hard to see which one would or could "get the job done." I quickly surveyed various obvious candidates for a third truth-value--gaps, gluts, undecidedness, on-the-border-between-true-and-falsiness and so on. In none of these cases does it seem plausible that any of these additions to the classical truth tables would make Distribution invalid.

Of course, the quantum logician could revise the classical truth tables instead of extending them--they could, for example, argue that it's sometimes possible for (P v Q) to be true even if neither P nor Q is true--but this makes them vulnerable to "change of meaning" charges in a way that, for example, as we saw in Part II, even as extreme a heterodox logician as the dialetheist is not vulnerable.

(In his argument against Putnam's early quantum logic stuff, Dummett claims that the whole notion of truth tables and truth-functionality subtly relies on Distribution and that, as such, the quantum logician is not entitled to it. This charge seems to me to rely on an illegitimate leap from the notion that Distribution fails in some contexts to the notion that it fails in all contexts. It is, in other words, like anti-dialetheist arguments that rely on equating dialetheism with trivialism.)

Now, it's certainly possible to concoct truth tables that fit the bill--retaining the classical lines while invalidating Distribution and keeping most of the rest of the classical laws we care about. I wrote a paper last year suggesting one scheme for that--treated as a formal exercise, because I've never been convinced by the quantum case against Distribution--and in the comments on Part II Brandon suggests another, while making clear that it's just an example, and raising the concern about his example that the "true if..." truth-values might be a bit too crudely rigged to get the desired result but speculating that there were probably more sophisticated ways to go about it. (In so far as I might have come off too strongly last time, sounding like there was literally no way to set up truth tables to get the desired results, Brandon's point was well-taken.) For the sake of elegance at least, my favorite way of invalidatng Distribution like this:

Assume there are three truth-values, 1, 0 and .5. For disjunctions to have value 1, the value of their conjuncts must add up to at least 1, for them to have value 0, the sum up of the conjuncts must be 0, and otherwise, they have value .5. Conjunctions have value 1 if the sum of their conjuncts is 2 and 0 otherwise. The value of the negation of any claim is the absolute difference between 1 and the value of the claim. Validity is preservation of value 1.

In terms of the most important rules about Disjunction and Conjunction, Disjunction-Addition and Conjunction-Elimination obviously both fall neatly out of that scheme. Disjunctive Syllogism is also safe. Here's the truth table for that:

( P v Q ) / ~ P // Q
1 1 1 0 1 1
1 1 0 0 1 0
1 1 .5 0 1 .5
.5 1 1 .5 .5 1
.5 1 .5 .5 .5 .5
.5 0 0 .5 .5 0
0 1 1 1 0 1
0 .5 .5 1 0 .5
0 0 0 1 0 0

Distribution, though, is invalid, the relevant line of the truth table being:

P & ( Q v R) // ( P & Q) v ( P & R )
1 1 .5 1 .5 1 0 .5 0 1 0 .5

A slightly less elegant way to do it is just to call the third value "O," not say much about it's relationship to T and F, and just choose a bunch of classical claims about truth-functionality while ignoring others--e.g. it will still be true that "conjunctions are only T if their conjuncts are both T", but it won't still be true that "conjunctions are only F if their conjuncts are both F," it'll still be true that "disjunctions are only F if their disjuncts are both F", but it won't still be true that "disjunctions are only T if their disjuncts are both T," etc.

So it's certainly possible to put together such truth tables, invalidating Distribution while retaining some plausible-sounding principles linking them to bivalent truth tables. The difficulty in each case is about putting conceptual meat on the bones, philosophically justifying the strange behavior of the new truth-value.

Intuitively, if .5 is read as "sort-of-true" or "half-true" or something, then, if P is completely true and Q is "sort-of-true" or "half-true" or something, then the joint statement of both should be "sort-of-true" or "half-true" or something as well. That won't do, though, in terms of the mathematicized truth tables given above, because if, when P is 1, Q is .5 and R is .5., (P & Q) and (P & R) both get value .5, their disjunction gets value 1 and Distribution goes through.

In general, however one sets things up, the sticky question is this:

What could we possibly say about the third truth-value that justifies its strange behavior here? How can we make sense of saying that the value is close enough to truth that two disjuncts that each have it add up to a disjunction which is fully true, but far enough from truth that one conjunct with the value, in combination with one fully true conjunct, add up to a conjunction that's not only untrue but fully false? Without this awkward combination, after all, Distribution goes through.

Again--if anyone has anything to offer in the comments by way of a plausible story about the nature of the proposed value, I'm all ears. I certainly don't take the limits of my creativity here to be the last word, and one of the joys of philosophy blogging is getting to brainstorm with people about this stuff, so seriously, if you've got something, I'd love to hear it. From my perspective, though, I don't see how it could work.

Of course, it could be that the crucial mistake happened at the beginning, when we assumed that validity was about truth-preservation, and so, for Distribution to be invalid, we had to find an instance of it where a true premise delivered a non-true conclusion. As widespread as this view of validity is--it certainly the official dogma of introductory logic textbooks, and it's a subject that Graham Priest and I agree about, which should indicate a certain kind of consensus--it's certainly not the only view out there. There is, for example, the truth-preservation-plus view often assumed by relevance logicians, according to which truth-preserving inferences can still fail to be valid if the premises and conclusion aren't sufficiently connected given some sort of relevance constraint (e.g. they don't share any non-logical terms in common). That clearly won't do, here, though, since instances of Distribution concerned with the behavior of quantum particles will pass those sorts of relevance tests with flying colors.

What, though, if we switched over to an important secondary tradition, the inferentialist one according to which certain inference rules are basic and analytically constitutive of the meaning of the relevant logical connectives?

One problem here will be a general one afflicting all change-of-logic proposals made by inferentialists--"given that you think that the inference rules are constitutive of the meaning of the connectives, and that you're changing your mind about the inference rules, aren't you just talking about new connectives, not making heterodox claims about the behavior of the old connectives?"

Now, I think this is a serious problem, and I don't want to dismiss it too quickly, but there is a way around it that's at least plausible enough to tentatively assume for the purposes of this discussion (as I assume for the sake of argument that--contrary to my views on the matter--inferentialism is correct):

We can bite half the bullet, in a way that makes room for arguing about logic, but saying that, for example, quantum-logical disjunction and classical disjunction really are two different connectives with two different meanings, but both represent analyses of our ordinary intuitive notion of (inclusive) disjunction. In general, we all start out with intuitive proto-versions of all the logical concepts, and the meaning-constituting inference rules of various formal systems all represent something like conceptual analyses of those proto-concepts.

(That was a bit rough, but it's probably good enough for our purposes here. Also note that a truth-preservationist quantum logician who was willing to simply revise the classical truth tables instead of expanding them could presumably make the same sort of move to justify themselves.)

Now, this sort of move may seem plausible when it comes to Van McGee's counter-example to Modus Ponens or relevance logicians' arguments against the classical notions of implication and entailment and so on, but it seems to me that the quantum context is the one in which this move falls the most spectacularly flat.

Surely, our ordinary-language, intuitive notions of "and" and "or" predate the discovery of subatomic particles, superposition and the rest by....

...well, by pretty much all of human history. These notions were developed to talk about the ordinary properties of ordinary "mid-sized dry goods."

Now, taking the "quantum leap" in our understanding of the physical world surely poses challenges to many of our ordinary intuitions about things like location, causation at a distance and so on. It's plausible that it turns out to upset all sorts of metaphysical applecarts. Some of these could even have consequences for logic, if logic is understood in truth-preservationist terms and odd possibilities, not accounted for in logical schemes developed in a previous era, turn out to sometimes be true if this strange domain of physical reality.

But. Surely the one thing that we won't discover from our experiments about the behavior of previously-unknown subatomic particles is that we were mistaken about what "and" and "or" meant in our ordinary pre-philosophical, pre-scientific discourse.

If revisionary moves about logic are justified as a result of the deliverances of quantum physics, it looks like we're going to have to justify them in terms of the truth-preservationist conception of validity. And the prospects for making sense of that are looking pretty dim as well.

So, if Distribution is safe after all, does that mean that classical logic has been comfortably and securely squared with quantum weirdness, that there are no remaining problems or challenges there?

Well....

Stay tuned for next week!