BINARY LOGIC v QUANTUM THEORY?

Great Debates
61

Heh, well, that’s alright. I probably wrote far more words than I needed to for the discussion I was trying to have.

Even better! :slight_smile:

I, uh, agree that we seem in agreement about most things. I don’t think there is any sense in talking about empirically determining the rules of logic or such things, for example. On this, I am completely with you.

The only point I would like to make which I suspect you may not entirely already agree with is that I think Boolean propositional logic is not in any way fundamentally privileged over other kinds of propositional logic (e.g., intuitionistic propositional logic or nondistributive propositional logic or such things). It is uniquely the logic of the two-element lattice {true, false}, but I think there are a number of things which we have traditionally viewed through the lens of the two values {true, false} which are perhaps better understood through some other lens instead. (Why should we say, of any judgement which we might in some cases consider ourselves warranted to make, that in all cases it has, by some inaccessible external hand’s assignment, associated with it either the value “true” or “false”? This unnecessary assumption is what the fixation on Boolean logic often leads to.)

62

Isn’t that exactly like saying that I can’t find a triangle whose angles sum to anything else but 180°? It’s only true if you assume that Euclidean geometry is the only true geometry, and that it defines what’s meant by points, angles, lines etc. But then, the real world would simply not contain points, angles, lines and all those things (at least, not in a straightforward way), but rather other kinds of things that aren’t those talked about by Euclid’s axioms. In the same way, then, properties of quantum systems would not be propositions, if you define propositions as those things propositional logic talks about, but some other kinds of things, and the real world would not contain propositions, but those other kinds of things.

No, I tried to emphasise that p & q and p & r (I wrote ^ for & in previous posts) are false, without attempting to put a definite truth value on r and q by themselves; since while the latter is dicey, the former is unambiguous. Similarly, r v q is true, without talking about the truth values of r or q. This is just where quantum mechanics differs from classical mechanics: due to uncertainty, the truth of propositions like r depends on the truth of propositions like p, because they aren’t compatible observables. So p & (q v r) is p, hence true if p is, while (p & q) v (p & r) is false.

In that example, the location was not completely undefined – it was constrained within an interval. So the fact was that it is certainly not anywhere outside of that interval, so it must be ‘here’, meaning within the interval. That doesn’t imply that it must be within any sub-interval of that interval – indeed, no such proposition is true. Again, that’s where the quantum differs from classical objects.

I don’t think that I want to say these triads are inconsistent – after all, it’s perfectly true that ‘the particle moves to the right, and is in interval A or interval B’, where A and B may be replaced by either set of intervals previously talked about – it’s just that it’s not therefore either true that ‘the particle is moving to the right and in interval A’ or ‘the particle is moving to the right and in interval B’. So an inconsistency is only arrived at if we assume distributivity holds – which is the reason distributivity is taken not to hold. And indeed, from the mathematical structure of quantum mechanics, that’s exactly what we would expect – the lattice of propositions, of properties of a quantum system, is not distributive.

Well, what makes the + of vector arithmetic ±ish?

In the case of quantum logic, it is just that propositions about quantum systems are equivalent to subspaces of the system’s Hilbert space – saying that ‘the particle moves to the right’ is equivalent to saying ‘the state of the particle is in the Hilbert subspace of right-moving states’. This is the same as properties of classical systems being equivalent to certain sets – a classical particle moving to the right is in the set of right-moving objects. The ‘and’ and ‘or’ of classical logic are the intersection and union of these sets; so the ‘and’ and ‘or’ in quantum logic are taken as the intersection and union of subspaces of Hilbert space. The lattice of sets with these operations and inclusion as ordering relation gives you a Boolean algebra, and hence, propositional logic; the lattice of Hilbert space subspaces, ordered by inclusion in the same way, is only an orthocomplemented one, where distributivity doesn’t hold.

So we say that the ‘and’ and ‘or’ operations of quantum logic are ‘and’ and ‘or’ operations proper because they have the same relation to the mathematical structure of the quantum world as the ‘and’ and ‘or’ of propositional logic have to the mathematical structure of the classical world. That the classical world can be framed in terms of sets is because one assumes that all properties are simultaneously and to arbitrary accuracy knowable; the failure of this assumption in the quantum world makes sets thus an inappropriate mathematical structure, and we look to Hilbert subspaces instead.

63

I guess what I’m saying here is – in agreement with Indistinguishable – that propositional logic isn’t special in its applicability to the world. It applies in a classical setting (or applies especially well there), because we can partition classical observables, classical properties some system under consideration may have – redness, rectangularity, heaviness, those kinds of things – into sets that any given system may either belong or not belong to. The membership function gives us the truth values of whether or not a system has a certain property; inclusion, union, intersection, complement etc. give us the operations of propositional logic.

But it’s not an a priori truth that the world fits neatly into sets. Indeed, there are circumstances where one might use different, more suitable mathematical objects. Mijin has pointed to fuzzy sets, where the membership is not sharply defined as in classical sets, which is useful in modelling properties that are not absolute, in a sense, but rather follow a gradation – there is no sharp cutoff between ‘tall’ and ‘not tall’, i.e. no set in which all tall objects go, but one can meaningfully reason about tallness if one uses a fuzzy set instead – i.e. if one uses sets that systems can be ‘more’ or ‘less’ a member of. A closely related example would be probabilistic logic, in which a system can have a certain property with some certain probability, i.e. is not definitely a member of either set, but can be ‘found’ in either with some likelihood.

Quantum systems and their properties, it turns out, are similarly not well modelled using sets. The reason for that is that into a set structure is built the assumption that each system has its properties in a sharply defined way independently of its other properties. This is an assumption, however, and might or might not hold – and in QM, it simply fails. So we look to another structure other than sets upon which to build our logic; this structure is given by Hilbert space subspaces, which better model quantum properties in the sense that they can deal with complementary, i.e. incompatible, observables (properties) – which is to say, properties which depend upon one another in a manner as given, for instance, by the uncertainty principle.

There’s no reason why properties must be things well described by sets – or, if you insist that this is the only proper meaning of ‘property’, then there is no reason that properties apply to all aspects of the world, any more than Euclidean lines and points do. It is not possible for observation to repeal the tenets of propositional logic; but it is very well possible to call into question its applicability.

64

To answer you both in one fell swoop:

One of my overarching assumptions here, I guess, is that when you make an assertion, you’re either right or wrong, and not both. That’s controversial though! But arguing for about this claim specifically is really difficult (I don’t even know if I’m up to the task at all) and instead, I’m arguing for it by proxy here, when I see people saying things which I take to require that one drop that claim.

Another overarching assumption I have is that to make an assertion is simply to express a proposition (together with a certain attitude toward that proposition)–and also, every proposition is something that someone somewhere could assert under the right circumstances.

While I’m woefully short on arguments for these assumptions, what I can do is say why no argument against them seems sufficient.

For example, does someone think the Liar shows not all assertions are right or wrong, or that not all propositions are assertable?

Well, I’ve got an answer for that–in the form of an argument I could make on independent grounds that sentences like the liar, when uttered with the attitude mentioned above, fail to constitute assertions.

Does someone think “x is bald” is not “right” or “wrong” but rather has a fuzzy in-between value?

I’ve got an answer for that as well–albeit not well developed at all–in the form of an argument that the use of fuzzy predicates requires an infinite heierarchy of meta-meta-fuzzy predicates, and that this is a problem because it turns out to make fuzzy predicates unassertable, while the logic is supposed to explain how they’re assertable. In light of this, I’d argue, a better approach is to think that assertions of baldness assume a clear delineation (and refer indirectly to "how things would turn out once we had the delineation specified) even where none exists. This saddles me with the view that many “baldness”-like predicates turn out to always be false when asserted of a thing, but everyone’s saddled with something.

Does someone think “The particle is in position x” is neither right nor wrong in Quantum Mechanics? I’ve got an argument for that as well. In QM, as far as I can tell from everything I’ve ever read (which it should be stipulated goes no farther than the Popularization level) “the particle is in position x” isn’t “neither right nor wrong” but rather, is simply wrong. The particle has no position. Instead, a set of positions are assigned to the particle, and a probability assigned to each of those positions. And this assignment of probabilities to positions is perfectly well defined. (And the set of positions may in some cases be null.)

So while I find it difficult to argue for the assumptions I gave above, I also find nothing in any of the traditional “problems” for these assumptions which seems particularly problematic to me. (The baldness problem seems most serious to me. More serious than the QM problem!)

I think that propositions are assertables, (where “assertable” is understood as a function of accuracy and nothing else–throw in other factors for “assertability” and you have to something more complicated of course) and that PL is the best logic for assertables. (Predicate logic, modal logic, etc are all logics of assertables as well, but they go further and model things that are more detailed than assertables. Propositional is the one that just draws boundaries between things that can be asserted and leaves it at that.) Since I think this, I also think that if someone says something (X) “violates” propositional logic, then what they’re saying implies that X literally can’t be talked about. Yet there they are, typically, talking about it–so I go in and figure out how PL does apply to what they’re talking about after all.

What “violates” means here might not be clear–but then the burden is on the one making the claim. Certainly I think you’ll never find a set of three assertables which violate distributivity. If the things you’re talking about violate distributivity, then they’re not assertable. Again–that’s not something I know how to argue for yet, but what I’m prepared to do is put the challenge on the table (as I’ve done in this thread)–give me three assertables you think violate distributivity, and I’ll show that they don’t. (Or I’ll show they’re not assertable.)

65

This is of course a valid position to take – analogous to ‘if the things you’re talking about don’t sum to 180°, then they’re not the angles of a triangle’ – but it just means that the things interesting about the world, especially in the domain of quantum mechanics, are not exhausted by what you call ‘assertables’.

It still seems perfectly clear to me – the three following things:

p: The particle is moving to the right
q: The particle is in [-1, 1]
r: The particle is in [-∞, -1) U (1, ∞]

do not distribute. (q v r) is true, so p & (q v r) is p. (p & q) and (p & r) are both false, so (p & q) v (p & r) is similarly false, thus not equivalent to p & (q v r). You may well say that this means that p, q and r are not assertable – though I seem to be able to assert either fairly well --, or that they don’t constitute propositions, though I would think of them that way, as talking about properties of a quantum system, but then this just means that the world isn’t limited to propositions or assertables, just as it isn’t limited to points and lines if you insist that points and lines are only what Euclid’s axioms talk about.

66

I’m going to insist that if the position is undefined, then q v r is not true. The argument for this is: A thing with an undefined position has no position. But a thing that has no position is in no interval. So q is not true, and r is not true.

Remind me where you’re saying that argument fails?

67

The particle’s position is constrained to lie within an interval, but not more sharply than that – it’s just that the interval happens to be infinite, which isn’t very intuitive. This is the reason I offered up a version with a finite interval ∆x; it’s certainly true that the particle’s position lies within ∆x. The particle does have a position, just not a sharp one.

68

So you’re saying:

  1. Every particle lies within some interval.
  2. This particle does not lie within interval x.
  3. This particle does not lie within the complement of interval x.

You’re saying all of the above are true, correct?

69

Not really. The problem is still with failing to take into account the complementary nature of quantum mechanical observables. I’m saying:

  1. a particle’s position is in ∆x, if its momentum is in ∆p
  2. it’s not within ∆x[sub]1[/sub] (which is a sub-interval of ∆x), if its momentum is in ∆p
  3. it’s not within ∆x[sub]2[/sub] (the complement of ∆x[sub]1[/sub] in ∆x), if its momentum is in ∆p

All of these must be true in quantum mechanics; 1. I think you grant me, 2. and 3. follow from the fact that if either were false, uncertainty would be violated.

And yes – this is classically contradictory, because it does not work if we think about things in terms of sets, where properties are well defined independently of one another, and particles have both definite momentum and position. But since quantum mechanics tells us that particles don’t have simultaneously definite momentum and position, we should stop doing that, renounce our bias towards ordinary sets, and use the more appropriate Hilbert subspaces, and thus, quantum logic, instead!

70

Okay, got it.

Taken classically, the three propositions aren’t contradictory–rather, they are a proof that the particle’s momentum isn’t in delta-p.

These four are contradictory, though:

  1. The particle’s momentum is in delta-p
  2. If the particle’s momentum is in delta-p, then its position is in delta-x.
  3. If the particle’s momentum is in delta-p, then its position is not in delta-x[sub]1[/sub]
  4. If the particle’s momentum is in delta-p, then its position is not in the complement of delta-x[sub]2[/sub]

(Where the two sub-x’s are supposed to exhaust x itself, as you said.)

So are you saying there are quantum phenomena concerning which all of the above 4 statements are true?

Have to dash but I think I’ve understood you correctly and I’ll think about how to put a response down later on. But I’ll check in on my Kindle on occasion to see if you wanted to say the above isn’t accurate after all.

I think it will help me if you tell me what exactly “position” is supposed to mean in the above statements.

71

Yes. It’s just the uncertainty principle: there’s a minimum value for the product of ∆x and ∆p; if ∆x and ∆p saturate that value, then it follows that states corresponding to ‘momentum in ∆p, particle in ∆x[sub]1[/sub]’ and ‘momentum in ∆p, particle in ∆x[sub]2[/sub]’ underrun it, and thus, are not states a quantum system can have.

To me, position is where you’ll find the particle if you look. Position may be constrained to (almost) a single point – then it’s sharply defined, and an experiment will find the particle at that point. More typically, it may be defined only ‘up to’ a certain interval; then, experimentally, the particle will be found in that interval.

Of course, an experiment finding the particle within ∆x will find it either within ∆x[sub]1[/sub] or within ∆x[sub]2[/sub]; but concluding from there that the particle had that position prior to the experiment runs into problems, i.e. the famous ‘no hidden variables’-theorems.

Still, this doesn’t mean that you’re forced towards quantum logic in order to describe quantum theory – it’s just more appropriate, or perhaps more natural. You can constrain what you want to call propositions, perhaps, or you can cook up some hidden variables model (which however entails giving up other nice things, like locality or non-contextuality); but with the same justification you might want to continue using classical logic in a quantum context, I could use quantum logic to describe a classical context – the two stand on equal footing (I’d have it easier than you, though, as quantum theory reduces to classical theory in the appropriate limit, and thus, all observables are at least very nearly compatible, and quantum logic just reduces to classical logic – the latter is a special case of the former, for propositions that distribute).

72

Alright, then the four propositions are to be interpreted as follows:

  1. The particle’s momentum is in delta-p
  2. If the praticle’s momentum is in delta-p, then when you look, the particle’s position will be in delta-x.
  3. If the particle’s momentum is in delta-p, then when you look, the particle’s position will not be in delta-x[sub]1[/sub]
  4. If the particle’s momentum is in delta-p, then when you look, the particle’s position will not be in the complement of delta-[sub]1[/sub].

Four contradictory assertions. But they’re not all true. Either 3 or 4 is false, though we won’t know which one til we actually look.

Do you disagree?

And I’m not arguing that the particle had a position prior to the experiment. You’ve explained what you mean by position, and I intend to mean nothing more or less than that. Position is where you’ll find the particle when you look. For all I know, on this notion of position, particles simply don’t have positions when people aren’t looking at them. (Which I know is what many physicists do in fact say.)

73

These two things:

Are contradictory. If in fact either 3 or 4 is false, which we’ll find out when we do the experiment, then they must have a truth value before the experiment, and then you do argue that the particle had a position (more sharply defined than allowed by uncertainty) prior to measurement.

74

Yep, when you get to the nitty-gritty of it the quantum formalism just doesn’t allow you to assign a truth value to 3 or 4. You could perform a measurement that would allow you to assign a truth values to 3 and 4 (and clearly if 3 is true after a measurment then 4 must be false and vice versa), but the measurement would alter the state of the quantum system such that you would then not be able to assign a truth value to 1.

As HMHW says you could assume hidden variables and assert that 1,2,3,4 always do have truth values and they are simply unknown. But hidden variables are outside of the formalism, so as well as having the problems of assuming hidden variables (e.g. non-locality) you haven’t altered the fact that within the formalism 3 and 4 cannot be assigned truth values.

75

Put briefly, the fact that we will find the particle in a certain place upon experiment does not show that the particle currently has a position. I’m surprised if this seems strange or even false to you–in another thread you argued that outside of what we are observing, there is basically no saying what is taking place.

But hold on a sec’–do you mean that a particle’s current position is where we will find it if we look, or do you mean it’s where we would have found it if we had looked?

76

Before you can assert this in a compelling way, you need to give me an explication of what “position” means.

77

To add a second thought to the above:

If “X’s position is Y” =def “When you look, you will find X at Y,” then you’re right–by definition, when you do the experiment and find X at Y, then the proposition “X’s position is Y” is true then and was true all along.

As I said before, either 3 or 4 is false on this understanding of “position.” (So my challenge still isn’t met.)

But on this understanding of position, it is correct to argue that the particle had the position all along, even prior to experiment–for the very definition of “position” here guarantees it. What it means for a thing to have a position, on this definition, at time 1, is just for it to be there when you look at some time or other.

Not really a match for the common understanding of “position” but of course things get weird when quanta are involved.

But is this a good understanding of what “position” means? Or is it too simplified?

78

I don’t think I do need to explicitly define postion. All I need do is assert that it is quantum mechanical observable that has a certain relationship with another quantum mechanical observable (momentum) and the rest follows from the quantum formalism.

79

Well, remember that the challenge I offered (which you’re free to take or not of course!) is to provide three propositions for which distributivity fails. I’m claiming no such three can be found. Others in the thread have explicitly claimed that such triads can–and have–been found.

But if you give me items with undefined terms in them, these are not propositions but mere strings.

80

They’re not undefined, they’re defined within the quantum formalism. We just need to say this is the state space of this quantum mechanical system, these are two quantum mechanical observables x and p, these are the hermitian operators that represent them in the state space and this is the relationship between the two operators. That’s the beauty of abstraction, we needn’t get bogged down in the specifics.

I’m not massively up on formal logic or quantum logic so someone please correct me if I’m wrong, but the failure of distributivity can be seen that there are quantum states which represent some statements, but when you try to apply the principle of distributivity to a statement corresponding to a quantum state you can end up creating statement(s) that don’t correspond to a quantum state.