Wednesday, September 15, 2010

Quantum Logic, Part II of IV

Assume (until we get to Part III) that validity is a matter of truth-preservation.

Now, for any proposal that we "change logics" (i.e. revise our current ideas about what the true laws of logic are) to succeed in doing more than just changing the subject, we need to make sure the terms we're using mean the same things after the revision that they did before. For example, if I suggest that "P & Q" is true iff P is true, Q is true, or both, whereas "P v Q" is true iff P and Q are jointly true, my proposal amounts to reversing the usual notation for conjunction and disjunction, not changing our ideas about them. By contrast, if someone denies that the inference from any and every premise to any instance of the disjunction "P v ~P" is truth-preserving, and justifies it with some philosophical story about the unevenness with which vague predicates map onto the world, they really are rejecting classical logic.

Quine, in a famous passage in Philosophy of Logic, seems to suggest that all revisionary proposals about logic fall into the first category, that "the deviant logician" always succeeds merely in changing the subject. He argues, in particular, that when the dialetheist says that some contradictions are true and not everything follows from them, they have stopped using "not" and "~" in the usual ways.

For a variety of reasons, I don't this is one of Quine's better or more plausible moments. For my tastes, Quine is at his best in Two Dogmas of Empiricism, where he suggests that perhaps quantum physics will one day force us to abandon classic logic, and provides a holistic framework for thinking about questions of confirmation and belief revision within which that makes sense.

Without getting into a longer critique of Quine here, it's worth noting, as at least one major test of sameness of meaning, that the dialetheist, who takes some contradictions to be true and denies that anything and everything follows from any given contradiction--because she denies that Disjunctive Syllogism is universally truth-preserving--actually affirms all of the relevant classical truth tables. She just adds some extra lines. Where the classical logician says that ~P is true iff P is false and ~P is false iff P is true, the dialetheist agrees, and, given that they think that P can be both true or false, take the step they are required to take by the assumptions they share with the classical logician and say that ~P can be both true and false as well. Where the classical logician says that a disjunction is true iff at least one of the disjuncts is true--indeed, given that the disjunction symbol refers to "inclusive disjunction" (P or Q or both), "P v Q" and "at least one of these things is true: P, Q" seem to mean precisely the same thing--the dialetheist grants this, assents to all the lines of the classical truth table (where "F" is read as "just false") and adds some extra ones, getting the result that "at least one of these things is true: P, Q" can be true even if P and Q are both false and Q fails to be true, if P is both true and false. Once all that is in place, it's a straightforward, principled consequence of the view--with the truth tables for the two classical truth values (just true and just false) intact and the meaning securely unchanged--that, given the assumption that truth and falsity can overlap, Disjunctive Syllogism is not universally truth-preserving.

A similar story could be told about the inferences denied by deviant logicians who do things like deny the Law of the Excluded Middle, but the example of the dialetheist makes the point:

Even very radical proposals for logic change--and, in some ways, the claim that there are true contradictions seems much more radical than the quantum logician's rejection of Distribution--can pass the "not just changing the subject" test with flying colors. Whatever one thinks of their claim to have discovered new logical possibilities ignored by orthodoxy, their views about how the classical truth-values interact are thoroughly orthodox. Crucially, they respect the fact that "P or Q" in the sense captured by "P v Q" seems to mean the same thing as "at least one of these things is true: P, Q."

The quantum logician, on the other hand, seems to fail the test.

For the premise of some instance of Distribution--i.e. [P & (Q v R)]--to be true, surely P must be true. (If a conjunction can be true even though one of its conjuncts is not, the "change of meaning" charge starts to sound pretty convincing.) For the same reason, "Q v R" must be true. How, then, can the conclusion--[(P & Q) v (P & R)] be false? Well, we've already said that P is true. And if Q and R were both false, then (Q v R) would be false, and hence the premise as a whole would be false. Moreover, if one of them were true, the conclusion would be true!

In order for the premise to be true and the conclusion to have some other status, then, both Q and R must have some other truth-value. What other truth-value, though, would do the trick? If Q and R were both both true and false, the conclusion would be true. If Q and R were both neither true nor false, then how could "at least one of these things is true: P, Q" possibly be true? If Q and R were both somehow undecided or unsettled between truth and falsity, or on the vague borderline between them or something, why wouldn't their disjunction similarly be undecided or unsettled between truth and falsity, or on the vague borderline between them or something?

If someone has a proposal for what truth-value Q and R could have that would make the premise of non-truth-preserving instances of Distribution true without the conclusion being true, I'd love to hear about it in the comments, but right now it looks like nothing fits the bill.

But wait! Maybe we went wrong in our initial assumption that validity is truth-preservation. Perhaps, once we switch over to one of the other theories of validity, the quantum logician's move will start making more sense.

To which all I can say is...

Stay tuned!


Brandon said...

Suppose you allow conditional truth values, just as we can allow conjunctive or nondisjunctive truth values. Then you could have a truth table for disjunction that's something like:

[p][q] = [p v q]
[T][T] = [T]
[T][F] = [T]
[F][T] = [T]
[F][F] = [F]
[cT][T] = [T]
[cT][F] = [T]
[T][cT] = [T]
[F][cT] = [T]
[cT][cT] = [T]

where cT is that particular conditional truth value that means 'true if disjoined with only false fellow disjuncts'. Here we have a truth table in which all the classical lines correspond, disjunction can still be understood as 'at least one of these is true', and, if it were properly expanded, [P & (Q v R)] could be true without [(P & Q) v (P & R)] being true. That is, substituting the truth values for the propositions:

[T] & ([cT] v (cT)]

won't preserve truth if reordered to:

([T] & [cT]) v ([T] & [cT])

unless [cT] doesn't just mean 'true if disjoined with only false fellow disjuncts' but also has meaning in a conjunction.

This is a pretty crude way of doing it, since [cT] as defined here guarantees the truth of any disjunction to which it is assigned; I'm sure more sophisticated ways are possible.

Of course, I could be missing something.

Ben said...


Interesting thought, but of course, intuitively, whether a statement is true or false should be determined by whatever it is the statement refers to, rather than depending on why we are uttering it--if we're saying that it's true, if we're embedding it in a disjunction, or whatever. I'd definitely be interested, though, to see what the more sophisticated versions of the proposal would look like.

Brandon said...

It's not an intuition I have at all, but you're right that it certainly does seem to be common. It's much the same intuition, I suppose, that wants to rule out inter-referring statement pairs like

(1) Statement (2) is false.
(2) Statement (1) is false.

(Since a natural way to assign truth values, if they are allowed to be assigned, is as conjoined conditional truth values.)

In any case, it was just an attempt to rise to the challenge and show that there is at least one deviant truth value that preserves the classical structure but breaks Distribution.

Ben said...

Fair enough.

Could you say a bit more about your analysis of the inter-referring sentences? I'm not sure I follow.

Brandon said...

One possible way to analyze them is to regard them as having truth values conditionally linked to each other, so that the truth value of (1) is [T if (2) is F and F if (2) is T] and of (2) is [T if (1) is F and F if (1) is T]. Only one possible way, of course, although if you give people the statements as a true/false quiz the conjoined conditional truth value assignment tends to capture their initial response fairly well.