Is Bell's Logic Aimed at Decoupling Correlated Outcomes in Quantum Mechanics?

[Gordon Watson]

I am hoping it may be helpful to separate Bell's logic from Bell's mathematics
https://www.physicsforums.com/showthread.php?t=406372.

Understanding one may better help us understand the other.

In Bell's Bertlmann's socks paper (http://cdsweb.cern.ch/record/142461/files/198009299.pdf), page 15, second paragraph, he says:

To avoid the inequality, we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other end.

Thank you Bill.

In the language that is evolving at "Understanding Bell's mathematics", https://www.physicsforums.com/showthread.php?t=406372, we have Alice with outcomes G or R (detector oriented a), Bob with outcomes G' or R' (detector oriented b).

H specifies an EPR-Bell experiment.

λ represents Bell's supposed [page 13] variables "which, if only we knew them, would allow decoupling ... " [of the outcomes].

Question: Why would Bell want to decouple outcomes which are correlated? Is he too focussed on separating variables?

Bell's λ would allow Bell to write -- consistent with with his (11) --

(11a) (P(GG'|H,a,b,λ) = P1(G|H,a,λ) P2(G'|H,b,λ).

So Bell's logic, as cited above in bold, leads him to suggest that

(11b) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a)

would avoid some well-known inequalities.

I do not follow Bell's logic. I do not see that his move avoids any inequalities.

Note 1: a and b are not signals.

Note 2: Probability theory, widely seen as the logic of science, would have --

(11c) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G).

So, by comparison [Bell's (11b) with (11c)], Bell's (11b) and his logic is equivalent to dropping G from the conditionals on G'.

Which is equivalent to saying that G and G' are not correlated?

 

[ThomasT]

I am hoping it may be helpful to separate Bell's logic from Bell's mathematics
https://www.physicsforums.com/showthread.php?t=406372.

Understanding one may better help us understand the other.



Thank you Bill.

In the language that is evolving at "Understanding Bell's mathematics", https://www.physicsforums.com/showthread.php?t=406372, we have Alice with outcomes G or R (detector oriented a), Bob with outcomes G' or R' (detector oriented b).

H specifies an EPR-Bell experiment.

λ represents Bell's supposed [page 13] variables "which, if only we knew them, would allow decoupling ... " [of the outcomes].

Question: Why would Bell want to decouple outcomes which are correlated? Is he too focussed on separating variables?

Bell's λ would allow Bell to write -- consistent with with his (11) --

(11a) (P(GG'|H,a,b,λ) = P1(G|H,a,λ) P2(G'|H,b,λ).

So Bell's logic, as cited above in bold, leads him to suggest that

(11b) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a)

would avoid some well-known inequalities.

I do not follow Bell's logic. I do not see that his move avoids any inequalities.

Note 1: a and b are not signals.

Note 2: Probability theory, widely seen as the logic of science, would have --

(11c) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G).

So, by comparison [Bell's (11b) with (11c)], Bell's (11b) and his logic is equivalent to dropping G from the conditionals on G'.

Which is equivalent to saying that G and G' are not correlated?

Ok, so Bell's logic was flawed. This was demonstated in the thread "Understanding Bell's Mathematics".

This has been known, and demonstrated, years ago.

Bottom line, few people care. If Bell's logic was flawed and if violations of Bell inequalities don't tell us anything about nature then ... so what.

 

[Gordon Watson]

Ok, so Bell's logic was flawed. This was demonstated in the thread "Understanding Bell's Mathematics".

This has been known, and demonstrated, years ago.

Do doubters understand the full implication of Bell's lambda, and therefore the full implication of his logic?

Do supporters?

Probabilistic refutations do not impress his myriad supporters.

The difference might be in how one views the logic attached to Bell's lambda.

Bottom line, few people care. If Bell's logic was flawed and if violations of Bell inequalities don't tell us anything about nature then ... so what.

Ether-logic was flawed. Stomach-ulcer logic was flawed. ...

Much was learned from the related experiments.

Including that the logic was flawed.

C'est la vie.

That's what.

 

[ThomasT]

Do doubters understand the full implication of Bell's lambda, and therefore the full implication of his logic?

Not to devalue your efforts, but my apprehension of the view of the physics community at large (garnered from conversations with dozens of working physicists over the years) is that Bell's theorem just isn't important.

If Bell was right then we have nonlocal or ftl influences that can't be detected or used for any conceivable purpose. If Bell was wrong, well, then he was just wrong. Nothing is affected either way (except wrt the agendas of a very small minority of physics professionals).

Nevertheless, it is satisfying to periodically revisit and dispell myths. And, I think that you and billschnieder have done a nice job in that regard.

I sensed that there was something not quite right about Bell's LR ansatz from the first time I saw it. But, lacking the requisite skills to communicate this clearly, I was only able to talk about my apprehension of it in rather vague terms.

So, I thank you. And don't let my previous post in this thread tarnish your efforts, or diminish the admiration I have wrt your ability to elucidate something which I intuitively saw but was unable communicate.

 

[Gordon Watson]

Not to devalue your efforts, but my apprehension of the view of the physics community at large (garnered from conversations with dozens of working physicists over the years) is that Bell's theorem just isn't important.

If Bell was right then we have nonlocal or ftl influences that can't be detected or used for any conceivable purpose. If Bell was wrong, well, then he was just wrong. Nothing is affected either way (except wrt the agendas of a very small minority of physics professionals).

Nevertheless, it is satisfying to periodically revisit and dispell myths. And, I think that you and billschnieder have done a nice job in that regard.

I sensed that there was something not quite right about Bell's LR ansatz from the first time I saw it. But, lacking the requisite skills to communicate this clearly, I was only able to talk about my apprehension of it in rather vague terms.

So, I thank you. And don't let my previous post in this thread tarnish your efforts, or diminish the admiration I have wrt your ability to elucidate something which I intuitively saw but was unable communicate.

Dear Thomas,

Ok. Thank you. No worries at all. And please ...

Do not devalue your own efforts.

You and your P(AB|H) are the catalysts that prompted me to present my similar intuition, backed by some knowledge of probability theory, etc.

So thank you again,

and prepare for the storm,

Jenni

 

[DrChinese]

Ok, so Bell's logic was flawed.

No, there is no known flaw in his logic. Please provide a peer reviewed reference that states this if you believe I am wrong. As I say over and over again, you must read it in the context he wrote it. If you don't like his derivation, there are plenty of other peer reviewed versions of it available. For example, Mermin. Or Aspect. Or Zeilinger.

Or even better, derive it for yourself. You will see that you can do it a variety of ways. You always use some variation of the following:

a) The setting at a does not affect the outcome at B, and vice versa.
b) P(A)+P(~A)=100%, and all variations of this with A, B and C simultaneously.
c) The QM prediction is cos^(theta).

Folks, please get a grip on this subject. Genovese does a review of Bell tests periodically, and his last review had over 500 peer-reviewed references in 100+ pages.

Research on Hidden Variable Theories: a review of recent progresses,
Marco Genovese (2005)
http://arxiv.org/abs/quant-ph/0701071

Do you seriously think that they just happened to overlook these "flaws" in Bell? If you do, publish a paper on it. Otherwise, I am going to point you back to Forum guidelines on personal theories. If you have a question, ask it. But quit making statements that are your pet opinions.

 

[JesseM]

Note 2: Probability theory, widely seen as the logic of science, would have --

(11c) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G).

So, by comparison [Bell's (11b) with (11c)], Bell's (11b) and his logic is equivalent to dropping G from the conditionals on G'.

Which is equivalent to saying that G and G' are not correlated?

No, it's not equivalent to saying G and G' are not correlated in a general sense, it just means that if you already know λ then learning G will give you no further information about the probability of G'. For example, if G and G' are correlated, but this correlation is explained entirely by a common cause that lies in the region where the past light cones of the two measurements overlap (the common cause might be that the two particles sent to either experimenter are always created with an identical set of hidden variables by the source), then if you already have the information about the common cause contained in λ (which could detail the set of hidden variables associated with each particle, for example) then in this case it will be true that learning G won't change your estimate of the probability of G'. This is precisely the sort of physical logic that Bell was using, and my argument in which λ was made to stand for all facts in the past light cones of the measurement events was an attempt to make this a bit more rigorous.

 

[billschnieder]

No, it's not equivalent to saying G and G' are not correlated in a general sense, it just means that if you already know λ then learning G will give you no further information about the probability of G'. For example, if G and G' are correlated, but this correlation is explained entirely by a common cause that lies in the region where the past light cones of the two measurements overlap (the common cause might be that the two particles sent to either experimenter are always created with an identical set of hidden variables by the source), then if you already have the information about the common cause contained in λ (which could detail the set of hidden variables associated with each particle, for example) then in this case it will be true that learning G won't change your estimate of the probability of G'. This is precisely the sort of physical logic that Bell was using, and my argument in which λ was made to stand for all facts in the past light cones of the measurement events was an attempt to make this a bit more rigorous.

You are wrong, and JenniT is correct dropping G from
(P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G)
means clearly that in the probability space defined by (H, a,b,λ) G and G' are not correlated. In other words under a given set of specific conditions ("H", "a","b","λ"), there will be no correlation between G and G'. It is a simple exercise to see if this is consistent with the EPR situation Bell was attempting to model. Your misunderstanding is fueled by a confusion between functional notation and probability notation. P(G'|H,b,λ,a,G) does not mean P2 is a function of (H,b,λ,a,G). It simply means the specific conditions (H,b,λ,a,G) define the probability space in which P(G') is calculated.

 

[JesseM]

You are wrong, and JenniT is correct dropping G from
(P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G)
means clearly that in the probability space defined by (H, a,b,λ) G and G' are not correlated. In other words under a given set of specific conditions ("H", "a","b","λ"), there will be no correlation between G and G'.

Yes, but there can still be a correlation in the total probability space even if there is no correlation in any subset of trials where ("H", "a","b","λ") all have some fixed value. You already showed that you understood this distinction in this post on your old thread when you said:

In case you are not sure about the terminology, in probability theory, P(AB) is the joint marginal probability of A and B which is the probability of A and B regardless of whether anything else is true or not. P(AB|H) is the joint conditional probability of A and B conditioned on H, which is the probability of A and B given that H is true.

In the same way, P(GG') may be different than P(G)*P(G') in our probability space (so the 'marginal probabilities' of G and G' are correlated), while at the same time P(GG'|H,a,b,λ) = P(G|H,a,b,λ)*P(G'|H,a,b,λ).

It is a simple exercise to see if this is consistent with the EPR situation Bell was attempting to model.

In the EPR situation only the marginal probabilities, along with conditional probabilities which condition on observable conditions like the detector settings, are actually measurable. The λ is defined to represent hidden-variable states so conditional probabilities involving that term cannot be directly observed, although we can reason theoretically about some general properties of these conditional probabilities that must be true under the theoretical assumption of local realism.

Your misunderstanding is fueled by a confusion between functional notation and probability notation.

What misunderstanding would that be? You disagree that Bell's equation allows there to be a (marginal) correlation between G and G'? If not, that's all I was saying, and it should have been quite obvious from the context that I was talking about a marginal correlation and not a correlation conditioned on λ.

P(G'|H,b,λ,a,G) does not mean P2 is a function of (H,b,λ,a,G). It simply means the specific conditions (H,b,λ,a,G) define the probability space in which P(G') is calculated.

I never used any words like "is a function of", so I have no idea what this criticism is referring to. And I don't know that "function of" has some precise definition in probability theory that forbids you from saying that the expression P(A|B) "is a function of" A and B (even if there was this would be more of a semantic quibble than a substantive critique). Also, I think it is legitimate to say that the probability space used to calculate P(G'), which includes events where H,b,λ,a,G take different values, is the same as the probability space used to calculated P(G'|H,b,λ,a,G), where we assume H,b,λ,a,G all have some known values. It's just that the expression P(G'|H,b,λ,a,G) indicates we must look at a subset of events in the larger sample space where H,b,λ,a,G take these known values, and look at the the frequency of G' within that subset.

 

[Gordon Watson]

No, it's not equivalent to saying G and G' are not correlated in a general sense, it just means that if you already know λ then learning G will give you no further information about the probability of G'. For example, if G and G' are correlated, but this correlation is explained entirely by a common cause that lies in the region where the past light cones of the two measurements overlap (the common cause might be that the two particles sent to either experimenter are always created with an identical set of hidden variables by the source), then if you already have the information about the common cause contained in λ (which could detail the set of hidden variables associated with each particle, for example) then in this case it will be true that learning G won't change your estimate of the probability of G'. This is precisely the sort of physical logic that Bell was using, and my argument in which λ was made to stand for all facts in the past light cones of the measurement events was an attempt to make this a bit more rigorous.

Thank you JesseM. I appreciate this detail. I have some basic questions.

1. Could you define for me (briefly) and distinguish Bell's use of the words observable and beable? Is Bell's lambda an observable or a beable or something else -- like what? What size set might it be?


2. If Bell's lambda were an infinite set of spinors (because we want a realistic general "Bell" vector that applies to both bosons and fermions), then wouldn't we need aG to define the infinite subset of spinors that were relevant to the applicable conditional? You seem to require that we would know a priori which of that infinite set satisfied this subset aG conditional? This a priori subset being the lambda you would require here?


3. Beside which, if aG were implicit in your lambda, its restatement/extraction by me would be superfluous and not change the outcome that attaches to the disputed conditional? Note that you seem to require lambda to be an undefined infinite set, perhaps not recognizing that it is an infinite subset (selected by the condition aG, out of your undefined infinite set) which is relevant here?

4. As with the ether experiments and their outcome, don't Bell-tests show that Bell's supposition re Bell's lambda is false?

Thank you.

 

[billschnieder]

Yes, but there can still be a correlation in the total probability space even if there is no correlation in any subset of trials where ("H", "a","b","λ") all have some fixed value.

I'm not sure you understand the point at all.

if A and B are correlated marginally, then P(AB) > P(A)P(B)

If you collect data such that your data samples the entire probability space (that is what marginal probability is) , then the above expression is true. It is no different that defining "Z = All possible facts in the universe", and writing P(AB|Z). You are still dealing with a marginal probability.

Now, if there exists a certain factor C within Z such that the set (C, notC) is the same as Z, then if we say C is the cause of the marginal correlaction between A and B, it means within C, under certain circumstances it maybe correct to write P(AB|C) = P(A|C)P(B|C). It means that C screens-off the marginal correlation between A and B.

However, and please pay attention to this part, this means if data is collected in the full universe fairly sampling both situations where C is true and situations where C is not true (or notC is true), a correlation will be observed in the data, and if data is collected only under situations where C is True, there will be no correlation in the data.

1) You see therefore why it makes no sense to define C as vaguely as you are defining it
2) If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data. So no matter how hard you try, all your observed frequencies will always be conditioned on the actual contexts that created the data, whether you like it or not.
3) Following from (2), if hidden elements of reality exist, then the correlations observed in experiments exist even when conditioned on C. Because it is impossible to not condition the results on C. Therefore equations such as P(AB|C) = P(A|C)P(B|C) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(AB|C=42)
4) For the type of situation Bell is modelling, where he is assuming that hidden elements of reality exist. Marginal probabilities do not come into the picture because the existence of hidden elements of reality MUST always be a conditioning factors.
5) Therefore I hope it is clear to you now why it makes no sense to say the observed EPR correlations are caused by the hidden variables and yet write an equation such as P(AB|C) = P(A|C)P(B|C) in which means if the hidden elements of reality C are realized, no correlation between will be observed between A and B.

Again, just in case it wasn't clear the first time, by writing P(AB|C) = P(A|C)P(B|C), you are saying if the hidden elements of reality C exist, then no correlation will be observed between A and B. Yet Bell starts out by assuming that hidden elements of reality exists. Just because you drop C from the LHS of P(AB|C) does not enable you to escape this trap. The only escape is for you to show how it is possible in a real experiment to collect data fairly for situations where C is true and also for situations where C is not True.

In case you still insist on your approach, could you answer one simple question.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

 

[JesseM]

I'm not sure you understand the point at all.

if A and B are correlated marginally, then P(AB) > P(A)P(B)

If you collect data such that your data samples the entire probability space (that is what marginal probability is) , then the above expression is true. It is no different that defining "Z = All possible facts in the universe", and writing P(AB|Z). You are still dealing with a marginal probability.

Now, if there exists a certain factor C within Z such that the set (C, notC) is the same as Z, then if we say C is the cause of the marginal correlaction between A and B, it means within C, under certain circumstances it maybe correct to write P(AB|C) = P(A|C)P(B|C). It means that C screens-off the marginal correlation between A and B.

However, and please pay attention to this part, this means if data is collected in the full universe fairly sampling both situations where C is true and situations where C is not true (or notC is true), a correlation will be observed in the data, and if data is collected only under situations where C is True, there will be no correlation in the data.

1) You see therefore why it makes no sense to define C as vaguely as you are defining it

No, I don't. If local realism is true, then the random variable representing "the state of all local physical variables in the past light cones of the measurements" will have a perfectly well-defined value on each measurement. Are you planning on answering the question I asked you in my most recent post to you (post #80) on the "Understanding Bell's Mathematics" thread? Again:

While it makes sense to calculate the probability of an event at a space-time point given a specific set of well defined physical facts, I do not agree that it makes sense to calculate the probability of an event at a given space-time point conditioned on the vague concept of all possible values of all possible physical facts that could be realized at that position.

Why not? In any well-defined local realist fundamental theory, the complete set of possible physical facts that obtain at a given point in spacetime should be well-defined, no? If your fundamental theory involves M different fields and N different particles and nothing else, then by specifying the value of all M fields at a given point along with which (if any) of the N particles occupies that point, then you have specified every possible physical fact at that spacetime point. As long as there is some fundamental theory of physics and it is a local realist one, then the theory itself gives a precise definition of the sample space of distinct physical possibilities that can obtain at any given point in spacetime--do you disagree?

Of course, if you want a simpler example of a "C" you could also consider post #18 from the thread where we first got into the Bell discussion, either the scratch lotto card analogy or the flashlight analogy. Do you disagree that in both those examples, there would be a correlation in the marginal probabilities of different measurement outcomes, but if C represented the value of the "hidden" facts on each trial (the hidden fruits behind the cards in the lotto analogy, the fact about whether Alice got flashlight X or flashlight Y in the flashlight example), then conditioned on C there would be no correlation in measurement outcomes?

2) If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data.

C is a random variable which can take multiple values on different trials. The simplest type of hidden-variables theory would just say that on each trial, the particles have hidden variables that predetermine their spins on each of the measurement settings. For example, if there are three measurement settings a=0 degrees, b=120 degrees, and c=240 degrees, then on each trial the random variable C might take anyone of the 8 values c1, c2, c3, c4, c5, c6, c7, c8, defined as:

c1: spin-up on a, spin-up on b, spin-up on c
c2: spin-up on a, spin-up on b, spin-down on c
c3: spin-up on a, spin-down on b, spin-up on c
c4: spin-up on a, spin-down on b, spin-down on c
c5: spin-down on a, spin-up on b, spin-up on c
c6: spin-down on a, spin-up on b, spin-down on c
c7: spin-down on a, spin-down on b, spin-up on c
c8: spin-down on a, spin-down on b, spin-down on c

(note that these are directly analogous to the eight possible hidden-fruit states on the cards in the scratch lotto card analogy)

According to this type of hidden-variables theory, do you deny that on each trial C would have one of these values, and the complete sample space would include trials with all possible values of C?

So no matter how hard you try, all your observed frequencies will always be conditioned on the actual contexts that created the data, whether you like it or not.

Do the "actual contexts" include hidden variables? For example, consider again the flashlight analogy:

suppose we have two identical-looking flashlights X and Y that have been altered with internal mechanisms that make it a probabilistic matter whether they will turn on when the switch is pressed. The mechanism in flashlight X makes it so that there is a 70% chance it'll turn on when the switch is pressed; the mechanism in flashlight Y makes it so there's a 40% chance when the switch is pressed. The mechanism's random decisions aren't affected by anything outside the flashlight, so whether or not flashlight X turns on doesn't change the probability that flashlight Y turns on.

Now suppose we do an experiment where Alice is sent one flashlight and Bob is sent the other, by a sender who has a 50% chance of sending X to Alice and Y to Bob, and a 50% chance of sending Y to Alice and X to Bob. Let H1 and H2 represent these two possible sets of "hidden" facts (hidden to Alice and Bob since the flashlights look identical from the outside): H1 represents the event "X to Alice, Y to Bob" and H2 represents the event "Y to Alice, X to Bob". Let A represent the event Alice's flashlight turns on when she presses the switch, B represents the event that Bob's flashlight turns on when she presses the switch.

Here, P(A) = P(A|H1)*P(H1) + P(A|H2)*P(H2) = (0.7)*(0.5) + (0.4)*(0.5) = 0.55
and P(B) = P(B|H1)*P(H1) + P(B|H2)*P(H2) = (0.4)*(0.5) + (0.7)*(0.5) = 0.55

Since P(A|B) = P(A and B)/P(B), we must have P(A|B) = (0.7)*(0.4)/(0.55) = 0.5090909...
So you see that P(A|B) is slightly lower than P(A), which makes sense since if Bob's flashlight lights up, that makes it more likely Bob got flashlight X which had a higher probability of lighting, and more likely A got flashlight Y with a lower probability of lighting.

But despite the fact that B does give some information about the probability of A, it is still true that P(A|B and H1) = P(A|H1) = 0.7, since H1 tells us that Alice got flashlight X, and that alone completely determines the probability that Alice's flashlight lights up when she presses the switch, the fact that Bob's flashlight lit up won't alter our estimate of the probability that Alice's lights up. Likewise, P(A|B and H2) = P(A|H2) = 0.4.

So, would "C" include the fact about whether H1 or H2 obtain on each trial? If we do define it this way, do you agree that P(AB|C) = P(A|C)*P(B|C), even though P(AB) is not equal to P(A)*P(B)?

3) Following from (2), if hidden elements of reality exist, then the correlations observed in experiments exist even when conditioned on C. Because it is impossible to not condition the results on C.

I think you've confused yourself with purely verbal, nonmathematical arguments. If you actually examine one of my examples that involve elements hidden from the experimenters (and which help determine the measurement outcomes), you'll see that your general verbal arguments are giving you incorrect conclusions when applied to these examples.

Therefore equations such as P(AB|C) = P(A|C)P(B|C) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(AB|C=42)

Again, the whole idea is that the variable can take different values on different trials, like in the flashlight example where the random variable H could take value H1 or H2 on different trials? Do you disagree that this was meant to be true of Bell's λ, since he actually integrated over all possible values of λ in equation (2) in his paper?

5) Therefore I hope it is clear to you now why it makes no sense to say the observed EPR correlations are caused by the hidden variables and yet write an equation such as P(AB|C) = P(A|C)P(B|C) in which means if the hidden elements of reality C are realized, no correlation between will be observed between A and B.

C can take different values, and for any specific value, if you look only at the subset of trials where C took that trial, there will be no correlation between A and B, but if you look at the total collection of trials, there will be a correlation. Of course this is a theoretical conclusion based on the assumption that the universe obeys local hidden variables, since C represents hidden variables, even if such a theory was correct there would be no way for us to actually know the value of C on each trial (which is why it is helpful to think of all equations involving hidden variables as having precise values that would be known by an imaginary omniscient observer).

Again, just in case it wasn't clear the first time, by writing P(AB|C) = P(A|C)P(B|C), you are saying if the hidden elements of reality C exist, then no correlation will be observed between A and B.

Nope, a marginal correlation will be observed between A and B. By writing that equation I'm only saying that if hidden variables exist, then there would be no correlation between A and B in any subset of trials where the hidden variables all took the same value.

In case you still insist on your approach, could you answer one simple question.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

In reality, or under the assumption that we live in a universe with local realist laws? Bell's whole approach is to derive certain inequalities from the assumption of local realism, show these inequalities conflict with actual quantum-mechanical results, and therefore conclude (proof by contradiction) that real-world quantum physics is inconsistent with local realism.

If you're talking about reality, I think Bell's reasoning is correct and quantum mechanics rules out local realism, so I don't think there are any real local hidden variables you can condition measurement outcomes on to make the correlations disappear. If you're talking about what would be true theoretically in a universe obeying local realist laws, then in that case all correlations between spacelike-separated measurements could only be marginal ones, and conditioned on a sufficiently large set of local physical facts in the past light cones of the measurements there could be no correlations between measurements.

 

[JesseM]

Thank you JesseM. I appreciate this detail. I have some basic questions.

1. Could you define for me (briefly) and distinguish Bell's use of the words observable and beable? Is Bell's lambda an observable or a beable or something else -- like what? What size set might it be?

beables represent local hidden variables that are supposed to explain correlations seen in observables, and observables are just facts we can actually measure, like whether a particle gives result "spin-up" or "spin-down" when passed through a Stern-Gerlach device oriented at some angle. Take a look at my scratch lotto card analogy in this post (beginning with the paragraph that starts 'Suppose we have a machine that generates pairs...')--the observables would be the cherries or lemons that Alice and Bob actually see when they pick a single square to scratch, the beables would be the complete set of hidden fruits behind all three squares, which are used to explain why it is that they always find the same fruit whenever they scratch the same box (the assumption being that on each trial, the two cards have the same set of hidden fruits).

2. If Bell's lambda were an infinite set of spinors (because we want a realistic general "Bell" vector that applies to both bosons and fermions), then wouldn't we need aG to define the infinite subset of spinors that were relevant to the applicable conditional?

I don't know much about relativistic quantum theory which is where I think "spinors" appear--my question here would be, are spinors actually local variables associated with a single point in spacetime, or are they defined in some more abstract "space" like Hilbert space?

You seem to require that we would know a priori which of that infinite set satisfied this subset aG conditional?

Not clear what you mean by "this subset aG conditional", can you elaborate?

3. Beside which, if aG were implicit in your lambda

What do you mean by "implicit in"? Do you mean that the measurement a and the result G can be determined from the value of lambda? If so, I'm not sure why you think that, the measurement can be random and I told you in post #82 on this thread that the probabilities of different outcomes may be other than 0 or 1 in a probabilistic local realist theory.

Note that you seem to require lambda to be an undefined infinite set

Nothing "undefined" about it, as I said to billschnieder:

In any well-defined local realist fundamental theory, the complete set of possible physical facts that obtain at a given point in spacetime should be well-defined, no? If your fundamental theory involves M different fields and N different particles and nothing else, then by specifying the value of all M fields at a given point along with which (if any) of the N particles occupies that point, then you have specified every possible physical fact at that spacetime point. As long as there is some fundamental theory of physics and it is a local realist one, then the theory itself gives a precise definition of the sample space of distinct physical possibilities that can obtain at any given point in spacetime--do you disagree?

4. As with the ether experiments and their outcome, don't Bell-tests show that Bell's supposition re Bell's lambda is false?

Like I said to billschnieder in the last post, the basic logic of Bell's argument is a proof-by-contradiction. He starts only by assuming that the universe obeys local realist laws, and then shows that they produce predictions about the statistics of Aspect-type experiments that contradict the predictions (and experimental results) of QM, and so concludes that QM is incompatible with local realism (so if QM's predictions hold up to experimental tests, our own universe must not obey local realist laws).

 

[billschnieder]

JesseM,
With all due respect, I do not take you seriously because you completely ignore everything I say. You keep repeating points I have debunked and expect me to keep repeating myself. You keep dragging tangential discussions from thread to thread and I don't bother going down that rabbit trail because it hijacks the thread. You redefine everything I say so that it means something different and then you use the strawman to purport to be arguing against what I said. The recent one is your claim that C is a random variable. It is NOT.


My simple response to everything in your last post is that C is NOT a random variable so you are arguing against yourself. At best, A and B may be considered random variables but C is definitely positively NOT a random variable. It is a specific conditioning factor. You keep repeating the fautly idea that C has multiple values. C as it appears in the equation I wrote, is a specific set of elements of reality. C is NOT all possible sets of elements of reality. It can not be because some of those sets will be mutually exclusive and you can not condition a probability on mutually exclusive factors. As I have explained, in calculating a conditional probability everything after the "|" is assumed to be true simultaneously. It is therefore fallacious to suggest that a probability can be conditioned on mutually exclusive factors at the same time. Until you understand this simple point, you will be totally confused by everything I'm saying.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

You did not answer. Clearly correlations are calculated in Aspect type experiments otherwise there will be nothing to compare to Bell's inequalities. Are those correlations marginal or conditional on the hidden elements of reality causing them. The answer should be simple, although it is a trick question.

In reality, or under the assumption that we live in a universe with local realist laws?

Clearly you seem to be confused about what it means to make an assumption. Once you make the assumption, that the universe is local realistic, there is (and should) no longer any be any distinction between reality and local reality in all your equations. If you continue to make a distinction and have one equation for reality and a different one for local reality, then your claim of having assumed local reality is false.

So let us try again.

Are the correlations calculated in Aspect type experiments marginal or conditional?

 

[JesseM]

JesseM,
With all due respect, I do not take you seriously because you completely ignore everything I say. You keep repeating points I have debunked and expect me to keep repeating myself. You keep dragging tangential discussions from thread to thread and I don't bother going down that rabbit trail because it hijacks the thread. You redefine everything I say so that it means something different and then you use the strawman to purport to be arguing against what I said. The recent one is your claim that C is a random variable. It is NOT.

You never gave a clear definition of what C is. You did say "If hidden elements of reality C exist" which suggests it should be a random variable, since the value of the hidden variables would differ from one trial to another. But then you said "C will define the actual context of the data" which perhaps suggests you intended C to be a mere specification of the sample space of possible combinations of values that could obtain on any trial, similar to the way you were defining "z" in posts 55, 70, and 70 on the other thread. If you want C not to be a random variable, but simply a specification of the sample space which should be the same on every trial, then it's exceedingly weird notation to actually include that as a symbol in your equations, in any standard probability equation the sample space will be defined beforehand and it'll then be implicit in all the equations rather than represented using a symbol. And if your C is not a random variable, then it has nothing to do with Bell's λ or with the symbols that have played a similar role in my equations like H, since those symbols were random variables--do you disagree?

If you agree those were defined as random variables that could take different values on different trials, then perhaps you can see why your whole discussion becomes a totally irrelevant tangent: you triumphantly declared that P(AB|C)=P(A|C)*P(B|C) also implies P(AB)=P(A)*P(B) as if this somehow discredited my earlier arguments about there being a marginal correlation but no conditioned correlation, but while this might be true under your definition of C, it in no way shows there is anything wrong with my argument that P(GG'|H,a,b,λ)=P(G|H,a,λ,b)*P(G'|H,b,λ,a,G) and yet P(GG') is not equal to P(G)*P(G') (i.e. G and G' are marginally correlated but uncorrelated conditioned on H,a,b,λ), since here λ is obviously meant to be a random variable. Nor does it show there is anything wrong with Bell's equation (2) in his original paper, where λ was also a random variable. So sure, I agree with your statement that if C is a non-variable that simply represents the sample space, then your arguments in post #11 are correct, but it would be completely incoherent to use those arguments to try to discredit my arguments or Bell's, since your C is defined in a completely different way than the λ and H that appeared in the equations.

You keep repeating the fautly idea that C has multiple values.

"Faulty" only under your definition of C, which you did not actually make clear in your previous post. So, now I agree that your C cannot have multiple values, but hopefully you agree that Bell's λ is a random variable that does take multiple values (as made clear by the fact that he is integrating over all values of λ in equation 2), and likewise that when I and other defenders of Bell write equations like P(AB|H)=P(A|H)*P(B|H), the symbols playing a similar role there like H are also intended to be random variables that can take different values on different trials (different points in the sample space). Do you disagree with this?

As I have explained, in calculating a conditional probability everything after the "|" is assumed to be true simultaneously. It is therefore fallacious to suggest that a probability can be conditioned on mutually exclusive factors at the same time.

Hold on, are you saying that regardless of your own personal definition of C, there is something incorrect in general about writing a conditional probability equation where the conditioning factor is a random variable that can take different values on different trials? For example, if H is a random variable that can take values H1 and H2 (as in my flashlight example), and A is another random variable that can take values A1 and A2, are you claiming it would then be incorrect to write the equation P(A and H) = P(A|H)*P(H)? If so you are badly confused, when a probability equation is written with random variables, all that means is that the equation should hold for each possible combination of specific values of the random variables--for example, P(A and H) = P(A|H)*P(H) is true as long as it's true that P(A1 and H1)=P(A1|H1)*P(H1) and P(A1 and H2)=P(A1|H2)*P(H2) and P(A2 and H1)=P(A2|H1)*P(H1) and P(A2 and H2)=P(A2|H2)*P(H2). If all four of those equations involving all possible combinations of specific values of A and H are true, that means the general equation P(A and H) = P(A|H)*P(H) is also true.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

You did not answer.

I made clear that the question wasn't sufficiently well-defined when I asked for the clarification "In reality, or under the assumption that we live in a universe with local realist laws?"

Clearly correlations are calculated in Aspect type experiments otherwise there will be nothing to compare to Bell's inequalities.

Yes, this would be "reality".

Are those correlations marginal or conditional on the hidden elements of reality causing them. The answer should be simple, although it is a trick question.

The correlations measured in real experiments may be conditional on detector settings, but they are not conditional on any hidden elements of reality, regardless of whether such hidden elements exist or not (since even if they do exist we don't know their value on each trial so we can't measure a frequency which is conditioned on them).

Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables. Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.

Clearly you seem to be confused about what it means to make an assumption. Once you make the assumption, that the universe is local realistic, there is (and should) no longer any be any distinction between reality and local reality in all your equations.

None of the equations in Bell's derivation of the Bell inequality involve statements about what is true in our reality, they are only about what should theoretically be true in a universe obeying local realist laws. Some of these equations actually involve the hidden local variables, while others involve only things that would be observed by theoretical experimenters in such a theoretical universe (which can then be compared with the observations of real experimenters in the real universe), but either way we are assuming a universe whose laws are locally realistic. If you ever thought I was saying anything different, it's you who was confused about the basic logic of Bell's proof.

 

[DrChinese]

You never gave a clear definition of what C is. ...

Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables. Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.

None of the equations in Bell's derivation involve statements about what is true in our reality, they are only about what should theoretically be true in a universe obeying local realist laws. Some of these equations actually involve the hidden local variables, while others involve only things that would be observed by theoretical experimenters in such a theoretical universe (which can then be compared with the observations of real experimenters in the real universe), but either way we are assuming a universe whose laws are locally realistic. If you ever thought I was saying anything different, it's you who was confused about the basic logic of Bell's proof.

Well said, JesseM! I wish more people would listen to these words.

The local realist is making an "extra" assumption (or two). If the term local realist is to mean anything, then such assumption(s) should be spelled out. It is then subject to verification or rejection... or in this case to be shown to be incompatible with something else (QM).

I think any reasonable local realist can come up with a mathematical constraint or requirement that models locality and realism. Once that is agreed upon, I think the Bell program can be applied and the conclusion will simply match Bell. On the other hand, failure to provide such constraints for locality and realism would be tantamount to accepting the result prima facie.

 

[billschnieder]

You never gave a clear definition of what C is. You did say "If hidden elements of reality C exist" which suggests it should be a random variable, since the value of the hidden variables would differ from one trial to another.

Did you completely ignore my statement that the set ("C", "notC") is equivalent to Z. Since you insist that C must have multiple values, can you explain what "notC" will represent according to your understanding of what C is supposed to entail. Give a short example of the different values you think could represent C and at the same time specify clearly what "notC" represents in your example -- please no 15-page scratch lotto cards examples that will require me to respond to every sentence because I will just ignore it.

And if your C is not a random variable, then it has nothing to do with Bell's λ or with the symbols that have played a similar role in my equations like H, since those symbols were random variables--do you disagree?

Yes I disagree.

you triumphantly declared that P(AB|C)=P(A|C)*P(B|C) also implies P(AB)=P(A)*P(B)

Go back and read it. I said no such thing.

Nor does it show there is anything wrong with Bell's equation (2) in his original paper, where λ was also a random variable.

λ in Bell's paper is supposed to represent the EPR "elements of reality" which cause in the observed correlations. EPR elements of reality are not random variables no matter how loudly you shout that they are.

"Faulty" only under your definition of C, which you did not actually make clear in your previous post. So, now I agree that your C cannot have multiple values, but hopefully you agree that Bell's λ is a random variable that does take multiple values (as made clear by the fact that he is integrating over all values of λ in equation 2)

That is circular reasoning and I can use the same point against you -- each term in the integral can only have a single value of λ so in fact by integrating you are adding P(AB|a,b,λ1) + P(AB|a,b,λ2) + ... + P(AB|a,b,λn) where n represents the number of possible realizations of your λ. You still can not escape the fact that the conditioning elements can never be as broad as λ. In each case in which a joint conditional probability is calculated, λn is specific and definitely not a random variable. That is why I told you repeatedly that in calculating a conditional probability you can not condition on a vague concept such as λ with multiple values. This is the same reason why in such a case, the LHS of Bell's equation (2) can not be a probability conditioned on the vaquely defined λ. It clearly looks like a marginal probability.

So the claim that his inequalities derived from such an expression are based on the assumption that hidden variables exist is specious.

likewise that when I and other defenders of Bell write equations like P(AB|H)=P(A|H)*P(B|H), the symbols playing a similar role there like H are also intended to be random variables that can take different values on different trials (different points in the sample space). Do you disagree with this?

Absolutely disagree, see my response above, H can not have multiple values in that expression. If you want to write it as P(AB)=P(A|H1)*P(B|H1) + P(A|H2)*P(B|H2) + ... + P(A|Hn)*P(B|Hn) go ahead, but don't deceive yourself and others that you are calculating P(AB|H).

BTW The sample space for H1 is different from the sample space for H2 etc. They are not part of the same sample space. If H1 and H2 are mutually exclusive, your so-called H-sample space is undefined.

Hold on, are you saying that regardless of your own personal definition of C, there is something incorrect in general about writing a conditional probability equation where the conditioning factor is a random variable that can take different values on different trials?

I am definitely saying saying if H can take on multiple values H1 and H2 it is OK to write P(AB|H1) and P(AB|H2), but when you write P(AB|H) the only meaning here is that H is a placeholder for a specific value of H not all possible values of H simultaneously. ie, each concrete value of P(AB|H) you could ever calculate can only be valid for a specific H not the vague concept of being conditioned on "the H variable" or all values of H. For example if H represents the face of a coin and has two possible realizations in a toss "heads" or "tails", writing P(...|H) in which H includes all possible values is no different than writing P(...|heads, tails) But since heads and tails are mutually exclusive, your probability is undefined and meaningless if you insist on that definition. If H1 and H2 are mutually exclusive P(AB|H1H2) is undefined and meaningless. Therefore P(AB|H) can not imply that H is a variable with multiple values in a single expression.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

The correlations measured in real experiments may be conditional on detector settings, but they are not conditional on any hidden elements of reality, regardless of whether such hidden elements exist or not (since even if they do exist we don't know their value on each trial so we can't measure a frequency which is conditioned on them)

Can you explain to me how Aspect and others made sure in their experiments that IF hidden elements of reality exist, then the measured data will not depend on the their presence.

In other words, is it possible for hidden elements of reality to exist and not exist at the same time? Isn't it obvious that IF hidden elements of reality exist, then they govern the results observed in Aspect type experiments?

IF hidden elements of reality exist, then it is impossible for Aspect et al to collect data under circumstances in which hidden variables do not exist. Therefore your statement that the correlations they observed is "regardless of whether such hidden elements exist or not" is far off base.

Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables.

In other words you are saying the correlations Bell is calculating are those that should be seen in our universe if experiments are performed such that those variables (which we have assumed exist), should not affect the results.
Now can you point me to an experiment in which the experimenters made sure that IF hidden elements of reality exist, they should not affect the data measured? By your own admission, those are the only data that are comparable to Bell's inequalities.

Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.

Could you explain how Aspect et al made sure their data was collected in such a way that IF hidden elements of reality exist, they should not influence the results.

None of the equations in Bell's derivation of the Bell inequality involve statements about what is true in our reality, they are only about what should theoretically be true in a universe obeying local realist laws.

Once the assumption is made that our universe is local realistic, the distinction you are trying to make is artificial.

Some of these equations actually involve the hidden local variables, while others involve only things that would be observed by theoretical experimenters in such a theoretical universe (which can then be compared with the observations of real experimenters in the real universe), but either way we are assuming a universe whose laws are locally realistic. If you ever thought I was saying anything different, it's you who was confused about the basic logic of Bell's proof.

Until and unless you can demonstrate that the "theoretical experiments" are comparable to actual experiments performed in our universe, you can not use Bell's equations to say anything about our universe.

 

[JesseM]

Did you completely ignore my statement that the set ("C", "notC") is equivalent to Z.

I guess I did miss that when reviewing your post, but since you didn't clearly define C I have to guess at your meaning. You did say "If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data", which suggested you believed "notC" would be some impossible situation that would never hold on any possible trial, so focusing on that statement, I assumed that C represented some general facts which would be true on every possible trial (for example, a statement about all combinations of values for the hidden variables allowed by the laws of physics, without specifying which combination is found on any particular trial). I guess if this were the case "notC" would represent some logical possibilities which are ruled out as impossible by the actual laws of physics, like all the combinations of values for the hidden variables which were logically possible but not physically possible given whatever laws of physics govern these variables.

But it's true, this interpretation of your words doesn't really allow me to make sense of your claim that if Z="All possible facts in the universe" and that "if there exists a certain factor C within Z such that the set (C, notC) is the same as Z" and "C is the cause of the marginal correlaction between A and B". How can a set of opposite possibilities which can't both be simultaneously true be equivalent to "all possible facts in the universe"? I can't really make sense of this, please either define your terms more clearly, or give me some simple "toy model" of a universe where very few facts determine some outcomes A and B, like the 8 possible combinations of hidden fruits on each trial in the lotto card analogy, and explain precisely what C and notC and Z would represent in this toy model. You don't have to use any of the analogies I've already come up with for the toy model, but some specific example would certainly help in making your terms more intelligible to me.

Since you insist that C must have multiple values, can you explain what "notC" will represent according to your understanding of what C is supposed to entail. Give a short example of the different values you think could represent C and at the same time specify clearly what "notC" represents in your example -- please no 15-page scratch lotto cards examples that will require me to respond to every sentence because I will just ignore it.

The scratch lotto analogy was only a few paragraphs and would be even shorter if I didn't explain the details of how to derive the conclusion that the probability of identical results when different boxes were scratched should be greater than 1/3, in which case it reduces to this:

Perhaps you could take a look at the scratch lotto analogy I came up with a while ago and see if it makes sense to you (note that it's explicitly based on considering how the 'hidden fruits' might be distributed if they were known by a hypothetical observer for whom they aren't 'hidden'):

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get the same result--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a cherry too.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card always matches the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the same as the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must also have been created with the hidden fruits A+,B+,C-.

Is that too long for you? If you just have a weird aversion to this example (or are refusing to address it just because I have asked you a few times and you just want to be contrary), I suggest you come up with your own toy model since I don't know what would satisfy you. On the other hand, if you are willing to reconsider, then I can certainly explain what my hypothesis about what you mean by the symbols "C" and "notC" would say about the meaning of these symbols in this example.

And if your C is not a random variable, then it has nothing to do with Bell's λ or with the symbols that have played a similar role in my equations like H, since those symbols were random variables--do you disagree?

Yes I disagree.

Which part do you disagree with? You disagree that λ in Bell's equations and H in mine were supposed to represent variables that could take multiple values? Or do you agree with that part, but then disagree that the fact that your C is not a random variable implies that it's not relevant to a discussion of Bell's proof?

you triumphantly declared that P(AB|C)=P(A|C)*P(B|C) also implies P(AB)=P(A)*P(B)

Go back and read it. I said no such thing.

You said:

2) If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data. So no matter how hard you try, all your observed frequencies will always be conditioned on the actual contexts that created the data, whether you like it or not.
3) Following from (2), if hidden elements of reality exist, then the correlations observed in experiments exist even when conditioned on C. Because it is impossible to not condition the results on C. Therefore equations such as P(AB|C) = P(A|C)P(B|C) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(AB|C=42)

Since you said "it is impossible to collect data under situations where C is not True", I interpreted that to mean you're saying C is "always present", So P(AB) "is not different from" P(AB|C), and therefore if P(AB) is not equal to P(A)P(B) then that also implies that "equations such as P(AB|C) = P(A|C)P(B|C) are not accurate" (i.e. you're saying that because C is present and has the same value in all trials, then any probability which is conditioned on C will be the same as the marginal probability, so if P(AB|C)=P(A|C)P(B|C) that would automatically imply P(AB)=P(A)P(B)). Perhaps I misunderstood you, but if so you certainly aren't expressing yourself very clearly, I can't see how the above quote would be compatible with the idea that the value of P(AB) could be different from P(AB|C), or that P(B) could be different from P(B|C).

λ in Bell's paper is supposed to represent the EPR "elements of reality" which cause in the observed correlations. EPR elements of reality are not random variables no matter how loudly you shout that they are.

Perhaps you are focusing on the word "random"--as I said, I accept that 0 and 1 are still valid probabilities, so even if the value of the hidden variables λ on each trial was generated by a completely deterministic process I would still refer to λ as a "random variable" if its value could differ from one trial to another. So let's focus on the "variable" part--do you disagree that Bell was defining λ as a variable whose value could differ from one trial to another, with each possible value of λ expressing some combination of values for all the hidden variables? (for example λ=1 might be defined to mean "spin-up on 0-degree axis, spin-down on 120-degree axis, spin-up on 240-degree axis" while λ=2 might be defined to mean "spin-down on 0-degree axis, spin-up on 120-degree axis, spin-up on 240-degree axis")

If you disagree with the basic premise that λ is intended to be a variable whose value could differ from one trial to another, can you explain why you think Bell wrote equation (2) as an integral with respect to λ? Isn't it basic to the notion of an integral that the "variable of integration" is allowed to vary?

So, now I agree that your C cannot have multiple values, but hopefully you agree that Bell's λ is a random variable that does take multiple values (as made clear by the fact that he is integrating over all values of λ in equation 2)

That is circular reasoning

Well, no, it doesn't remotely resemble "circular reasoning" since I am not arriving at any conclusion by taking the conclusion as a premise.

and I can use the same point against you -- each term in the integral can only have a single value of λ so in fact by integrating you are adding P(AB|a,b,λ1) + P(AB|a,b,λ2) + ... + P(AB|a,b,λn) where n represents the number of possible realizations of your λ.

How is that using the same point against me?? I 100% agree with the above, and in fact I have tried to say exactly the same thing in a number of my previous posts to you. For example, in post #75 on the other recent thread I said:

When I talked about summing over all the different values of z, that was for the purposes of eliminating it from the equation to get P(A|abs). Suppose for example the random variable Z has only two possible values z₁ or z₂, so on a large set of N trials, we'd expect the number of trials with z₁ to be N*P(z₁), and the number of trials with z₂ to be N*P(z₂), with P(z₁) + P(z₂) = 1. Then if we want to know P(A|abs), do you disagree that the following equation would hold? P(A|abs) = P(A|abs, z₁)*P(z₁) + P(A|abs, z₂)*P(z₂)

You still can not escape the fact that the conditioning elements can never be as broad as λ. In each case in which a joint conditional probability is calculated, λn is specific and definitely not a random variable.

Well yes, that's exactly what "random variable" means, something that takes different specific values on each trial. For example, if I am flipping coins, I can define the random variable R to have value 1 if the coin comes up heads and 0 if it comes up tails...this would be a "discrete random variable". See wikipedia's random variable page:

There are two types of random variables: discrete and continuous.[1] A discrete random variable maps events to values of a countable set (e.g., the integers), with each value in the range having probability greater than or equal to zero. A continuous random variable maps events to values of an uncountable set (e.g., the real numbers).

Do you agree that my definition of R above constitutes a perfectly good random variable in experiments where a coin is flipped on each trial, with the value of R differing on different trials? If so, what's wrong with defining λ as a more complex random variable whose value also differs on different trials, depending on the specific values of whatever hidden variables exist?

That is why I told you repeatedly that in calculating a conditional probability you can not condition on a vague concept such as λ with multiple values.

Do you think in a coinflip experiment there would be something wrong with conditioning on R, which also takes multiple values depending on whether the coin comes up heads or tails on each trial? For example, if S is some other random variable representing some other set of mutually exclusive events which can happen on each trial, do you think it would be incorrect to write the equation P(R and S) = P(S|R)*P(R) ?

likewise that when I and other defenders of Bell write equations like P(AB|H)=P(A|H)*P(B|H), the symbols playing a similar role there like H are also intended to be random variables that can take different values on different trials (different points in the sample space). Do you disagree with this?

Absolutely disagree, see my response above, H can not have multiple values in that expression. If you want to write it as P(AB)=P(A|H1)*P(B|H1) + P(A|H2)*P(B|H2) + ... + P(A|Hn)*P(B|Hn) go ahead, but don't deceive yourself and others that you are calculating P(AB|H).

OK, take a look at section 13.1 of this book, titled "Conditioning on a random variable", where the author writes:

Given a random variable X, we shall consider conditional probabilities like P(A|X), and also conditional expected values like E(Y|X), to themselves be random variables. We shall think of them as functions of the "random" value X.

Do you think the author is making an error in saying that expressions like P(A|X), where X is a random variable, have a well-defined meaning in probability theory?

BTW The sample space for H1 is different from the sample space for H2 etc. They are not part of the same sample space. If H1 and H2 are mutually exclusive, your so-called H-sample space is undefined.

Huh? There is nothing stopping us from considering a set of trials which includes both trials where H1 was true and trials where H2 was true, even though they are "mutually exclusive" in the sense they can't both be true on any single trial. For example, if H1 represented the event of my coin coming up heads, and H2 represented the event of my coin coming up tails, then I could consider a sample space including instances of trials where H1 was true and H2 false, as well as instances of trials where H1 was false and H2 was true, but no trials where they were both simultaneously true or both simultaneously false.

I am definitely saying saying if H can take on multiple values H1 and H2 it is OK to write P(AB|H1) and P(AB|H2), but when you write P(AB|H) the only meaning here is that H is a placeholder for a specific value of H not all possible values of H simultaneously.

I have no idea what it would mean to say H stands for "all possible values of H simultaneously," you'll have to give me a definition or example. All I am saying is that if you write some equality or inequality involving random variables like A and H, like my example of P(A and H) = P(A|H)*P(H), then such an equation is understood to be equivalent to the statement that the equation holds for all possible combinations of specific values of A and H. If A can take only two values A1 and A2, and H can take only two values H1 and H2, then writing P(A and H)=P(A|H)*P(H) is simply a shorthand for the statement that all four of the following equations are true:

1. P(A1 and H1) = P(A1|H1)*P(H1)
2. P(A1 and H2) = P(A1|H2)*P(H2)
3. P(A2 and H1) = P(A2|H1)*P(H1)
4. P(A2 and H2) = P(A2|H2)*P(H2)

Would you say that by using P(A and H)=P(A|H)*P(H) as a shorthand for the idea that all for of these more specific equations are true, I am illegally using H to represent "all possible values of H simultaneously"?

For example if H represents the face of a coin and has two possible realizations in a toss "heads" or "tails", writing P(...|H) in which H includes all possible values is no different than writing P(...|heads, tails)

Yes, it is different. As noted in the textbook, P(...|H) would itself represent a random variable which can take different values on different trials. And if you write an equality involving random variables, like P(...|H) = P(... and H)/P(H), that means that even though the values of each side individually can vary from one trial to another, it must be true on every trial that the specific value of the left side in that trial works out to be equal to the specific value of the right side in that same trial.

 

[JesseM]

(continued from previous post)

The correlations measured in real experiments may be conditional on detector settings, but they are not conditional on any hidden elements of reality, regardless of whether such hidden elements exist or not (since even if they do exist we don't know their value on each trial so we can't measure a frequency which is conditioned on them)

Can you explain to me how Aspect and others made sure in their experiments that IF hidden elements of reality exist, then the measured data will not depend on the their presence.

Uh, why should they need to make sure of that? Saying the correlations "are not conditional on any hidden elements of reality" does not mean they are not causally influenced by hidden elements of reality, it just means the correlation that's calculated is not a conditional one that controls for those elements. For example, suppose I have a large population of people, each of whom is either a smoker or nonsmoker, each of whom either has lung cancer or doesn't, and each of whom either has yellow teeth or doesn't. I can certainly calculate the correlation between yellow teeth and lung cancer alone, i.e. find the fraction of people who satisfy (yellow teeth AND lung cancer) and compare it to the product of the fraction that satisfy (yellow teeth) and the fraction that satisfy (lung cancer), even if it happens to be true that the correlation can be explained causally by the fact that smoking increases the chances of both. That's all it means to say that the correlation I'm calculating is not "conditioned on" the smoking variable, that I'm just not bothering to include it in my calculations, not that it isn't causally influencing the correlation I do see between yellow teeth and lung cancer.

IF hidden elements of reality exist, then it is impossible for Aspect et al to collect data under circumstances in which hidden variables do not exist.

They don't need to! I can collect data on the marginal probability a random member of a population will have yellow teeth or will have lung cancer, and calculate the marginal correlation between these two variables, even if it happens to be true that each person either is or isn't a smoker and that this "hidden" variable is having a causal influence on the two variables I am measuring/correlating.

In other words you are saying the correlations Bell is calculating are those that should be seen in our universe if experiments are performed such that those variables (which we have assumed exist), should not affect the results.

Nope, you're just confused about the difference between saying a calculated correlation is not "conditioned on" some variable and saying it's not causally affected by that variable.

Once the assumption is made that our universe is local realistic, the distinction you are trying to make is artificial.

No one is assuming our universe is local realist. They are calculating what would be observed by experimenters in a theoretical local realist universe, then comparing that with actual observations by actual real experimenters in our own real universe. If they differ, that means the predictions of local realism are falsified, therefore the theory that our universe is local realist is falsified. That's how theory-testing works in all of physics--you do a theoretical analysis to figure out what would be observed if a certain theory were true, then you compare that with real-world observations.

Until and unless you can demonstrate that the "theoretical experiments" are comparable to actual experiments performed in our universe, you can not use Bell's equations to say anything about our universe.

What do you mean by "comparable"? You can show theoretically that, under local realism, if two experimenters each have a choice of three detector settings which they make randomly, and both their choices and measurements are made at a spacelike separation from one another, then that implies certain statistical conclusions about the results of their measurements, regardless of exactly what it is they are measuring. So if you set up some quantum-mechanical experiments satisfying these basic conditions, and you find that the statistical conclusions are false in these experiments, then you've falsified local realism.

 

[billschnieder]

You did say "If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data", which suggested you believed "notC" would be some impossible situation that would never hold on any possible trial

Exactly.

so focusing on that statement, I assumed that C represented some general facts which would be true on every possible trial (for example, a statement about all combinations of values for the hidden variables allowed by the laws of physics, without specifying which combination is found on any particular trial).

Yes, you can define C like that. But No, if you define C like that, you can not use it as a condition in calculating a conditional probability because those values could be mutually exclusive. Any such probability will be an impossibility.

I guess if this were the case "notC" would represent some logical possibilities which are ruled out as impossible by the actual laws of physics, like all the combinations of values for the hidden variables which were logically possible but not physically possible given whatever laws of physics govern these variables.

The point is that once you define C as such, there is no difference between C and Z. In other words the P(notC) is zero and P(C) = 1. Do you see now why it makes no sense to talk of a conditional probability while defining the condition the way you do?

But it's true, this interpretation of your words doesn't really allow me to make sense of your claim that if Z="All possible facts in the universe" and that "if there exists a certain factor C within Z such that the set (C, notC) is the same as Z" and "C is the cause of the marginal correlation between A and B". How can a set of opposite possibilities which can't both be simultaneously true be equivalent to "all possible facts in the universe"? I can't really make sense of this, please either define your terms more clearly

I gave that example specifically to show you how absurd the implications of your approach are. I am happy you now see it.

Which part do you disagree with? You disagree that λ in Bell's equations and H in mine were supposed to represent variables that could take multiple values?

I disagree with the idea that such variables could represent the EPR hidden elements of reality. I disagree with the idea that any probabilities calculated with such constructions could be compared to actual experiments IF hidden variables do exist.

Since you said "it is impossible to collect data under situations where C is not True", I interpreted that to mean you're saying C is "always present", So P(AB) "is not different from" P(AB|C), and therefore if P(AB) is not equal to P(A)P(B) then that also implies that "equations such as P(AB|C) = P(A|C)P(B|C) are not accurate" (i.e. you're saying that because C is present and has the same value in all trials, then any probability which is conditioned on C will be the same as the marginal probability, so if P(AB|C)=P(A|C)P(B|C) that would automatically imply P(AB)=P(A)P(B)). Perhaps I misunderstood you.

Yes you did. If hidden elements of reality exist, and C represents those hidden elements of reality
then P(AB|C) = P(A|C)P(B|C) implies that there is no correlation between the A and B. Therefore such an equation can not model the EPR situation in which correlations are in fact observed between A and B. You claimed earlier that the correlation exists marginally even if it does not exist conditionally. But I just showed you that by defining C the way you do, there is no difference between marginal and conditional. So then IF hidden elements of reality exist, and correlations are still observed in their presence, the equation
P(AB|C) = P(A|C)P(B|C) will not appropriately model the situation.

Do you think in a coinflip experiment there would be something wrong with conditioning on R, which also takes multiple values depending on whether the coin comes up heads or tails on each trial? For example, if S is some other random variable representing some other set of mutually exclusive events which can happen on each trial, do you think it would be incorrect to write the equation P(R and S) = P(S|R)*P(R) ?

When you write a term such as P(S|R), where R = ("heads","tails"), R can not represent "all possible" values in a single term. R can only be a place-holder for one of the possible values of R. Sure, you can add up many separate probability terms involving the different instances, Rn, of R but the result you get can not said to be conditioned on R, the so-called "random variable".
So in a universe in which only one of those instances of R are actually realized, whatever probability you obtain in your summation calculation can not be compared to anything measurable in such a universe. Because their marginal probability is NOT defined on the same probability space as your marginal probability. In other words, if in their universe only R="tails" is possible, then their marginal probabilities are the same as your P(...|tails). Just because their universe was one of the terms in your probability does not mean your probability is conditioned on the existence of their universe.

 

[JesseM]

The point is that once you define C as such, there is no difference between C and Z. In other words the P(notC) is zero and P(C) = 1.

If P(C)=1, and C is a fixed statement of facts rather than a variable that can take different values on different trials, then presumably whatever facts are referred to by "C" are true on every possible trial. Is that correct?

Do you see now why it makes no sense to talk of a conditional probability while defining the condition the way you do?

No, because your definition of C (assuming my understanding above is right) has nothing to do with the hidden-variable terms that appeared in either my or Bell's equations.

I gave that example specifically to show you how absurd the implications of your approach are. I am happy you now see it.

But it isn't my approach, since none of the terms in my equations are like your C. All you've shown is that if you define your terms in silly ways it won't be very useful.

Which part do you disagree with? You disagree that λ in Bell's equations and H in mine were supposed to represent variables that could take multiple values?

I disagree with the idea that such variables could represent the EPR hidden elements of reality.

You're not answering my question. Do you agree or disagree with my claim that, unlike your C which takes the same value in every trial (assuming I got that part right), the λ in Bell's equations and H in mine were supposed to represent variables that could different values on different trials?

I disagree with the idea that any probabilities calculated with such constructions could be compared to actual experiments IF hidden variables do exist.

Yes, you "disagree", but so far on this thread you haven't actually presented any argument as to what's wrong with Bell's reasoning, instead you've just come up with symbols and definitions of your own that have nothing to do with how Bell defined the symbols that appear in his own equations, and acted as though this somehow scores points against Bell's arguments.

Also, you refuse to engage with the simple numerical examples I give that show exactly how we can reason about unseen hidden variables (like hidden fruits imagined to exist behind the three boxes of the lotto cards) to draw conclusions about the statistics that should be seen in experiments (observations about the frequencies that Alice and Bob get the same fruit depending on whether they choose to scratch the same box or different boxes) under the assumption that the results of the experiments are determined by the hidden variables. It seems to me like you aren't really making a good-faith effort to understand and engage with the arguments of people you disagree with, but are just trying to use rhetorical strategies to "win" and make the opposing side look bad.

Yes you did. If hidden elements of reality exist, and C represents those hidden elements of reality
then P(AB|C) = P(A|C)P(B|C) implies that there is no correlation between the A and B.

But if your C is defined to be a non-variable that must hold on every single trial, then the meaning of the equation P(AB|C)=P(A|C)P(B|C) is completely different from any equation that appeared in Bell's paper, or any equation I have written down during the course of our discussion like P(AB|H)=P(A|H)P(B|H). So, it would simply be a strawman to claim that with your definition of C, either I or Bell would assert that P(AB|C)=P(A|C)P(B|C) in the first place.

But I just showed you that by defining C the way you do

Er, what? I don't define C any way, I'm just trying to understand your definition, which is still rather unclear to me. But if my basic understanding is correct that you are defining C so that it is a non-variable which is true on every single trial, then I would certainly not assert that the correlation conditioned on C is any different than the marginal correlation. When I said that the correlation conditioned on some other symbol like H or λ could be different than the marginal correlation, I was always using a symbol that was supposed to represent a variable which could take different values on different trials, like a variable H in the lotto card example that would take the value H=1 if the hidden fruits were cherry-cherry-cherry, and would take the value H=2 if the hidden fruits were cherry-cherry-lemon, etc. If A and B represent some observable measurements, it's certainly possible to come up with an example where P(AB|H)=P(A|H)*P(B|H) is true for every possible value of H (i.e. where it's true that P(AB|H=1) = P(A|H=1)*P(B|H=1), and also true that P(AB|H=2) = P(A|H=2)*P(B|H=2), and so on for all possible specific values of H), and yet where P(AB) is not equal to P(A)*P(B) -- do you disagree that this is possible?

When you write a term such as P(S|R), where R = ("heads","tails"), R can not represent "all possible" values in a single term.

I don't know the meaning of the phrase "represent all possible values in a single term" (a phrase I never used), and I suspect you don't either and are just making lofty-sounding assertions which have no clear meaning. If you think the phrase has a well-defined meaning, so that it is meaningful to assert that R cannot "represent all possible values in a single term", then please give me a precise definition.

R can only be a place-holder for one of the possible values of R. Sure, you can add up many separate probability terms involving the different instances, Rn, of R but the result you get can not said to be conditioned on R, the so-called "random variable".

When I say that there is no correlation between A and B when "conditioned on R", all I mean is that for any possible specific value R_n that R can take, it will be true that P(AB|R_n) = P(A|R_n)*P(B|R_n). For example, if R can take only two values, R=1 and R=2, then saying there's no correlation between A and B conditioned on R would just be shorthand for the claim that both the following equations hold:

1. P(AB|R=1) = P(A|R=1)*P(B|R=1)
2. P(AB|R=2) = P(A|R=2)*P(B|R=2)

Hopefully you agree that the claim that both these specific equations hold is perfectly well-defined as statistical claims go? If so, then even if you don't like using the phrase "no correlation between A and B conditioned on R" to describe this claim, that is merely a semantic quibble, now you hopefully understand what I mean even if you don't like the vocabulary I use to describe it (and hopefully you agree that the above two equations can hold even in a situation where the marginal probability P(AB) is different from the product of the marginal probabilities P(A)*P(B)). In terms of pure semantics, I think the statistics community would side with me on this, not you; after all, I just linked to a textbook which explicitly talks about conditioning on a random variable.

So in a universe in which only one of those instances of R are actually realized

But I'm not talking about a universe where only one value of the random variable λ is actually realized, and neither was Bell. Again, in a local hidden variables theory it is quite possible that if you do multiple trials with different pairs of particles, the hidden variables associated with the pair on one trial may be different than the hidden variables associated with a different pair on a different trial. For example, the simplest local-hidden-variables theory to try to explain quantum experiments would just say that if the particles always have the same spin whenever they're measured on the same axis, that must mean each pair is created with the same predetermined answers to what spin they'll give if measured on a given axis, so λ=1 might represent the hidden variable state "spin-up on axis 1, spin-up on axis 2, spin-up on axis 3" while λ=2 could represent the hidden variable state "spin-up on axis 1, spin-up on axis 2, spin-down on axis 3", and so on and so forth. If on some trials an experimenter picks axis 3 and gets spin-up, while on other trials the experimenter picks axis 3 and gets spin-down, then in this simple hidden-variables theory it must be true that the value of λ differs from one trial to another.

 

[Maaneli]

Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables. Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.

Actually, this isn't Bell's approach. Bell does not speak anywhere of 'local realism'. He speaks only of local causality as his starting theoretical assumption. As evidence of this, see Bell's papers entitled, La Nouvelle Cuisine, Free Variables and Local Causality, The Theory of Local Beables, and Bertlemann's Socks and the Nature of Reality. Also have a look at Norsen's paper,

Against `Realism'
Authors: Travis Norsen
Journal reference: Foundations of Physics, Vol. 37 No. 3, 311-340 (March 2007)
http://arxiv.org/abs/quant-ph/0607057

What Norsen essentially points out is that there isn't some additional notion of realism in Bell's theorem, over and above the notion of realism already implicit in Bell's definition of locality. This is a crucially important point to recognize in any discussion about what Bell actually said and did.

 

[DrChinese]

What Norsen essentially points out is that there isn't some additional notion of realism in Bell's theorem, over and above the notion of realism already implicit in Bell's definition of locality. This is a crucially important point to recognize in any discussion about what Bell actually said and did.

Maaneli, I disagree strongly with your point; Norsen is wrong too. I can point out the exact spot in Bell in which realism is introduced, as it is explicit:

After (14): It follows that c is another unit vector... [in addition to the a and b of Bell (2)]

Bell did not hightlight this as the introduction of hidden variables, realism, counterfactuality or whatever one may choose to call it. But there it is, and it is quite impossible to derive the Bell result without dear old c. Please don't cheat c of her 15 minutes...

And as you are undoubtedly aware, there are many who feel that - and with substantial justification from other arguments against realism I might add - that there is no possibility of a realistic (non-contextual) theory under any circumstances. I realize that you tend towards the non-local side of things and don't follow that line of thinking.

 

[billschnieder]

(continued from previous post)

Uh, why should they need to make sure of that? Saying the correlations "are not conditional on any hidden elements of reality" does not mean they are not causally influenced by hidden elements of reality, it just means the correlation that's calculated is not a conditional one that controls for those elements. For example, suppose I have a large population of people, each of whom is either a smoker or nonsmoker, each of whom either has lung cancer or doesn't, and each of whom either has yellow teeth or doesn't. I can certainly calculate the correlation between yellow teeth and lung cancer alone, i.e. find the fraction of people who satisfy (yellow teeth AND lung cancer) and compare it to the product of the fraction that satisfy (yellow teeth) and the fraction that satisfy (lung cancer), even if it happens to be true that the correlation can be explained causally by the fact that smoking increases the chances of both. That's all it means to say that the correlation I'm calculating is not "conditioned on" the smoking variable, that I'm just not bothering to include it in my calculations, not that it isn't causally influencing the correlation I do see between yellow teeth and lung cancer.

You do not understand the difference between a theoretical exercise and an actual experiment. If I am trying to study the relative effectiveness of two possible treatments T = (A, B) against a kidney stone disease. Theoretically it is okay to say that you randomly select two groups of people from the population of people with the disease, give treatment A to one and B to the second group and then calculate the the relative frequencies of those who recovered in group 1 after taking treatment A. Theoretically speaking, you can then compare that value with relative frequency of those who recovered in group 2 after taking treatment B. This is fine as a theoretical exercise.

Now fast-forward to an actual experiment in which the experimenters do not know about all the hidden factors. What does "select groups at random" mean in a real experiment? Say the experimenters select the two groups according to their best understanding of what may be random. And then after calculating their relative frequencies, they find that Treatment B is effective in 280 of the 350 people (83%), but treatment A is only effective in 273 of the 350 people (78%). So they conclude that Treatment B is more effective than treatment A. Is this a reasonable conclusion according to you?

Now suppose the omniscient being, knowing fully well that the size of the kidney stones is a factor and after looking at the data he finds that if he divides the groups according to the size of kidney stones the patients had the groups break down as follows

Group 1 (those who received treatment A): (87-small stones, 263-large stones)
Group 2( those who received treatment B): (270-small stones, 80-large stones)

He now finds that of the the 81 of the 87 (93%) in group 1 who had small stones were cured by treatment A, and 192 of the 263 (73%) of those with large stones in group 1 were cured by treatment A.
For group 2, he finds that 234 of the 270 (87%) with small stones were cured and 55 of the 80 (69%) with large stones were cured.

Clearly, when all the hidden factors are considered, Treatment A is more effective than than treatment B contrary to results obtained by the experimenters. Does this then mean there is some spooky business happening? This is called Simpson's paradox and I believe I have pointed this to you not too long ago.

As you can hopefully see here, not knowing about all the hidden factors at play, the experiments can not possibly collect a fair sample, therefore their results are not comparable to the theoretical situation in which all possible causal factors are included. That is why I have repeatedly pointed out to you that in order to collect a fair sample comparable with Bell's inequalities, experimenters in Aspect type experiments must design their experiments such that, not only should all possible "values" of the hidden elements of reality are realized, but they should be realized fairly. In the kidney stone example above, all possible values were realized but not fairly. As a result, the conclusions were are odds with what is known by the omniscient being and therefore not comparable. In other words, all values observed by the experimenters is conditioned on their assumptions about what is causing the results. Their definition of random in this case was flawed.

So again, do you have a reference to any Aspect type experiment in which they ensured randomness with respect to all possible hidden elements of reality causing the results? By comparing observed correlations to Bell's inequalities, you are claiming that they are in fact comparable.

No one is assuming our universe is local realist. They are calculating what would be observed by experimenters in a theoretical local realist universe, then comparing that with actual observations by actual real experimenters in our own real universe. If they differ, that means the predictions of local realism are falsified, therefore the theory that our universe is local realist is falsified.

Huh? The break down of a conclusion can only be taken to imply a failure of one of the premises of that conclusion. The argument usually goes as follows:
(1) Bell's inequalities accurately model local realistic universes
(2) Our universe is locally realistic
(3) Therefore actual experiments in our universe must obey Bell's inequalities.

However, we now know that actual experiments in our universe do not obey Bell's inequalities. It therefore follows that either (1) is false or (2) is false. If your claim now is that (2) is not a premise in that argument, then you are admitting that (1) is false. There is no escape here.

 

[Maaneli]

Maaneli, I disagree strongly with your point; Norsen is wrong too. I can point out the exact spot in Bell in which realism is introduced, as it is explicit:

After (14): It follows that c is another unit vector... [in addition to the a and b of Bell (2)]

Bell did not hightlight this as the introduction of hidden variables, realism, counterfactuality or whatever one may choose to call it. But there it is, and it is quite impossible to derive the Bell result without dear old c. Please don't cheat c of her 15 minutes...

Sorry, but you are the one who is wrong. The introduction of the unit vector c is not where realism is initially introduced, nor does c contain within it some independent and additional assumption of 'realism', over and above the notion of realism that is already implicitly introduced by Bell's condition of local causality. In other words, all the realism in Bell's theorem is introduced as part of Bell's definition and application of his local causality condition. And the introduction of the unit vector, c, follows from the use of the local causality condition. Indeed, in La Nouvelle Cuisine (particularly section 9 entitled 'Locally explicable correlations'), Bell explicitly discusses the relation of c to the hidden variables, lambda, and the polarizer settings, a and b, and explicitly shows how they follow from the local causality condition. To summarize it, Bell first defines the 'principle of local causality' as follows:

"The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light."

In fact, this definition is equivalent to the definition of relativistic causality, and one can readily see that it implicitly requires the usual notion of realism in special relativity (namely, spacetime events, and their causes and effects) in its very formulation. Without any such notion of realism, I hope you can agree that there can be no principle of local causality.

Bell then defines a locally causal theory as follows:

"A theory will be said to be locally causal if the probabilities attached to values of 'local beables' ['beables' he defines as those entities in a theory which are, at least, tentatively, taken seriously, as corresponding to something real, and 'local beables' he defines as beables which are definitely associated with particular spacetime regions] in a spacetime region 1 are unaltered by specification of values of local beables in a spacelike separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a spacetime region 3 [he then gives a figure illustrating this]."

You can clearly see that the local causality principle cannot apply to a theory without local beables. To spell it out, this means that the principle of local causality is not applicable to nonlocal beables, nor a theory without beables of any kind.

Bell then shows how one might try to embed quantum mechanics into a locally causal theory. To do this, he starts with the description of a spacetime diagram (figure 6) in which region 1 contains the output counter A (=+1 or -1), along with the polarizer rotated to some angle a from some standard position, while region 2 contains the output counter B (=+1 or -1), along with the polarizer rotated to some angle b from some standard position which is parallel to the standard position of the polarizer rotated to a in region 1. He then continues:

"We consider a slice of space-time 3 earlier than the regions 1 and 2 and crossing both their backward light cones where they no longer overlap. In region 3 let c stand for the values of any number of other variables describing the experimental set-up, as admitted by ordinary quantum mechanics. And let lambda denote any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR. Suppose that the c and lambda together give a complete specification of at least those parts of 3 blocking the two backward light cones."

From this consideration, he writes the joint probability for particular values A and B as follows:

{A, B|a, b, c, lambda} = {A|B, a, b, c, lambda} {B|a, b, c, lambda}​

He then says, "Invoking local causality, and the assumed completeness of c and lambda in the relevant parts of region 3, we declare redundant certain of the conditional variables in the last expression, because they are at spacelike separation from the result in question. Then we have

{A, B|a, b, c, lambda} = {A|a, c, lambda} {B|b, c, lambda}.​

Bell then states that this formula has the following interpretation: "It exhibits A and B as having no dependence on one another, nor on the settings of the remote polarizers (b and a respectively), but only on the local polarizers (a and b respectively) and on the past causes, c and lambda. We can clearly refer to correlations which permit such factorization as 'locally explicable'. Very often such factorizability is taken as the starting point of the analysis. Here we have preferred to see it not as the formulation of 'local causality', but as a consequence thereof."

Bell then shows that this is the same local causality condition used in the derivation of the CSHS inequality, and which the predictions of quantum mechanics clearly violate. Hence, Bell concludes that quantum mechanics cannot be embedded in a locally causal theory.

And again, the variable c here is nothing but part of the specification of the experimental set-up (as allowed for by 'ordinary quantum mechanics'), just as are the polarizer settings a and b (in other words, a, b, and c are all local beables); and the introduction of c in the joint probability formula follows from the local causality condition, as part of the complete specification of causes of the events in regions 1 and 2. So, again, there is no notion of realism in c that is any different than in a and b and what already follows from Bell's application of his principle of local causality.

So there you go, straight from the horses mouth. I hope you will have taken the time to carefully read through what I presented above, and to corroborate it for yourself by also reading (or re-reading) La Nouvelle Cuisine. It's really important, at least for the sake of intellectual honesty, to understand this point about what Bell said, and to not misrepresent what he claimed and what he actually proved.

And as you are undoubtedly aware, there are many who feel that - and with substantial justification from other arguments against realism I might add - that there is no possibility of a realistic (non-contextual) theory under any circumstances. I realize that you tend towards the non-local side of things and don't follow that line of thinking.

No, you misrepresent what I think. I am well aware that there is no plausible possibility of a realistic non-contextual theory, and I have even stated so many times on this forum before. But (A) this is not relevant to my point of disagreement with you regarding what Bell actually said and proved, and (B) the inability to retain contextuality in an empirically adequate realistic theory is not a 'problem' for realistic theories, in any scientifically meaningful sense. It may be a problem for some people's naive intuitions about how a realistic theory of the physical world should work, but that's completely subjective, and in any case, I myself have never found contextuality to be a counter-intuitive or paradoxical or inelegant notion. Also, I have strong reasons to think that contextuality is already a property of measurements in classical nonequilibrium statistical mechanics, in which case, the usual assumption that non-contextuality is a fundamental property of measurements in classical realistic physical theories, is just wrong.

 

[DrChinese]

Sorry, but you are the one who is wrong. The introduction of the unit vector c is not where realism is initially introduced, nor does c contain within it some independent and additional assumption of 'realism', over and above the notion of realism that is already implicitly introduced by Bell's condition of local causality. In other words, all the realism in Bell's theorem is introduced as part of Bell's definition and application of his local causality condition. And the introduction of the unit vector, c, follows from the use of the local causality condition.

Well, my intent wasn't to debate the point. We can agree to disagree. So I won't get into a point by point rebuttal.

However, for anyone following this argument: There is a very specific reason why counterfactual setting c is explicitly required to make the Bell argument work. The a and b settings discussed early in Bell can be experimentally tested. And guess what, there is absolutely NOTHING unusual about these. They can be modeled within a local realistic theory.

On the other hand, once you assume the existence of c, everything falls apart pretty quickly. Anyone who attempts to derive the Bell result will see this. No c, no Bell result. So c must be important. And the generally accepted view is that this is the realism requirement. So don't take my word for it, try it yourself. Or ask Mermin.

 

[billschnieder]

I would like to go back to what started this thread. In Bell's Bertlmann's socks paper, he writes the following (pages 12-13):

For example the statistics of heart attacks in Lille and Lyons show strong correlations. The probability of M cases in Lyons and N in Lille, on a randomly chosen day, does not separate

P(M,N) /= P1(M)P2(N)

In fact when M is above average N also tends to be above average.
You might shrug your shoulders and say "coincidences happen all the time", or "that's life". Such an attitude is indeed sometimes advocated by otherwise serious people in the context of quantum philosophy. But outside that peculiar context, such an attitude would be dismissed as unscientific. The scientific attitude is that correlations cry out for explanation. And of course in the given example explanations are soon found. The weather is much the same in the two towns, and hot days are bad for heart attacks. The day of the week is exactly the same in the two towns, and Sundays are especially bad because of family quarrels and too much to eat. And so on. It seems reasonable to expect that if sufficiently many such causal factors can be identified and held fixed, the residual fluctuations will be independent, i.e.,

P(M,N|a,b,λ) = P1(M|a,λ)P2(N|b,λ) ... (10)

where a and b are temperatures in Lyons and Lille respectively, λ denotes
any number of other variables that might be relevant, and P(M,N|a,b,λ) is the
conditional probability of M cases in Lyons and N in Lille for given (a,b,λ).
Note well that we already incorporate in (10) a hypothesis of "local causality"
or "no action at a distance". For we do not allow the first factor to depend
on a, nor the second on b. That is, we do not admit the temperature in Lyons
as a causal influence in Lille, and vice versa.

Just in case anyone wants to distract by bringing up marginal probabilities, Note that this is presented before Bell has introduced any integration or marginalization, so he is talking about a specific set of (a,b,λ) here, note the underlined given present in his original.

Right here in plain sight (bold) is Bell's definition of local causality. I assume Bell supporters will agree with this definition. According to Bell, making P1 depend on b, and P2 depend on a or M implies non-locality. Do Bell supporters agree with this?

1) In other words, writing the equation the following way:

P(M,N|a,b,λ) = P1(M|a,b,λ)P2(N|a,b,λ,M) ... (x)

according to the universally valid chain rule of probability theory, would violate Bell's "local causality" condition. Do Bell supporters agree with this?

2) What is the probability rule or law that allows equation (10) to be written? As far as I know, The only rule for expressing joint probabilities is the chain rule and equation (10) can only be correct, if it is a simplification of (x). i.e,

P(M,N|a,b,λ) = P1(M|a,b,λ)P2(N|a,b,λ,M) = P1(M|a,λ)P2(N|b,λ,M) = P1(M|a,λ)P2(N|b,λ) ... (z)

is a valid reduction, if and only if P1(M|a,b,λ)P2(N|a,b,λ,M) = P1(M|a,λ)P2(N|b,λ) (Duh!)
That means the resulting probability calculated from both expressions MUST be the same. Do Bell supporters agree with this? If you don't agree, then there is no other justification for making the reduction.

3) Therefore if using (x) results in different values from using (10), there must be something wrong with the reasoning behind the reduction from (x) to (10). Is this unreasonable?

4) Now fast-forward in the same paper to where Bell is talking about the contradictions (Page 15):

So the quantum correlations are locally inexplicable. To avoid the inequa-
lity we could allow P1 in (11) to depend on b or P2 to depend on a. That
is to say we could admit the signal at one end as a causal influence at the other
end. For the set-up described this would be not only a mysterious long range
influence - a non-locality or action at a distance in the loose sense - but one
propagating faster than light

Bell is saying here that using an equation like (x) or any of the intermediate terms in (z), would result in a different result than the one he chose (10). Do Bell supporters agree with this? An equation derived as a simplication of the chain rule result in a different result than a direct application of the chain rule, unless the assumptions used in the simplication are false. The only reason a simplication can be made in certain situations is precisely because in those situations, they give the same result! But clearly that is not the case here, therefore Bell's simplification is unfounded.

5) Again in the snippet quoted above under point (4), we see again that Bell's definition of "local causality" does not allow any form of "logical dependence". Do Bell supporters agree with this? So If I can come up with an example, any locally causal example for which logical dependence is present, it effectively invalidates Bell's local causality definition. Is that unreasonable?

 

[JesseM]

Uh, why should they need to make sure of that? Saying the correlations "are not conditional on any hidden elements of reality" does not mean they are not causally influenced by hidden elements of reality, it just means the correlation that's calculated is not a conditional one that controls for those elements. For example, suppose I have a large population of people, each of whom is either a smoker or nonsmoker, each of whom either has lung cancer or doesn't, and each of whom either has yellow teeth or doesn't. I can certainly calculate the correlation between yellow teeth and lung cancer alone, i.e. find the fraction of people who satisfy (yellow teeth AND lung cancer) and compare it to the product of the fraction that satisfy (yellow teeth) and the fraction that satisfy (lung cancer), even if it happens to be true that the correlation can be explained causally by the fact that smoking increases the chances of both. That's all it means to say that the correlation I'm calculating is not "conditioned on" the smoking variable, that I'm just not bothering to include it in my calculations, not that it isn't causally influencing the correlation I do see between yellow teeth and lung cancer.

You do not understand the difference between a theoretical exercise and an actual experiment. If I am trying to study the relative effectiveness of two possible treatments T = (A, B) against a kidney stone disease. Theoretically it is okay to say that you randomly select two groups of people from the population of people with the disease, give treatment A to one and B to the second group and then calculate the the relative frequencies of those who recovered in group 1 after taking treatment A. Theoretically speaking, you can then compare that value with relative frequency of those who recovered in group 2 after taking treatment B. This is fine as a theoretical exercise.

Now fast-forward to an actual experiment in which the experimenters do not know about all the hidden factors. What does "select groups at random" mean in a real experiment? Say the experimenters select the two groups according to their best understanding of what may be random. And then after calculating their relative frequencies, they find that Treatment B is effective in 280 of the 350 people (83%), but treatment A is only effective in 273 of the 350 people (78%). So they conclude that Treatment B is more effective than treatment A. Is this a reasonable conclusion according to you?

It depends on how "random" the selections were. If they were just looking at people who were already on one of the two treatments, it might be that there are other factors which influence the likelihood that a person will choose A vs. B (for example, socioeconomic status) and these factors might also influence the chances of recovery independent of the influences of the treatments themselves. On the other hand, if they picked a large population and then used a truly random method to decide which members received treatment A vs. treatment B, and no one chose to drop out of the experiment, then this would be a reasonable controlled experiment in which the only reason that other factors (like socioeconomic status) might vary between group A and group B would be a random statistical fluctuation, so the larger the population the less likely there'd be significant variation in other factors between the two groups, and thus any differences in recovery would be likely due to the treatment itself.

Now suppose the omniscient being, knowing fully well that the size of the kidney stones is a factor and after looking at the data he finds that if he divides the groups according to the size of kidney stones the patients had the groups break down as follows

Group 1 (those who received treatment A): (87-small stones, 263-large stones)
Group 2( those who received treatment B): (270-small stones, 80-large stones)

This might be possible if the groups were self-selecting, for example if people with low socioeconomic status were both more likely to have large kidney stones (because of diet, say) and more likely to choose treatment A (because it's cheaper), but if the subjects were assigned randomly to group A or B by some process like a random number generator on a computer, there should be no correlation between P(computer algorithm assigns subject to treatment A) and P(subject has large kidney stones), so any difference in frequency in kidney stones between the two groups would be a matter of random statistical fluctuation, and such differences would be less and less likely the larger a population size was used.

He now finds that of the the 81 of the 87 (93%) in group 1 who had small stones were cured by treatment A, and 192 of the 263 (73%) of those with large stones in group 1 were cured by treatment A.
For group 2, he finds that 234 of the 270 (87%) with small stones were cured and 55 of the 80 (69%) with large stones were cured.

The relevance of your example to what we were debating is unclear. I was making the simple point that there can be situations where there is a marginal correlation between two random variables, but it the correlation disappears when you condition on some other set of facts (I won't say 'condition on another random variable' because that would probably lead to more semantic quibbling on your part--I'm just talking about a situation where if you condition on each specific value of some other random variable, in each specific case the correlation disappears, as I illustrated at the end of post #21). But your example isn't like this--in your example there seem to be two measured variables, T which can take two values {received treatment A, received treatment B} and another one, let's call it U, which can also take two values {recovered from disease, did not recover from disease}. Then there is also a hidden variable we can V, which can take two values {large kidney stones, small kidney stones}. In your example there is a marginal correlation between variables T and U, but there is still a correlation (albeit a different correlation) when we condition on either of the two specific values of V. So, let me modify your example with some different numbers. Suppose 40% of the population have large kidney stones and 60% have small ones. Suppose those with large kidney stones have an 0.8 chance of being assigned to group A, and an 0.2 chance of being assigned to group B. Suppose those with small kidney stones have an 0.3 chance of being assigned to group A, and an 0.7 chance of being assigned to B. Then suppose that the chances of recovery depend only one whether one had large or small kidney stones and is not affected either way by what treatment one received, so P(recovers|large kidney stones, treatment A) = P(recovers|large kidney stones), etc. Suppose the probability of recovery for those with large kidney stones is 0.5, and the probability of recovery for those with small ones is 0.9. Then it would be pretty easy to compute P(treatment A, recovers, large stones)=P(recovers|treatment A, large stones)*P(treatment A, large stones)=P(recovers|large stones)*P(treatment A, large stones)=P(recovers|large stones)*P(treatment A|large stones)*P(large stones) = 0.5*0.8*0.4=0.16. Similarly P(treatment A, doesn't recover, small stones) would be P(doesn't recover|small stones)*P(treatment A|small stones)*P(small stones)=0.1*0.3*0.6=0.018, and so forth.

In a population of 1000, we might then have the following numbers for each possible combination of values for T, U, V:

1. Number(treatment A, recovers, large stones): 160
2. Number(treatment A, recovers, small stones): 162
3. Number(treatment A, doesn't recover, large stones): 160
4. Number(treatment A, doesn't recover, small stones): 18
1. Number(treatment B, recovers, large stones): 40
2. Number(treatment B, recovers, small stones): 378
3. Number(treatment B, doesn't recover, large stones): 40
4. Number(treatment B, doesn't recover, small stones): 42

If we don't know whether each person has large or small kidney stones, this becomes:

1. Number(treatment A, recovers) = 160+162 = 322
2. Number(treatment A, doesn't recover) = 160+18 = 178
3. Number(treatment B, recovers) = 40+378 = 418
4. Number(treatment B, doesn't recover) = 40+42=82

So here, the data shows that of the 500 who received treatment A, 322 recovered while 178 did not, and of the 500 who received treatment B, 418 recovered and 82 did not. There is a marginal correlation between receiving treatment B and recovery: P(treatment B, recovers)=0.418, which is larger than P(treatment B)*P(recovers)=(0.5)*(0.74)=0.37. But if you look at the correlation between receiving treatment B and recovery conditioned on large kidney stones, there is no conditional correlation: P(treatment B, recovers|large stones) = P(treatment B|large stones)*P(recovers|large stones) [on the left side, there are 400 people with large stones and only 40 of these who also received treatment B and recovered, so P(treatment B, recovers|large stones) = 40/400 = 0.1; on the right side, there are 400 with large stones but only 80 of these received treatment B, so P(treatment B|large stones)=80/400=0.2, and there are 400 with large stones and 200 of those recovered, so P(recovered|large stones)=200/400=0.5, so the product of these two probabilities on the right side is also 0.1] The same would be true if you conditioned treatment B + recovery on small kidney stones, or if you conditioned any other combination of observable outcomes (like treatment A + no recovery) on either large or small kidney stones.

So do you agree that with my numbers, we find a marginal correlation between the observable variable T (telling us which treatment a person received, A or B) and U (telling us whether they recovered or not), but no correlation between T and U when we condition on any specific value of the "hidden" variable V (telling us whether the person has large or small kidney stones)? Please give me a yes or no answer to this question. If you agree that this sort of thing is possible, why do you think the same couldn't be true in a local hidden variables theory where the two observable variables represented measurements (each under some specific detector setting) at different locations, and each specific value of the variable λ represents a specific combination of values for various local hidden variables?

As you can hopefully see here, not knowing about all the hidden factors at play, the experiments can not possibly collect a fair sample, therefore their results are not comparable to the theoretical situation in which all possible causal factors are included.

When you say "fair sample", "fair" in what respect? If your 350+350=700 people were randomly sampled from the set of all people receiving treatment A and treatment B, then this is a fair sample where the marginal correlation between the treatment and recovery outcome variables (T and U above) in your group should accurately reflects the marginal correlation that would exist between these same variables if you looked at every single person in the world receiving treatment A and B. The problem of Simpson's paradox is that this marginal positive correlation between B and recovery does not tell you anything about a causal relation between these variables ('correlation is not causation'), because the positive correlation might become a negative correlation (as in your example) or zero correlation (as in mine) when you condition on some other fact like having large kidney stones.

If you think this somehow suggests a problem with Bell's reasoning, you are really missing the point of his argument entirely! Bell does not just assume that since there is a marginal correlation between the results of different measurements on a pair of particles, there must be a causal relation between the measurements; instead his whole argument is based on explicitly considering the possibility that this correlation would disappear when conditioned on other hidden variables, exactly analogous to my example where there was no correlation between treatment group and recovery outcome when you conditioned on large kidney stones (or when you conditioned on small kidney stones). The whole point is that in a local realist universe, marginal correlations between any spacelike-separated events cannot represent causal influences, the correlations must disappear when you condition on the state of all local variables in the past light cones of the two spacelike-separated events. And that assumption, that marginal correlations between spacelike-separated events must not be causal influences under local realism, so that local realism predicts these correlations must disappear when conditioned on the values of other variables in the past light cones, is exactly what is represented by equation (2) in his paper (or by equation 10 on page 243 of Speakable and Unspeakable in Quantum Mechanics). So criticizing Bell by comparing his argument to that of an imaginary fool who thinks the marginal correlation between treatments and recovery outcomes does indicate a causal influence between the two, failing to consider that the correlation might reverse or disappear when you condition for some other hidden variable like large kidney stones, indicates a complete lack of understanding of Bell's argument!

So again, do you have a reference to any Aspect type experiment in which they ensured randomness with respect to all possible hidden elements of reality causing the results? By comparing observed correlations to Bell's inequalities, you are claiming that they are in fact comparable.

Bell's inequalities deal with marginal correlations, the ones that are seen when you don't condition on hidden variables (though of course they are derived from the assumption that any such marginal correlations must disappear when conditioned on the proper hidden variables). Experiments also deal with the same marginal correlations. So, your request makes absolutely no sense.

Huh? The break down of a conclusion can only be taken to imply a failure of one of the premises of that conclusion. The argument usually goes as follows:
(1) Bell's inequalities accurately model local realistic universes
(2) Our universe is locally realistic
(3) Therefore actual experiments in our universe must obey Bell's inequalities.

No, it doesn't; no mainstream physicist argues that way, either you're engaging in pure fantasy or you've completely misread whatever papers gave you this idea (if you think any actual mainstream papers make this sort of argument, why don't you link to them and I can point out your mistake). The actual argument is as follows:

(1) The theoretical postulate of local realism implies Bell's inequalities should be satisfied
(2) In real experiments, Bell's inequalities are violated
(3) Therefore, the theoretical postulate of local realism has been falsified in our real universe

Obviously you are not convinced of point (1), but do you dispute that if (1) were theoretically sound and (2) is true of actual experiments, then (3) must follow?

 

[JesseM]

Just in case anyone wants to distract by bringing up marginal probabilities, Note that this is presented before Bell has introduced any integration or marginalization, so he is talking about a specific set of (a,b,λ) here, note the underlined given present in his original.

Mentioning marginal probabilities is not a distraction. His argument must be treated as a whole, the fact that this appears "before Bell has introduced any integration or marginalization" does not change the fact that he brings them up later (starting in equation 12 on p. 14 of the Bertelman's socks paper, where the left side of the equation deals with observable quantities only). As I pointed out in the previous post, the logic is pretty simple--the postulate of local realism implies that there can be no correlations between spacelike-separated events when conditioned on the right set of facts about the past light cones of these events, then we can use that to derive certain statistical rules that should hold (if local realism is true) for the marginal correlations between these same spacelike-separated events, like the rule derived in equation 14 on p. 14 of the Bertelman's socks paper.

Right here in plain sight (bold) is Bell's definition of local causality. I assume Bell supporters will agree with this definition. According to Bell, making P1 depend on b, and P2 depend on a or M implies non-locality. Do Bell supporters agree with this?

It depends on how much information is included in λ. Bell is saying that according to local realism, it is always possible to define a variable λ such that the specific value of λ on each trial encodes a sufficiently large amount of information about local facts in the past light cone of M on that trial that it's guaranteed that P1(M|a,λ) will be equal to P1(M|a,b,λ), for each specific value of λ (e.g. if λ can take three possible values λ₁, λ₂, and λ₃, then over a large number of trials the three equations P1(M|a,λ₁)=P1(M|a,b,λ₁) and P1(M|a,λ₂)=P1(M|a,b,λ₂) and P1(M|a,λ₃)=P1(M|a,b,λ₃) will all be valid). Obviously this would not be true for any arbitrary definition of λ, but under local realism there should always be some way to define λ such that this is true (like the definition involving complete information about local facts in the cross-sections of the past cones indicated on p. 242 of Speakable and Unspeakable in Quantum Mechanics).

1) In other words, writing the equation the following way:

P(M,N|a,b,λ) = P1(M|a,b,λ)P2(N|a,b,λ,M) ... (x)

according to the universally valid chain rule of probability theory, would violate Bell's "local causality" condition. Do Bell supporters agree with this?

There's nothing incorrect about that equation, it's just that given the proper definition of λ it must be true that P1(M|a,b,λ)=P1(M|a,λ) and P2(N|a,b,λ,M)=P2(N|b,λ) if local realism holds (i.e. these equations hold for each specific value of λ, if you have a semantic objection to the notion of conditioning on a random variable). If it is impossible to define any λ encoding information about the past light cones of M and N such that the above two equalities hold, then local realism would be false.

2) What is the probability rule or law that allows equation (10) to be written?

No general probability law can be used to derive it, instead it is derived from the specific meaning of the symbols in the physical scenario we are considering. Similarly, there is no general probability law allowing us to write P(F₁)=P(F₂) for any arbitrary random variable F that has possible values F₁ and F₂, but if we know we are dealing with a problem where a fair coin is being flipped and F₁ represents the outcome "heads" while F₂ represents the outcome "tails", then that equation should hold and we could substitute P(F₂) into any equation involving the term P(F₁). Analogously, P1(M|a,b,λ)=P1(M|a,λ) and P2(N|a,b,λ,M)=P2(N|b,λ) are not general equations that would hold regardless of the meaning of the symbols, but given the specific physical meaning Bell assigns them these equations should hold in a local realist universe, and thus we can substitute P1(M|a,λ) in for P1(M|a,b,λ) and P2(N|b,λ) in for P2(N|a,b,λ,M) in your equation P(M,N|a,b,λ) = P1(M|a,b,λ)P2(N|a,b,λ,M).

As far as I know, The only rule for expressing joint probabilities is the chain rule and equation (10) can only be correct, if it is a simplification of (x). i.e,

P(M,N|a,b,λ) = P1(M|a,b,λ)P2(N|a,b,λ,M) = P1(M|a,λ)P2(N|b,λ,M) = P1(M|a,λ)P2(N|b,λ) ... (z)

is a valid reduction, if and only if P1(M|a,b,λ)P2(N|a,b,λ,M) = P1(M|a,λ)P2(N|b,λ) (Duh!)

Yes.

That means the resulting probability calculated from both expressions MUST be the same.

If we are dealing with the same physical scenario where the symbols have the same physical meaning, then sure, the actual values of the probabilities will end up being the same. But if we are doing a general algebraic proof that should hold in a large class of possible physical situations where the specific values can be different, then an equation we derive using the substitutions P1(M|a,b,λ)=P1(M|a,λ) and P2(N|a,b,λ,M)=P2(N|b,λ) may not be derivable if we don't use those substitutions. See the point I made about each step in a mathematical proof being necessary if you want to get to the conclusion, despite the fact that each new theorem may be "equivalent" in some sense to earlier ones, at the end of post 63 from our first discussion.

3) Therefore if using (x) results in different values from using (10), there must be something wrong with the reasoning behind the reduction from (x) to (10). Is this unreasonable?

"Different values" if the physical scenario gives us precise values for all the specific probabilities, or are you talking about deriving general equations that cover a broad class of possible probability distributions, as Bell was doing? If you know the specific values for terms like P1(M|a,b,λ) then in any local realist theory the specific values won't differ from the specific values for equal terms like P1(M|a,λ), but if you are just talking about a general class of probability distributions that would be compatible with local realism, getting to the final equation like equation 14 on p. 14 of the Bertelman's socks paper may depend on making such substitutions.

4) Now fast-forward in the same paper to where Bell is talking about the contradictions (Page 15):

Bell is saying here that using an equation like (x) or any of the intermediate terms in (z), would result in a different result than the one he chose (10).

No he isn't. The only "contradiction" he mentions is between the predictions of quantum mechanics (equation 17 on p. 14) and the results he derives from the assumptions of local realism (equation 14 on p. 14), he isn't suggesting any internal contradictions in the equations used as steps in the derivation of equation 14. If you disagree, please give a specific quote where you think he is talking about an internal contradiction in the derivation.

5) Again in the snippet quoted above under point (4), we see again that Bell's definition of "local causality" does not allow any form of "logical dependence".

Presumably you're referring to this snippet:

To avoid the inequality we could allow P1 in (11) to depend on b or P2 to depend on a. That is to say we could admit the signal at one end as a causal influence at the other
end.

...it's clear from the context he means that P1(M|a,b,λ) cannot have a different value than P1(M|a,λ). He would still certainly allow for forms of "logical dependence" involving different equations, like P(M|a,b) being different from P(M|a).

So If I can come up with an example, any locally causal example for which logical dependence is present, it effectively invalidates Bell's local causality definition.

It would only invalidate Bell's reasoning if it were impossible to find a conditioning variable λ (dealing with facts in the past light cones of the events) such that the logical dependence (between spacelike-correlated events) is removed when conditioned on that variable. So, your challenge is only reasonable if I (or anyone else accepting the challenge) is allowed to define λ using any facts in the past light cones of the events that I want, merely showing that some poor choice of λ causes the equations to fail would not contradict Bell's reasoning.

 

[Maaneli]

Well, my intent wasn't to debate the point. We can agree to disagree.

Sorry, I can't let you off the hook that easily.

You keep making this claim that Bell's theorem refutes 'local realism'; and you will most likely continue to do so if no one continues to challenge you on it. Why are you all of a sudden unwilling to debate this issue and address the evidence I provided? When I initially asserted (without evidence from Bell) that Bell did not invoke any concept of 'local realism', you 'strongly disagreed' with me, and even claimed to point out exactly where Bell smuggled in 'the' realism assumption. Now that your claims has been challenged directly with evidence from Bell's own writings, I think the least you can do (not just for me, but for other people reading this thread) is to try and defend your claim. Or, if you feel that your claim is no longer tenable, then why not just concede that Bell did not talk at all of 'local realism', but rather local causality?

On the other hand, once you assume the existence of c, everything falls apart pretty quickly. Anyone who attempts to derive the Bell result will see this. No c, no Bell result. So c must be important.

Nobody claimed that c isn't important in Bell's derivation. The point, again, is that c *is not* where the realism assumption is initially introduced nor does the introduction of c have any implications for realism that are any different than what follows from Bell's local causality condition. This is plainly seen from what Bell actually wrote. And to reiterate, the variable c here is nothing but part of the specification of the experimental set-up (as allowed for by 'ordinary quantum mechanics'), just as are the polarizer settings a and b (in other words, a, b, and c are all local beables); and the introduction of c in the joint probability formula follows from the local causality condition, as part of the complete specification of causes of the events in regions 1 and 2. (As Bell also notes, one can think of c as specifying the source that produces the particles in the experimental set-up). So, again, there is no notion of realism in c that is any different than in a and b and what already follows from Bell's application of his principle of local causality.

And the generally accepted view is that this is the realism requirement. So don't take my word for it, try it yourself. Or ask Mermin.

The generally accepted view? I've never heard this claim before. Please cite a reference (or references) which supports the claim that the generally accepted view is that the invocation of c is the realism requirement. Also, please don't make appeals to authority or majority. Those are fallacies of logic, and a generally accepted view can certainly still be wrong.

 

[JesseM]

Maaneli, as for the issue you bring up of whether "realism" means anything separate from local causality, I think the most common idea is that in a philosophical view of physics that abandons the idea that the universe has some objective state at all times and just talks about measurement results (an instrumentalist or positivist view of science), we can still distinguish between theories that obey "locality" in a more limited sense and those that don't...comment 58 here says:

David, it depends what you mean by “local.” You can think of the Copenhagen interpretation as “local but not realistic” in the following sense: nothing Alice can do to her half of an EPR pair can possibly affect Bob’s density matrix, and in Copenhagen the density matrix is all there is.

Of course, if you want a density matrix describing Alice’s and Bob’s systems jointly, then it has to be entangled. But maybe that’s not so bad, since even in the classical world, we know that a joint probability distribution over two systems in general has to be correlated.

I also think there's a sense in which the MWI can be viewed as a "nonrealistic" theory that obeys locality, if you take "realism" to imply that any event at a single point in spacetime (like a measurement) must have a single unique outcome. Certainly a lot of MWI advocates say the theory obeys locality (see here for example), and in this post I gave a simple toy model showing how if you allow each experimenter to locally split into different copies when they make a measurement, with copies of an experimenter at one location not matched up into a single "world" with copies of an experimenter at a different location until there's been time for a signal to pass from one to the other, then you can have a situation where a randomly-selected copy will see statistics that violate Bell inequalities even though this scenario could be simulated on classical computers at different locations (one computer for each collection of copies of a simulated experimenter, with an actual spacelike separation between the event of each computer simulating the measurement-event). I don't think physicist who talk about "local realism" are normally thinking about "realism" as meaning that each event has a unique outcome, but you can see that the assumption is implicit in Bell's proof.

 

[Maaneli]

Bell is saying that according to local realism...

Bell is not saying this. As I've already stressed, he does not use this phrase 'local realism' (which, incidentally, has no clear meaning). He speaks only of local causality.

 

[JesseM]

Bell is not saying this. As I've already stressed, he does not use this phrase 'local realism' (which, incidentally, has no clear meaning). He speaks only of local causality.

When I say "Bell is saying that according to local realism" I do not mean to imply Bell used that exact phrase, just that the assumptions that Bell makes in his proof are equivalent to what later physicists mean by the phrase "local realism", a phrase which is commonly used in describing Bell's proof (see the comments in my most recent post #31 about what I think 'realism' is intended to denote)

 

[Maaneli]

Maaneli, as for the issue you bring up of whether "realism" means anything separate from local causality, I think the most common idea is that in a philosophical view of physics that abandons the idea that the universe has some objective state at all times and just talks about measurement results (an instrumentalist or positivist view of science), we can still distinguish between theories that obey "locality" in a more limited sense and those that don't...comment 58

But when you are talking about Bell's theorem and what Bell actually said and proved, then you should talk about the definition of locality that he specifically used in his theorem, and not some other more limited definition of "locality".

BTW, even the definition of locality implied by the equal-time commutation relation in QFT still assumes a notion of realism. So it is questionable whether there exists any physical definition of locality that doesn't involve some notion of realism. It is even more questionable whether any physical definition of locality could exist at all, without some implicit assumption of realism.

 

[Maaneli]

When I say "Bell is saying that according to local realism" I do not mean to imply Bell used that exact phrase, just that the assumptions that Bell makes in his proof are equivalent to what later physicists mean by the phrase "local realism", a phrase which is commonly used in describing Bell's proof

This is why I cited Norsen's paper earlier. Norsen methodically goes through various uses in the literature of the phrase 'local realism' by many prominent physicists in the field, and shows that the phrase has no clear meaning, and is certainly not equivalent to Bell's definition of local causality. I will also say that in my personal experience of talking with many quantum opticists (most notably, Joseph Eberly, Alain Aspect, and Pierre Meystre), I have not seen any evidence that they are aware of Bell's definition of local causality, or that they have a sharp definition of what 'realism' means in the phrase 'local realism'.