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1 Tomasz Bigaj Is quantum mechanics incomplete or non-local? Counterfactual perspective on the locality, the EPR argument and the Bell theorem Introduction The paper explores the possibility of applying modal logic of counterfactuals to some foundational issues in quantum physics. This is by no means a novel approach: there have been several attempts to use David Lewis’ formal semantics of counterfactuals in analyzing such problems as the EPR argument and Bohr’s response to it, Bell’s theorem and its later versions, and so on.1 For example, some authors try to strengthen the result of Bell’s original theorem by using the logic of counterfactuals to show that from the locality assumption together with quantum-mechanical predictions a contradiction can be derived. This claim, if justified, would constitute the strongest argument yet for the non-locality of quantum mechanics.2 Counterfactual statements prove themselves useful in many ways in the analysis of philosophical problems of quantum mechanics. The most direct application of counterfactuals in this context lies in the analysis of possible measurements which could have been performed in place of the actual ones (this use of counterfactuals is most extensively employed in the original EPR argument.) Counterfactuals also provide us with a method of formalizing the so- called locality requirement, which is a focal point of all discussions concerning entangled quantum states. As it is known, the locality condition encompasses the general idea that causal influences do not propagate at arbitrarily high speeds. Because causal correlations are often interpreted in terms of counterfactual dependence (see Lewis 1986b), it is natural to expect that logic of counterfactuals can shed a new light on the locality issue. The third, and in my opinion most promising application of counterfactuals is when they serve as explications of the so-called property attribution statements, i.e. statements which attribute 1 To mention only few: (Stapp 1997), (Ghirardi, Grassi 1994), (Bedford, Stapp 1995), (Redhead, La Riviere 1997), (Griffiths, 1999), (Dickson, 2002) 2 This argument is presented in many versions in works of H. Stapp. See (Stapp 1971), (Stapp, 1997), (Bedford, Stapp 1995). Stapp’s arguments have been criticized in (Redhead 1987), (Redhead et al., 1990), (Unruh 1999), (Mermin 1999), and, most recently, in (Shimony, Stein 2001). I have analyzed in depth the alleged formal proof of non-locality of quantum mechanics given in (Bedford, Stapp 1995), and argued that this proof is faulty (Bigaj, 2002). 2 objective measurable properties to quantum systems even at times when these systems do not undergo any quantum measurements. In this paper I am going to touch upon all these three areas of applicability of counterfactual statements. The structure of the paper is as follows. The first two section of the paper will be devoted to the issue of clarification of the concept of counterfactuals and the notion of locality built upon it. Two interpretations of counterfactual conditionals will be presented and analyzed, together with two corresponding formulations of the locality condition known in the literature. In the second section a proof of the equivalence of these locality conditions will be presented, and the final version of the locality assumption will be accepted. The third section contains a counterfactual reconstruction of the EPR argument, which in its original form purported to show the incompleteness of quantum mechanics. The result of the presented analysis is that under both readings of counterfactuals the EPR argument turns out to be faulty, and therefore its alleged consequences are avoidable even without a violation of the locality assumption. The last section applies the same counterfactual procedures to the analysis of the initial assumptions of Bell’s theorem. The main goal of this analysis is to explore an interesting possibility that when property attribution statements are interpreted counterfactually, the reality assumption (i.e. the assumption that quantum systems are fully and precisely characterized by their appropriate measurable properties at all times) might not lead to the Bell inequality (and, ultimately, to the clash with quantum-mechanical predictions). I will argue that under one interpretation of counterfactuals, Bell’s theorem is unavoidable, however the other interpretation affords a possibility, albeit an unlikely one, that the original Bell argument could be blocked. Spatiotemporal counterfactuals The most popular interpretation of counterfactual conditionals is in terms of possible worlds and the similarity relation with respect to the actual world. Roughly speaking, according to this interpretation, a counterfactual A B is true, if B is true in the possible world in which the antecedent A holds and which is most similar to the actual one among all A-worlds. In order to give this formal definition the substance, Lewis proposes to interpret the similarity relation in question with the help of a complicated, multi-leveled set of criteria, involving differences in particular facts as well as differences in laws of nature (Lewis 1986a). However, his analysis is for many reasons unsatisfactory. Firstly, it introduces possible worlds in which current laws of nature can be violated, which unnecessarily 3 complicates the picture. And more importantly, when comparing possible worlds, Lewis takes into account the entire spatiotemporal universe (‘block-universe’), including future events. This makes Lewis’ counterfactuals useless for the task of explicating quantum property attribution statements, for the results of future quantum experiments would count towards similarity. Therefore, it is often argued that in the context of quantum mechanics the following, temporal interpretation of counterfactual statements is more appropriate: the counterfactual A B is true iff B is true in all possible worlds which are exactly like ours up to the point t of the occurrence of A, in which A occurs, and in which all actual laws of nature are true. The only problem with this formulation is that it is not Lorentz-invariant, as it implicitly assumes the existence of the absolute hypersurface of simultaneity. Now we have to address the issue of making this definition relativistically correct. There are two basic relativistic counterparts of the pre-relativistic notion of the past for an event e. One of them refers to the set of events which form the so-called past (backward) light cone of the event e, and the other candidate picks out as past events all events, which are not contained in the future (forward) light cone of e. The first spatiotemporal region can be defined as follows: it contains all events which are earlier than e in all inertial frames of reference. The second, on the other hand, consists of all and only those events, which are earlier than e at some inertial frame of reference. Having drawn this distinction, we can now formulate two different, relativistically invariant notions of counterfactuals. I will present them in the version restricted to antecedents describing free-choice events: (C1) A counterfactual A B is true iff B is true in all possible worlds which are exactly like the actual one in the entire region outside the future light cone of the event A. (C2) A counterfactual A B is true iff B is true in all possible worlds which are exactly like the actual one in the entire region inside the past light cone of the event A.3 In the present context by a “free-choice event” I will mean an event which is not nomically connected with any event outside its future light cone. Especially in the context of quantum mechanics, it is important to make sure that there are no correlations between our event A and events space-like separated from it. For example, when we have two EPR particles (i.e. particles prepared in the so-called singlet state, in which the total spin equals 0) and we 3 These two ways of reading spatiotemporal counterfactuals are explicitly stated in (Finkelstein 1998). 4 perform two space-like separated measurements of the same spin component for two distant particles, the results of these measurements will be strictly correlated. The unsatisfactory feature of both truth conditions (C1) and (C2) is that their applicability is restricted in two important ways. Firstly, the antecedent A of the counterfactual is assumed to refer to a localized point-event, and secondly, this event has to be free-choice, or chance. Now we face the challenge to generalize both truth conditions (C1) and (C2), so that they become applicable in cases when A is whatever sentence we please. Fortunately, it appears that at least in the case of the interpretation given by (C1), the required generalization is easy to obtain. I will follow here the proposal put forward by J. Finkelstein in (1999), with some corrections suggested and explained in (Bigaj, forthcoming). Finkelstein has noticed that we can introduce quite straightforward similarity relation between possible worlds, which together with the appropriate Lewis-style truth conditions for counterfactuals will yield (C1) as a special case. This similarity relation can be defined as follows. First, for each possible world j we define a set Dj of primary points of divergence, which are characterized as the earliest spatiotemporal points at which a difference between j and the actual world occurs. Then we consider the closure Dj* of the set Dj of points with respect to their absolute future, i.e. the set of all spatiotemporal points such that they are absolutely later than some primary point of divergence. Intuitively Dj* represents the entire region of all (possible) differences between the world j and the actual one (see Fig. 1). Now the definition of the similarity relation can be easily presented as follows: (SIM) world j is at least as similar to the actual one as k iff Dj* Dk*. time the region of difference D* primary points of divergence Figure 1. An example illustrating Finkelstein’s criterion of comparative similarity between possible worlds. 5 It can be quickly checked that the above similarity relation reproduces (C1) as a special case, when A refers to a localized free-choice event. For in that case the set Dj* reduces to the future light cone of that event A, and hence in evaluating the counterfactual A B we should take into account every possible world k such that Dk* = Dj*. But this in turn means that we should keep everything which happens outside the future light cone of A exactly the same as in the actual world. On the other hand, the truth conditions based on the proposed definition of the similarity relation give us the possibility of evaluating counterfactuals not restricted to those mentioned in (C1). For example, we can easily handle cases where the antecedent A is equivalent to a finite disjunction or conjunction of sentences A1, A2, ..., An, each of which picks up one particular localized free-choice event. In both cases we should obviously consider all possible worlds j such that the antecedent holds in them and they are closest to the actual one according to the relation given by Dj* Dk*. So, for example, in case of a disjunction, when A1, A2, ..., An refer to events mutually space-like separated, all we need to do is consider all the worlds j such that Dj consists only of one single spatiotemporal point picked up by some Aj. In the second case the only possible world we need consider is the world k for which Dk includes all spatiotemporal points to which some conjunct in A refers. Next, it is possible to extend our analysis of counterfactuals to cases where A describes an event which is not a free-choice event. For example, when there are correlations between A and another event C space-like separated from A, all we need do is consider possible worlds which have not one but two primary points of divergence: namely points A and C. And so on and so forth. Now, what about the second intuition (C2)? Is it possible to cook up an analogous relation of comparative similarity, which would yield (C2) as a special case? Unfortunately, and surprisingly, the answer is “no”. Here I am going to report the result which I achieved and presented elsewhere. I was able to prove that there is no similarity relation, satisfying some reasonable conditions, which would produce (C2) as a special case (see Bigaj, forthcoming). The constraints put on the similarity relations are the following: first, we should always treat one possible world j in which some particular departure from reality occurs absolutely later than in the other one k, as strictly more similar to the actual world than k. The second assumption states that when we decide to call the world j strictly more similar to the actual one than k, than the total region of divergence at j should be properly included in the region of divergence at k. These two conditions are sufficient to show that definition (C2) is impossible to satisfy. 6 In spite of this negative result, it is still possible to give a generalization of the conditions given in (C2) applicable to a broader class of antecedents. However, because in the present analysis we are concerned only with antecedents referring to spatiotemporally localized events, we will only attempt to generalize (C2) in order to cover the cases in which the antecedent doesn’t refer to a free-choice event. But this is an easy task. Let us assume that A describes a point event, which can be nomically connected with some other events from its ‘present’ (i.e. which are space-like separated from A). Actually, this case is already included in the way we evaluate counterfactuals according to (C2), for we are not forced to keep the area outside the past light cone of A fixed, as in (C1). So all we have to do is to consider, as it is prescribed by (C2), all possible A-worlds which share the absolute past of A with the actual one, and in which laws of nature are observed. If there are true lawful correlations between A and some events space-like separated from it, these events will automatically ‘pop up’ in our possible worlds. The only case which cannot be accommodated by this strategy is when we want to consider lawful correlations between a counterfactual event A and its past (so-called ‘backtracking’). But for now there is no need to introduce this possibility into the picture, as long as we don’t want to talk about counterfactual dependence of the past on the future. Three notions of locality In all considerations related to such problems as the EPR paradox, Bell’s inequalities etc., one notion is of extreme importance: the notion of locality. The general idea of locality encompasses the intuition that distant events cannot influence, or cause, what is happening now and here. In other words, there is no action-at-a-distance: all physical influences are transmitted at a finite speed. A typical illustration of this general understanding of the notion of locality can be found in (Ghirardi, Grassi 1994), where the authors characterize the locality as follows (italics mine): An event cannot be influenced by events in space-like separated regions. In particular, the outcome obtained in a measurement cannot be influenced by measurements performed in space-like separated regions; and analogously, possessed elements of physical reality referring to a system cannot be changed by actions taking place in space-like separated regions. Unfortunately, the notions like “influence”, “cause”, or “change” are notoriously resistant to explications. However, in the context of the above-mentioned quantum issues we would need a much more precise notion of locality. And it is often conjectured that counterfactuals could afford us the opportunity to make the notion of locality sufficiently 7 precise. I am going to argue that the way we formulate counterfactually the locality principle should depend on our choice of one of two available interpretations (C1) or (C2). This is a crucial thing, for when we use one formulation of the locality principle with the inappropriate interpretation of counterfactuals, we could end up in a conceptual confusion. However, in spite of their surface differences, the two formulations of the locality assumption working under two different interpretations of counterfactuals will appear to be fundamentally equivalent. I will show that there is a third, conceptually simple way of interpreting locality in terms of possible worlds only, which is equivalent to both counterfactual versions of the locality condition. Let us start with the truth conditions (C1), and consider a factual event B. Now the idea of locality can be presented as follows: if we assumed that things “out there” were different, this shouldn’t change at all the fact that B is present. More precisely, locality guarantees that no matter what counterfactual assumption A we make, if A refers only to a region space-like separated from B, B should remain intact. This principle can be given a precise formulation as follows: (LOC1) For all A, B, if A and B describe events mutually space-like separated, then B (A B).4 Symbol represents strict implication, and the formula above says essentially that in all possible worlds in which B is true it is so that if a distant event A occurred, still B would hold. And now observe that this intuition is in perfect agreement with the truth condition (C1). For in order to evaluate the counterfactual A B we must take into account worlds which agree with the actual one everywhere outside the future light cone of A, including B. So (C1) makes (LOC1) true for free-choice events, and that’s exactly what we should expect. However, when combined with the approach (C2) to counterfactuals, (LOC1) appears to imply more than the intuition of non-influence at a distance. For suppose that B denotes an event which is a chance event, so that B is not determined by its absolute past. Now, according to (C2), we have no right to claim that the counterfactual A B will be true for A space-like separated from B, because B lies outside the past light cone of A, and therefore 4 This version of the principle of locality was adopted in all of Stapp’s works mentioned earlier, which strongly suggests that he accepts (C1) as an intuitive way of evaluating counterfactuals. Let me add that in this formulation and in all subsequent versions of locality, we will assume that B cannot include any counterfactual statements. This restriction will prove important later. 8 cannot be taken for granted while counterfactually considering A. This means that when we assert the truth of B (A B), we are saying more than merely that A cannot influence B: we are assuming implicitly that B is determined by the absolute past of A (or, even worse, that there is a non-local influence between A and B which “keeps” B true). And this is devastating to the issues connected with Bell’s theorem, for we should clearly separate two different problems: the problem of locality and the problem of determinism, or the hidden-variable hypothesis. Therefore we must come up with another notion of locality, appropriate for the intuition (C2). What can such a notion of locality look like? Here is one possibility, suggested by some proponents of (C2).5 To say that an event A cannot influence distant regions is to say that assuming counterfactually A we cannot derive anything definitive about these regions: all possibilities are still open. So someone can conjecture that the right way of expressing the idea of locality would be to stipulate that the counterfactual A B should be false, for it carries the supposition that A influences, or “brings about” B. However, it would be definitely too quick to jump to this conclusion. Recall that the counterfactual A B carries not only information about the necessary connections between A and B, but also some information about the actual world. If in the actual world B is causally determined by A’s absolute past, then we should accept A B as true, in spite of the non-existence of any influences between A and B. Therefore we should be more cautious, and the idea is to do the following: there is no causal influence or dependence between A and B, if at least one of the following counterfactuals is false: A B or ~A ~B (alternatively, this intuition can be presented with the help of “might” counterfactuals: either it is so that if it were A, then it might be not-B, or if it were not-A, it might be B.) Let us write this down: (LOC2) For all A, B, if A and B describe events mutually space-like separated, then either A B is false, or ~A ~B is false. Admittedly, (LOC2) is not as clear and intuitively appealing as (LOC1), due to the presence of a disjunction of two negated counterfactuals. Let us then try to reformulate it somehow, in order to capture the essence of the condition expressed. First of all, let us consider separately cases in which A is true and cases in which A is false in the actual world. If A is true in w0, then the first counterfactual in (LOC2) becomes a null counterfactual, i.e. a 5 For example by Redhead and others; see (Redhead et al. 1990, p. 34-35). 9 counterfactual with a true antecedent. For the sake of this analysis, let us adopt Lewis’ way of interpreting null counterfactuals, namely as material implications (see Lewis 1973, pp. 26-31). Then immediately, if B were false in w0, the first counterfactual would be false and the entire locality condition satisfied. So suppose that B is true. Then the first counterfactual becomes trivially true, and the only way of satisfying (LOC2) would be that the second counterfactual should be false. An analogous reasoning can show that when A and B are false, the first counterfactual should turn out to be false. Hence we can write the following equivalent formulation of (LOC2): (LOC2) For all A, B, if A and B describe events mutually space-like separated, then (a) if A and B are true in the actual world, then ~A ~B is false, and (b) if A and B are false in the actual world, then A B is false. Actually, assuming that A and B can refer to negative states of affairs we can eliminate condition (b), for it would be contained in (a) by a stipulation that A = ~C and B = ~D, for some C and D. All in all, the meaning of this locality condition is the following: a contrary-to- fact assumption of A cannot create counterfactually any new event B in the region distant from A. In spite of its intuitiveness, condition (LOC2) has one flaw which later will prove itself rather crucial, and which needs to be corrected. Let us start with an observation that there might be cases which justifiably deserve the name of “non-local correlations” and which nonetheless satisfy (LOC2). Consider for example the famous GHZ case of quantum entanglement. Here we have three spatially separated particles labeled 1, 2 and 3, and three measurements X1, X2 and X3, each with two possible results 1 or –1. If the particles are prepared in a specific quantum state, then the following relation must be observed: the product of all three results must give –1 (see Greenberger et al. 1990). Suppose then that the actual results obtained are the following: for X1: 1, for X2: 1, and for X3: –1. And now we can ask, what would have been the case, if the result for the first particle had been –1 instead of 1. In order to accommodate the above-mentioned quantum-theoretical relation, exactly one of the other two results must be changed. But which one? On the basis of quantum-mechanical predictions we can say neither that if the result of the measurement of X1 were –1, the result of X2 measurement would be –1, nor that if the result of X1 were –1, the result of X3 would be 1. Therefore condition (LOC2) is not directly violated. However, it is still true that under the counterfactual supposition that the result of X1 measurement were –1, it must be that either X2 10 or X3 would give a result different than in actuality. And this clearly constitutes a case of a non-local correlation, for both measurements X2 and X3 are space-like separated from X1. In order to cope with cases like these, we have to introduce the following correction to (LOC2): (LOC2’) For all A, B, if A describes a point-event and B is equivalent to a disjunction of sentences, each of which describes an event space-like separated from A, then either A B is false, or ~A ~B is false. Again, as in the previous case, this condition can be presented equivalently as follows: (LOC2’) For all A, B, if A describes a point-event and B is equivalent to a disjunction of sentences, each of which describes an event space-like separated from A, then (a) if A and B are true in the actual world, then ~A ~B is false, and (b) if A and B are false in the actual world, then A B is false. Now I’d like to raise the question of a logical relation between (LOC1) and (LOC2’). On the surface they are definitely not equivalent, but remember that in both formulations symbol is used in a slightly different sense. So it is possible that those differences will effectively cancel each other out, and that ultimately (LOC1) and (LOC2’) represent one and the same intuition concerning the notion of locality. In what follows I am going to show that this is indeed the case. I will prove the equivalence of (LOC1) and (LOC2’) by means of showing that both formulations are equivalent to yet another formalization of locality condition, which doesn’t use the logical connective of counterfactual conditionals at all, but instead talks only about possible worlds. I’ll call this version “semantic condition of locality” (SLOC): (SLOC) For all point-events A, there is a possible world w such that A occurs in w and w is exactly the same as the actual one everywhere outside the future light cone of A. The meaning of this formulation should be clear. It states that whatever counterfactual situation we are entertaining, it is admissible by the laws of nature that the world ‘outside’ this situation will remain intact. I’d like to emphasize that it wouldn’t be appropriate to claim in 11 (SLOC) that all A-worlds should be exactly the same outside the future light cone of A as the actual one, for there can be factual chance events involved, which might or might not occur under the counterfactual supposition of A. Now let us move to the task of proving that (SLOC) is logically equivalent to (LOC1) under the (C1) interpretation of counterfactuals, and to (LOC2’) under the (C2) interpretation. The following proof will proceed under the so-called limit assumption (see Lewis 1973, pp. 19-20), i.e. the assumption stating that for all antecedents A of considered counterfactuals the following holds: for every A-world w there is an A-world w’ such that w’ w and w’ is the closest possible A-world to the actual one (there is no other A-world w’’ such that w’’ < w’). Now we will prove the implication (SLOC) (LOC1). Let us then take two propositions A and B describing space-like separated events, and let’s assume that B is true in the actual world. According to (SLOC), there is a possible A-world w* which is exactly the same as the actual world w0 outside the future light cone of A, and hence B holds in w*. But w* is obviously the closest possible world to the actual one among all A-worlds, according to (SIM). Hence the counterfactual A B must be true. The proof of the reverse implication (LOC1) (SLOC) is no more difficult. Suppose, then, that for all A’s the counterfactual A B is true, where B denotes an event space-like separated from A, or in the absolute past of A, and B is true in the actual world. Now consider the set of all possible A-worlds. This set is (partially) ordered by the relation of comparative similarity (closeness) . According to the limit assumption, invoked earlier, there must be an A-world w* such that there is no other A-world w for which w < w*. And obviously, if counterfactual A B is to be true, B must be true in this world w*. Thence, all B’s space- like separated from A and true in the actual world must be true in w* as well. But this means that w* is an A-world which is exactly the same as the actual world everywhere outside the future light cone of A, which shows the fulfillment of (SLOC). Incidentally, let us observe that the limit assumption played an important role in the above derivation. For if it hadn’t been true, (SLOC) could have been false in spite of (LOC1) being true. To see this, suppose that there is no possible A-world which is identical to the actual one in the entire spatiotemporal region outside A’s absolute future, but for every spatiotemporal point p outside the absolute past in future of A there is a possible A-world with two initial points of divergence from the actual world: A and p. It appears that in such a set of A-worlds all counterfactuals of the form A B, where B is true in the actual world, turn out to be true according to the generalized Lewis-style truth conditions for counterfactuals, given 12 in (Bigaj, forthcoming). In essence, given any B, for every A-world w you can find a world w’ which is strictly closer to the actual one than w: w’ < w, and in which B is true, and moreover such that whatever world w’’ w’ we take, B remains true in w’’. And yet those A-worlds have no lower limit with respect to . It might come as a surprise that all sentences of the form A B can be true for B’s taken from a given set, and yet there could be no possible A-world in which all such B’s were true together, but this is perfectly possible in the light of our conceptual analysis. If we don’t accept the limit assumption, the world in which all B’s are true can be seen as a non-existent limit which A-worlds can only approach infinitesimally close. Let us proceed to the equivalence of (SLOC) and (LOC2’), of course this time under the interpretation of counterfactuals encapsulated in (C2). First, suppose that (SLOC) is true. Now let us consider any A and B such that they are both false in the actual world w0, and B is equivalent to a (finite) disjunction of sentences describing events space-like separated from A. Locality condition (LOC2’) states that in such a case counterfactual A B should be false. And that’s exactly the case, which can be shown in the following way. According to (SLOC) there is a possible world wA* which differs from the actual world only within the confine of the future light cone of A. Therefore, B must be false in wA*. But now it means that there is a possible world which is exactly the same as the actual one within the past light cone of A, in which A is true and yet B is false. This, according to (C2), means that counterfactual A B is going to be false, which was to be proved. In exactly the same way we can prove that when A and B are true, counterfactual ~A ~B has to be false. This shows that (SLOC) really implies (LOC2’) under the interpretation given in (C2). Finally, we should attempt at proving the opposite implication, leading from (LOC2’) to (SLOC). To achieve this, we will need yet another assumption, this time the finiteness assumption, saying that for all antecedents A there is only a finite number of all A-worlds which agree with the actual one in the spatiotemporal region of A’s absolute past.6 I believe that in the context of quantum mechanical experiments, such as the EPR example with spins, or the GHZ example, or the Hardy case, this assumption is a reasonable one, and it doesn’t seriously affect the generality of the counterfactual analysis. However, for now I am not going to discuss this problem in depth. So let us assume that (SLOC) is violated, i.e. that for some A there is no possible world wA* such that wA* agrees with the actual one everywhere outside the future light cone of A. This implies that for every possible A-world wAi which keeps fixed 6 From the finiteness assumption the limit assumption obviously follows. 13 the absolute past of A there is an event Bi such that Bi is space-like separated from A, and Bi is true in the actual world w0 but false in wAi. From the finiteness assumption it follows that there is a finite number of such Bi-s, so we can consider the following disjunction: ~B1 ~B2 ~Bn. This sentence is obviously false in w0, but true in all A-worlds preserving the absolute past of A, and therefore the counterfactual A ~B1 ~B2 ~Bn must be true, according to the method of evaluation prescribed in (C2). But this contradicts (LOC2’), therefore the proof is complete. It might be instructive to see that in contrast to the above result, implication (LOC2) (SLOC) doesn’t hold, which shows the real necessity of the correction we have done to (LOC2). According to (LOC2), when space-like separated A and ~B are false in the actual world, counterfactual A ~B must be false as well, but this is true only for B describing a single spatiotemporally localized event. This implies that for every event B space-like separated from A and occurring in the actual world, there must be a possible A-world with the same absolute past of A as in the actual world and such that B holds in it. But this fact by no means guarantees that there is a single A-world in which all such B-s hold, and therefore falls short of proving that there is a world wA* described in (SLOC). Let us sum up the results which have been reached. We have been able to prove the equivalence of the semantic locality condition (SLOC) with the locality (LOC1) under the interpretation (C1) of counterfactuals, using the limit assumption. Similarly, we have succeeded in proving the equivalence of (SLOC) and (LOC2’) interpreted under (C2), this time making the finiteness assumption. Because the finiteness implies the limit assumption, we can now conclude that provided that the finiteness condition is satisfied, condition (LOC1) under (C1) is equivalent to (LOC2’) under (C2). This fact can be seen as an independent argument for the thesis that locality conditions (LOC1) and (LOC2’) correctly capture our deep and essential intuition concerning the notion of locality, for if (LOC1) and (LOC2) were only ad hoc conditions created in the course of a philosophical debate for the sole purpose of defending particular views on non-locality and indeterminism of quantum mechanics, their equivalence would be quite a surprising coincidence. Moreover, in any future use of counterfactuals in the conceptual analysis of quantum-mechanical correlations, we can now safely resort to the ‘neutral’ version of (SLOC), no matter which reading of counterfactuals we decide to stick to. I am going now to make use of this formulation of locality in my counterfactual analysis of the EPR argument and the Bell theorem. 14 The EPR argument We will be considering here a generic EPR case with two distant particles L (left) and R (right), and a pair of non-commuting observables for each of them: XL, YL and XR, YR. Moreover, we assume that XL commutes with XR and YL commutes with YR. Two particles are prepared in an initial state such that the probability of a joint occurrence of any two given results xL, xR (yL, yR) equals either 0 or 1 (a case of perfect correlations). The original EPR argument was formulated in a non-relativistic setting, assuming implicitly the existence of a preferred frame of reference, and therefore the existence of an absolute time frame (see Einstein et al.). The argument can be briefly presented as follows. Suppose that the measurement of XL was performed on L at time t giving a particular result xL. Using the assumption of perfect correlations we can infer that at time t particle R has a determined value of XR (let’s call it xR). By appealing to the locality principle, which roughly states that the measurement on L cannot instantaneously change a physical state of R, and by the criterion of physical reality, it is argued that observable XR had to have its value determined even before time t. But this contradicts the initial assumption that before t both particles were jointly in state which is not an eigenstate for observable XR, and therefore the incompleteness of the quantum-mechanical description follows. As we see, in the above reconstruction of the EPR argument the second observable Y doesn’t play any role. Nevertheless, the original argument does not stop at the above conclusion but continues with an introduction of observable Y into the picture, possibly to strengthen the case against standard quantum mechanics, or to make the conclusion even more astounding. Basically, the above derivation is repeated, this time under a counterfactual supposition that instead of XL, YL was chosen for a measurement, leading to the conclusion that before t YR had to have its value determined. And now, appealing once again to the locality principle it can be argued that the experimenter’s decision as to which observable measure on the left-hand side particle should not change the state of the remote particle R. Therefore, what was true of particle R at t in the actual world, should remain true under the counterfactual assumption that YL had been chosen. Combining both parts of the argument, we arrive at a conclusion that in the possible world in which YL was measured, the right-hand side particle has to have determined values for both observables XR and YL, which contradicts the quantum-mechanical assumption that these observables are non-commuting. Undoubtedly the reasoning sketched above requires thorough and critical examination. It contains dubious transitions from actuality to potentiality, and also appeals to the idea of the 15 locality without giving it a rigorous and unambiguous formulation. Our ultimate goal here will be to improve the original EPR argument in two respects: with respect to the notion of counterfactual conditionals used and the locality condition assumed. But before this, we have to address the question of how to present the EPR argument in a relativistic setting. Under a relativistic interpretation we cannot introduce an absolute ‘time slice’ t. Instead, we can talk about space-like separated events, i.e. events such that each of them lies outside the future light cone and the past light cone of the other. Suppose, then, that the measurement of XL was done at a certain spatiotemporal point p, and consider any point p in the history of particle R which is space-like separated from p (a relevant spatiotemporal diagram is presented on Fig. 2). As before, a consequence of quantum-mechanical predictions seems to be that at p particle R should have its definite value for observable XR determined, the reason being that this value can be inferred from the measurement on L without physically interfering with R. And obviously this assessment is valid for all spatiotemporal points p space-like separated from p. At this moment we can again choose two strategies known from the previous case. First, appealing to the locality principle we can trace the attribution of the particular value xR back to the common past of particles L and R, arguing that the measurement on L cannot physically create any new physical state in the entire space-like separated region of spacetime. As a result, we would contradict our assumption that in their common past the particles were in state . But a more common strategy is to consider counterfactually a measurement of incompatible observable YL at p, and to argue in a similar vein that the right-hand side particle ought to have determined values for both non-commuting observables XR and YR at point p. At this juncture let us present all presuppositions and terminological stipulations on which our counterfactual reconstruction of the relativistic EPR argument will be based. We will consider separately two counterfactual reconstructions, based on two notions of space- time counterfactuals introduced and analyzed above. In each particular reconstruction we will explicitly adopt the semantic formulation of locality condition (SLOC), which was shown to be equivalent to the versions (LOC1) and (LOC2’) under appropriate readings of the counterfactual conditional. And finally, I am arguing, following the approach proposed in (Ghirardi, Grassi 1994, p. 404), that the property attribution statement, claiming that at t an observable A characterizing a quantum system x possesses a definite value a (in short: PxA=a(t)), should be interpreted counterfactually as (we will call this interpretation Counterfactual Property Attribution, or CPA for short): 16 (CPA) If the measurement of A were performed at t, the outcome would be a, or in symbols: MxA(t) OxA=a(t). Let us start our reconstruction with the (C1) interpretation of counterfactuals, formally generalized with the help of the similarity relation given by (SIM). In the simplest case, when A represents a chance event, evaluating counterfactual A B reduces to the following: we take all the possible worlds which are identical with the actual one everywhere outside the future light cone of A, and we check whether B holds in them. In general, there might be no such a possible world, because of some non-local correlations between A and some other event space-like separated from it. But if we assume (SLOC), then because the existence of such a world is now guaranteed, it follows that this procedure is always executable. The first step of the EPR argument is the following: (1.1) MLX(p) OLX=a(p) PRX=a’(p’) where p and p’ denote appropriate space-time locations space-like separated from each other. Using our rule of interpretation (CPA), we can reformulate (1.1) counterfactually, by replacing the property attribution statement with an appropriate counterfactual: (1.2) MLX(p) OLX=a(p) [MRX(p’) ORX=a’(p’)] In words (1.2) says that if in the actual world the left-hand side particle underwent the measurement of observable X at p, and the result of this measurement was a, then if X had been measured on the right-hand side particle at p’, the result would have been a’. Now the question is whether the truth of (1.2) can be guaranteed by the truth conditions for counterfactuals (C1) and by the assumption (SLOC). The truth conditions dictate that the counterfactual MRX(p’) ORX=a’(p’) is true when the consequent ORX=a’(p’) is true in the possible world closest to the actual one with respect to the similarity relation (SIM), and in which the antecedent holds. According to (SLOC), there is a possible world w(XR) in which MRX(p’) is true, and w(XR) is identical to the actual one everywhere outside the future light cone of p’ (see Fig. 2 and 3), hence in w(XR) it must be that MLX(p) OLX=a(p), and moreover w(XR) is obviously the closest possible world according to the criterion (SIM). Finally, appealing to the quantum-mechanical predictions we can derive that at w(XR) ORX=a’(p’) must 17 hold. Hence step (1.2) of the EPR argument was proven to be correct under the counterfactual interpretation (C1), however, the assumption of locality was already necessary in the derivation (the necessity of appealing to the locality condition in the derivation (1.1) is also acknowledged in Ghirardi, Grassi 1994, p. 409). In other words, in order to argue that particle R has the objective property XR = a’, we have to assume that a measurement which would aim at revealing this property cannot change anything in the distant system L, and thereby affect the derivation of the property attribution PRX=a’(p’) from the actual outcome OLX=a(p). OLX=a X p’ p ML L R Fig. 2. The actual world ORX=a’ OLX=a X X MR p’ p ML L R Fig. 3. The possible world w(XR). Shaded area indicates the region shared with the actual world. From (1.2) we can now try either to project the property attribution statement PRX=a’(p’) into the common past of p and p’, or to counterfactually consider an alternative measurement of observable YL. However, the first strategy has no obvious backing in the locality condition (SLOC). When we consider a spatiotemporal point p’’ in the history of particle R lying in the absolute past of p, there is obviously no immediate reason to believe in 18 counterfactual MRX(p’’) ORX=a’(p’’), for in this case the spatiotemporal region which should be kept fixed in the evaluation of this counterfactual does not include point p, and therefore we cannot take the outcome of the measurement MLX(p) for granted. Now, in the original EPR argument it was the criterion of physical reality, together with the locality assumption that was used to argue for tracking the property attribution PRX=a’ back to the common past of p and p’. But appealing to the locality principle in such a way is justifiable only when we talk about genuine physical states of affairs, or events7. In other words, if PRX=a’(p’) denoted a genuine physical event taking part in p’, then we could argue on the basis of (SLOC) that there is a possible world in which measurement MLX(p) is not performed, but the fact expressed in PRX=a’(p’) remains unchanged. Using further the plausible assumption (close in spirit to Einsten’s criterion of physical reality) that genuine property attribution facts cannot be created ‘out of nowhere’ without any direct physical cause, we could finally conclude that PRX=a’ had to be true even at the moment of the creation of both particles L and R. But this strategy is not open to us when we decide to interpret property attributions as appropriate counterfactual conditionals. Remember that we have explicitly excluded counterfactuals facts from the formulation of the locality condition (see footnote 4). Therefore the proposed reformulation of the EPR argument failed to produce a contradiction with any quantum-mechanical principle. Now let us consider the second strategy, more commonly associated with the original EPR argument, of deriving the conclusion about the incompleteness of quantum mechanics, based on the introduction of the incompatible observable Y. The counterfactual claim which has to be argued for is the following: if we consider a counterfactual situation in which YL was measured instead of XL, this should not affect the property attribution on particle R. We can write this step formally as follows: (1.3) MLX(p) OLX=a(p) [MLY(p) PRX=a’(p’)] and using (CPA) we can rewrite it as: (1.4) MLX(p) OLX=a(p) {MLY(p) [MRX(p’) ORX=a’(p’)]} 7 By genuine physical events in this context I understand fact whose full characteristics is included in the complete description of the actual world. This means that no reference to possible worlds other than the actual is allowed in the description of such events. Therefore ‘counterfactual events’ do not count as genuine physical events (see also footnote 4). 19 In terms of possible worlds, this multiple counterfactual statement can be read in the following manner. In order for (1.4) to be true, statement MRX(p’) ORX=a’(p’) has to hold in the possible world which is closest to the actual and in which the measurement of YL was performed. Let us call this world w(YL). According to the locality assumption (SLOC), w(YL) has to look like this: it should be exactly the same as the actual world everywhere outside the future light cone of p, and at p the measurement of YL should take place (see Fig. 4). And now, in order to evaluate counterfactual MRX(p’) ORX=a’(p’) in this world w(YL), we have to invoke yet another possible world w(XR, YL), in which MRX(p’) takes place, and which is closest to w(YL) (and not to the actual one!). Again, according to (SLOC) this will be the world which shares with the actual one the entire spacetime excluding two light cones: one with its initial point at p and the other at p’ (see Fig. 5). But what is more important is that in this world w(XR, YL) the left-hand side particle L does not undergo any measurement of XL, and consequently no outcome of this measurement can be given at p, which cuts off the derivation leading in (1.2) to the conclusion that the outcome of the measurement of XR is ORX=a’(p’). There is no guarantee that counterfactual MRX(p’) ORX=a’(p’) has to hold in w(YL), and therefore the entire multiple counterfactual (1.4) is unwarranted. It appears that the principle of locality (SLOC) is not sufficient to derive any contradiction with standard quantum mechanics under the counterfactual interpretation of the EPR argument based on the reading (C1) of spatiotemporal counterfactuals. p’ p MLY L R Fig. 4. Possible world w(YL) 20 Y p’ p MLY MR L R Fig. 5. Possible world w(XR, YL) Ghirardi and Grassi follow a slightly different strategy in their counterfactual reconstruction of the EPR argument (1994). Because their conclusion is that the counterfactual EPR argument is valid and that the locality assumption leads to the incompleteness of quantum mechanics, it could be instructive to bring up the points of disagreement between their approach and mine. Ghirardi and Grassi’s reconstruction of the EPR argument does not introduce an incompatible observable Y. Rather, it aims at showing that in the possible world where measurement MLX(p) is not performed, the right-hand side particle will have still the same property as when MLX(p) has a specific outcome. In other words, they argue that the following holds: (1.3’) MLX(p) OLX=a(p) [~MLX(p) PRX=a’(p’)] However, rather then further analyzing the claim described in (1.3’) with the help of the counterfactual reformulation of the property attribution statement PRX=a’(p’), they simply support it by referring to the locality condition, stating that “the element of physical reality existing at the right-hand side system is independent of whether or not the measurement at L has been performed”. This makes clear that they adopt much stronger interpretation of the locality condition than our (SLOC) or, in the current context, (LOC1). Ghirardi and Grassi’s locality condition implies that when a counterfactual claim B C is true in the actual 21 world, it will remain true in all possible worlds which diverge from the actual one with the occurrence of an event A space-like separated from the events described by B and C.8 My main objection to this interpretation of the locality condition is that it begs the question. Counterfactuals are not statements which can be directly verified in the actual world, and therefore we can hardly see them as describing “the element of physical reality” only. Rather, they reflect a complicated and subtle web of interconnections between what is real and what we see as possible according to our best knowledge. Now, it seems to me that the only acceptable way of arguing for the Ghirardi-Grassi locality condition is to start with the more basic assumption (SLOC), and to show that the stronger locality encompassing counterfactual facts has to follow because of certain restrictions on the similarity relation and/or the set of accessible worlds. And it can be easily showed that in order to accept the Ghirardi-Grassi locality in our context, we have to eliminate certain possible worlds – for example the possible world in which MLX(p) is not performed, but MRX(p) is, and it gives the result different than a’. On the surface I cannot see any reasons why this possible world should not exist, or why its existence would violate our intuitions concerning locality condition. The EPR argument using the second interpretation (C2) of counterfactuals fails even more spectacularly. It can be easily verified that when we interpret counterfactual claim MRX(p’) ORX=a’(p’) as stating that ORX=a’(p’) has to hold in all possible antecedent-worlds which are identical to the actual one only within the confinement of the past light cone of p’, there is no reason to claim that (1.2) should be true. The locality condition (SLOC) assures us that there is a possible MRX(p’)-world which shares with the actual one more than only the absolute past of p’; namely it shares the absolute past and the absolute ‘elsewhere’. But the existence of such a world is not enough, for there are still possible worlds which can differ ‘elsewhere’ with respect to some undetermined events while keeping the absolute past of p’ intact. And if we endorse the standard Copenhagen interpretation of the quantum measurement, then an obvious candidate for such an undetermined event would be a result of the measurement MLX(p) on the left-hand side particle. So, because there is a possible MRX(p’)-world in which the outcome of the left-hand side measurement is different than a, and this world retains the same absolute past of point p’ as the actual one (see Fig. 6), according to the truth conditions given in (C2) counterfactual MRX(p’) OLX=a(p) has to be 8 Note that the same dubious version of the locality condition was adopted by H. Stapp in his article (1997). This “counterfactual locality” was subsequently criticized by his opponents (Unruh 1999, Mermin 1999). 22 false, which in turn renders our counterfactual MRX(p’) ORX=a’(p’) unjustified. If this analysis is correct, it appears that according to the second reading of counterfactuals we cannot even claim that when the outcome of the measurement of XL is known to be a, the property attribution for particle R should be PRX=a’(p’), i.e. as inferred by ordinary rules of quantum mechanics.9 OLX=b X p’ p MLX MR L R Fig. 6. A possible world in which the outcome of the measurement of XL is different than a. Shaded area again indicates the region which is kept identical to the actual world. I believe that this last consequence will be blatantly unacceptable in the eyes of many readers. Apart from the fact that it renders all EPR-like arguments unjustified at their starting point, it makes all inferences based on EPR correlations and proceeding from a given property of one system to the property attribution of the other, correlated system, totally invalid. But how could it be? After all, experience tells us that the purported correlation really exists: each time when we measure two correlated properties, the outcomes agree with quantum- mechanical predictions. So it might look like the above-mentioned consequence is at odds with experience, and therefore the particular reading of counterfactual conditionals on which the entire reasoning was based should be abandoned, as it provably leads to a contradiction not only with quantum-theoretical predictions, but with the experimental results as well. 9 My result at this point is in full agreement with the result of the counterfactual analysis of the relativistic EPR argument proposed by M. Redhead and P. La Rivere in (1997), which is a response to (Ghirardi, Grassi 1994). The authors implicitly assume a method of evaluating counterfactuals encapsulated in (C2) (see p. 212), and then conclude that in order to infer the value of the counterfactual measurement from the actual, we would have to assume determinism (stating that the value obtained in the actual experiment remains the same in all relevant possible worlds). While I agree with their general conclusion that the EPR argument allows for a “peaceful coexistence” between quantum mechanics and relativity, I think that in their attempt to improve Ghirardi and Grassi’s derivation, Redhead and La Rivere should underscore that both interpretations rely on different readings of counterfactuals. 23 Yet the above argument is incorrect. The second interpretation (C2) of counterfactuals leads neither to a contradiction with experimental facts, nor with theoretical predictions. The rules of quantum mechanics tell us only that when we conduct two experiments on two separate but entangled particles, the results of these experiments will be correlated. In our situation it means that the following counterfactual must be true: (1.2’) MLX(p) OLX=a(p) {[MRX(p’) MLX(p) OLX=a(p)] ORX=a’(p’)} However, in (1.2’), as opposed to (1.2), the condition MLX(p) OLX=a(p) is explicitly introduced into the antecedent of the counterfactual. This forces us to consider possible worlds in which the result of the left-hand side experiment is XL = a, and obviously in these possible worlds the result of the measurement of XR can be no other than a’. But in (1.2) the only reason for believing that the result of the counterfactual measurement on particle R would be a’ is that the corresponding result a was obtained in the actual world. However, this fact does not have to be ‘transferred’ to the closest possible world, if it is not salient with respect to the similarity criteria adopted. In order to reinforce this point, let us suppose that point p lies in fact in the absolute future of p’. In such a situation formula (1.2’) would still hold: obviously in the possible world in which we ‘manually’ ensure that the result of the measurement if XL is a, the outcome of the measurement of XR has to be appropriate. But does it guarantee that once we know that the outcome at p is a, we can infer that back in time at p’ the right-hand side particle already possessed objective property XR = a’, as it is suggested by (1.2)? Surely the answer is no. To conclude this discussion, I am not denying that the probability of getting outcome XR = a’ conditional on the result XL = a equals 1 for space-like separated measurements, as well as for measurements being time-like separated. But this fact does not imply that while one experiment is performed, the other particle has the corresponding property. Neither it implies that if we counterfactually consider a measurement on this particle, the result is guaranteed to be this-and-this. To answer the question what would have been the case, had we decided to measure XR at point p’, we have to let the world run its course once again up to point p’, and according to the ontological intuition behind interpretation (C2) of counterfactuals, the world ‘reaching’ point p’ in its temporal evolution does not yet contain events taking part at point p, hence in this situation the measurement at p’ can reveal any admissible value. 24 The final result of the presented analysis is that the EPR argument, interpreted consistently with the help of two notions of counterfactuals, in both cases appears to be unwarranted. Moreover, there is no need to question the locality principle in order to rescue the completeness of quantum mechanical description. This, I believe, is an interesting result which promises that the consistent application of counterfactual conditionals may help in throwing new light on some other notorious problems in the foundation of quantum mechanics.10 Counterfactual Bell’s theorem As the final step, we’ll attempt to give a counterfactual reconstruction of the derivation leading to Bell’s theorem. The logical structure of the standard Bell argument is very often presented as follows. The argument deals with a pair of ½-spin particles (L, R) in the EPR- Bohm state described as = |SxL = ½ |SxR = -½ – |SxL = -½ |SxR = ½, where Sx stands for the x-component of spin of particles L and R respectively. Let us now choose three components of spin Sa, Sb and Sc such that directions a, b and c are coplanar and the angle between a and b equals the angle between b and c, which in turn equals 60. The Bell argument can be seen as an indirect proof of the fact that the hidden-variable hypothesis (the realism assumption) and the locality condition cannot be jointly satisfied, so (at least) one of them should be abandoned. The proof proceeds under the supposition of the realism assumption and the locality condition, and leads to a conclusion which is incompatible with quantum-mechanical predictions and with experimental results. The realism assumption in our case amounts to the assertion that all observables involved (i.e. SaL, SbL, ScL, SaR, SbR and ScR) have their values determined even when no measurements are performed. Now, the crucial step in the proof is the assumption that all objective values assigned to the observables should obey quantum- mechanical predictions stemming from the analysis of quantum state and concerning 10 Yet another counterfactual interpretation of the EPR argument and Bohr’s response to it has been proposed in (Dickson 2002). Dickson bases his interpretation on the distinction between two types of the locality assumption (he calls them “non-disturbance principles”): a strong and a weak one. The strong locality claims that when we assume counterfactually than instead of observable X, an incompatible observable Y was measured on particle L, the property attribution for particle R should remain the same as when inferred from the result of the actual measurement of X on L. On the other hand, the weak locality holds that the above counterfactual inference is valid only under the supposition that no measurement is performed on particle L. According to Dickson’s interpretation, Bohr’s response to the EPR amounts to the rejection of the strong locality, while retaining the weak one. The main problem I see with this interpretation is that even the weak locality leads ultimately to the incompleteness of quantum mechanics, contrary to Bohr. For the weak locality implies that there is a possible world in which there is a pair of EPR particles not subject to any measurement, and yet one of these particles possesses a physical property not included in its standard quantum-mechanical description. 25 probabilities of measurement outcomes. These predictions imply for example that when observable SaL is revealed to posses value ½, the measurement of the counterpart observable SaR has to yield the opposite value –½. Once these conditions are imposed, it is a matter of relatively simple derivation to arrive at the Bell-Wigner inequality, which connects together relative frequencies of obtaining the following combinations of experimental outcomes: (SaL = ½, SbR = ½), (SbL = ½, ScR = ½) and (SaL = ½, ScR = ½). How can we argue that the objective values for all observables involved will obey quantum-mechanical restrictions? After all, all we can derive from the mathematical form of the EPR state are probabilities of certain combinations of measurement outcomes, not objective properties. More specifically, the mathematical form of the state together with the statistical algorithm of quantum mechanics imply that when a joint measurement (SiL, SiR) is performed, the results will be always opposite. In order to argue that the objective values of SiL, SiR should be opposite too, even when no measurement is performed, we have to rely on two additional premises. One of them, called “the principle of faithful measurements” states that if an observable possesses a definite value right before measurement, this measurement is bound to reveal exactly this value. Without the principle of faithful measurements any talk of objective values of quantum observables would be useless, as we could assume whatever we please without any regard for experiments whatsoever. The second assumption necessary to extend quantum-mechanical correlations onto the possessed values is the notorious principle of locality. It assures that once both particles are sufficiently far away, the fact that a particular measurement is performed on one of them cannot change objective values of observables pertaining to the other. With these two conditions in place, we can finally apply quantum-mechanical restrictions to the possessed values, and then derive the Bell inequality. Our counterfactual reconstruction of Bell’s argument will essentially rely on the same set of assumptions as the previous counterfactual analysis of the EPR paradox. We will use counterfactual conditionals to interpret property attribution statements, and also to explicate the locality condition. As before, let us interpret a statement saying that an observable Si characterizing a particle x has an objective value a at a spatiotemporal point p with the help of the following counterfactual (2) M(Six, p) O(Six = a, p), 26 where M(Six, p) means that a measurement of Six was performed at p, and O(Six = a, p) that its outcome was a. Subsequently, the symbol will be interpreted twofold, according to the definitions of spatiotemporal counterfactuals (C1) and (C2) given above. The counterfactual interpretation of property attribution statements, together with the space-time reading of counterfactuals which emphasizes the future’s dependence on the past, already ensure that the principle of faithful measurements will be fulfilled. For if at a certain point a counterfactual of the type (2) is true for a given a, then according to both readings (C1) and (C2) it means that in all worlds which are identical to the actual one up to the specified moment, and in which the measurement is performed, the laws of nature guarantee that the result can be no other than a. Now we have to formulate the locality condition, and again drawing on the previous analysis we will adopt the semantic version of locality condition (SLOC). It should be emphasized that the identity of a particular region with the actual world is to be judged entirely on the basis of the comparison of spatiotemporal genuine events, which don’t include any ‘counterfactual’ events, expressed with the help of counterfactual conditionals. Otherwise the circularity would threaten, as we would have to know in advance what are the truth values of counterfactual statements in the actual world in order to evaluate the very same counterfactual conditionals. If we assumed that the property attributions themselves constitute genuine events, or states of affairs, then (SLOC) together with the principle of faithful measurement would obviously imply that objective values should obey quantum-mechanical restrictions on possible results of experiments. For suppose that the two particles L and R were created with a complete set of quantum properties, including definite values for the observables Sa, Sb and Sc. We obviously assume that those properties cannot ‘disappear’ spontaneously during the subsequent evolution of the particles, which means that we do not admit the existence of two possible worlds which would differ from each other only in that in one of them one particular property ceased to characterize a given particle at a certain moment of the evolution of that particle. And now, all we have to show is that under the counterfactual assumption that a pair of observables (SiL, SjR) was chosen for measurement at space-like separated points pL and pR, the results would be exactly as given in the initial property attribution. According to (SLOC) there is a possible world in which SjR is performed and which is otherwise identical to the actual one, including the property attribution for particle L. By itself, this does not exclude the possibility that there is another possible world in which some property attribution of L would spontaneously disappear. But because we have explicitly rejected this, the world predicted by 27 (SLOC) is actually the only one admissible by the laws of nature. Therefore we can see that right before the measurement of SiL particle L has to have exactly the same value of SiL as at the moment of its creation, hence according to the principle of faithful measurements this value is bound to be revealed. Due to symmetry the same goes for particle R, which finally proves that the properties possessed will be also revealed in any possible experiment. The crucial thing about the above reasoning is that we have treated property attributions as if they constituted genuine events. But now we are going to interpret property attributions counterfactually, and (SLOC) does not include explicitly counterfactual events. So there might be a possibility that with the counterfactual interpretation of property attributions the reality assumption plus the locality principle will not lead to a contradiction with quantum mechanics. In other words, an interesting possibility of avoiding the no-go Bell theorem while keeping some basic intuitions concerning the objectivity of quantum properties presents itself. Now we are going to analyze this possibility using two available interpretations of spatiotemporal counterfactuals. We are then assuming that for every observable Si in question there is some value a for which counterfactual (2) turns out to be true. A pivotal move necessary in a derivation of the Bell inequality can be now presented as follows. We have to be able to prove that once two counterfactuals M(SiL, pL) O(SiL = a, pL) and M(SjR, pR) O(SiR = b, pR) are true, the following counterfactual has to be true as well: [M(SiL, pL) M(SjR, pR)] [O(SiL = a, pL) O(SiR = b, pR)]. In words, we have to make sure that when a counterfactual measurement on the left-hand particle is bound to reveal a particular value, and a counterfactual measurement on the right-hand side particle also has to yield a specific value, then the combined measurement for two particles should reveal the same values for each particle. In such way we can argue that the counterfactual values of observables should respect quantum- mechanical restrictions regarding joint probabilities of outcomes. Let us tackle this issue generally. Suppose then that we have two events A and B space-like separated, and that we know that the following counterfactuals hold: A A, B B. Now we are asking the question whether this can guarantee that the counterfactual (A B) A will be true (obviously, by symmetry it means that (A B) B would have to be true as well). Let us write down the purported implication: (3) (A A) (A B) A 28 It is well known that (3) is not a logical truth – counterfactual conditionals do not obey the law of monotonicity. In order to prove (3) we have to use the locality assumption (SLOC) together with the fact that A and B are space-like separated. Let us start then with the first interpretation of counterfactuals (C1). According to the truth criteria given with the help of the similarity relation (SIM), the counterfactual A A is true if and only if A is true in the possible A-world which has the ‘smallest’ region of divergence D*. Taking into account the locality condition (SLOC) we see that there must be a possible world for which the region of divergence D* is identical with the absolute future of A, and obviously this world will be the closest with respect to the (SIM) relation. Hence the truth of A A amounts to the fact that A is true in every possible world wA in which A holds and which is exactly identical to the actual one everywhere outside the absolute future of A. On the other hand, the required counterfactual conditional (A B) A will be true if A is true in all AB-worlds wAB such that they agree with the actual one everywhere outside two future light cones: one starting in A, and one with its apex at B. Now, we could try to prove that this is the case in the following way. Let us consider any of the worlds wA. According to (SLOC) there exists a possible B-world wB such that wB is identical to wA throughout the entire space-time except the absolute future of B. This means that in wB both A and A hold, as they are true in wA and they take place outside the future light cone of B (recall that A and B are space-like separated). So it seems that we have almost achieved our goal, because the world wB turned out to be one of the wAB worlds, and moreover A holds in it. But unfortunately it is too early to pronounce the victory. We have proven that in some of wAB worlds A has to hold, but it does not mean that A is true in all of them. It is still possible that there exists another possible world wAB in which A is not true. But wouldn’t the existence of such a world contradict the locality principle (SLOC)? The answer is no, because wAB is not the closest B-world to any of wA worlds. From the perspective of wA world wAB has two primary points of divergence (one at the point where B occurs, and the other where A is no longer present), so this world is not salient with respect to evaluating the sentence A (B A), at least as long as the principle (SLOC) guarantees that there are possible B-worlds more similar to wA. However, world wAB is perfectly salient with respect to evaluating the conditional (A B) A, as it is one of the closest AB-possible worlds to the actual world. Without any additional reasons for believing that wAB should not exist we cannot claim that the counterfactual (A B) A has to be true. 29 To recapitulate: the locality principle (SLOC) guarantees that there is a possible AB- world closest to the actual, in which A holds, but it cannot ensure that A will hold in all such AB-worlds. The latter would follow if we accepted that one of the following has to be true: either (A B) A, or (A B) ~A. For we know that (A B) ~A cannot be true, because A holds true at least in some possible worlds wAB. In our situation this additional assumption would boil down to the thesis that the result of the left-hand measurement has to be counterfactually determined even when the right-hand side measurement is performed. We can dub this new assumption “the principle of the definiteness of outcomes of joint measurements” (DJM): (DJM) For some values a and b the following counterfactual holds: [M(SiL, pL) M(SjR, pR)] [O(SiL = a, pL) O(SiR = b, pR)] Now the locality principle (SLOC) entails that those values a and b have to be exactly identical to those figuring in the original counterfactual property attributions, and therefore finally the road to the Bell inequality is open. But still the question remains whether (DJM) is a necessary assumption for somebody who believes in the existence of objective quantum properties defined in terms of appropriate counterfactuals. For now I will leave this question open, but I don’t see any decisive arguments to exclude the possibility that each quantum system treated separately (‘locally’) might have counterfactually determined all its properties, whereas when analyzed ‘globally’ these properties would cease to exist. Unlikely as it might be, the only way to avoid the consequence of Bell’s theorem while sustaining some counterfactual form of the realism assumption together with the locality condition is to reject the principle of the definiteness of outcomes of joint measurements (DJM). Suppose now that we interpret logical connective according to the second proposal (C2), which stresses that in order to evaluate a sentence A B we have to consider all possible worlds sharing with the actual one only the past light cone of the event described by A. Therefore, it means that a property attribution statement of the form (2) can be restated in terms of possible worlds as follows: in all possible worlds (with exactly the same set of laws of nature) that are identical to the actual one in the entire absolute past of the measurement of Si, the outcome of this measurement is always the same and equals a. And now, it appears that this interpretation of property attribution statements is so strong that under the assumption of the objective existence of all parameters characterizing both EPR 30 particles the Bell inequality follows even without the use of the locality condition! The locality condition is, so to speak, already included in our way of counterfactual reading of property attributions, for in order to say that a particular object possesses an objective property given counterfactually, we have to be sure that this property is bound to be revealed no matter what happens in the region space-like separated from the measurement. To be more precise, let us prove that the condition (3), necessary for the derivation of the Bell inequality, is guaranteed by the interpretation (C2) of counterfactuals. The antecedent (A B) A is true, according to (C2), if A is true in all possible worlds in which A and B are true and which are the same as the actual one in the region being the sum of two past light cones: one with its apex at A and the other at B.11 But from this it follows that each such world is at the same time a world in which A holds and which is identical with the actual one throughout the absolute past of A, and therefore it is one of the worlds which must be taken into account while evaluating counterfactual A A. On the assumption that this counterfactual is true, we can conclude that A has to hold in all of the above-mentioned AB-worlds, which completes the proof of (2). The above result is somewhat surprising, especially when we compare it with the previously achieved conclusion concerning the EPR argument. There it was shown that under the interpretation (C2) of counterfactuals the EPR argument fails already at its initial step. Now we are claiming that the Bell argument can be completed equally easily under the same interpretation. Given that the structures of both arguments are analogous, one might wonder whether our results are compatible with one another. But actually they are perfectly compatible. The EPR argument aimed at showing that it is possible to derive a statement attributing to a distant system a property not included in its quantum-mechanical description. However, the counterfactual interpretation of property attribution based on (C2) appears to be too strong for this derivation to go through. On the other hand, the initial assumption of the Bell argument is that quantum systems already possess all the properties involved. With the strength of this assumption under (C2), it is not surprising that a contradiction with quantum- mechanical predictions is so easy to obtain, even without any help from the locality condition. 11 Actually, the truth condition (C2) applies only when the antecedent of the counterfactual refers to a single localized point-event. So in order to evaluate the conditional (A B) A we have to extend the criterion given in (C2). One intuitive extension which can be adopted here is as follows: when the antecedent of the counterfactual is equivalent to a finite conjunction of sentences, each of which singles out a particular event, the consequent has to be true in all possible worlds which keep all the absolute pasts of these events exactly the same as in the actual world. 31 To sum up, our analysis brought us to the following conclusions. The only possibility of avoiding Bell’s no-go result opens up when we interpret property attribution statements as counterfactual conditionals, and counterfactuals in turn are defined according to the “all differences only in the absolute future” reading. However, while it is possible to claim that properties interpreted in such a way characterize objectively separate quantum systems, we have to reject the supposition that these systems considered jointly have also all their properties determined. On the other hand, under the second available interpretation of spatiotemporal counterfactuals, the hypothesis that quantum systems could have all their properties determined leads directly to the Bell inequality. Bibliography Bedford, D and Stapp, H.P.: 1995, “Bell Theorem in an Indeterministic Universe”, Synthese 102, 139-164. Bigaj, T.: 2002, “Counterfactuals and Non-locality of Quantum Mechanics”, Philosophy of Science Archive, Pittsburgh, http://philsci-archive.pitt.edu/documents/disk0/00/00/09/61/PITT-PHIL-SCI00000961- 00/Article1.pdf. Bigaj, T.: 200?, “Counterfactuals and Spatiotemporal Events”, Synthese, forthcoming Dickson, M: “Bohr on Bell: A Proposed Reading of Bohr and Its Implications for Bells Theorem” in: T. Placek, J. Butterfield (eds.) Non-Locality and Modality, Kluwer, Dordrecht 2002, 19-35. Einstein, A., Podolsky, B., and Rosen, N.: 1935, “Can quantum-mechanical description of physical reality be considered complete?”, Physical Review 48, 696-702. Finkelstein, J.: 1998, ‘Yet Another Comment on Nonlocal Character of Quantum Theory’, preprint quant- ph/9801011 Finkelstein, J.: 1999, ‘Space-time Counterfactuals’, Synthese 119, 287-298. Ghirardi, G. and Grassi, R: 1994, “Outcome Predictions and Property Attribution: the EPR Argument Reconsidered”, Studies in the History and Philosophy of Science, 25 (3), 397-423. Greenberger, D.; Horne, M.; Shimony, A., and Zeilinger, A.: 1990, ‘Bell’s Theorem without Inequalities’, American Journal of Physics, 58, 1131-43. Griffiths, R.B.: 1999, “Consistent Quantum Counterfactuals”, Physical Review A, 60 (1). Lewis, D.: 1973, Counterfactuals, Harvard Univ. Press, Cambridge (Mass.) Lewis, D.: 1986a, “Counterfactual Dependence and Time’s Arrow” in: Philosophical Papers Vol II, OUP, Oxford. Lewis, D.: 1986b, “Causality” in: Philosophical Papers Vol II, OUP, Oxford. Lewis, D.: 1986c, “Counterfactuals and Comparative Possibility” in: Philosophical Papers Vol II, OUP, Oxford. 32 Maudlin, T.: 1994, Quantum Non-locality and Relativity, Blackwell, Oxford. Mermin, D.: 1998, ‘ Nonlocal Character of Quantum Theory?’, American Journal of Physics 66(10), 920-924. Redhead, M.: 1987, Non-locality, Incompleteness and Realism, Oxford Univ. Press, Oxford. Redhead, M., Butterfield, J. and Clifton R.: 1990, “Nonlocal Influences and Possible Worlds”, Brit. J. Phil. Sci. 41, 5-58. Redhead, M., La Riviere, P.: 1997, “The Relativistic EPR Argument” in R.S. Cohen, M. Horne and J. Stachel (eds.), Potentiality, Entanglement and Passion-at-a-Distance, Dordrecht. Shimony, A., Stein, H.: 2001, “Comment on ‘Nonlocal character of quantum theory’ by Henry P. Stapp”, American Journal of Physics 69, 848-853. Stapp, H.: 1971, “S-matrix Interpretation of Quantum Theory”, Physical Review 3D, 1303-20. Stapp, H.: 1997, ‘Nonlocal Character of Quantum Theory’, Am. J. Phys. 65, 300-304. Stapp, H.: 1999, ‘Comments on ‘Non-locality, Counterfactuals, and Quantum Mechanics’, Physical Review A 60(3), 2595-2598. Unruh, W.: 1999, ‘Nonlocality, Counterfactuals, and Quantum Mechanics’, Physical Review A, 59, 126-130.