Thoughts about Bell, Bohm, Christian, et al.

[AnotherGuest]

Jumping the gun here..., if we put the inequality like this,



Then one can see by simple inspection that the inequality is false when the three terms on the RHS are independent.

This is correct. But if the three terms are defined by where the three functions , and have the usual properties and and integrates to 1, then the three terms are clearly not independent because, by some simple algebra, (15) is true.

[minkwe]

Experiments can only verify what was actually measured, not what could have been measured but wasn't. It is impossible to verify Counterfactual Definiteness experimentally for the EPRB scenario.

The particle measured by Bob along axis "b" is already destroyed. Therefore it is impossible to verify experimentally if Bob would have obtained had he measured along "a".

Of course Bob can pick the same axis as Alice, and verify that He always gets if he actually measures along "a". But this is not the same as an experimental verification of what he did not measure but could have.

In other words CFD is not experimentally falsifiable.
Similarly, Bell's inequality is not experimentally falsifiable, since it contains 2 counterfactual terms, which are impossible to measure. A valid Bell test experiment cannot be done. Once you measure P(a,b) the particles are destroyed, it is therefore impossible to measure the counterfactual, P(b,c) and P(a,c) needed for Bell's inequality. Separate independent measurements P(b,c) and P(a,c) of course would not be the counterfactual counterparts of the first P(a,b), as they should be. The counterfactual results are not independent from the actual one, but three seperate actual measurements are all independent. Fred has already shown that the inequality is invalid for independent results.

Millions of taxpayer dollars are being spent to do the impossible so-called " Bell test experiment ".

[Joy Christian]

A valid Bell test experiment cannot be done. Once you measure P(a,b) the particles are destroyed, it is therefore impossible to measure the counterfactual, P(b,c) and P(a,c) needed for Bell's inequality. Separate independent measurements P(b,c) and P(a,c) of course would not be the counterfactual counterparts of the first P(a,b), as they should be. The counterfactual results are not independent from the actual one, but three seperate actual measurements are all independent. Fred has already shown that the inequality is invalid for independent results.

Millions of taxpayer dollars are being spent to do the impossible so-called " Bell test experiment ".

And everyone is ignoring the only experiment that can actually verify the counterfactual results (because they would be available in that experiment for eternity).

This relatively-cheap-to-do experiment --- as some of you already know --- is the one I have proposed: http://libertesphilosophica.info/blog/e ... taphysics/.

This experiment has been bogusly criticised by Richard Gill, but his criticisms have been answered here: http://arxiv.org/pdf/1501.03393.pdf.

[minkwe]

This is correct. But if the three terms are defined by where the three functions , and have the usual properties and and integrates to 1, then the three terms are clearly not independent because, by some simple algebra, (15) is true.

True for an inequality containing actual P(a, b) with counterfactual P(b,c), and P(a,c), like Bell's, but false for independent separate measurements. For example, Bell's inequality can not be derived, starting with independent measurements:

[Heinera]

Experiments can only verify what was actually measured, not what could have been measured but wasn't. It is impossible to verify Counterfactual Definiteness experimentally for the EPRB scenario.

The particle measured by Bob along axis "b" is already destroyed. Therefore it is impossible to verify experimentally if Bob would have obtained had he measured along "a".

Of course Bob can pick the same axis as Alice, and verify that He always gets if he actually measures along "a". But this is not the same as an experimental verification of what he did not measure but could have.

In other words CFD is not experimentally falsifiable.

As your earlier wrote yourself, CFD is a logical property about a theory, not a physical one. It is a concept that can only be meaningfully applied to theories, not experiments. It is not something you can experimentally falsify, it is something you find out by investigating the theory in question. Does the theory allow you to compute predictions for counterfactual outcomes? That is the issue, not whether these predictions are experimentally correct or not (you can never find that out).

Similarly, Bell's inequality is not experimentally falsifiable, since it contains 2 counterfactual terms, which are impossible to measure.

Being a theorem, Bell's theorem is obviously not empirically falsifiable. Nor are any of these experiments performed in order to check whether Bell's theorem is correct or not. The theorem says that a LHV-model cannot produce the same correlations that QM can. Obviously there is no physical experiment that can verify or falsify that claim. Take the theorem that in Euclidean space the angles in a triangle sums to 180 degrees. If you draw up a triangle and find that the sum is not 180, you obviously can not conclude that the theorem is wrong. You must conclude that you live in a non-Euclidean space.

[Schmelzer]

Experiments can only verify what was actually measured, not what could have been measured but wasn't. It is impossible to verify Counterfactual Definiteness experimentally for the EPRB scenario.

Indeed, counterfactual definiteness is a theoretical property. In local realistic theories, the EPR criterion allows to derive it in certain circumstances.

And, then, given CFD, one can derive Bell's inequality.

Of course Bob can pick the same axis as Alice, and verify that He always gets if he actually measures along "a". But this is not the same as an experimental verification of what he did not measure but could have.

And it also does not prove counterfactual definiteness if one could measure in all directions. Because in a nonlocal theory, it would remain possible to choose all the measurement results using some random (not counterfactually predefined) values and send the chosen values to the other side so that they would be able the corresponding results.

True for an inequality containing actual P(a, b) with counterfactual P(b,c), and P(a,c), like Bell's, but false for independent separate measurements.

Depends on how one understands "independent separate measurements". If one makes separate measurements, each with different predefined settings a, b, then, of course, this gives the loophole of using, in fact, different preparation procedures for different choices of a, b, and with this loophole it is easy to violate Bell's inequalities.

But the point of Bell's theorem is a different one. It is theoretically derived (using locality and realism) that



Now, one simply derives properties of the E(AB|a,b), using this assumption. And after this one measures E(AB|a,b) and compares. There is no reason at all to expect that the inequalities hold somehow theory-independent. The possibility to compare the expressions E(AB|a,b) for different pairs a, b is a consequence of the particular theoretical assumptions, which lead to CFD.

[FrediFizzx]

Experiments can only verify what was actually measured, not what could have been measured but wasn't. It is impossible to verify Counterfactual Definiteness experimentally for the EPRB scenario.

Indeed, counterfactual definiteness is a theoretical property. In local realistic theories, the EPR criterion allows to derive it in certain circumstances.

And, then, given CFD, one can derive Bell's inequality.

Of course Bob can pick the same axis as Alice, and verify that He always gets if he actually measures along "a". But this is not the same as an experimental verification of what he did not measure but could have.

And it also does not prove counterfactual definiteness if one could measure in all directions. Because in a nonlocal theory, it would remain possible to choose all the measurement results using some random (not counterfactually predefined) values and send the chosen values to the other side so that they would be able the corresponding results.

True for an inequality containing actual P(a, b) with counterfactual P(b,c), and P(a,c), like Bell's, but false for independent separate measurements.

Depends on how one understands "independent separate measurements". If one makes separate measurements, each with different predefined settings a, b, then, of course, this gives the loophole of using, in fact, different preparation procedures for different choices of a, b, and with this loophole it is easy to violate Bell's inequalities.

But the point of Bell's theorem is a different one. It is theoretically derived (using locality and realism) that



Now, one simply derives properties of the E(AB|a,b), using this assumption. And after this one measures E(AB|a,b) and compares. There is no reason at all to expect that the inequalities hold somehow theory-independent. The possibility to compare the expressions E(AB|a,b) for different pairs a, b is a consequence of the particular theoretical assumptions, which lead to CFD.

LOL! The whole point that Michel made is that Bell's inequalities are "rigged" against LHV and not QM.

[Heinera]

LOL! The whole point that Michel made is that Bell's inequalities are "rigged" against LHV and not QM.

They are, indeed (sorry for the late reply, but it is only now that I managed to stop laughing).

The theorem says that no LHV model can generate the correlations that QM can generate. The theorem says absolutely nothing about the empirical correctness of any theory whatsoever. The QM correlations could be completely wrong empirically (i.e., measured correlations in experiments could be something entirely different from QM); the theorem would still be true.

[minkwe]

I see that there's still confusion between "Bell's inequality" and "Bell's theorem".

"Bell's inequality" is equation (15) in his paper. "Bell's theorem" is a result of combining "Bell's inequality" with the predictions of Quantum Mechanics for and then concluding that since the results violate the inequality, one of assumptions required to obtain the inequality must be false. I haven't said anything about Bell's theorem yet, we will address this next, because the argument has huge holes in it too.

First, to summarize what we have revealed so far:
1. There is no "locality assumption" in Bell's derivation of his inequalities. None whatsoever, despite repeated noises about "locality".
2. The counterfactual definiteness assumption invoked by Bell applies to QM as well. Bell himself uses it on page 1 to make a QM argument.
3. Bell's inequality relates one actual measurement, to two counterfactual measurements which could have been done but weren't.
4. Bell's inequality can not be derived if we do not assume counterfactual results, but instead use actual independent measurements .

because:

- Since , we can not factorize as Bell did at the top of page 406.
- Bell's condition applies for the counterfactual measurements but does not apply for independent separate measurements .

5. It is impossible to experimentally verify a counterfactual argument such as Bell's inequality, therefore those spending millions of taxpayers' money trying to do same are, are either perpetrating a massive fraud, or have no clue what they are doing.
6. The definitive Bell-test experiment is logically impossible. It cannot be done, and will never be done. No experiment which uses only actual outcomes can test Bell's inequality. Just like no experiment in which Bob and Alice always measure along "a", can test what the counterfactual result along "a" would have been had Bob actually measured along "b". Even in the alternate universe in which it is possible to experimentally test Bell's inequality (which is impossible), the results of this test will have no impact on locality (since it is irrelevant to the derivation). It can only have an impact on CFD. But we have found that without CFD, we can't even discuss scientific predictions of scientific theories, including QM. This is not just my opinion:

In my opinion the present unfashionableness of counterfactual reasoning in the philosophy of science is quite misguided. We would not have ethics, justice, or science, without it.

One other thing which Gill got spot on in that article:

Bell offered four quite different positions which one might like to take compatible with his mathematical results.
...
In my opinion he missed an intriging[sic] fifth position:
5. A decisive experiment cannot be done

Although Gill's reasons for thinking about that possibility were off target.

Since CFD is required to obtain the inequalities, some often interpret violations of Bell's inequality by an experiment as evidence that CFD is false. Duh, CFD is false in the experiment because none of the terms from the experiment being used to compare with the inequality are counterfactual! Think think think!

As for QM, none of the terms calculated from QM and being used to compare with the inequality are counterfactual. Therefore Bell's theorem is unfounded.

[Schmelzer]

"Bell's inequality" is equation (15) in his paper. "Bell's theorem" is a result of combining "Bell's inequality" with the predictions of Quantum Mechanics for and then concluding that since the results violate the inequality, one of assumptions required to obtain the inequality must be false.

Of course, Bell's theorem is the theorem that local realistic theories fulfill the Bell's inequalities. Considerations about QM are, of course, part of Bell's paper, but not of Bell's theorem.

First, to summarize what we have revealed so far:
1. There is no "locality assumption" in Bell's derivation of his inequalities. None whatsoever, despite repeated noises about "locality".

Wrong. The proof starts with the EPR argument, which requires locality. Without locality there would be also no reason to restrict to .

2. The counterfactual definiteness assumption invoked by Bell applies to QM as well. Bell himself uses it on page 1 to make a QM argument.

No. In the minimal interpretation there is no CFD.

3. Bell's inequality relates one actual measurement, to two counterfactual measurements which could have been done but weren't.

False. BI relates averages. The proof uses CFD at intermediate steps, but the result is not about single measurements.

4. Bell's inequality can not be derived if we do not assume counterfactual results, but instead use actual independent measurements .

Even this is wrong, because we can derive counterfactual results, so that we do not have to assume them. Which is what is done using the EPR argument.
And, of course, nobody claims that one can derive the BI "using actual measurements". The BI are derived from assumptions about the general theory (that it is a local realistic one). Nor locality, nor realism can be derived from considering actual measurements.

5. It is impossible to experimentally verify a counterfactual argument such as Bell's inequality, therefore those spending millions of taxpayers' money trying to do same are, are either perpetrating a massive fraud, or have no clue what they are doing.

Wow, a correct claim - of course, a CFD is named "counterfactual" because it is not about an observable fact. But this is immediately followed by nonsense. Because the experiments try to falsify, not to verify, the BI.

6. The definitive Bell-test experiment is logically impossible. It cannot be done, and will never be done. No experiment which uses only actual outcomes can test Bell's inequality.

Also false, because BI relates average values, and these average valued can be measured without problem.

But we have found that without CFD, we can't even discuss scientific predictions of scientific theories, including QM. This is not just my opinion:

In my opinion the present unfashionableness of counterfactual reasoning in the philosophy of science is quite misguided. We would not have ethics, justice, or science, without it.

While positivism, which is the base for the rejection of counterfactual reasoning, is indeed deeply misguided and anti-scientific, the claim that "we cannot even discuss scientific predictions" is an overexaggeration too. One can make a reasonable argument that, if we would apply positivism consistently, not only doing science, but applying everyday common sense would be impossible. But, of course, even the most dogged positivist does not do this.

Instead, a particular CFD claim, like the claim that the outcomes of the Bell experiments are CFD, can be simply wrong even in causal realist theories. For example, they are wrong in dBB theory, where the wave function of the system and the hidden trajectory of the system are insufficient to predict the outcome of experiments, because it also depends on the hidden trajectory of the measurement device. Thus, the outcome of the experiment is not defined for experiments which are not actually done.

To summarize:

First, to summarize what we have revealed so far:

Revelations, indeed. Amen

[minkwe]

For those who failed the exam, let us do a review the main points:

1. There is no "locality assumption" in Bell's derivation of his inequalities. None whatsoever, despite repeated noises about "locality".

- implies settings independence, not locality. It is possible to have , where represents a non-local hidden variable. Moreover, it is possible to have settings dependence even if the whole experimental setup is local and within each others light cones such that , as explained in this post
- Secondly, setting indepdendence is irrelevant for the derivation anyway because one could eliminate hidden variables (local or non-lcoal) completely and still proceed to derive the inequality, since the expectation value of the paired product can be calculated for any valid probability measure. This has been clearly explained in this post.



, Since
and after factorizing out , remembering that we get


The second term on the right is .

Therefore, any insistence that locality is required is simply poppycock, and a waste of time.

2. The counterfactual definiteness assumption invoked by Bell applies to QM as well. Bell himself uses it on page 1 to make a QM argument.

- Counterfactual Definiteness is defined as: Wikipedia: In quantum mechanics, Counterfactual definiteness (CFD) is the ability to speak meaningfully of the definiteness of the results of measurements that have not been performed (i.e. the ability to assume the existence of objects, and properties of objects, even when they have not been measured).
Gill (http://arxiv.org/pdf/1207.5103v6.pdf): Its formulation refers to outcomes of measurements which are not actually performed, so we have to assume their existence, alongside of the outcomes of those actually performed: the principle of realism, or more precisely, counterfactual definiteness.
http://arxiv.org/pdf/1007.4281.pdf: the assumption that a measurement that was not performed had a single definite result..
- Bell's 2nd and 3rd sentence in paragraph 2 of page 1, applies CFD to quantum mechanics as follows: If Alice measures along "a" and obtained +1, then if Bob were to measure the sister particle along "a" he must obtain -1. Clearly, Bell believes and QM states that the measurement that was not performed has a single definite value. In fact, CFD applies to any theory that makes predictions. Any suggestion otherwise reflects lack of thinking ability, as it can easily shown that a prediction for an experiment which ends up not being done, is counterfactually definite. No theory is immune to this, including QM.

3. Bell's inequality relates one actual measurement, to two counterfactual measurements which could have been done but weren't.

- Bell invokes CFD in his derivation starting on Page 406 where he states that "it follows that c is another unit vector". Note, we have a pair of particles, one is measured along "a", the other is measured along "b". Bell says it follows that "c" is another unit vector, but we don't have any other particles to measure along "c". Therefore "c" is a counterfactual axis (what we could have measured but didn't). This is also revealed in the above derivation, by the use of the same index for all the summation terms, reflecting the fact that Bell is always looking at the outcomes for the same pair of particles along the three axes, one of which must be counterfactual. Futhermore, the factorization of outcomes from both terms in the summation/integral again reveals that it is always the same pair of particles measured at 3 axes, one of which must be counterfactual. Therefore, any terms in the final inequality which contain "c", MUST be counterfactual terms, ie .
- Therefore Bell's inequality is a relationship between one actual expectation value and two counterfactual expectation values .

4. Bell's inequality can not be derived if we do not assume counterfactual results, but instead use actual independent measurements .

- It does not apply to three actual expectation values because it can not be derived starting from those.
--> DEAD END. No way to derive P(b,c) on the RHS therefore no way to derive the inequality

Therefore, the following is also poppycock:

Starting from realism and Einstein locality, we can prove Bell's inequality.
Bell's inequality is violated.
Thus, "local realism" (Einstein-local realism would be more accurate) is falsified.

If locality is not required to obtain the inequalities, there is no reasonable way for a reasonable person to conclude that violations of the inequalities falsifies locality. [emphasis on reasonable] Thus poppycock.

[FrediFizzx]

"Bell's inequality" is equation (15) in his paper. "Bell's theorem" is a result of combining "Bell's inequality" with the predictions of Quantum Mechanics for and then concluding that since the results violate the inequality, one of assumptions required to obtain the inequality must be false.

Of course, Bell's theorem is the theorem that local realistic theories fulfill the Bell's inequalities. Considerations about QM are, of course, part of Bell's paper, but not of Bell's theorem.

From https://en.wikipedia.org/wiki/Bell's_theorem,

"Bell himself wrote: "If [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says." John Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987, p. 65."

So you are wrong again.

[FrediFizzx]

If locality is not required to obtain the inequalities, there is no reasonable way for a reasonable person to conclude that violations of the inequalities falsifies locality. [emphasis on reasonable] Thus poppycock.

Didn't De Raedt, et al, show that Boole obtained the inequality from a purely mathematical standpoint?

[Schmelzer]

From https://en.wikipedia.org/wiki/Bell's_theorem,

"Bell himself wrote: "If [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says." John Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987, p. 65."

So you are wrong again.

This is a nice informal summary of the consequences of Bell's theorem, given that everybody knows what QM predicts.
But the theorem is the part which proves the BI (15), thus, is about local realistic theories, and there is no reason to even mention that such an animal as QM exists for proving the theorem.

Therefore, any insistence that locality is required is simply poppycock, and a waste of time.
...
Bell's 2nd and 3rd sentence in paragraph 2 of page 1, applies CFD to quantum mechanics as follows: If Alice measures along "a" and obtained +1, then if Bob were to measure the sister particle along "a" he must obtain -1. Clearly, Bell believes and QM states that the measurement that was not performed has a single definite value. In fact, CFD applies to any theory that makes predictions.

Let's see what Bell writes:

Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of , by previously measuring the same component of , it follows that the result of any such measurement just actually be predetermined.

So, Bell makes an assumption about locality, and, using this assumption, derives that the value is predetermined.

Then, you obviously did not get the point, of course Bell uses the CFD in his proof. But the result is about averages, and these averages can be measured without measuring P(a,b), P(a,c) and P(b,c) on the same particles. So, it is the proof which uses CFD, but the BI is not about them.

Any suggestion otherwise reflects lack of thinking ability, as it can easily shown that a prediction for an experiment which ends up not being done, is counterfactually definite. No theory is immune to this, including QM.

The Gurus of the new, revolutionary science obviously cannot argue without primitive insults, ok, I'm used to such things, modern democratic education leads to similar problems in all other domains too. You can, of course, show whatever you want about predictions, but QM does not predict the results (except in very special situations) but only probabilities.

[Schmelzer]

Didn't De Raedt, et al, show that Boole obtained the inequality from a purely mathematical standpoint?

I haven't read de Readt and Boole about this, but of course, it is quite trivial in a classical situation where you have no doubt that the third value c exists to prove the BI, without any reference to any locality.

As I have explained, the question which assumptions are "physically necessary" to prove the BI is nonsensical. And there is no question that one can invent assumptions so that one can prove the BI which do not use whatever you don't want to have used. The only reasonable question is if Bell has used locality, and this is obvious from the text.

[FrediFizzx]

From https://en.wikipedia.org/wiki/Bell's_theorem,

"Bell himself wrote: "If [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local. This is what the theorem says." John Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987, p. 65."

So you are wrong again.

This is a nice informal summary of the consequences of Bell's theorem, given that everybody knows what QM predicts.
But the theorem is the part which proves the BI (15), thus, is about local realistic theories, and there is no reason to even mention that such an animal as QM exists for proving the theorem.

Do you have a reference for your statement, "But the theorem is the part which proves the BI (15)..."? I gave a reference for mine. So Michel is right, you (and others) are confused as to what Bell's theorem actually is it would seem. I don't see how any definition of what the theorem is can be better than what Bell claims it to be.

[minkwe]

If locality is not required to obtain the inequalities, there is no reasonable way for a reasonable person to conclude that violations of the inequalities falsifies locality. [emphasis on reasonable] Thus poppycock.

Didn't De Raedt, et al, show that Boole obtained the inequality from a purely mathematical standpoint?

Indeed. It was Boole who discovered those inequalities not Bell, and from a purely mathematical standpoint, which means their violation points to mathematical error and not such physical concepts as locality.

You can find more details in :
J.Comp.Theor.Nanosci. 8, 1011 (2011)
http://arxiv.org/pdf/0901.2546v2
This is a huge, and very thorough paper.

The central result of this paper is that the necessary conditions and the proof of the inequalities of Boole for n-tuples of
two-valued data (see Section II) can be generalized to real non negative functions of two-valued variables (see Section III) and to quantum theory of two-valued dynamical variables (see Section IV). The resulting inequalities, that we refer to as extended Boole-Bell inequalities (EBBI) for reasons explained in the Introduction and in Section III, have the same form as those of Boole and Bell. Equally central is the fact that these EBBI express arithmetic relations between numbers that can never be violated by a mathematically correct treatment of the problem: These inequalities derive from the rules of arithmetic and the non negativity of some functions only. A violation of these inequalities is at odds with the commonly accepted rules of arithmetic or, in the case of quantum theory, with the commonly accepted postulates of quantum theory
...
A violation of the EBBI cannot be attributed to influences at a distance. The only possible way that a violation could arise is if grouping is performed in pairs (see Section VII A).

In the original EPRB thought experiment, one can measure pairs of data only, making it de-facto impossible to use Boole’s inequalities properly. This obstacle is removed in the extended EPRB thought experiment discussed in Section VI C. In this extended EPRB experiment, one can measure both pairs and triples and consequently, it is impossible for the data to violate Boole’s inequalities. This statement is generally true: It does not depend on whether the internal dynamics of the apparatuses induces some correlations among different triples or that there are influences at a distance. The fact that this experiment yields triples of two-valued numbers is sufficient to guarantee that Boole’s inequalities cannot be violated

And a different perspective in :
Quantum Matter, Volume 3, Number 6, December 2014, pp. 499-504(6)
https://hal.archives-ouvertes.fr/hal-00824124/document

It was shown in [1], cited in the sequel as DRHM, that upon a correct use of the respective statistical data, the celebrated Bell inequalities cannot be violated by quantum systems. This paper presents in more detail the surprisingly elementary, even if rather subtle related basic argument in DRHM
...

The inequalities (17) are purely mathematical. In particular, their proof depends in absolutely no way on anything else, except the mathematical
properties of the set Z of positive and negative integers, set seen as a linearly ordered ring, [9].
As for the inequalities (16), they are a direct mathematical consequence of the inequalities (17), and thus again, their proof depends in absolutely no way on anything else, except the mathematical properties of the set R of real numbers, set seen as a linearly ordered field, [9].
It is, therefore, bordering on the amusing tinted with the ridiculous, when any sort of so called “physical” meaning or arguments are enforced upon these inequalities - be it regarding their proof, or their connections with issues such as realism and locality in physics - and are so enforced due to a mixture of lack of understanding of rather elementary and quite obviously simple mathematics

For those who have been following along, note that the Bell inequalities require triples of outcomes for each event , one of which is counterfactual, in order to derive the inequality, while for all claimed violations by QM and experiments, they always use pairs of outcomes for each event: they measure one set of pairs to calculate , a different set of pairs to calculate to calculate , and yet a different set of pairs to calculate to calculate . This is the mathematical error De Raedt et al are talking about. And it is the same mathematical error as deriving an inequality using counterfactual expectations, and trying to test it using actually measured expectation values. It is as stupid as expecting that when if Alice measures along "a" and obtains +1 for event , When Bob measures along "a" for event he MUST observe -1, and vice versa. It really is that stupid.

[minkwe]

But the theorem is the part which proves the BI (15)

I see that there are still some who do not know what "Bell's theorem" is.

Bell's theorem states that since the predictions of Quantum Mechanics violate Bell's inequalities, it is impossible to find a local theory which reproduces those predictions of quantum mechanics.

To prove Bell's theorem, first the inequalities are derived, claiming that only local theories can give those inequalities. Then, the expectation values are calculated from QM, and substituted into the inequalities, and a violation is obtained. Then, it is claimed that since local theories imply the inequalities, and QM violates the inequalities, therefore local theories and QM are incompatible.

However, we now know that even though the inequalities are valid, and Bell claimed to be invoking locality, no concept of locality carries forward into his equations. Simply invoking "locality" in the text is not enough if what is encoded in the equations, is completely devoid of a locality assumption as we have seen. Besides, we have seen that locality is not required at all to derive the inequalities. And basic logic tells us, if it is not required, then the negation of the inequalities has absolutely no impact on locality.

[Schmelzer]

This is a nice informal summary of the consequences of Bell's theorem, given that everybody knows what QM predicts.
But the theorem is the part which proves the BI (15), thus, is about local realistic theories, and there is no reason to even mention that such an animal as QM exists for proving the theorem.

Do you have a reference for your statement, "But the theorem is the part which proves the BI (15)..."?

What about common sense?

The BI are (15), and the theorem is what proves them. Simple common sense.

Anyway, it is irrelevant what you name "Bell's theorem". But I can explain you why I think it is reasonable not to incorporate QM into the theorem. The very point of the theorem is that the BIs can be falsified by observation. And that after this one does not have to believe that QM is true to conclude that Einstein-causal realism is dead.

The Wiki page you have posted makes, BTW, the usual error to present the proof as if CFD was assumed instead of derived from the EPR argument.

[FrediFizzx]

This is a nice informal summary of the consequences of Bell's theorem, given that everybody knows what QM predicts.
But the theorem is the part which proves the BI (15), thus, is about local realistic theories, and there is no reason to even mention that such an animal as QM exists for proving the theorem.

Do you have a reference for your statement, "But the theorem is the part which proves the BI (15)..."?

What about common sense?

The BI are (15), and the theorem is what proves them. Simple common sense. Anyway, it is irrelevant what you name "Bell's theorem".

Michel already has shown that you are wrong about that. Correct, Bell's theorem and the inequality (15) are indeed irrelevant for physics. Pretty much common sense.