## Thursday, 5 May 2011

### Coherence not measured by probability

Mark Siebel has an interesting paper in the new Analysis arguing that the popular programme of measuring coherence by probability cannot succeed. The essence of his argument is that if G explains E better than H does, then {G,E} is more coherent than {H,E}, so any measure of coherence must give these two distinct measures. Yet there are such cases in which G and H are logically equivalent and for which the joint probability distributions of G and E and H and E will be the same, a consequence of which is that no probability measure will distinguish {G,E} and {H,E}. Seems pretty cogent.

1. Thanks very much for the pointer.

Frankly, from Siebel's paper I have been unable to see how one of two logically equivalent hypotheses can explain a given datum while the other does not.

Also, recent approaches seem to take for granted that probabilistic models could represent explanatory power GIVEN THAT an explanans-explanandum relationship exists, the latter requiring an independent characterization (e.g., Schupbach, J.N. and Sprenger, J., 2011, “The Logic of Explanatory Power”, Philosophy of Science, 78: 105-127).

Finally, I did not get a very clear idea about the connection between explanation and coherence, and thus why the probabilistic program for coherence should itself suffer from "devastating" effects (Siebel, p. 266). For one thing, coherence must be symmetric - Coh(A,B) = Coh(B,A) - while explanation surely is not.

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4. Ah. Finally got it to work right (why can't we edit comments?)

Hi Vincenzo. I see you got him to send you a copy. OK, well I'm not really seeing how you think your remarks are a problem for his argument.

Regarding your first point, what was wrong with his example?

I'm not seeing how your second point bears on his argument. The point that being an explanation requires characterisation independent of probability seems rather to support it. Furthermore, Siebel conclusion might also be a problem for Schupbach and Sprenger

On your third, the fact that explanation is asymmetric doesn't seem a problem for the claim that 'explanatory relations...increase...coherence' p266, and hence not relevant to the significance of there being {G,E} more coherent than {H,E} but no probabilty measure can distinguish them.

5. Hi Nicholas. Yes, Mark Siebel has been so kind to send me the paper. (By the way, I also pointed out this post to him.)

Let me start from the beginning and check out if I get it right. Suppose N = Newtonian celestial mechanics and M = Mars moves on an elliptic orbit. Leaving various subtleties aside, also assume that N implies M. Then N&M is logically equivalent to N.

If I get Siebel correctly, he would say that N explains M but N&M does not. Why? Because "one cannot explain an event by itself" (p. 265). Maybe so, *if* that means that X does not explain X. But surely that's not what happens here. N&M seems no more than an idle way to state N, which of course is far from trivially identical to M.

Or am I missing some serious difference between my example and his own (the barometer)?

6. > The point that being an explanation requires
> characterisation independent of probability
> seems rather to support it.

It does support the conclusion, yes. But I think that knock-out arguments already exist. Say, probabilistic relevance is symmetrical, explanation is not. Done :-)

> Furthermore, Siebel conclusion might also be
> a problem for Schupbach and Sprenger

As I see it, it would surely be a problem *if* logical equivalence did not preserve explanatory power. But that's part of what I doubt in the first place.

7. I think you need to address his actual example, since he only needs one and his does not have the logical form of yours, which may be why it has more intuitive appeal than yours. Yours works because we don't need M in M&N to get the entailment of M (and hence gives a route to feeling that M&N can explain M...because the M isn't needed), whereas in his we cannot remove his Gm from his H2 and still entail Gm.

My own thought against that premiss is that the example underpinning it is using explanation by entailment, which is a rather special case when we are considering coherence. Although explanatory relations generally increase coherence, entailment relations are so strong, we might say, that their contribution to coherence swamps any other factors and hence it is no surprise if probability cannot distinguish coherence in such cases. E.g. although G explains E and H doesn't, because of the entailment relations {G,E} and {H,E} are both maximally coherent anyway and so any probability measure of coherence should assign them the same maximum measure.

So for his argument to really go through requires an example in which G and H are logically equivalent but do *not* imply E, and G explains E but H does not. The example would have to work differently, since his example works by appealing to the principle that an event can't be self explanatory and then including the event in H but not in G, whereas if G and H are l.e. and one implies E the other will. But given the hyperintensionality of explanation (which you seem to be denying) he ought to be able to work one out.

8. Thanks for the reply, Nicholas.

> Yours works because we don't need M in M&N to
> get the entailment of M

Great to know my example works :-) No, seriously, this helps me pinpoint where the disagreement may lie.

> whereas in his we cannot remove his Gm from
> his H2 and still entail Gm.

Is it not the case that we do remove Gm from his H2 precisely by transforming H2 into the logically equivalent H1? It seems to me that Gm is not more "needed" to derive Gm from H2 (as proven by the fact that H1 suffices) than M is "needed" in my example. (Alternatively, I would need some more clarification on what "needed" means here, which I can't find in the paper, anyway.)

So I agree that Siebel needs no more than one compelling example for his case. But I still fail to see in what relevant respect his and mine are *not* on a par. And if they *are* on a par, and it so happens that someone's intuition seem to fail in mine and to apply in his, then... well, woudn't we challenge the intuition?

9. >Great to know my example works :-)

As I think was pretty obvious, I'm not conceding that yours works in that sense, merely pointing out the mechanism by which yours has the appearance of getting round the objection that events are not self-explanatory.

>Is it not the case that we do remove Gm from his H2 precisely by transforming H2 into the logically equivalent H1?

No. Removing Gm from H2 leaves (FmvGm)&Ax(Fx->Gx) (Removing M from N&M leaves N)

>It seems to me that Gm is not more "needed" to derive Gm from H2 (as proven by the fact that H1 suffices) than M is "needed" in my example. (Alternatively, I would need some more clarification on what "needed" means here, which I can't find in the paper, anyway.)

(FmvGm)&Ax(Fx->Gx) does not entail Gm (Whereas N does entail M)

> But I still fail to see in what relevant respect his and mine are *not* on a par. And if they *are* on a par, and it so happens that someone's intuition seem to fail in mine and to apply in his, then... well, wouldn’t we challenge the intuition?

Instead of doing what is required to advance the argument...saying what is wrong with his example... you are wasting my time by repeatedly arguing about whether your example is adequately analogous.

I have already pointed out quite clearly the main relevant failing of your example. Furthermore, when you recall how much you have packed into unexpressed background conditions, initial conditions etc, it should be obvious that all of those factors make it an unperspicuous example to be considering for the very simple reason that such tacit factors make your example indefinite. His example, by contrast, does not rely on disputable tacit factors but is wholly explicit. Because your example is indefinite there are indefinitely many disputes that could be had over analogy and disanalogy, all of which are time wasting. That is why there is no gain to be had from your repeated attempts to embroil us in prolonged argument over whether yours and his are analogous or not.

10. > Is it not the case that we do remove Gm from
> his H2 precisely by transforming H2 into the
> logically equivalent H1?

Sorry, I thought this was clear. I was asking whether the following conditional is true: if we transform H2 in H1, then Gm is removed. As Gm does not appear in the statement of H1, this is clearly the case. Unless more is said about "removing" and the like. That was part of my point on the relationship between the two examples.

I did think it could advance the argument, but please do not waste any more of your time.

11. I have just seen that this idiotic blogging system removed all the negation signs from my last comment. Each case of (FmvGm) should have read (Fmv¬Gm).

So it should be

No. Removing Gm from H2 leaves (Fmv¬Gm)&Ax(Fx->Gx) (Removing M from N&M leaves N)

(Fmv¬Gm)&Ax(Fx->Gx) does not entail Gm (Whereas N does entail M)

12. I've now found out (by it repeating when I tried to put in the following comment) what the problem is. It appears that if you write in Word and then paste it into the comment box the idiotic system removes all negations.

13. You were perfectly clear, and so was my reply (modulo this idiotic blogging system removing the negation sign) that removing Gm from H2 leaves (Fmv¬Gm)&Ax(Fx->Gx). That is because, as was clear from when I first mentioned it, all I meant by removal was the removal of a conjunct, and H2 is Gm&(Fmv¬Gm)&Ax(Fx->Gx). But you attempted to introduce some other sense of removal and some undefined sense of sentence transformation under which transforming H2 into H1 (H1 is Fm&Ax(Fx->Gx)) is a removal of Gm. That is just equivocating on 'removal' rather than addressing the issue that I raised. That issue is now effectively summarised in the preceding comment with negations corrected. To spell it out: The disanalogy is made obvious by considering what happens when you remove the explanans from H2 and the explanans from N&M. In the first case what remains does not entail the explanans whilst what remains in the second does. That is why in your example N&M continues to appear to explain whereas H2 doesn’t. In your example each conjunct independently entails the explanans M , and so there is more going on if you think N&M explains M than an event explaining itself. The more going on is that the remainder, N, entails M. Whereas in H2, since the remainder, (FmvGm)&Ax(Fx->Gx), does not entail Gm, the entailment of Gm by H2 depends on the presence of Gm in H2 and so all attempts to take H2 as explanandum of Gm amount to taking an event as explaining itself.

14. Hi again, Nicholas.

Clearly, I have never "attempted to introduce" any "equivocation". I have just posed some questions (three on the whole, if I'm not mistaken) that I think were useful. But I will do both of us the favor to leave aside this and other pending issues about your manners, and try to go ahead with the topic.

Here is what we end up with, as far as I can see. We consider H, D and H* such that H (deductively) explains D and H and H* are logically equivalent. Allegedly, if
(i) H* is a conjunction of which D is a conjunct, and
(ii) the statement obtained by deleting "D &" from H* does not (deductively) explain D,
then H* does not (deductively) explain D.

Based on your latest comment, I take it that this captures your position. This is because it lets Siebel's example in while it leaves out my own, and does that for the reasons that you advocate.

So, for example, let the following be given:
H = plants emit oxygen during the day;
H* = (plants that are in my flat in Munich emit oxygen during the day) & (plants that are not in my flat in Munich emit oxygen during the day);
D = plants that are in my flat in Munich emit oxygen during the day.
And the above principle would now tell us that H* does not (deductively) explain D no matter that H does.

As it happens, my intuition remains entirely unmoved by all the foregoing (more or less as by Siebel's original barometer example, as clever as it is), meaning that I fail to grasp its "intuitive force" and "cogency" (your terms). Alas, this also prevents me from appreciating the ensuing "devastating effects" (Siebel, p. 266). But maybe I just need some time to think it through.

I wish you a very nice week.