## Scholarpedia article on Bell's Theorem

 Quote by billschnieder This is a very simple question. Why don't you answer what $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ implies. I'm not asking you to define the symbols. What does it mean for the LHS to be different from the RHS in the above "definition" of local causality. You say local causality implies $P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 )$. So I'm simply asking you what $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ imples? Very simple question.
Oh, now I get the question. I thought you were asking what it *meant*, but your just trying to get me to say that $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ implies non-locality.

Yes, it does. $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ implies nonlocality.

But you have to read and understand and remember the words -- in particular that B_3 denotes a complete description of the physical state of a certain spacetime region, that b_2 has to live in a certain spacetime region (and can't be just any old extra thing you want to throw in), and that the P's are the fundamental dynamical probabilities assigned by some physical theory (as opposed to the kinds of probabilities that are based on ignorance of various things, etc.). If you actually hold all this in mind, it's trivial to see why all your examples, with the balls in the urns and whatnot, don't show what you think they show. Seriously, you have to actually slow down and read and process Bell's formulation. Let it marinade. (Sorry, I'm watching American Idol in the background.) Understand and appreciate what he's doing. Bell is not a dummy and he didn't formulate "locality" in a way that would diagnose, as nonlocal, trivial cases of correlation-without-causation like the ones you bring up. If you think it's so easy to refute -- if you think Bell is a dummy -- it only shows that you haven't taken the time to understand and appreciate what he accomplished.

Here, I'll put it as a challenge. State clearly, for your balls and urns or whatever example you want, what b_1, b_2, and B_3 are. Convince yourself and me that these satisfy all the conditions Bell laid down. (So, for example, oh, i dunno, B_3 better not turn out to be something like "what somebody who doesn't know the color of the first ball pulled knows about the state of the urn", and b_2 better not turn out to be in the past of b_1 rather than at spacelike separation and also outside the future light cone of region 3.) Then see if you still think there is some counter-example to Bell's formulation here.

 Quote by ttn No, they don't. I don't think you understood either what I wrote, or what Bell wrote.
Bell's words are clear as to what he meant, I'm not even interpreting his words, I quote them to you verbatim. You haven't provided any quote to support your claim just a pronouncement without evidence that I'm wrong.

 No, I don't say that at all. (See what I meant just above...) I say that if somebody refuses to accept the possibility of an irreducibly stochastic theory -- i.e., if they assume that determinism is true, such that stochasticity already implies incompleteness of the descriptions -- then they will think (erroneously) there is some kind of problem with the formulation. But that's their problem (indeed, your problem, since this seems to be your view!) not Bell's.
Now these are your words which you are now trying to undo:

 Quote by Travis Norsen Of course, if one insists that any stochastic theory is ipso facto a stand-in for some (perhaps unknown) underlying deterministic theory (with the probabilities in the stochastic theory thus resulting not from indeterminism in nature, but from our ignorance), Bell’s locality concept would cease to work.
(1) A *deterministic local hidden variable theory* which attempts to complete QM, is in fact making the *assumption* that the stochastic properties of QM simply arise due to incompleteness of QM, and such incompleteness can be completed by a "more complete specification". Now read Bell's original paper, excepts of which I posted above which clearly state this.
(2) It makes no sense for a *deterministic local hidden variable theory* to allow for the possibility of an irreducable stochastic theory, which is completely contrary to the concept of a *deterministic local hidden variable theory*.

THEREFORE, if you *assume a deterministic local hidden variable theory*, your statement implies that Bell's locality concept would cease to work in the narrow confines of your assumption. Maybe you misspoke in the article but this is clearly the meaning conveyed by the text.

 Quote by ttn Yes, it does. $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ implies nonlocality.
To be more precise then you are saying the above implies non-local causality. What is causing what in the above? Is it your claim that b_1 and b_2 are simultaneous? My next question would be for you to define what you understand by "cause".

 But you have to read and understand and remember the words -- in particular that B_3 denotes a complete description of the physical state of a certain spacetime region, that b_2 has to live in a certain spacetime region (and can't be just any old extra thing you want to throw in), and that the P's are the fundamental dynamical probabilities assigned by some physical theory (as opposed to the kinds of probabilities that are based on ignorance of various things, etc.).
Please define what you mean by fundamental dynamical probabilities.

 If you actually hold all this in mind, it's trivial to see why all your examples, with the balls in the urns and whatnot, don't show what you think they show.
I do not believe that anyone who understands probability theory can hold all of those things in their mind while being intellectually honest as will soon be evident.

 Seriously, you have to actually slow down and read and process Bell's formulation.
I hope you will be patient enough to go through the process with me and we'll see in the end who is right and who has no clue what they are saying. This is my challenge, answer the questions I have given above.

 Here, I'll put it as a challenge. ... better not turn out to be something like "what somebody who doesn't know the color of the first ball pulled knows about the state of the urn".
I'm happy you are posting this challenge because now it turns out the issue is about the meaning of probability. So let us start there. I will provide defintion of what probability means, and you will provide yours. then I will provide my definition of "cause" and you will provide yours and then we can discuss who is being consistent and who is not. I'm also happy that you like the urn example because we can use it to illustrate our meanings of probability. Feel free to do so.

So here is my definition of "probability":
a probability is a theoretical construct, which is assigned to represent a state of knowledge, or calculated from other probabilities according to the rules of probability theory. A frequency is a property of the real world, which is measured or estimated.
And my definition of "cause":
To say "C" (a cause) causes "E" (an effect) means that whenever C occurs, then E follows. Therefore we can not say "C" causes "E" if the two events are simultaneous. Similarly if "E" occurs before "C", then "C" can not be the cause of "E"
 While waiting for your definitions I thought I should also point out the following mathematical contradictions. NOTE, the following is simply a mathematics exercise, no physics whatever, but it clearly shows the problem Bell proponents are still unable to see: Consider the CHSH inequality: |E(a)E(b) - E(a)E(c)| + |E(d)E(b) + E(d)E(c)| ≤ 2, where E(a), E(b), E(c), E(d) ∈ [−1,1] This inequality is violated IFF (1) |E(a)E(b) - E(a)E(c)| + |E(d)E(b) + E(d)E(c)| > 2 We are interested to understand the mathematical properties of the 4 terms E(a), E(b), E(c), E(d) when this violation happens From (1) we have via factorization (2) |E(a)||E(b) - E(c)| + |E(d)||E(b) + E(c)| > 2 However, since E(a), E(b), E(c), E(d) ∈ [−1,1], it follows that |E(b) - E(c)| ≤ 2 and |E(b) + E(c)| ≤ 2 Let us consider the different possible extremes of the values for E(b) and E(c). If E(b) = E(c) then |E(a)||E(b) - E(c)| = 0 and |E(d)||E(b) + E(c)| must be greater than 2 for equation (1) to hold. But we know that |E(b) + E(c)| ≤ 2 which means |E(d)| must be greater than 2 which is impossible given that E(d) ∈ [−1,1]. If E(b) = -E(c) then |E(a)||E(b) + E(c)| = 0 and |E(a)||E(b) - E(c)| must be greater than 2 for equation (1) to hold. But we know that |E(b) - E(c)| ≤ 2 which means |E(a)| must be greater than 2 which is impossible given that E(a) ∈ [−1,1]. Therefore (1) is mathematically impossible. It is not possible mathematically to violate the CHSH inequality even before we start talking about any physics and what the terms might mean in any physical situation. This is the simple fact that Bell proponents are blind to. I challenge anyone to find values for E(a), E(b), E(c), E(d) ∈ [−1,1] that violate the above inequality from any source whatsover using any means whatsoever. You can even assume that E(a) are averages over many runs or whatever you like.

 Quote by billschnieder To be more precise then you are saying the above implies non-local causality. What is causing what in the above?
I say explicitly in my papers that you can't say, merely from the failure of this condition, what is causing what. You just know that there is some nonlocality somewhere.

 Is it your claim that b_1 and b_2 are simultaneous?
No.

 My next question would be for you to define what you understand by "cause".
It's increasingly clear with every question that you haven't read or processed what Bell wrote, or what I've written about what he wrote. I'm not going to play your games if you won't do your homework first.

 Please define what you mean by fundamental dynamical probabilities.
That's explained in my papers. It is really simple (though I'm sure this won't satisfy you): it means the probabilities that some candidate fundamental theory attributes to an event.

 I do not believe that anyone who understands probability theory can hold all of those things in their mind while being intellectually honest as will soon be evident. I hope you will be patient enough to go through the process with me and we'll see in the end who is right and who has no clue what they are saying. This is my challenge, answer the questions I have given above.
Right, so accuse me of being intellectually dishonest, and then literally in the next sentence ask me to please be patient enough to answer all your questions (the ones you have because you won't read or can't understand things that you've been referred to). No thanks.

(Your def'n of "probability" is inapppropriate in this context, as I've explained. And your definition of "cause" smuggles in the presupposition of determinism, which is a problem for the reasons I've explained.)

Now I give up.

 Quote by ttn That's explained in my papers. It is really simple (though I'm sure this won't satisfy you): it means the probabilities that some candidate fundamental theory attributes to an event.
That is an incomplete definition. What does probability mean in that phrase, that was my question. Define probability.

 Right, so accuse me of being intellectually dishonest,
No I'm saying *I* will have to be intellectually dishonest to believe all the things you want me to believe at the same time, in other words, that you do not understand probability theory. Prove me wrong by defining the terms I asked.

 (Your def'n of "probability" is inapppropriate in this context, as I've explained. And your definition of "cause" smuggles in the presupposition of determinism, which is a problem for the reasons I've explained.)
You don't have to agree with my definitions but I've clearly explained to you what *I* mean when *I* say "cause", and "probability". You haven't provided any alternate definitions of your own which you think are more appropriate.

After your article "Against Realism" in explained that many people arguing about Bell do not know what "realism" means, I would have thought you would understand the importance of clear definitions of terms. Once, you provide your definitions it would become evident that you do not know what you are talking about. All your claims about having explained things clearly in your articles, when you don't even have consistent definitions of terms will become evident.

I'm still waiting for your definitions for "probabilities" and "cause".

 Quote by billschnieder I'm still waiting for your definitions for "probabilities" and "cause".
Sorry, I'm really done. You'll have to get the answers you seek from my papers, or better, Bell's. ("La Nouvelle Cuisine" is particularly strongly recommended.) It's just frankly no fun talking with you.

 Quote by ttn Sorry, I'm really done. You'll have to get the answers you seek from my papers, or better, Bell's. ("La Nouvelle Cuisine" is particularly strongly recommended.) It's just frankly no fun talking with you.
So you are unable to define here what you mean by "probability" and "cause". Now hopefully you can point exactly to somewhere else where they are defined the way you like. Please provide a reference to a book, or article and specify a page number and paragraph where those terms are defined in a way you approve. This is a simple request. Simply saying, "read all my papers" or "read La Nouvelle Cuisine" would not cut it. Provide a specific location where the definition can be found.

Getting to the truth is not always fun if you are on the wrong side. This is not an entertainment exercise.

Recognitions:
Gold Member
 Quote by ttn It's just frankly no fun talking with you.
In my experience with billschnieder, I would say it is more fun to trim my nails with a hacksaw than to discuss anything with him on a good day.

 Quote by DrChinese In my experience with billschnieder, I would say it is more fun to trim my nails with a hacksaw than to discuss anything with him on a good day.

 Quote by DrC In my experience with billschnieder, I would say it is more fun to trim my nails with a hacksaw than to discuss anything with him on a good day.
I agree, arguing against the truth is very uncomfortable.
 Quote by ttn
Still unable to find a specific reference pointing to a definition of "probability" and "cause" that you agree with? (assuming they exist).

Since you like my questions so much, I thought I should add another. You say:
 Quote by ttn Yes, it does. $P(b_1 |B_3 , b_2 ) \neq P (b_1 |B_3 )$ implies nonlocality.
So you perform an experiment (X) and you measure P(b_1|X), P(b_2|X) from your experiment in just the manner in which it is done in Bell-test experiments, you also calculate P(b_1|X, b_2) or P(b_2|X, b_1) and lo and behold you find that P(b_1|X) ≠ P(b_1|X, b_2) and P(b_2|X) ≠ P(b_1|X, b_1). You quickly jump to the conclusion that the results imply non-locality. My question to you is

How did you make sure in your experiment that X is a complete specification?
In other words:
How have experimenters performing Aspect type experiments made sure that X is a complete specification?

You admit in your article that $P(b_1 |B_3 , b_2 ) = P (b_1 |B_3 )$ ONLY implies local causality if B_3 is a complete specification and b_2 adds nothing. Therefore unless you can make sure in an EXPERIMENT (X) that everything relevant for the outcome is specified, you can not reject local-causality on the basis of such a violation.

Put simply, "it is impossible to screen off a variable with another variable you know nothing about".

Still waiting for your definition for "probability" and "cause" after which we will examine if your idea of "complete specification" is consistent with the definitions.
 Highly enjoyable post ttn, and I want to respond to it with a full answer, but am currently snowed under. Will do so tomorrow.
 Recognitions: Science Advisor It provoked me to buy the second edition of "Speakable and Unspeakable", just for the "Nouvelle Cuisine" essay. This was money well-spent.