John Bell Workshop 2014

Are there really two di fferent Bell’s theorems?

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  • #1651

    Hi Travis,

    This is a belated comment to say thanks for your thought-provoking paper.

    In particular, you paper has changed my mind on one point: it is wrong to say, as Howard did, that the theorem Bell proved in 1964 uses [what is now often called] parameter independence. Bell’s locality assumption is more accurately captured by your equation 3, i.e. the 1964 locality assumption cannot even be stated without the determinism assumption.

    Hence we can only speculate on how the 1964 Bell would have formally defined locality in a stochastic hidden variable theory. Since both parameter independence and local causality both reduce to exactly your equation (3) in the deterministic case, the mathematically rigorous part of the 1964 paper does not distinguish between those two possibilities. Hence any arguments that pick out one of those will always involve “pretty creative interpretation”.

    You may not be surprised to hear that I don’t agree with everything you say. Perhaps my strongest disagreement is that I read the conclusion of the argument in Einstein’s autobiographical notes to be ψ-incompleteness (i.e. physicist B on your p7 is wrong) rather than determinism (i.e. physicist A is right). This root of this disagreement is your assumption that the A and B on page 7 are exhaustive possibilities. As far as I can see, Einstein doesn’t argue that they are exhaustive, and indeed they aren’t. For example, a third possibility is:
    C – the real factual situation of the individual system allows us to predict measurement outcomes better than ψ, but still not with certainty.



    Thanks for the comments, Matt. I’m of course happy to hear that you found the paper thought-provoking, and in particular happy to hear that it helped you realize that there is no basis for thinking Bell meant Parameter Independence by “locality”. As should be clear, I completely agree with what you say about the “mathematically rigorous part of the 1964 paper”.

    Of course, as I imagine must also be clear, I would want to stress that this “mathematically rigorous part” is the second part of an overall two-part argument, the first part of which is indeed treated in a disappointingly non-mathematically-rigorous way. But just because the first part is not laid out very rigorously, doesn’t mean it isn’t there. As I have tried to argue, I think it is quite reasonable for an author who takes himself to be adding a further step to something that has been previously established by others, to merely summarize (and cite) the relevant earlier work, without feeling the need to rehearse it in meticulous detail. That’s how I read Bell’s 1964 paper. Of course, with the benefit of hindsight, we can now say that this was a huge mistake: Bell was quite naive to think that others would agree with him about Einstein/EPR having really previously *established* that *only* pre-determined values can explain the perfect correlations locally. But however clear it is that the first part of Bell’s 2-part-argument was presented in a disappointingly sketchy way, however clear it is that Bell was naive or foolish, etc., I think it is equally clear — indeed, perfectly and totally clear — that Bell in 1964 did take Einstein as having previously established that deterministic hidden variables are *required* by locality, such that one could not legitimately claim to avoid nonlocality by rejecting determinism. So if one simply ignores this part one is fundamentally misunderstanding what Bell did in 1964.

    I stress this background point here because it seems extremely relevant to the question of deciding what Bell meant by “locality”, which seems to be your primary interest here. You write that “we can only speculate on how the 1964 Bell would have formally defined locality”. I disagree. I think he provided a quite explicit general definition of locality three times in his 1964 paper, by citing the quoted sentence from Einstein’s Autobiographical Notes. Maybe that’s not formal enough for your tastes (and of course I agree that it was really good progress when Bell later formalized this!) but you can’t reasonably deny that Bell intended this Einstein quote as providing a generalized definition of “locality”. Something like equation (3) in my paper — i.e., what he actually uses in the “mathematically rigorous part” of his paper — is merely an *implication* of this generalized concept of locality to the particular kind of deterministic theory that is under investigation in the “mathematically rigorous part”.

    As to your last point, about whether Einstein’s A and B exhaust the possibilities, I think it’s pretty clear that he intended for the two options to be jointly exhaustive (and that they are!). Here is Einstein: “But what about the single measured value of q? Did the respective individual system have this q-value even before the measurement?” Note that it’s a straightforwardly yes-or-no type question. And then the options A and B are clearly identified with these respective answers: “A. The individual system (before the measurement) has a definite value of q … for all variables of the system, and more specifically, *that* value which is determined by a measurement of this variable.” And then: “B. The individual system (before the measurement) has no definite value of q…”

    So I don’t see how there could be some third possibility. Either (to use more contemporary terminology) there is a hidden variable that determines the measurement outcome, or not.

    I don’t really understand your concrete proposal for a third alternative (“the real factual situation of the individual system allows us to predict measurement outcomes better than ψ, but still not with certainty”) since it is phrased in terms of our ability to predict measurement outcomes. But if what you have in mind is a kind of model in which there is some hidden variable which “tilts the balance” in favor of a certain q value, but without determining it (say, it makes the probability 90% that a certain outcome will be realized if a measurement is performed), I just don’t think that will work. Such a model, if local, would fail to predict *perfect* correlations.




    I agree that Bell was probably taking the Einstein quote to be the definition of locality, and that it is stronger than your equation (3), as it applies to any “real factual situations”, not just pre-determined measurement outcomes. However, to my mind the quote is not totally unambiguous in all cases (particularly when probabilities are involved), which still leaves us with “creative interpretation”.

    Thanks for the comments on physicists A and B, I hadn’t read the autobiographical notes closely enough. You’re quite right that the fundamental distinction is between

    A: q has a predetermined value
    B: q does not a have predetermined value

    which are self-evidently exhaustive options. However, the discussion of physicist B ends with a stronger statement

    B’: The ψ-function is an exhaustive description of the real situation.

    Einstein says that physicist B “will (or, at least, he may) state” B’. My physicist C was supposed to be somebody who believes in B but not B’, which shows that Einstein’s parenthetical proviso is essential.

    I now see that Einstein is actually pretty careful to say that he is only ruling out B’: “B will have to give up his position that that the ψ-function constitutes a complete description of a real factual situation.” There is no mention of B giving up his B-defining belief that q does not have a predetermined value.

    Your interpretation of my C in terms of “tilting the balance” is fine. It is indeed the case that such a model, if locally causal, will not predict perfect correlations. But the autobiographical notes neither give an unambiguous general definition of local causality nor make any mention whatsoever of perfect correlations! Doesn’t it therefore make more sense to read Einstein as giving a perfectly clear and logically watertight refutation of B’ than as giving an uncharacteristically sloppy refutation of B that doesn’t so much as hint at a key premise (perfect correlation)?



    Hi Matt. Yes, I agree with you. Refuting B’ (i.e., refuting Bohr’s “completeness doctrine”) and refuting B (i.e., establishing deterministic hidden variables) are distinct, though of course closely-related. (In case it’s not obvious, here by “refuting” I mean “subject to the assumption of locality”. Post-Bell — i.e., once it is established that locality is just false — one would no longer say that any EPR-ish argument actually proves that the completeness doctrine is false or that determinism is true.)

    Anyway, note that just after the bit you quoted, Einstein goes on to say: “The statistical character of the present theory would then have to be a necessary consequence of the incompleteness of the description of the systems in quantum mechanics, and there would no longer exist any ground for the supposition that a future basis of physics must be based upon statistics.” I read this as implying that rejecting B’ somehow leads inevitably to rejecting B (and accepting A, i.e., a deterministic — not “statistical” — hidden variable model). So I think Einstein is not quite as careful as you suggest, “to say that he is only ruling out B’.”

    In any case, the missing link is just the EPR paper/argument, which does of course explicitly involve the perfect correlations and makes it easy to see that, indeed, locality requires abandoning not just B’ but B. Your type of model “C” provides a nice way to see why. So consider a hidden variable type of theory in which there is some residual, local, randomness. That is, assume each particle (in some appropriately entangled and spatially separated pair) has its own “real factual situation” that generates probabilities for possible outcomes of a measurement (independently, of course, of what is done with the distant particle). Then the probability for a certain joint outcome will just be the product of the individual probabilities (independence). But we can only get perfect correlations out this way (i.e., joint probabilities of either 1 or 0) if all of the individual probabilities are themselves 1 or 0. We really need local hidden variables to *determine* the individual outcomes. Their merely “tilting the balance” in a certain direction (tilting, that is, “past” whatever probabilities would have been assigned on the basis of some quantum state) doesn’t help, if the goal is to explain the perfect correlations locally.

    Of course, there are lots of ways to express that same basic argument. There’s a somewhat convoluted version of it in the actual EPR paper. It’s the same as the “Einstein’s boxes” argument that Einstein (and others) give in various places. And of course — most relevantly here — it’s the argument that Bell sketches in the first paragraph of section 2 of his 1964 paper.



    Hi Travis,

    I guess B’ was (and still is) usually what drives people to B, so that “refuting” B’ certainly undermines the case for B, which I think is what Einstein was getting at in your quote.

    Of course I agree that the EPR paper contains a valid argument from their background assumptions + perfect correlations to determinism, and that Bell attempted to review that argument in his 1964 paper.


    P.S.: I can’t resist putting on the record my strong objection to your parenthetical remark “Post-Bell — i.e., once it is established that locality is just false — one would no longer say that any EPR-ish argument actually proves that the completeness doctrine is false or that determinism is true.”. Whilst (translating “locality” into, say, “local causality”) this is certainly the right thing to say about any argument based on local causality, I personally find EPR-ish arguments based on “localistic” premises weaker than local causality to be among the most compelling reasons to reject the reality of the quantum state. But I know you don’t have much time for any contemporary use of “localistic” assumptions that don’t amount to local causality, so I’ll leave it there.

    Howard Wiseman

    Hi all,
    Just a quick comment to say that, as per my original paper, and my reply paper, I do not at all agree that
    “… we can only speculate on how the 1964 Bell would have formally defined locality in a stochastic hidden variable theory. Since both parameter independence and local causality both reduce to exactly your equation (3) in the deterministic case, the mathematically rigorous part of the 1964 paper does not distinguish between those two possibilities. Hence any arguments that pick out one of those will always involve “pretty creative interpretation”.”

    The reason is simply that we have Bell’s own words for what he meant by locality:
    i) locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty
    ii) Now we make the hypothesis … that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.
    iii) The vital assumption [2] is that the result $B$ for particle 2 does not depend on the setting $a$, of the magnet for particle 1, nor $A$ on $b$.
    iv) [one assumption is that] the setting of one measuring can[not] influence the reading of another instrument, however remote.

    None of these statements require determinism to make sense, and are unequivocally about the effect of an agent choice on one side affecting the measurement result on the other. That is, PI. They certainly cannot be read as a statement of local causality. If Bell had said “locality, or more precisely that the result of a measurement on one system be unaffected by distant events”, or “Now we make the hypothesis … that if the two measurements are made at places remote from one another the orientation of one magnet and the result obtained using it do not influence the result obtained with the other.” things would be very different. But he does not.


    Howard Wiseman

    Sorry, another comment: Travis admits that Bell’s words in the EPR paragraph cannot be read as being a statement of local causality. That is why he says the paragraph “leaves something to be desired,” that it “disappoints” and is “problematic”. If it was all plausible that Bell’s words could be interpreted as being “local causality” then Travis would of course used that to tear down my whole case. But they cannot.

    I hope that convinces you Matt that it requires no special creativity to see that Bell’s localistic notion in 1964, interpreted to be applicable to probabilistic theories (as Bell thought it was), is PI, not LC.


    Hi Howard,

    I can’t deny that the “operationalist” in me jumps to the parameter independence conclusion when reading any of your four quotations. Indeed that is why I didn’t question your interpretation until I read Travis’ paper. But, outside of the deterministic case, that interpretation requires a certain style of thinking about causation in probabilistic theories that may not have been very common in 1964, and is still not universal today. I don’t think it’s fair to assume Bell would necessarily have thought in that way (especially given his general scepticism of operationalist thinking).

    To give one example of an alternative interpretational path, the most fully developed framework for formal causal reasoning is the one described in Pearl’s book Causation, based on directed acyclic graphs. If we (rather naturally) interpret the requirement of the setting not influencing the remote outcome as being the absence of a directed path from the setting to the remote outcome, add in the usual free will assumption (the settings are parentless) and the fact that the setting does influence the local outcome, we are led to the DAG a -> A <- λ -> B <- b which implies local causality.

    I think you agree that, rightly or wrongly, 1964 Bell took EPR to have already established the need for determinism. It therefore seems probable that 1964 Bell simply didn’t think much, if at all, about exactly what his definition of locality would be in a stochastic theory. If somebody had demanded that 1964 Bell explain exactly what the quotes i-iv mean in a stochastic theory, I find both of the following responses plausible:
    1) “Well I guess it means the probability of the outcome, given any hidden variables, doesn’t depend on the remote setting.”
    2) “Good question, let me think about it. [Disappears for a few days.] It means … [local causality]”.

    Even if you judge one response vastly more likely than the other, do you not agree that at least some amount of speculation is involved?


    Howard Wiseman

    Hi Matt,

    I agree with some your comments here. But:

    1. Your discussion about DAGs is not really relevant to the physics community in 1964. Pearle’s first book was only published in the 1980s. No-one in physics was thinking this way in the 1960s. And in any case, the DAG model of causality is one that orthodox quantum mechanics fails to obey. So it would hardly be fair to allow it as an *implicit* assumption in Bell’s 1964 theorem!

    2. Your hypothetical is an interesting one. What would have happened would depend very much on the circumstances (Bell’s mood, how much free time he had, what else he had been thinking about …). So we’ll never know. I said in my 2014 paper: ” have no doubt that anyone familiar with Bellʼs later work could have educed from Bell in 1964 the precise notion of LC … with little effort on eitherʼs part.” I am not building my interpretation on the basis that your option (1) was correct. My approach is not to try to guess what Bell might have come up with given the opportunity, but to make sense of the paper AS IT IS WRITTEN.

    3. I am actually being as *generous* as possible in allowing that Bell’s locality = PI. If someone wanted to argue “No, Bell made an even worse blunder than you think. The only concept of locality he had in mind was one that already presumed determinism.” then that would not change my basic position at all, which is that 1964 locality is not sufficient by itself to obtain a contradiction with QM, and so Bell’s 1964 theorem is not the same as his 1976 theorem.



    Matt, re: #1866, good, it seems like we’re basically on the same page. I suspect there remains some lingering disagreement having to do with whether Bell (in ’64) meant to define locality with his Einstein quotations and/or how similar the Einstein quotation (what Howard calls “no telepathy”) is to (Bell’s later formalized) “local causality”. But I can’t quite put my finger on what gives me the feeling there’s some unresolved dispute there, and, well, I’m a little exhausted by all of this. Oh, I also don’t quite understand what you’re alluding to in the PS. If at some point you want to take time to explain the “EPR-ish arguments based on `localistic’ premises weaker than local causality” business, I’d be quite interested.

    Howard, Bell’s paper “AS IT IS WRITTEN” includes repeated references to Einstein’s Autobiographical Notes. Prima facie, Bell intends these to explain what he means by “locality”. I agree that he says things elsewhere in the paper that could also be interpreted as attempts to define “locality”, and that are definitely different, so there is something of an interpretive puzzle. I’ve repeatedly offered a candidate solution: the Einstein quote captures his *generalized* notion of “locality”, whereas the other remarks merely characterize the specific implication of locality that he applies to deterministic theories. It continues to baffle me that you can claim to be trying to understand Bell’s paper “AS IT IS WRITTEN”, but simply ignore the repeated Einstein citations. You pretty much said, on the other thread, that if you’d had a chance to edit the paper you would have simply removed those citations. It just increasingly seems to me as if you’ve done this editing in your mind and are working hard to interpret that fantasy version of the paper instead of the version Bell wrote.

    I mention this here because, while I agree completely with Matt that there is an ambiguity about how to formalize and generalize some of Bell’s other remarks in the context of not-necessarily-deterministic theories, I don’t think that’s the important issue to be arguing about. It’s a pointless little side show. The crucial question is whether one regards these other remarks (which are unambiguous *only* for deterministic theories), or instead Bell’s repeated quoting of Einstein’s statement from Autobiographical Notes, as capturing his basic, generalized notion of “locality”. What exactly is your argument, based on the paper “AS IT IS WRITTEN”, against taking the Einstein citations as expressing a generalized notion of “locality” that Bell means to endorse?

    Howard Wiseman

    Hi Travis,

    I don’t have anything to add to my reply under my article about why I think Bell gives his own definition of `locality’, and uses the referencing of Einstein as an appeal to authority to justify the reasonableness of making an assumption like this.

    In my formal reply I will give the quotes and both our opinions, and readers can make up their own mind.



    Howard #1872,

    Of course nobody would have put it quite like I did in 1964. But the DAG is an attempt to formalise fairly natural and long-standing way of thinking about causality (as evidence: two different formulations based on functions and probabilities respectively, turn out to be completely equivalent), the basic ideas of which (e.g. correlations need causal explanation) were clearly in the minds of Einstein and Bell. So a somewhat more heuristic version of what I said would not have been completely beyond Bell’s reach in 1964.

    I’m not claiming that any of this is an implicit assumption in Bell’s 1964 theorem, which I still largely agree with your formulation of. I was really just trying to show by counterexample that your four Bell quotes do not unambiguously mean PI (whereas once determinism is assumed they do easily translate into Travis’ equation 3). But it does also suggest an alternative to your theory (i.e. that Bell misunderstood the implications of PI) about why 1964 Bell thought that his remarks and quotations on locality were sufficient to, at least informally, capture the (most important?) assumptions needed to run the EPR argument for determinism.



    Hi all,

    I can’t help wondering if a few tweaks to Howard’s nomenclature might help bridge much of the remaining divide. How about something like this:

    Deterministic Locality (DL): A somewhat clunky name for Travis’ equation 3.
    A 1964 Bell’s theorem: any theorem of the form “There exist quantum phenomena for which there is no theory satisfying [Localist Assumption] and determinism.” where [Localist Assumption] is any reasonable “localist” notion that manifestly reduces to DL in the deterministic case.
    Perhaps the three most important flavours are:
    The 1964 Bell’s theorem [minimal]: Take [Localist Assumption] = DL.
    The 1964 Bell’s theorem [modern]: [Localist Assumption] = PI.
    The 1964 Bell’s theorem [EPR]: [Localist Assmption] = EPR’s premises.
    The EPR+Bell theorem: “There exist quantum phenomena for which there is no theory satisfying EPR’s premises” (Proof: Combine EPR’s argument for determinism with the 1964 Bell’s theorem [EPR].)

    With this terminology in hand, I would hope that the following statements are fairly uncontroversial:

    1. When a typical “realist” says “Bell’s theorem”, they mean either the 1976 Bell’s theorem and/or the EPR+Bell theorem.
    2. When a typical “operationalist” says “Bell’s theorem”, they mean a 1964 Bell’s theorem.
    3. Bell’s purpose in writing the 1964 paper was to establish the EPR+Bell theorem.
    4. Anybody should be able to reconstruct a rigorous version of the EPR+Bell theorem by reading the EPR paper and then Bell’s 1964 paper.
    5. Trying to reconstruct a rigorous version of the EPR+Bell theorem from Bell’s 1964 paper alone would, to put it mildly, require quite a lot of work.
    6. The EPR+Bell theorem is only of historical interest, having been superseded by the 1976 Bell’s theorem which serves essentially the same purpose but has both better-motivated premises and a more transparent derivation.
    7. The 1964 Bell’s theorem [modern] is what a modern physicist is most likely to take away from reading Bell’s 1964 paper in isolation, and is also the most “practically applicable” form of Bell’s theorem.
    8. The 1964 Bell’s theorem [minimal] is the sole theorem stated and proven in full rigour in Bell’s 1964 paper.
    9. Exactly which flavour of Bell’s 1964 theorem is the “official” one is a speculative or perhaps even meaningless question (but recall point 3).

    In particular, I think the first two points give a fuller understanding of why the realists and operationalists tend to talk past each other – even if they agree to discuss only “the theorem proven in 1964”, one can mean the EPR+Bell theorem and the other a 1964 Bell’s theorem!


    Howard Wiseman


    Sure, I totally agree that the DAG intuition is why Bell thought he had proven determinism from predictability. But I guess we both agree that it would be completely unreasonable to allow that as an implicit assumption in a theorem, since orthodox quantum theory violates it. As I’ve always said, I am interested in Bell’s *theorem* of 1964, what he actually proved then.

    Your “Deterministic locality” is usually called “local determinism”, no? It includes both assumptions. “Separable predetermination” was actually the phrase Bell used in 1964. So I don’t think it makes sense to say that Bell assumed determinism and local determinism.

    Apart from that, your terminological suggestions are good. Except that you are leaving out Travis’ preferred option, that locality = the Einstein quote Bell gives. This is *not* the same as EPR’s assumptions. The EPR assumptions are quite convoluted. Read the Appendix of my 2014 paper if you want to get an idea how much work it is to actually prove things. There is no single localistic assumption in EPR, and at least one is implicit (i.e. never recognized by them).

    If we are going to talk about *theorems* then there should be some reasonably rigorous assumptions. Bell is really the first person to do that, when he writes down A(a,lambda). EPR come close, but, as I said, they sometimes use localistic assumptions implicitly. Einstein 1949 is clearer about locality, but vague. (He never says what it means for a system to have a “real physical situation”).

    So, from your “uncontroversial” list
    1. agree
    2. agree
    3. I don’t agree. I think he was probably aware that nobody had rigorously set out the assumptions needed for an EPR-argument, and so the only thing he would have claimed as a theorem was his own result from LD. This would explain why he always stated his theorem in terms of two assumptions in 1964 (and even 1971).
    4. I don’t agree. See above about how much work it is to make the EPR assumptions rigorous, and how there are missing assumptions.
    5. Yes. Bell’s version of EPR just doesn’t make sense.
    6. Yes. (Which is why I’m actually more interested in the future of Bell’s theorem, rather than the past, and I’ll have another paper out soon on causation).
    7. Yes. (And this is the message realists have to understand).
    8. Yes.
    7. Yes, but given I disagree with your point 3. So with my argument that Bell clearly intended his several-times-state notion of “locality” to be applicable to non-deterministic theories, I think my case for the [modern] version is very strong.



    Hi Howard,

    I agree with your first paragraph.

    “Local determinism” (LD) indeed encompasses everything you need for the 1964 Bell’s theorem [minimal]. I was attempting to reserve DL just for the formulation of locality once determinism is already in place. I think it makes phrases like “any reasonable “localist” notion that manifestly reduces to DL in the deterministic case” a bit easier to get your head round than the (formally equivalent) phrase using LD. But it’s not that important.

    I guess 7 out of 9 isn’t bad. Perhaps I’ll have a go at writing out an actual statement and proof of the EPR+Bell theorem some time – I’m hopeful it would be a bit less painful than you imagine (the difference from your appendix treatment would probably be that I’d use a different choice of primitive concepts).


    Howard Wiseman

    Well of course if you use different assumptions you can do the EPR argument very simply. The challenge is to read the EPR paper, and try to extract a rigorous argument from it, phrase by phrase, word by word, even. Or maybe I don’t understand what you mean by “primitive concepts”.

    Glad you agree with my own first paragraph. But I disagree with it. 🙂 What I should have said is that Bell knew he had an intuitive argument for determinism from predictability, not that he thought he had a proof.


    Hey, I agree with Howard about something! Namely: the actual EPR paper is quite convoluted. Re: Matt’s #s 4 and 5, I actually think somebody would have a much easier time reconstructing a rigorous version of EPR+Bell by reading Bell alone (and perhaps the cited Einstein!) than by reading Bell plus EPR. Bell’s recapitulation of the EPR-ish argument, while imperfect, gives a much clearer sense of “how it basically goes” than the EPR paper itself.

    Otherwise, I can agree with Matt’s 9 uncontroversial items, and I look forward to his writeup of EPR+Bell.

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