Conditionals and Bell’s Inequalities

NOTE: section 6 has recently been updated (July 2022)

1.  Introduction: An a priori proof of Bell’s Inequalities?

2. The logic CE and probability modeling

3. How the Bell Inequalities are tested

4. Setting up for the derivation

5. Modeling this with conditionals

6. An empiricist view of conditionals

1.  Introduction: An a priori proof of Bell’s Inequalities?

Bell’s Inequalities, which are satisfied in the probabilities of results, conditional on experimental set-ups of a traditional sort, are famously violated in the results of certain quantum mechanics experiments.  It is therefore remarkable that those inequalities could be deduced from certain putative principles about causality and locality.  It would seem that their violation could be taken as refuting those principles.

But even more remarkable were the signs that Bell’s Inequalities could be deduced logically, given certain principles seriously proposed for the logic of conditionals.  (See Bibliography: Stapp 1971, Eberhard 1977, Herbert and Karush 1978.) My project here is to examine that deduction, and the options there are for a philosophical response.  Is it the logic of conditionals that is at fault?  Or is it an understanding of conditionals and their logic that was loaded with philosophical realist presuppositions?

Bell’s Inequalities related conditional probabilities, of results given certain measurement set-ups.  Therefore, if they are to be approached in this way, it can only be in a logic compatible with a bridge principle that relates conditionals and conditional probabilities.  The bridge principle introduced by Robert Stalnaker, known generally as Stalnaker’s Thesis, or more recently as just The Thesis, was this:

Thesis.  P(A |B) = P(B |A)

Its combination with Stalnaker’s logic of conditionals turned out to be inadequate (it would allow only trivially small models).  David Lewis rejected the Thesis, but also pointed the finger at what he took to be a mistaken principle about conditionals

Conditional Excluded Middle (CEX)   [A –> (B v C)] is equivalent to [(A –>B) v (A –> C)]

The Thesis and CEX do indeed go together, for one needs to accommodate the fact about conditional probability that if B and C are contraries then P(B v C| A) = P(B |A ) + P(C | A).

(CEX is what Paul Teller called the Candy Bar Principle; see previous post about this.)

There is however a logic of conditionals, which I called CE, somewhat weaker than Stalnaker’s, which includes CEX yet combines successfully with the Thesis.  It has only infinite models, but ‘small’ situations, like experimental set-ups, can be modeled with partial structures that are demonstrably extendable to models of CE combined with the Thesis, for all sentences, regardless of complexity. (See previous posts on probabilities of conditionals.)

So, we are in a position to examine the arguments that putatively lead from the logic of conditionals to Bell’s Inequalities.

2. The logic CE and probability modeling

The first principle for any logic of conditionals is that if  A implies B then A–> B is true.  This yields at once the theorem that A –> A is always true. The second is Modus Ponens:  if A and A –> B are both true then B is true.  Beyond this, there be controversy.

The logic CE adds CEX, stated above, as well as two more principles about conjunction.  Stated as a theory of propositions (rather than the sentences that express them):

(I)  Conditional Excluded Middle       A –> (B v C)    =   (A –> B)  v   (A –> C)

(II) Conjunction Distribution             A –> (B & C)   =   (A –> B)  & (A –> C) 

(III) Modus Ponens Amplified            A & (A –> B)  =   (A & B)

The last includes Modus Ponens, but adds something:  if A and B are both true, then there is no question about whether B would be true if A were true, of course it would, because it is.

How are situations, like experimental set-ups, modeled?  The standard way of doing this in philosophical logic is to say that each proposition is either true or false in each possible world, and the proposition can be identified with (or at least, represented by) the set of worlds in which it is true.  Of course, ‘possible worlds’ is a metaphor, there is just the set-up and the possible results of doing the experiment.

As a simple example suppose a die is to be tossed.  We have a hypothesis: the die is fair, and all outcomes have the same probability.  So as possible worlds we just take the sequences <toss, outcome> of which there are six.  In <toss, 1> the statement ‘the outcome is 1’ is true, and so forth.

Now the conditionals we are interested in are such as these: 

            A –> B ‘if the outcome is odd, it is less than 4’. 

That is true in <toss, 1> and in <toss, 3>.  But where else is it true?   To satisfy Stalnaker’s Thesis, since the probability of ‘greater than four, conditional on even’ is 2/3, or 4/6. So the conditional must be true in two other worlds besides those.

So we have in the model a function s:  given non-empty antecedent A and world w, the world s(A,w) is a world in which A is true. Intuitively, s(A, w) is the way things would have been the case with A true.  The proposition A –> B is then the set of worlds {w: s(A, w) is in B}.

Elsewhere (see Notes at the end) you can see the details for this very simple experimental set up, modeled with conditionals and probabilities.  Just to give you the idea, there the proposition A –>B that we have here as example, is true in worlds <toss, 1>, <toss, 3>, <toss, 2>, <toss, 4>.  So, in this model, if the outcome is actually 4, then the outcome would have been less than 4 if it had been odd.  There is no rationale for this:  it is just how things are in that possible universe of worlds.  We might be living in it, or in another one, that is not up to us.

The important point is this:  for any such situation we can construct a representation that is extendible to a model of CE in which the Thesis holds for all propositions.

3. How the Bell Inequalities are tested

For the original Einstein-Podolsky-Rosen thought experiment David Bohm designed an experiment that would be technically feasible, in which a pair of photons are emitted from a source in an entangled state.

For us, what we need to display is only the ‘surface’ of the experimental set-up, with some notes to help the imagination; we do not need to look into the quantum mechanics.

Look at the left side, labeled (A) for ‘Alice’.  It is a device in which there is a polarization filter, which may or may not be passed by an incoming particle.  A red light goes on if it does pass, a green light if nothing passes.  That filter has three different orientations, and one is chosen beforehand by the experimenter, or by a randomizing device.  Similarly for the right hand side, labeled (B) for ‘Bob’.

The experimental facts are these: if we only look at one side, left or right, then regardless of the setting, the red light is going on in exactly 50% of the runs.  But if we look at both, we see that if the two settings are the same, then the red lights never turn on at the same time (Perfect (anti)Correlation.)  Furthermore, there are specific probabilities for the red lights turning on at the same time, for any pair of settings.  These are conditional probabilities:  P(red light for Alice and red light for Bob | Alice chose setting i and Bob chose setting j).   It is these for which the Bell Inequalities may or may not hold.

4. Setting up for the derivation

With reference to the above diagram let’s refer to Alice as the one on the left (L) and Bob as the one on the right (R) .  The two outcomes, red light on and green light on, I will refer to as outcomes 1 and 0.  And the settings are settings 1, 2, 3.  Let little letters like “a”. “b”, “i” and “j” be variables over {1,0}.  Then we can symbolize:

On the left the setting is i:    Li

On the left the setting is i and the outcome on the left is a:    Lia

On the right the setting is j and the outcome on the right is b:    Rjb

and then we can have sentences like (Lj1 –> Rj0) to indicate that with the same setting in on both sides, if the outcome is 1 on the left then it will be 0 on the right.  Also sentences like P(Lj0|Rk0) = 3/4 to indicate that if the left and right settings are j and k respectively, then the probability that the light is green on the left side, given that it is green on the right side, equals 3/4.

The conclusion drawn from many observations are the following two premises.

For i, j = 1, 2, 3 and a, b = 0, 1:

I. Perfect Correlation: If the setting is i on both sides then the probability of outcome a on both sides equals 0

II. Surface Locality: The probability of outcome Lia is the same conditional on Li as it is conditional on Li & Rj —

that is, the probability of an outcome on one side is unaffected by the setting on the other side.

Now the Bell Inequalities can be expressed in a simple way.  Let us abbreviate the special case of the probability of outcome 1 happening on both sides, for specific settings, as follows:

p(i; j)  = the probability of (Li1 & Rj1) given settings Li and Rj

The Bell Inequalities can then be expressed in a set of ‘triangle inequalities’:

p(1;2) + p(2;3)  ≥  p(1;3)

and so forth.  There is no reference in these inequalities to any factor which may be hidden from direct measurement — any violation can be found on the observable level.

So it would be very disconcerting if there were a proof that there could not be any violations!

5. Modeling this with conditionals

Naturally, some idealization will be involved.

The entry moves

We fix on some entailments implied in the experimental set-up, assuming the sort of perfection not found in an actual lab.  So we take it that the settings being chosen, and the experiment initiated, entails that there will be an outcome, and it will be red light or green light:

Li entails (Li1 v Li0), hence Li –> (Li1 v Li0) is true

Similarly for the other similar points; for example, (Li &Rj) –> [(Li1 & Rj1) v …. v (Li0 & Rj0)], all logically possible combinations listed in the blank here.  

Moreover, we take it as necessary, that is true in all worlds, that outcomes are unique.  That is, the conjunction (Li1 & Li0) is never true; similarly for R.

Finally, the modeling must accommodate this:  Lia and Rjb are each, taken individually, possible, that is, there is a possible world in which it is true. 

Consequences of the entry moves  

Starting with the entry move, and using Conditional Excluded Middle, we infer accordingly:

Either Li –> Li1 or Li –> Li0 is true,

more generally,

One of the conditionals (Li & Rj) –> (Lia & Rjb) is true, i, j = 1, 2, 3 and a, b = 1, 0.

Note that this is a finite set of conditionals, since there are only finitely many combinations of settings and outcomes.

I have just written “is true”, as if we are only interested in the actual world.  Of course we have to be interested in all the possible worlds in the modeling set-up, each characterized by specific settings and specific outcomes.  But the above reasoning holds for any world in the model. 

We note also that as long as an antecedent of a conditional is itself possible, there cannot be a conflict in the consequents:

            if  A –> B and A –> ~B are true, or if [A –> (B & ~B)]is true, then A is false.

This follows from the Conjunction principles, and applies to our case because with our entry moves Li1 implies the falsity of Li0, and so forth.

The hidden variable

Which counterfactual conditionals are true in a given situation is not something empirically accessible.  But by the above reasoning, in any given world α in the model, there is a set of true propositions of form Li –> Lia, Rj –>Rjb, (Li & Rj) –> (Lia & Rjb) which characterizes that world. 

Call that set A(α).  It is a hidden factor which completely determines what the outcomes will or would be, whatever setting the experimenter chooses or would have chosen if he had chosen differently. A(α) represents the world’s hidden dynamical state.

NOTE: At this point we have to raise a question not answerable in that simple artificial language: what makes those conditionals true? In order for the discussion to have any bite, with respect to the experiment, whatever makes them true must not be, for example, the actual but unknown future — it has to be something determined before the experiment has its outcomes. That set A(α) of statements has to represent something characterizing the particle-pair at the outset: that hidden dynamical state has to be a physical feature. To this we will come back below.

(Historically minded logicians will be reminded here of the objections to Diodoros Chronos’ Master Argument.)

Importing Perfect Correlation

This is a Surface principle that must govern the modeling.  It can be extrapolated, because there is nothing you could add to the antecedent that would raise a probability zero to a positive probability.  So we conclude

For i = 1, 2, 3, P(Li1 & Ri1| Li & Ri  & X) = 0, and so [(Li & Ri & X) –> (Li1 & Ri1)] is false in all worlds, regardless of what X is, unless (Li & Ri & X) is itself impossible.

Consider now world α and let X be its hidden state A(α).  Suppose Li and Ri are both true.  In that case, since A(α) is also true, it follows that Li1 and Ri1 are not both true.  So Li –> Li1 is in A(α) if and only if Ri –> Ri1 is not in A(α), which entails that Ri –> Ri0 is in A(α).

Thus for each i = 1, 2, 3 we need only know whether Li –>Li1 is in A(α), we need not add a conditional with antecedent Ri.  So really, all that matters in A(α) are three conditionals:  L1 –>L1a, L2 –>L2b, L3 –>L3c.  And it is the triple <a, b, c> that summarizes what matters about A(α), a triple of numbers each of which is 0 or 1.  In some worlds the hidden state is thus summarized by <1,0,1>, in others by <0, 1, 1>, and do forth.  Let’s introduce a name:  

            Cabc = the set of worlds β such that the hidden state of β is summarized by <a, b, c>

That set of worlds has a probability, P(Cabc).

Suppose now that we have chosen settings L1 and R2 in world α.  What is the probability that the red light will turn on, on both sides — i.e. the probability of L11 & R21?  It is the probability that A(α) is either of type <1, 0, 1> or type <1, 0, 0>, hence P(C101)+P(C100).

So that sum equals the conditional probability p(1; 2) = P(L11 & R21 | L1 & R2).  Similarly for the other terms in Bell’s Inequalities. Now we can see whether they follow from what we have arrived at so far:

p(1; 2) = P(C101) + P(C100)

p(2; 3) = P(C110) + P(C010)

p(1; 3) = P(C110) + P(C100)

and so we calculate:

p(1; 2) + p(2; 3)          = P(C101) + P(C100) + P(C110) + P(C010)

                                    = P(C101) + p(1; 3) + P(C010)

                                    ≥  p(1; 3)

as required.

Similarly for the other triangle inequalities that make up Bell’s Inequalities.

Thus, we have deduced the Bell’s Inequalities, and this implies that in the experimental set-up we predict that those inequalities will not be violated. But in certain set-ups of this form, they are violated.

Our task: to show just what is wrong with the above deduction, what hidden assumptions it must have that are disguised by our traditional ways of thinking about counterfactual conditionals or even more hidden assumptions underlying those.

6. An empiricist view of conditionals

Faced with the above result, and given the attested phenomena that violate Bell’s Inequalities, the first temptation is surely to conclude that the logic of conditionals is at fault, with the main suspects being CEX and or Stalnaker’s Thesis.

The problem with this is that if we reject CEX or Stalnaker’s Thesis, we no longer have any way to relate conditionals to Bell’s Inequalities, which deal with conditional probabilities.  So the conversation ends there.

I propose that the fault lies rather in the philosophical background, with realism about conditionals.  That is a metaphysical position, even if it mimics common sense discourse oblivious to its own presuppositions.  On such a realist view, conditionals, even when counterfactual, are factually true or false.  On that view, what I called the hidden state is real, an aspect of the objective modalities in nature. 

The are different options for empiricist/nominalist views about conditionals. Elsewhere I have explored a switch from semantics to pragmatics, by moving the focus from truth conditions to felicity conditions for assertion.

But here I will explain an alternative that seems pertinent for the present topic, the analysis of the sort of reasoning that surrounds the Einstein-Podolsky-Rosen paradox and the violation of Bell’s Inequalities. (I first suggested this in in a Zoom lecture in March 2022 to a German student organizaton, and have since then worked out the technical details in the post called A Rudimentary Approach to the True, the False, and the Probable.)

Let us start from the core description of the experiment (or any experiment or situation of this sort) which involves the assertions about the actual settings and the conditional probabilities of outcomes or different settings.  As far as the exposition of the phenomena is concerned, that suffices.  All relevant information for the discussion of Bell’s Inequalities’ violation by certain phenomena can be expressed here.

Now suppose that into this language we introduce the arrow, the ‘conditional’ propositional operator, with Stalnaker’s Thesis as the basic principle governing its meaning:

P(B | A) =   P(A –> B) 

Extending the language cannot by itself create new information, let alone new facts!  So we should insist that the right-hand part of the equation contains no more information about the experiment than the left-hand part does.

A realist interpretation denies this, in part at least:  it insists that we, the observers, have no more information than what is there on the left, but this merely a limitation to our knowledge.  In fact, those conditionals are divided into the true and the false, in accordance with facts not describable in the original language.  

What alternative can we offer?  We can submit that the conditional A –> B is true only if P(B | A) = 1 and false only if P(B | A) = 0.  In that case there is nothing more to be known about the truth values of conditionals beyond what we can gather from the probabilities.

It follows of course that many of these conditionals are neither true nor false.  Indeed, in the original set-up, one of the remarkable facts is that we know that P(L11|L1) = 0.5, and so (L1 –> L11) is not true, and not false.  The true conditionals, do not tell us what will definitely happen, though they tell us something about what will definitely not happen.   An example is [(L1 & R1 &R10) –>  L11], because P(L11| L1 & R10) = 1.

Specifically, the Candy Bar Principle is correct in one sense and not in another. In general, A –> (B or ~B) is true, that is part of CEX, and the fact that P(B or ~B | A) = 1. But still in general, neither P(B|A) nor P(~B|A) will have probability 1, so neither A –> B nor A –> ~B will be true. Conditional Excluded Middle is valid, but the Principle of Bivalence fails. (And this is a familiar phenomenon in various places in philosophical logic.)

My suggestion is therefore that factual statements about preparation, measurement, and outcomes are true or false always, while the subjunctive conditionals about them are true or false only when the conditional probabilities are 0 or 1. This does not make much sense in the usual approaches to semantic analysis of modals, which are realist at least in form. How could there be such a difference between the evaluation of conditionals and the evaluation of statements lacking any such modal connectives? 

It can be done in the way I’ve presented in the post “A Rudimentary Algebraic Approach to the True, the False, and the Probable”.

NOTES

A bit of history. In a seminar in 1981, after teaching about Bell’s Inequalities, I handed out a small addition called “The End of the Stalnaker Conditional?” It contained a sketch of the argument I presented here. This paper was widely distributed though never published.  John Halpin (1986), who cites this paper, elaborated on it with reference to the defense and modifications Stalnaker offered in his (1981), to show that Bell’s Inequalities would still be derivable after that defense.  Both Halpin and I were addressing Stalnaker’s logic, which is stronger than CE, and could not be successfully combined with Stalnaker’s Thesis.  So the point was really moot, unless the argument concerning Bell’s Inequalities could be shown to use no resources going beyond CE.

The logic CE and its combination with Stalnaker’s Thesis, with proofs of its adequacy, were presented in my (1976).  Essentially the same theory was developed by  I. R. Goodman and H. T. Nguyen, independently; for a quick look see the Wikipedia articles “Conditional Event Algebra” and “Goodman-Nguyen-van Fraassen algebra”.   The ideas of my 1975 were also developed further by Stefan Kaufman in a series of more recent papers; see especially Kaufman (2009), and still more recently by Goldstein and Santorio (2021).

Above I wrote about CE, combined with the Thesis, that it has only infinite models. But ‘small’ situations, like experimental set-ups, can be modeled with partial structures that are demonstrably extendable to models of CE combined with the Thesis, for all sentences. Details about this can be found in my previous posts in this blog, “Probabilities of Conditionals” (1) and (2). The examples with die tosses, and their details, can be found there.

BIBLIOGRAPHY

Eberhard, P. H. (1977) “Bell’s Theorem without hidden variables”. Il Nuovo Cimento 38B(1): 75-79.

Goldstein, S. and P. Santorio (2021) “Probability for Epistemic Modalities”. Philosophers’ Imprint 21 (33): 1-34.

Halpin, J. (1986) “Stalnaker’s Conditional and Bell’s Problem”.  Synthese 69: 325-340.

Herbert, N. and J. Karush (1978) “Generalizations of Bell’s Inequalities”. Foundations of Physics 8: 313-317.

Kaufmann, S.  (2009) “Conditionals Right and Left: Probabilities for the Whole Family”. Journal of  Philosophical Logic 38: 381-353.

Stalnaker, R.  (1981)  “A Defense of Conditional Excluded Middle”.   Pages 87-104 in Harper, Stalnaker, and Pearce (eds.) Ifs. Dordrecht: Reidel.

Stapp, Henry P. (1971) “S-matrix interpretation of quantum theory”. Physical Review D3: 1303-1320.

van Fraassen, B. C. (1976)  “Probabilities of Conditionals”.  Pages 261-308 in W. Harper and C.A. Hooker (eds.) Foundations of Probability and Statistics, Volume l.  Dordrecht: Reidel.