BINARY LOGIC v QUANTUM THEORY?

[Frylock]

@Mijin

Well, I’ve come across lots of examples, but two come to mind:

Evaluating “this statement is false”.
This might be considered a paradox by many who assume every statement can be classified as either true or false (a claim which boolean logic doesn’t make, but an awareness of tri-state logic can make this more obvious).

Try this on for size:

This statement is not true.

(Where “not true” covers false as well as whatever additional truth-values you’d like to throw in.)

Incidentally, the issue with “this statement is false” is not so much the contradiction, it’s that it’s purely a self-reference and there’s nothing to evaluate.

I agree, but you still get a paradox on the traditional way of analyzing statements like “this statement is not true.”

My own idea is that we shouldn’t think it’s licensed to “suppose the sentence is true” or “suppose the sentence is false.”

Another is all the “heap” paradoxes: e.g. since we can’t identify a particular number of hairs at which someone transitions from being “non-balding” to “balding”, it is impossible to become balding (or bald).
This is one where the “right” solution is debatable, but most agree that fuzzy logic is one possible solution (i.e. the loss of every hair increases one’s “balding-ness” and something like fuzzy collapse happens in trying to classify someone as balding or not).

I’d have to know more about how fuzzy logic works exactly, but I’m skeptical that this provides an outright solution to the heap paradoxes. There will always be a boundary between the clear and the fuzzy, and that boundary itself is fuzzy in a way not measured by the fuzziness already incorporated into the logic.

[Frylock]

@Half_Man_Half_Wit

Wiki gives the following example:

p: the particle is moving to the right (I think this would be better phrased as ‘the particle’s momentum is p’, or something like that)
q: the particle is in [-1, 1]
r: the particle is not in [-1, 1]

Clearly, q v r = T, the particle is somewhere – i.e. an experiment will find it at some location, to steer clear of hidden variable accusations – thus p ^ (q v r) = p; however, p ^ q and p ^ r are both false, since if the particle’s momentum is known well, there is some uncertainty to its position, and it will have nonvanishing amplitude to be found both in and out of the interval – thus, p ^ q v p ^ r is false.

q v r is true, not simply as a result of the fact that A v -A is always true, but because in this specific case, r is true. The particle is not in [-1, 1]–because if the momentum is well-defined (which it must be for p to be true) then the position is not. (And if position is not well-defined, then any statement of the form “the particle is not at position x” is true.)

[Frylock]

(This is not to say that there’s no point in building a many-valued quantum logic. My point is just the narrow one that it’s a mistake to think quantum phenomena provide an empirical counterexample to propositional logic. Nothing provides such a counterexample, because propositional logic is basically jury-rigged to come out right every time. Propositional (and predicate) logic is custom-made to model everything.)

[Half_Man_Half_Wit]

@Frylock

q v r is true, not simply as a result of the fact that A v -A is always true, but because in this specific case, r is true. The particle is not in [-1, 1]–because if the momentum is well-defined (which it must be for p to be true) then the position is not. (And if position is not well-defined, then any statement of the form “the particle is not at position x” is true.)

Not sure I follow. If momentum is sharply defined, r isn’t true – an experiment may find the particle within [-1, 1].

But really, the example is just fluff – the non-distributivity of quantum logic follows more fundamentally from the fact that projection operators in Hilbert space, which correspond to propositions like ‘the value of some physical quantity lies in a certain range’, form an orthocomplemented lattice, i.e. a ‘logic’ without distributivity. That this should be thought of as a true logic is implied by the quantum formalism, i.e. the probability theory built upon this logic, yielding true probabilities – relative frequencies of actual measurement outcomes.

[Half_Man_Half_Wit]

@Frylock

(This is not to say that there’s no point in building a many-valued quantum logic. My point is just the narrow one that it’s a mistake to think quantum phenomena provide an empirical counterexample to propositional logic. Nothing provides such a counterexample, because propositional logic is basically jury-rigged to come out right every time. Propositional (and predicate) logic is custom-made to model everything.)

Well, of course, Putnam thought differently – the idea being that, like with geometry, it’s an empirical question whether our world is governed by classical rules (i.e. propositional logic, or Euclidean geometry) or non-classical ones (like quantum logic, Riemannian geometry, etc.).

It’s widely accepted that general relativity showed that our universe is (pseudo-)Riemannian, rather than Euclidean; thus, or so the argument goes, it shouldn’t be too hard to imagine that quantum mechanics shows that our universe is fundamentally described not by classical logic, but rather just by quantum logic. After all, both (i.e. geometry and logic) can in the end be framed in the form of axiomatic systems, which may or may not apply to reality. Why should one be subject to empirical evaluation, while the other isn’t?

(I should admit to playing the devil’s advocate a little here: I’m not actually convinced that ‘logic is empirical’. But I’m interested to see your reasoning first!)

[Mijin]

@Frylock

Try this on for size:

This statement is not true.

(Where “not true” covers false as well as whatever additional truth-values you’d like to throw in.)

Well, the third option I was envisioning is not technically a truth value at all: something like “undefined”.

So statements comprising just a self-reference are “undefined”. “This statement is not undefined” is undefined. And there is no contradiction in this.

There will always be a boundary between the clear and the fuzzy, and that boundary itself is fuzzy in a way not measured by the fuzziness already incorporated into the logic.

No, it doesn’t work like that. In the hair example, there might be a cut off (;)) such that any number of hairs above, say, 100,000 will trivially defuzzify to “not-balding”.
But equally you could model the system in such a way that for any finite number of hairs there is a non-zero probability of defuzzifying to “balding”.

[Frylock]

@Half_Man_Half_Wit

Not sure I follow. If momentum is sharply defined, r isn’t true – an experiment may find the particle within [-1, 1].

“An experiment will find the particle within [-1, 1]” != “The particle is within [-1, 1].”

If the three sentences are:

p = The particle is moving to the right
q = An experiment will find the particle within [-1, 1]
r = An experiment will not find the particle within [-1, 1]

then p, q and r are a clear instance of distributivity, not a counterexample.

But really, the example is just fluff – the non-distributivity of quantum logic follows more fundamentally from the fact that projection operators in Hilbert space, which correspond to propositions like ‘the value of some physical quantity lies in a certain range’, form an orthocomplemented lattice, i.e. a ‘logic’ without distributivity. That this should be thought of as a true logic is implied by the quantum formalism, i.e. the probability theory built upon this logic, yielding true probabilities – relative frequencies of actual measurement outcomes.

I don’t think that adds up to showing that quantum phenomena are a counterexample to propositional logic. It does show that there’s a logic without distributivity which models QM well, but it does not show that propositional logic fails to model QM.

[Half_Man_Half_Wit]

@Frylock

If the three sentences are:

p = The particle is moving to the right
q = An experiment will find the particle within [-1, 1]
r = An experiment will not find the particle within [-1, 1]

then p, q and r are a clear instance of distributivity, not a counterexample.

What’s your reasoning here? It’s clearly not the case that the particle is within [-1, 1]; it’s also clearly not the case that the particle is not within [-1, 1] (if, in both cases, the particle’s momentum is p); nevertheless, it is the case that the particle is within [-1, 1] or not within [-1, 1]. For the original three propositions, distributivity is violated. Look at the following phase space diagram:

 p
  |
*p* |                    
  |
  |
  |
  |
  |____________________ 
         '   '         x
        -1   1

The line represents the state of the particle; its momentum is exactly known, so its position is maximally unknown (technically, a particle can’t be in an infinitely sharp momentum (or position) state, but we’ll gloss over that for now). Clearly, ‘the particle has momentum p and is in [-1, 1] or not in [-1, 1]’ is true. However, neither ‘the particle has momentum p and is in [-1, 1]’ nor ‘the particle has momentum p and is not in [-1, 1]’ is true.

I don’t think that adds up to showing that quantum phenomena are a counterexample to propositional logic. It does show that there’s a logic without distributivity which models QM well, but it does not show that propositional logic fails to model QM.

The problem is that in QM, not all observables are compatible. In classical mechanics, you can simultaneously measure anything that can be measured to arbitrary precision; this isn’t possible in general in QM (there do exist subsets of observables for any system that are compatible – staying within these subsets recovers classical logic, just as it recovers classical physics). If you’re willing to admit that propositions regarding these observables are propositions in the logical sense, then the logic you get out of that is one in which distributivity doesn’t hold; this isn’t controversial at all, as far as I know.

[The_Other_Waldo_Pepper]

@Frylock

This statement is not true.

“This statement IS true!”

Is that true or false?

[Frylock]

@Half_Man_Half_Wit

What’s your reasoning here? It’s clearly not the case that the particle is within [-1, 1]; it’s also clearly not the case that the particle is not within [-1, 1] (if, in both cases, the particle’s momentum is p); nevertheless, it is the case that the particle is within [-1, 1] or not within [-1, 1].

In the above, you’re talking about where the particle is. But in my p, q and r, I was talking about where an experiment will find the particle. They are two different statements. My p, q and r (which I took to be yours as well) are an instance of distribuitivity, not a counterexample.

Now what if they are (as you originally said, and as Putnam had them)

p = The particle is moving to the right
q = The particle is within the interval
r = The particle is not within the interval.

My response is just what I said before–if p is true, then the momentum is defined, and so the position is not defined, so r is true, because if the position is not defined, then “the particle is not within interval x” is true for every x.

To that you replied by switching from talk of where the particle is to talk of where an experiment will find the particle. Hence my move to the new p, q and r. But now you want to go back to the original p, q and r. So I repeat my response!

For the original three propositions, distributivity is violated. Look at the following phase space diagram:

 p
  |
*p* |                    
  |
  |
  |
  |
  |____________________ 
         '   '         x
        -1   1

The line represents the state of the particle; its momentum is exactly known, so its position is maximally unknown (technically, a particle can’t be in an infinitely sharp momentum (or position) state, but we’ll gloss over that for now).

It’s position isn’t unknown–its known to be undefined.

The problem is that in QM, not all observables are compatible.

Do I understand you correctly when I think you’re saying that in QM, there are observations which contradict each other?

[Half_Man_Half_Wit]

Hence my move to the new p, q and r. But now you want to go back to the original p, q and r. So I repeat my response!

Really, don’t get too hung up on the example if it’s not clear. It’s a completely unambiguous matter – the lattice of propositions formed by projection operators is not distributive; since these encode properties a quantum mechanical system may have (such as, having momentum or position within a certain interval), the logic used to reason about these properties can’t be distributive. That’s just what the math says – rejecting this amounts to rejecting quantum mechanics. Which of course you may well do, but then that’s a whole 'nother discussion. Within QM, in any case, propositions given by projectors don’t follow a distributive logic.

Do I understand you correctly when I think you’re saying that in QM, there are observations which contradict each other?

No, there are observations that can’t be simultaneously made, such as precise measurements of both momentum and position.

[Frylock]

@Half_Man_Half_Wit

Really, don’t get too hung up on the example if it’s not clear. It’s a completely unambiguous matter – the lattice of propositions formed by projection operators is not distributive;

Correct: There is a logic which QM is a good model of, and for which a distributive property doesn’t hold.

This does not mean that QM is a counterexample to propositional logic. It just means you can model QM using logics other than propositional logic. (And you probably should for most important purposes in physics.)

I’ll explain what I mean when I say propositional logic is jury-rigged to have no (I guess I should add “consistent” here) counterexamples. It’s just this: Given what “and,” “or,” “not”, and “if-then”* mean in propositional logic, it is absolutely guaranteed that any tautologous statement you make in propositional logic will be true on any assignment of truth values to propositions. This isn’t a fact about the world, it’s simply a fact about how the language of propositional works. A tautology in propositional logic is always true by virtue of the definitions of the four logical connectives.

In other words, given what “v” means in propositional logic, the statement “A v -A” cannot be assigned any value (in propositional logic) other than “true.” If someone thinks they can find a counterexample–a case in the actual world where “A v -A” is false, (and they intend it to be interpreted as a statement of propositional logic,) they must invariably have either misunderstood what “A” means in the interpretation, misunderstood what “-” means in propositional logic, or misunderstood what “v” means in propositional logic. Because in propositional logic, A v -A is true by definition. Calling it “false” means you’ve either misunderstood or you’re not doing propositional logic. But in both of those cases, you’re not dealing with a counterexample to PL. Rather, you’re simply not doing PL.

It’s all a set-up. The definitions of the connectives force tautologies to be true. If you think you’ve found a false tautology, you have simply misunderstood the meaning of one or more connectives (or the meaning of one of the interpreted propositions).

Now, it turns out that propositional logic isn’t the most useful model to use for some purposes. For example, it’s not a useful model for dealing with arguments like “Socrates is a man and all men are mortal, so Socrates is mortal.” You need predicate logic for that. Predicate logic lets you do somethings that propositional logic doesn’t let you do. But, importantly, the Socrates argument I just gave doesn’t constitute a counterexample to propositional logic. It’s not a case, in other words, where the premises are true, the conclusion false, and the argument propositionally valid.

Distributivity still holds in predicate logic, but meanwhile, there’s this other kind of logic–“quantum logic”–in which distributivity doesn’t hold. And it turns out QL lets you do some things that PL doesn’t let you do, and it turns out these are useful things. But importantly, this isn’t because there are any counterexamples to PL to be found in quantum phenomena. Rather, it’s just that PL doesn’t have the necessary expressive power to handle some valid arguments that QL does have the expressive power for.

You’re not going to find an example of a quantum phenomenon where it really is true that P & (Q v R) and it really is false that (P & Q) v (P & R) (and where these two expressions are to be interpreted as statements of propositional logic.)

Putnam’s example doesn’t work–the argument that it works relies on a misunderstanding of the meaning of one of the interpreted propositions. And no other example will work either, because, as I said, PL is rigged. The meanings of the connectives force tautologous statements in PL to be true by definition.

Show me a P, Q and R that you think constitute a counterexample to distributivity, and I’ll show you how you’ve either equivocated on P Q or R, misunderstood one of them, or misunderstood one of the connectives involved.

[Frylock]

@The_Other_Waldo_Pepper

“This statement IS true!”

Is that true or false?

People don’t talk about this one as much because it doesn’t seem to lead to a contradiction like “this statement is false.” But I think it’s actually a better illustration of what’s wrong with statements like these–the liar sentence isn’t defective because it leads to a contradiction, it’s defective because it never gets around to actually saying anything.

The “truth-teller” sentence illustrates this even better, since there’s no question of a contradiction following from it.

Anyway, just saying “it never gets around to saying anything” doesn’t solve the problem, because you then get cheeky people who say “Oh yeah, then what about this?”

Either this statement is false, or it doesn’t ever get around to saying anything.

(If it’s false, then it’s true. But if it’s true, then it doesn’t say anything, so it’s not true. Contradiction either way… So yet again, I’ll insert my view that I think it’s not legitimate in the first place to “suppose it’s false” or “suppose it’s true.” I don’t think anything’s actually getting supposed there, but we can be tricked into thinking we’ve supposed something because we’re used to doing this formal “supposition” operation with strings on paper and we forget to check to make sure the “supposition” operation actually means something. We forget to do this because in non-troublesome cases, being able to do it formally always does guarantee that the operation actually means something.)

[Frylock]

@Mijin

Well, the third option I was envisioning is not technically a truth value at all: something like “undefined”.

If a statement is undefined, then it’s not true.

So the “strengthened liar” as it’s called still presents a problem for you. To make it more explicit, try this:

This statement is either false or undefined.

If this statement is false, then it’s true–contradiction.

But if it’s true, then it’s undefined. Since an undefined statement can’t be true, we have another contradiction.

The strengthened liar always wins!

No, it doesn’t work like that. In the hair example, there might be a cut off (;)) such that any number of hairs above, say, 100,000 will trivially defuzzify to “not-balding”.
But equally you could model the system in such a way that for any finite number of hairs there is a non-zero probability of defuzzifying to “balding”.

It’s implausible to think that a man with a full head of hair is possibly (however improbably) balding.

[Frylock]

@Half_Man_Half_Wit

(I should admit to playing the devil’s advocate a little here: I’m not actually convinced that ‘logic is empirical’. But I’m interested to see your reasoning first!)

I’ve said a lot of it–admittedly not well since this is the first time I’ve ever thought any of this!–but to add a little more that is relevant to the post the above is clipped from, an important difference between geometry and propositional logic is that geometry starts from axioms–and must–while propositional logic need not have any axioms.

So you can have a counterexample to euclidean geometry by finding something that falsifies one of its axioms. An axiom of euclidean geometry isn’t true by definition, it’s just assumed.

But tautologies in propositional logic aren’t simply assumed to be true–they’re definitionally true. Given the meanings of the logical connectives in PL, any tautology in PL can not be understood as other than true. If a tautology appears false, then the relevant string is not actually being regarded as a statement of propositional logic–though the person regarding it may mistakenly think he’s regarding it as a statement of propositional logic. He’s failed to do so either because he’s not consistently interpreting the atoms, or he’s not interpreting the connectives correctly. (Or he may indeed be regarding it as a sentence of propositionally logic and he’s simply wrong about the observables.)

It’s a trap! No true scotsman is written right into the rules!

[Half_Man_Half_Wit]

@Frylock

You’re not going to find an example of a quantum phenomenon where it really is true that P & (Q v R) and it really is false that (P & Q) v (P & R) (and where these two expressions are to be interpreted as statements of propositional logic.)

Well, yes, hence, one typically takes these to be statements in quantum logic, since it describes QM better than predicate logic does. That’s what I’ve been saying! I’m rather certain I’ve nowhere said that any such example shows propositional logic to be wrong, but merely that it shows that quantum logic may be more applicable.

Nobody is claiming that any empirical observation can refute a mathematical (or logical) structure – that’s nonsense. But observation can decide which structure to use for modelling!

Just like no observation can falsify that in Euclidean geometry, parallels don’t intersect (or diverge), observation can’t falsify that in the propositional calculus, distribution holds. But Euclidean geometry does not describe our world very well, and similarly, it may be the case that propositional logic doesn’t, either – it might be the case that what we would like to call parallels end up diverging, and it might be the case that what we would like to call propositions don’t distribute. So, we conclude that our world is better described by Riemannian geometry, or by quantum logic.

[Half_Man_Half_Wit]

@Frylock

I’ve said a lot of it–admittedly not well since this is the first time I’ve ever thought any of this!–but to add a little more that is relevant to the post the above is clipped from, an important difference between geometry and propositional logic is that geometry starts from axioms–and must–while propositional logic need not have any axioms.

But propositional logic is built over a mathematical structure (a Boolean lattice) which satisfies certain axioms that may or may not be satisfied by, uh, ‘reality’. Besides, the proposition ‘propositions don’t distribute’ may be true and hence necessarily an axiom or theorem of every mathematical model you propose to apply to reality – since it isn’t true in the propositional calculus, it may well not apply.

[Frylock]

@Half_Man_Half_Wit

Well, yes, hence, one typically takes these to be statements in quantum logic, since it describes QM better than predicate logic does. That’s what I’ve been saying! I’m rather certain I’ve nowhere said that any such example shows propositional logic to be wrong, but merely that it shows that quantum logic may be more applicable.

Point of order: I said QM doesn’t violate propositional logic, and you replied in disagreement, which is most naturally interpreted as you claiming QM does violate propositional logic. But QM being a model for other logics doesn’t mean it “violates” propositional logic. And people very often do talk about QM providing an actual counterexample to PL because of the distributivity thing. If that’s not what you were saying, then at the very least my misunderstanding was very understandable.

Nobody is claiming that any empirical observation can refute a mathematical (or logical) structure – that’s nonsense. But observation can decide which structure to use for modelling!

I haven’t read Putnam directly, but I’ve certainly seen plenty of people make exactly this claim–that PL turns out to be literally wrong in some sense, because of QM. Whether Putnam was saying this or not, I don’t know.

Just like no observation can falsify that in Euclidean geometry, parallels don’t intersect (or diverge), observation can’t falsify that in the propositional calculus, distribution holds. But Euclidean geometry does not describe our world very well, and similarly, it may be the case that propositional logic doesn’t, either – it might be the case that what we would like to call parallels end up diverging, and it might be the case that what we would like to call propositions don’t distribute. So, we conclude that our world is better described by Riemannian geometry, or by quantum logic.

I can’t see how the bolded part is supposed to work.

Do you still think that Putnam’s p, q and r are examples of things we’d like to call propositions which don’t distribute?

[Mijin]

@Frylock

If a statement is undefined, then it’s not true.

So the “strengthened liar” as it’s called still presents a problem for you.

You’re still not grasping that “undefined” is not another truth value as such.

For example: NOT undefined = undefined
(and I’m capitalizing logical NOT to disambiguate it from the English word “not”).

It’s implausible to think that a man with a full head of hair is possibly (however improbably) balding.

Not in fuzzy logic.

And the point is, there need be no discrete cut-off. It’s one of the main reasons for using fuzzy logic in the first place, and why it’s one solution of the heap problems.

[Half_Man_Half_Wit]

@Frylock

Point of order: I said QM doesn’t violate propositional logic, and you replied in disagreement, which is most naturally interpreted as you claiming QM does violate propositional logic.

Well, I said that ‘in quantum logic, distributivity doesn’t hold’ – not that it matters terribly much. But I feel I’ve been rather clear that I’ve been talking about the applicability of propositional logic, not its being right or wrong in itself; if not, then let me emphasize this now.

I can’t see how the bolded part is supposed to work.

Well, if we want to consider things like ‘the system’s A-value is in B’ to be a proposition, where the system is a quantum mechanical one, A is an observable, and B a range of possible values the observable may attain, this proposition corresponds to a projector, and projectors don’t distribute.

Do you still think that Putnam’s p, q and r are examples of things we’d like to call propositions which don’t distribute?

Yes, of course!