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It should suffice to say that in the case of the broad fixed past analysis it is relatively simple to come up with a similarity relation based entirely on the comparison of areas of perfect match with respect to particular facts for more on that see Finkelstein ; Bigaj On the other hand, if the introduction of the antecedent-event requires a modification of its broad past, the no-miracle approach to counterfactuals will diverge from the miracle-based analysis.
The appropriate similarity relation will force us to consider possible worlds which lawfully accommodate the contrary-to-fact antecedent event, and which diverge from the actual world at the latest possible moment in a suitable relativistic sense of the word. The task of finding a suitable extension of the narrow fixed past semantics presents us with a greater challenge. I have proven in Bigaj that there is no similarity relation of the Lewisian type even if we admit partially ordering relations which could reduce to this method of evaluation for indeterministic events.
The best we can do is selecting, for each antecedent-event separately, a set of possible antecedent-worlds in which the consequent has to be true in order for the counterfactual to be satisfied. It can be easily verified that both no-miracle semantics produce the same valuation for alternative-outcome counterfactuals connecting space-like separated measurements under the assumption that these outcomes are truly indeterministic events.
While the sets of possible worlds in which we should carry out such an evaluation differ in both cases, the net result is the same: the counterfactual comes out true. This result is to be expected: after all, in the currently considered approaches no law-breaking worlds are permitted, and thus an alteration of the outcome of one measurement must be associated with a corresponding change in the other outcome.
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The price we have to pay for eliminating law-violating possible worlds is that counterfactual dependence no longer implies causal dependence. It is now possible to have two events such that if one did not occur, the other would not occur either, without any causal link connecting the two. This may happen if both events have a common cause in their joint past, in which case the relevant-antecedent worlds will have a divergence point in the past, and the counterfactual will be evaluated as true.
In order to derive conclusions regarding the existence of a causal link we have to make sure that such a situation is excluded. Thus, if we have two events A and B , and there is a non- A -world such that B does not occur in it either, and moreover both absolute pasts of the locations of A and B are exactly as in the actual world, we can conclude that there is a causal link between A and B. Unfortunately, the very fact that there is no such world does not conversely imply that there is no causal link—it is still possible that either event is separately caused by some occurrence in their joint past without there being a common cause.
Thus the currently considered conceptions of counterfactuals can supply us with a sufficient, but not necessary condition for causality. In the next section we will shift our attention to the problem of causation within quantum entangled systems. As we have seen, the jury is still out regarding the truth of the counterfactual connecting alternative outcomes of space-like separated measurements. Even though the majority of the considered interpretations of counterfactuals imply that the distant outcomes are indeed counterfactually dependent on each other, still there are available conceptions that can cast doubt on this conclusion.
In this section we will limit ourselves to the interpretations that imply the existence of counterfactual dependence between distant outcomes, in order to be able to discuss question 1. Before we can attempt to formulate even tentative answers to these questions, we should revisit the relevant facts. We know that counterfactual dependence entails causality only for the right type of counterfactual statements.
It seems that indeed they are. However, in the specific case that we are considering, an additional assumption is believed to be satisfied: no adjustment of the joint absolute past of both measurements is necessary in order to introduce alternative outcomes of the experiments. Therefore no explanation of the outcome-to-outcome counterfactual dependence in terms of a common cause is available, and the path to direct causation stands wide open.
But are we sure that there is no common cause? Given the existence of perfect correlations between outcomes, the only possibility for a local, common-cause explanation is in the case when an event in the joint past of the measurements determines both outcomes beforehand. But either way, it seems that non-local causal links are unavoidable in both approaches.
Or are they? Here philosophers are likely to make the following complaint. We have proven so far that according to several compelling accounts of counterfactuals, the alternative-outcome counterfactuals come out true in a way that makes it almost inevitable that their truth should be underpinned by a legitimate causal link. But we have ignored one inconvenient fact: the counterfactual outcome-outcome dependence goes in both directions.
The situation is entirely symmetric: we may equally well say that had the L -outcome been 0, the R -outcome would have been 0, or vice versa. Consequently, we have to accept the fact that the L -outcome causes the R -outcome, and the latter reciprocates, causing the former in turn. But is this even intelligible? And when we add to that the additional assumption of transitivity which, it has to be admitted, has been questioned by several authors , we end up with a clear case of causa sui , so dreaded by all metaphysicians.
Some authors try to skirt this problem by doing a bit of terminological maneuvering. Instead of talking about causal links, let us say that distant outcomes in an entangled system are causally implicated with one another. However, this solution strikes me as being rather disingenuous.
One is when the events in question are not directly causally linked, but are part of a broader causal network. A clear example of such a case is the common cause scenario, and of course there can be more complex causal connections involving the events in question. But clearly there is no reason to hedge our bets in such a way in the case of quantum non-local correlations between outcomes.
But again the ignorance scenario does not apply to our case, as we already know given all the required assumptions that one event causes the other, while the other causes the first event back. At this moment we should recall one particular interpretation of counterfactuals that may be able to break the impasse here.
On quantum entanglement, counterfactuals, causality and dispositions
Namely, it is the interpretation which relativizes the truth value of a given counterfactual to a frame of reference. According to this approach, it is never the case that both counterfactuals connecting alternative outcomes are true in the same frame. If we select a particular frame of reference in which the L -measurement temporarily precedes the R -measurement, the counterfactual dependence goes from the former to the latter, and so does the causal link.
However, in a different frame where the temporal relation between measurements is reversed, it is the outcome of the R -measurement that causes the other outcome.
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So there is no description of the entire system which would require us to admit that there is a causal loop there. Promising as this strategy may be, it is nevertheless far from being entirely immune from legitimate objections. But this is a mere word-play. It remains the case that in one frame of reference event A causes event B , while in another frame A is no longer a cause of B.
Given these quandaries, it may be advisable to look again at the purported connection between counterfactual dependence and causality.