Question about the "catch-all" hypothesis as a response to the "bad lot" objection

Question

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.

Answer 1

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.

Answer 2

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.