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In response to van Fraassen's "bad lot" objection, I have seen in multiple papers (and in the response of Lipton, from what I remember) refer to the idea of the use of a 'catch-all' hypothesis which states the negation of all other hypotheses.

e.g. we can have some set of hypotheses:

H1, H2, H3, H4

where H4 is simply the statement "H1, H2 and H3 are false" or something along those lines, such that the set of hypotheses fills out the entire 'theoretical space'. Or, as a Bayesian might put it, the total probability of considered hypotheses sums to 1. Because we can rank this catch-all hypothesis in with all of the other considered hypotheses, we can have confidence that the highest ranked hypothesis is most likely true of all hypotheses making up the theoretical space.

I understand what the catch-all hypothesis does and I understand that the theoretical space is 'filled'. I also understand how evidence would effect this catch-all hypothesis (confirmation of any other hypothesis would mean disconfirmation for the catch-all and vice versa).

What I'm having trouble with is the idea that this catch-all statement may very well contain a hypothesis which would be well-confirmed if properly conceived even though we consider the catch-all disconfirmed when alternatives are confirmed. I'm not sure how to get this idea across effectively. Let's take the most simple example possible, where we have one hypothesis and then the catch-all which is simply its negation.

Example

I have some fact that I wish to explain: My couch pillow is torn to pieces on the floor.

As of yet, I have only one hypothesis which I have come up with to explain the data and I know fully well that there may be other hypotheses able to explain it.

My Hypothesis (H1): The dog ripped the pillow to shreds while I was out at work.

The catch-all negation (Hc): The dog did not rip the pillow to shreds whilst I was out at work.

After further investigation, I find my dog in the other room looking extremely guilty. He usually looks guilty when he's done something wrong. This confirms hypothesis H1 and disconfirms Hc. Therefore, I have reason to affirm that H1.

So as it stands P(H1) > P(Hc)

And perhaps I end my investigation there, deciding that I have solved the case.

But perhaps someone else investigates also and gives me some new hypothesis (H2).

H2: The cat ripped up the pillow while you were at work and the dog misbehaved in some other way which you did not notice.

Now, obviously, there are many more "parameters" involved in H2 and rightly we will judge its complexity to be a detrimental flaw. But let's ignore complexity for a minute. My observation that my dog looked guilty should have confirmed H2 (however ad-hoc H2 is) but this hypothesis was "contained" within Hc. So when we disconfirmed Hc by observing the guilty dog, were we wrongly disconfirming H2 since H2 was contained within Hc?

This seems very strange to me. Is my concept of alternative hypotheses being "contained" within the negation somehow flawed or is my example not a coherent account of how IBE should be performed using a catch-all hypothesis? If any answer could be given in terms of this simple example or one similar, that would be extremely helpful.

Thanks.

P.S.

A better way to phrase all of that would have been to say that H2 entails Hc so any confirmation of H2 should be a confirmation of Hc. But when we observed a fact that did confirm H2, we considered Hc disconfirmed.

Cheers.

  • Why does a guilty dog confirm H2? P(H2 | Guilty Dog) = P(H2 | No Guilty Dog). How the dog looks does not affect the likelihood that the cat did it under H2. – Nuclear Wang Oct 22 '18 at 17:59
  • Well, I suppose we're assuming my dog is most likely to look guilty if he's done something bad. H2 states that he has done something bad (we just don't know what) so it predicts that he will look guilty. Therefore the fact that he does look guilty confirms H2 as well as H1. – Joe Lee-Doktor Oct 22 '18 at 18:05
  • @NuclearWang The whole of H2 is "The cat ripped up the pillow and the dog misbehaved in some other way". Of course, this hypothesis is highly ad-hoc and not very parsimonious but that doesn't seem to effect the problem at hand. This was just the first example that came to mind. – Joe Lee-Doktor Oct 22 '18 at 18:07
  • When you add H2 about the cat doesn't that mean you changed Hc as well to be neither H1 nor H2? As I see it H1 is "The dog did it." H2 is "The cat did it." Hc is now neither H1 nor H2, that is, "Something else than the dog or cat did it." This way H2 is not contained in Hc. – Frank Hubeny Oct 26 '18 at 13:38
  • @FrankHubeny Yeah, Hc would change. But it still seems as though ~H1 is disconfirmed incorrectly. If some evidence confirms both H1 and H2 but H2 implies ~H1, it seems that ~H1 is confirmed at the same time as H1? – Joe Lee-Doktor Oct 26 '18 at 21:10
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You change the theoretical space. The two Hcs are different, the two hypothesis spaces are different. You begin with: dog vs not dog, then go dog vs cat vs not dog minus cat, a new set of three, only 'dog' remaining the same.

Consider for the purpose of generalising, the Buddhist device of the catuskoti, which sets aside the principle of the excluded middle, and of non-contradiction, and makes truth into a relatiin instead of a function. In this way we can see the logical possibilities as directions to face, 'four corners', and you stand in two different places in the two scenarios.

You missed, both cat and dog ripped up the pillow, also. And might wish to consider degrees of certainty, which you could put into some Bayesian statistics for expressing that. Can you see claw or bitemarks? Saliva? DNA test of that? CCTV from security system? I would think of that as how fine-grained and clear the view is, with more time and looking you can get more clear, if additional details are there.

So, there is where you stand in the logical space, and how clearly you define your divisions and how to distinguish them.

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I worry a bit about the term "confirm," since it sounds so final. Your evidence isn't decisive, it's only adjusting the probabilities. If you think of your catch-all as being composed of an limitless set of statements linked together, it's entirely possible for a piece of evidence to increase the likelihood of a subset of those, while at the same time decreasing the likelihood of the catch-all as a whole.

In light of your evidence, it is more likely that the dog did the deed (than without the evidence). It is also more likely (in light of the evidence) that the dog did something different that it feels guilty for. But it is not more likely that the dog did not do the deed.

That's because there are other scenarios also contained in the catch-all, such as "the dog is not feeling guilty about anything," whose likelihood have also diminished, and their impact outweighs the (comparatively) unlikely alternate scenario where the dog just happens to feel guilty about something completely unrelated.

  • Given this, is this response successful? Could it not be then that the catch-all contains some better hypothesis? It seems as though, given this, we ought not believe our current best hypothesis since we fear that there may be better ones contained within the catch-all. – Joe Lee-Doktor Oct 29 '18 at 16:24
  • I think you're being led astray by the term "confirm." You're dealing with probabilities here. The catch-all MIGHT contain some "better" hypotheses, but those hypotheses are not more likely than your initial one, because they have more assumptions. I'm not much of a Bayes expert, but I do know this is the very kind of situation his theorem was intended to address. – Chris Sunami Oct 29 '18 at 16:42
  • There's no problem with the word confirm. "Confirmation" refers to the increase of the probability of a given hypothesis and "disconfirmation" refers to the opposite. That's how the terminology is used in philosophy of science and in discussions about "theories of confirmation". – Joe Lee-Doktor Nov 1 '18 at 21:42
  • @JoeLee-Doktor I get that. But there's no paradox if you're not viewing confirmation as an absolute. That's why I say the terminology is leading you astray. – Chris Sunami Nov 2 '18 at 1:12
  • So hang on, if the catch-all contains some better hypothesis, how could it possibly have a lower probability than the initial one which is confirmed by the evidence? – Joe Lee-Doktor Nov 3 '18 at 21:59

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