Wes Salmon thinks the following "Leibniz Principle" is incompatible with the explanation of indeterministic phenomena:

if, on one occasion, the fact that circumstances of type C obtained is taken as a correct explanation of the fact that an event of type E occurred, then on another occasion, the fact that circumstances of type C obtained cannot correctly explain the fact that an event of type E' (incompatible with E) occurred.Salmon thinks that the Leibniz Principle is incompatible with explanations in indeterministic cases and hence false.

I don't know if the Leibniz Principle is false. But I do have an argument that it is compatible with explanations in indeterministic cases.

Consider an electron in a mixed (3/5)|*u**p*>+(4/5)|*d**o**w**n*> state. The electron then undergoes a process whereby it is measured whether it is in an up or down state, thereby requiring collapse. It has probability 9/25 of collapsing into |*u**p*> and 16/25 of collapsing into |*d**o**w**n*>. In fact it collapses into |*u**p*>. This is pretty much the hardest kind of real-life case for explanations of indeterministic cases, since it is the less likely outcome that happens. But it is also one where an explanation can be given that satisfies the Leibniz Principle.

Consider now the following circumstances:

- The electron is in a state such that the squared modulus of its probability amplitude for |
*u**p*> is at least 9/25, and it was collapsed into |*u**p*> or |*d**o**w**n*>.

*u*

*p*>. Salmon should agree with me on this point, since given what he thinks about indeterministic explanation, he will agree that

- The electron is in a state such that the squared modulus of its probability amplitude for |
*u**p*> is 9/25 and the squared modulus of its probability amplitude for |*d**o**w**n*> is 16/25, and it was collapsed into |*u**p*> or |*d**o**w**n*>.

*d*

*o*

*w*

*n*> component is not relevant to explaining that the electron collapsed into |

*u*

*p*>, and the information that the squred modulus of the probability amplitude for |

*u*

*p*> is at most 9/25 does not help explain why the electron collapsed into |

*u*

*p*>. When we remove this irrelevant information (and Salmon famously said that irrelevant information is fatal to explanations) from (2), we get (1).

Now, if we accepted (2) as an explanation of the electron's collapsing to |*u**p*>, we would also have to accept it in another case as an explanation of the electron's collapsing to |*d**o**w**n*> (an even better one, since that is a likelier result), contrary to the Leibniz Principle. This is the sort of reason for which Salmon rejects the Leibniz Principle.

But (1) has no such unfortunate result. For while (1) does explain why the electron collapsed into |*u**p*>, it cannot explain why an electron collapsed into |*d**o**w**n*>.

One could also weaken the Leibniz Principle and take it to be a constraint on *contrastive* explanation (cf. this paper). If so, then the above would show that we can satisfy at least one desideratum for contrastive explanation in indeterministic cases (for a different approach, see this paper).

## 21 comments:

I have other reasons for thinking that such cases pose no challenge to the Leibniz Principle, and which also serve as reasons to be dissatisfied with the line that you give in this post. I'd be very curious to hear your thoughts. [Incidentally, I had written a short section of my dissertation on this topic (section 6.1.1), and so I'm pulling from there -- I haven't actually gone and dug up all of the quotes below in response to your blog post : ) ]. In your example (and I think in all of the cases of indeterministic explanation that Salmon, Jeffrey and others commonly point to), we put forward a “statistical explanation of an event [which] exhibits that event as the result of a stochastic process from which such events arise with some probability whose degree may be high, middling, or even very low” (Salmon, 1971, p. 9). In effect, we respond to a why query by reciting the chances of the explanandum’s occurrence. Richard Jeffrey (1969, p. 24) puts this point very clearly: “The knowledge that the process was random answers the question, ‘Why?’ –- the answer is, ‘By chance’. Knowledge of the probabilistic law governing the process answers the question ‘How?’ -- the answer is, ‘Improbably, as a product of such-and-such a stochastic process’.”

The key question that comes to my mind with regards to this type of "explanation" is whether the hypotheses in these examples really offer potential explanations of the evidence in question at all. It seems to me that these are cases where the hypotheses just fail to offer such an explanation -- in such cases, we are denying that there is any explanation to be had. As such, they are cases that the Leibniz Principle just does not speak to. As noted above, in these examples, “there’s no reason for the fact: it came about by chance” (Jeffrey 1969, p. 24). But when we can only appeal to the stochastic facts of a scenario, we are effectively throwing our hands up and saying, “the explanandum just happened, and there is nothing further to say about it other than how likely its chance occurrence was.” If we gain no “reason” for the explanandum, as Jeffrey puts it, or any other information about the explanandum other than knowledge of its likeliness, then it is unclear at best why we would think we have gained a potential explanation. Any psychological relief that such a move may give us in a particular instance is not, I suggest, due to the fact that we now have a deeper understanding of the explanandum but rather to the fact that we are no longer unsettled in our search for one; we have decidedly given up on our search for understanding in this case.

[Apparently, I have to continue on another comment given blogger's comment size restrictions (?)]

[continued from the last comment]

Another way to think about this is that when we are faced with a ‘why?’ question, we may respond either by giving an explanation or by saying that there is none available. In the former case, we -- at least typically -- will cite causes, reasons, laws, or the like that go some way to showing that the explanandum was actually not so unexpected as previously thought. In the latter case, on the other hand, we can effectively say that there is no such explanation simply by saying that the explanandum event just happened by chance; we can give a more informative response of this sort by saying that the explanandum event just happened by chance, and by citing the probability of its occurrence -- if we know it. It is the latter sort of move that Salmon and Jeffrey exploit, and so it seems that they are pointing to a case where one denies the possibility of an explanation.

Back to the example in your post: the following sort of dialogue seems likely to ensue in response to someone who puts the stochastic facts of the matter forward as an explanation:

Person 1: I wonder why the electron collapsed into |up>.

Person 2: Well, it’s because the electron was in a state such that it had a probability of 9/25 of collapsing into |up>, and it was either collapsed into |up> or |down>.

Person 1: That’s not much of an explanation. Now I’m even more curious why the electron collapsed into |up>!

Person 2: Well, I wish I could tell you a quantum story -perhaps involving some hidden variable theory -that would show you just why this electron collapsed in the way that it did. But since I don't have such a story to tell, we really just have to accept the brute fact that it did.

Thanks for the quick response!

I don't share the intuition that it's not much of an explanation. We've given the relevant cause--that sure seems like an explanation to me.

Maybe what you're puzzled by is the question of why it is that,

given the facts about the wavefunction, the electron ended up in an up state? But if that's the question you want an answer to, of course there isn't going to be an answer (cf. Hitchcock 1999). However, we shouldn't, I say, expect an answer to an explanatory question relative toeverypossible presupposition, and the denial of answers relative to some presuppositions doesn't imply bruteness.Where did we give the relevant cause?

There are two causes working together:

1. The electron's having a wave-function such that the up component has squared modulus at least 9/25

2. The electron's state being collapsed into up or down.

(By the way, if we coarsely individuate events, event 1 is identical with the event of the wave-function being thus-and-so, and explaining in terms of the latter will violate the Leibniz Principle. But even if events are ontologically individuated coarsely, they might enter into explanations only under some descriptions.)

By the way, the explanation I give has positive explanatory power in the sense of your dissertation, since the priors of the electron ending up in a pure up state are low. (I take it that in the real world there almost always is some low degree of entanglement.) That may just be an artifact of the case.

Well, we're quickly getting into unfamiliar territory for me, so please let me know what's wrong with saying the following:

I guess I find it odd to think of either of those things as being causes. 1. Citing the wave function that spits out the relevant probability seems hardly to be citing a cause (my mental heuristic for saying this: I could fiddle with the function

itselfand it won't make a bit of difference to what actually happens with regards to whether the electron's state ends up collapsing up or down.) The wave function, in fact, hardly seems like thetypeof thing that could be a cause of some physical event.2. I find it even more odd to think that a cause of the electron's collapsing in a particular way is just that it collapsed one way or another.

What am I misunderstanding here?

I am assuming some kind of a realism about wave functions. For instance, maybe, it is a real property of the electron that it has a wave function with such-and-such components.

While I don't know what the ontological status of wave functions are (are they properties of particles? are wave function facts just facts about the underlying physical fields?), whatever it is, facts about components of wave functions are real explanatory facts, and are not just statements of probabilities. Facts about wave functions affect how it is that stuff in the world interacts.

One way to see that wave functions aren't just summaries of probability facts is that the

phaseof the wave function matters in some experiments (cf. this paper).By "that it was collapsed into up or down", I didn't mean "that it was collapsed into up or it was collapsed into down" (though I grant that that is a natural way to read it). Read the "or" here as similar to the "or" in "she chose whether to do A or B" (that she chose whether to do A or B entails that she chose to do A or she chose to do B, but is not entailed by it).

In other words, I am describing the collapse, that it's a collapse induced by measuring the electron's spin. I could also have replaced this by the statement that the electron was sent through the Stern-Gerlach apparatus.

If you want to avoid quantum stuff, take the case of an indeterministic die. What explains why the die landed on six? That the die was tossed, and had at most six sides, and the side with the six was no less biased-for than any other side.

Question, for my own edification: is the following a correct explanation of why a coin lands heads? “It’s a fair coin, and I flipped it.”

If that is a correct explanation, then the same explanation will work for the next time when the coin lands tails, and we have a counterexample to the Leibniz Principle. (I am assuming you meant the LP is compatible with all indeterministic explanations, not merely some.)

If that is not a correct explanation, I am not clear on how it differs (in any important way) from the explanation proferred in the quantum example (or the fair die example). Does it?

Heath:

The orthodoxy among people working on probabilitistic explanation is that the explanation you offer is correct.

If "fair coin" just means "coin that has probability 1/2 of heads and probability 1/2 of tails", then I think it may be subject to "virtus dormitiva" types of worries, where the content of the explanans is too closely tied to the content of the explanandum. I would prefer it if "fair coin" was just elliptical for a description of a physical description like: "The coin is a very short circular cylinder, with heads on one side and tails on the other, with a center of mass in its geometric center, and no significant forces acting on it besides the initial impulsion, air friction, gravity and the final impact."

Now, the "with a center of mass in its geometrical center" clause is equivalent to the conjunction of three clauses:

1. with the center of mass on its axis (where the axis of a circular cylinder goes through the centers of the two circular faces)

2. with the center of mass at least as close to the tails side as to the heads side

3. with the center of mass at least as close to the heads side as to the tails side

Now in these three clauses, 1 is relevant to explaining why the coin landed heads because it helps explain why the coin did not land on its side. 2 is relevant because it helps explain why the coin did not land tails. But 3 is not explanatorily relevant with respect to explaining why the coin landed heads.

Now, when we want something that's strictly an explanation, we should remove irrelevancies. So a better or stricter explanation is more like: "The coin is a very short circular cylinder, with heads on one side and tails on the other,

such that 1 and 2 hold, and no significant forces acting on it besides the initial impulsion, air friction, gravity and the final impact."And this explanation explains heads, but it wouldn't explain tails. What would explain tails is: "The coin is a very short circular cylinder, with heads on one side and tails on the other,

such that 1 and 3 hold, and no significant forces acting on it besides the initial impulsion, air friction, gravity and the final impact."Now back to the Leibniz Principle.

I could make one of two claims about the symmetric putative explanation.

A. It's not a correct explanation, because it contains irrelevant information.

B. It is a correct explanation, but it could be improved

quaexplanation by removal of irrelevant information.If we go for A, then Salmon's argument against the Leibniz Principle fails. If we go for B, then Salmon's argument succeeds and the Leibniz principle is false, but we still have the interesting fact that it is possible to give "Leibnizian explanations" of at least some indeterministic phenomena, where Leibnizian explanations are ones that couldn't explain an incompatible phenomenon.

I am inclined to B.

And then I could endorse a weakened version of the Leibniz Principle: if C explains E, then there an explanation C* of E such that (a) all the explanatorily relevant information from C is contained in C* and (b) all the explanatorily irrelevant information from C* is contained in C, and (c) C* couldn't explain any incompatible E'.

Or more weakly: if C explains E, then for any incompatible E', there an explanation C* of E such that (a) all the explanatorily relevant information from C is contained in C*, (b) all the explanatorily irrelevant information from C* is contained in C, and (c) C* couldn't explain E'.

We will need such a weakening if we're going to be willing accept as explanations the context-laden explanations that ordinary language offers. "Sam ate the banana rather than the apple because the banana was tastier" seems a perfectly good explanation. But suppose it's Lent and Sam has resolved to prefer less tasty foods. Then: "Sam ate apple rather than the banana because the apple was tastier" also seems a perfectly good explanation.

But we can avoid this by moving from the explanation "the banana was tastier than the apple" in the original case to the fuller explanation "the banana was tastier than the apple and Sam had a reason to eat for tastier foods as such and Sam had no reason to eat less tasty foods as such."

Thatis a Leibnizian explanation.Actually, I'm wrong. That last is not a Leibnizian explanation!

Here's something incompatible with Sam's eating the banana that could be explained by the banana being tastier, and Sam preferring tastier foods: Sam was murdered before he could start eating by a crazy guy who murders people who are about to eat bananas.

It may be that such a crazy case can

alwaysbe constructed.If so, then I can only defend the weaker version of my weakened Leibniz principle, the one where C* depends on E'.

Actually, I intended "fair coin" to mean "coin that has probability 1/2 of heads and probability 1/2 of tails," in the same objective sense of probability that we invoke in the quantum case. If there are

virtus dormitivaworries about this, then can we not invent a “quantum quarter” which, um, collapses into the heads state with probability one-half and into the tails state with probability one-half? And rerun the argument.Ah. So, I say that the better explanation (in respect of omitting irrelevant stuff) is that the coin had collapse-to-heads tendency of at least 1/2. And that couldn't explain tails.

You write, "By the way, the explanation I give has positive explanatory power in the sense of your dissertation, since the priors of the electron ending up in a pure up state are low."

That depends, of course, on how the story gets told. You go on to build details into the story (low degree of entailment) which then set things up in such a way that it would indeed seem plausible to assign a low prior to the explanandum. The way that Salmon and Jeffrey typically tell the story makes it the case that the probability actually decreases. This is true of the quantum examples they put forward, but let me cite another example here (from Salmon, 1970):

"Suppose [...] that a game of heads and tails is being played with two crooked pennies, and that these pennies are brought in and out of play in some irregular manner. Let one penny [penny A] be biased for heads to the extent that 90 percent of the tosses with it yield heads; let the other [penny B] be similarly biased for tails.

Furthermore, let the two pennies be used with equal frequency in this game, so that the overall probability of heads is one-half.[...] Suppose a play of this game results in a head; the prior [probability] of this event is one-half. [Now suppose] that the toss was made with the penny biased for tails [penny B]; [the probability of the explanandum] is decreased from 0.5 to 0.1."In this case, my measure says that the hypothesis that penny B was flipped has negative explanatory power over the result. Just for the record, however, I think that is exactly the right result. This is because it seems to me that the hypothesis that B was flipped does a better job of explaining the falsity of our explanandum (it does a better job of explaining a tails than a heads). Even if, in fact, the hypothesis that B was flipped was true, I would still maintain that this is the inferior explanation. A more powerful explanation of the result of our flip is that A was flipped.

Another thing to consider here is the question of whether or not giving a cause is sufficient for giving an explanation (as you seem to assume, but perhaps I'm misreading). A typical move in the phil science literature is to separate the notions of cause and causal explanation. One version of this (something like Strevens's Depth) says that the set of causes that we may refer to in causal explanations is a proper subset of the complete set of causes (for any one event). He gives a long description of a formal algorithm that allows him to select the members of the former out of the latter. The upshot is that even if we're all realists about wave functions (in such a way that we believe that citing details about an electron's wave function describes a cause of how that electron collapses) that doesn't mean that we ought to take an electron's wave function as causal-explanatory of that electron's collapse.

Jonah:

Yes, this part does depend on how the story is told. I still think that even if you tell the story in such a way that the probability goes down, you can give an explanation. The explanation is: a penny was flipped which has a bias that results in a probability of at least 1/10 of landing heads.

I agree that the connection between causes and explanations is tricky. But I think that if something is caused, and you've given all the relevant set of causes, you've either given an explanation or a polluted explanation (where a polluted explanation is something that will be an explanation once you remove some irrelevancies).

Sorry, one other point: another way that someone could criticize the move you're trying to make is by saying the following. "You mention the following hypothesis:

H: The electron is in a state such that the squared modulus of its probability amplitude for |up> is 9/25 and the squared modulus of its probability amplitude for |down> is 16/25, and it was collapsed into |up> or |down>

"I'm convinced that H carries with it some explanatory irrelevancies, but that doesn't mean that H is explanatorily irrelevant. Even with the irrelevancies included, H has positive explanatory power over E. And this same H would have (other) irrelevancies included were we to try to explain ~E, but it would still have positive explanatory power over this one too (for all the reasons Salmon gives)."

The point is that you might escape the untoward consequence in the case where you consider the explanation minus irrelevancies. But that is just to ignore the original problem which had to do with the hypothesis, irrelevancies and all! The latter does seem to have positive explanatory power over E and ~E (according to the proposal) and so it presents a problem for the Leibniz Principle.

PS: I'm just now looking at your last comment about "polluted explanations", and I'm not sure whether that bears on what I just wrote. Curious to hear your thoughts.

Right: if the Salmon-style explanations work, then LP is toast.

And my post for today shows that LP is toast.

Moreover, my explanation in the post here might not actually cohere with LP. For the fact that there is such-and-such an up-component could perhaps explain why the electron collapses into down, in a world where there are demons buzzing around, making electrons in up states collapse into down states. (Or in a world with different laws, but I don't know if electrons could exist in a world with different laws.) I am not SURE of this counterexample, though, because I don't know if anyone can cause a quantum system in a mixed state to collapse into a particular state. (A supernatural being presumably could make the system transition into either pure state, but not every transition is a collapse.)

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