99% of A’s are B’s
Therefore
the next A is also a B

can be rewritten, with no loss of any formal property, as:

99% of A’s are B’s
therefore
probably the next A is also a B,

you can also think that an argument of the form:

P&Q
therefore
P

can be rewritten as:

P&Q
therefore
necessarily P

So, the probably/necessarily attribute indicates what kind of vero-functional support there is between premises and conclusion – and that could mean they imply two different kinds of inferential patterns.

As for the other case, I was trying to think of a simple situation where all the available internal tests were corrupted in the same way. I figured that the simplest way to do so was to give you good evidence that an outside intelligence was directly and deliberately misleading you in all of your intellectual endeavors. I was assuming that in this case your belief was correct; the demon really is messing with your mind, and the one thing he lets you know is that he is doing so. Anti-invincibility seems to require that we can never be put into an epistemically doomed situation and still become aware of this fact. I can’t see why we should think that isn’t possible, though.

It’s worth pointing out in this case, though, that the reason the defeater is invincible isn’t merely because of its generality; it’s because it is a sufficiently broad and specific type of defeater to cut off all avenues of approach for fixing the problem.

1. Characteristically inductive arguments take us from samples to populations (as GRW suggests). Often, we add two features to the conclusions of such inferences — some vagueness and some probability. For example, in the school-sampling case, we might look at a few children and then say that with probability p, the average height of the children is between h1 and h2. But those two features need not be added in order to have the *form* of an induction. You could simply conclude with a point estimate: “The average height of the children is h.” (The case is parallel to singular statistical syllogisms, like: 90% of the balls in the bucket are red; this ball was drawn from the bucket; therefore, this ball is red.)

2. Some people have claimed that there are inductive arguments with conclusions that cannot be assigned any probability. Peirce seems to have thought this about some inductive arguments with respect to infinite populations and with respect to extrapolations to the future. John Norton gives a general argument against probability as the one true logic of induction. (See this paper (pdf), for example.) If their arguments are right, then induction is not likely to be replaceable with deduction plus a probability function.

That’s very helpful. I now see the problem with the drug/mathematics case as it was described, but I don’t think it matters for the main argument of the post. What you (and perhaps James) point out is that one could rely on the coherence of the different mathematical checks to determine that it’s unlikely that I’ve been given the drug. I see the point. But relying on the coherence of mathematical checks doesn’t violate Independence, because, by engaging in coherence-based reasoning, I’m relying, in part, on a type of non-mathematical reasoning. So this case doesn’t seem to motivate Independence very well. But, since the point of the post was to object to Independence, I’m not sure why the main argument would inherit the defects of my motivation for Independence. (I’m not sure I understand the final three sentences of your first paragraph. If you still think the overall argument has a problem, could you clarify?)

As far as Anti-Invincibility goes, I don’t share the intuitions in your cases. If I acquire evidence E1 that an evil demon is going to cause me to make mistakes quite often, and I acquire evidence that E1 is misleading, then wouldn’t E1 be at least partially defeated? As long it is possible for some subject to have E1 and acquire evidence that E1 is misleading, then E1 doesn’t seem invincible to me in the sense I intended. A similar point applies to the “screwing with your mind” case. Now, the word “know” features prominently in your two cases (the demon let you know…, and you know that someone is screwing with you). Was that fact that you know about the demon activity–rather than just having evidence about it–supposed to do some important work that I’m missing?

The non-monotonic reasoning literature is a rich one, dating from the 1970s and christened in an AI journal special issue from 1980. But, for a (non-technical) reference which anticipates the idea of a logic for non-demonstrative inference, have a look at (Fisher 1922, 1936), referenced and discussed (briefly) in this review of the lottery paradox.

Best, GRW

Thanks for those comments. The overall point about non-monotonicity is helpful and (to me) convincing. Just the sort of thing I was hoping for!

Mark

However, you don’t seem to need a universal version of Independence to create an inconsistency. If there are specific types of undercutting defeaters that can’t be overridden by evidence internal to the method, then there could still be examples of undercutting defeaters that violate Anti-invincibility. However, in a case where you add in enough features to get around all these internal measures of confirmation, I don’t see why we should accept Anti-Invincibility. Suppose an evil demon let you know that he existed and that he was going to cause you to have erroneous beliefs quite often. In that situation, there doesn’t seem to be anything you could do to fix this problem. If you know that someone is actually directly screwing with your mind, and you can’t do anything to stop them, there’s probably nothing you can do to overcome this sort of defeating evidence. So, if you’re trying to show that you can’t create an Invincible defeater just by creating a general defeater, that seems correct. But I don’t see why we should think that there can’t be invincible defeaters.

In textbooks one sees the proviso put in positive terms, say, that the sample is random. But, what one wants is assurances that the sample is representative. But if you had the assurance that the sample was representative, had the grounds to put that in positive terms, and thus to put the argument in deductive form, you would not be in the position of needing to draw the inference: for you would effectively be in a position of describing (or reasoning about) the population. All statistical reasoning would be descriptive, and science would be much easier than it in fact is.

As for a form of this kind of reasoning, you can think of this as a Reiter default rule: the antecedent is the descriptive bit about the sample, the conclusion applies the description to the conclusion, and the ‘justifications’ are conditions which, so long as none are provable given what you know, none, in other words, tip you off to the sample being unrepresentative, allows you to draw the conclusion.