Sensitivity and Evidence Quality

Here’s a puzzle for the E=K (i.e. all and only knowledge is evidence) theory.

Jack is inspecting a new kind of balance made by Acme Corporation. He thoroughly inspects the first 10 (out of a batch of 1,000,000) that come off the assembly line. And each of them passes the inspection with flying colours. Each of them is more accurate than any balance Jack had tested before that day. So Acme is making good balances. He knows, by observation, that the first 10 balances are reliable. He also knows, by induction, that the next balance will be reliable. It’s not obvious that he knows the next one will be phenomenal, like the ones he has tested, but he knows it will be good enough for its intended usage. But he doesn’t know they all will be that good. Surprisingly, it will turn out that every balance made in this assembly line will be reliable. But you’d expect, given what we know about assembly lines, for there to be a badly made machine turn up somewhere along the way.

So Jack knows the first 11 are reliable, and doesn’t know the first 1,000,000 are reliable. Let n be the largest number such that Jack knows the first n are reliable. (I’m assuming such an n exists; those who want to hold on to E=K by giving up the least number theorem are free to ignore everything that follows.) For any x, let R(x) be the proposition that the first x are reliable. So Jack knows R(n). Hence by E=K R(x) is part of his evidence. But he doesn’t know R(x+1). This is extremely odd. After all, R(x) is excellent evidence for R(x+1), assuming it is part of his evidence. And R(x+1) is true. Indeed, by many measures it is safely true. So why doesn’t Jack know it?

It seems to me there is a mystery here that, given E=K, we can’t explain. If we have a more restrictive theory of evidence, then it is easy to explain what’s going on. If, for instance, evidence is perceptual knowledge, then Jack’s evidence is simply R(10). And it might well be true, given the correct theory of what hypotheses are supported by what evidence, that R(10) supports R(84) but not R(85). That explanation isn’t available to the E=K theorist. And we might well wonder what explanation could be available.

I have one idea that saves the letter of E=K, though at some cost I think to the spirit of it. Let’s say that evidence can be of better or worse quality. If you don’t know p, then p is of no evidential use to you. But even if you do know it, how much evidential use is might depend on how you know it. For instance, if you infallibly know p, then p is extremely useful evidence. More relevantly for today’s purposes, if you have sensitive knowledge that p, then p is more useful than if you have insensitive knowledge that p.

Let’s go through how this plays out in Jack’s case. Although he knows R(11), this knowledge is insensitive. If R(11) were false, he would still believe it. Had the production system malfunctioned when making the 11th balance, for instance, then the 11th machine would have been unreliable, but Jack would have still believed it. The only sensitive evidence he has is R(11). By the time he gets to R(n), his knowledge is extremely insensitive. There are all sorts of ways that R(n) could have been false, in many fairly near worlds, and yet he would still have believed it.

So here’s a hypothesis. The more insensitive your evidence is, the less inductive knowledge it grounds. If Jack had sensitive knowledge that R(n), he would be in a position to infer, and thereby know R(n+1). The reason he can’t know R(n+1) is not that he doesn’t have enough evidence, but rather that the evidence he has is not of a high enough quality. That’s an explanation for why Jack can’t infer R(n+1) that neither leads to inductive scepticism, nor violates the letter of E=K. I’m not sure that E=K was meant to go along with the view that how you know something is evidentially relevant, not just whether you know it, so I don’t think this keeps the spirit of E=K. But perhaps the letter of E=K is more defensible than the spirit of it.

10 Replies to “Sensitivity and Evidence Quality”

  1. This is cool.

    One of the commitments you don’t explicitly mention is that there can be inductive knowledge. So an E=K theorist who thinks knowledge must be based on entailing evidence can get out of the puzzle.

    I think there’s something pretty plausible about insisting that you can’t know the eleventh one to be reliable merely on the basis of the first ten. But then, I think there’s something pretty plausible about insisting that sensitivity is necessary for knowledge…

  2. I agree that this argument isn’t particularly telling against infallibilists! It’s really meant as an argument against Williamson – you have to have quite a lot of the views he has in order to be vulnerable to it. In particular, it relies on:

    1. E=K
    2. No sensitivity requirement
    3. Epistemicism, or some broadly classical theory of vagueness (so n can be a referring expression)
    4. All that matters evidentially supervenes on what evidence you have.

    There is a quote where he seems to strongly commit to 4, but I can’t find it in my files right now. If I find it, I’ll add it to the post, because I think 4 is the mistake here. But it’s really an argument from 1, 2 and 3 to not-4, and if you reject 2, it won’t look very plausible.

  3. Sensitivity might sort of be a side issue. I think what you’re really relying on here, to which I was trying to draw attention, is that that we have knowledge based on non-entailing evidence. One could deny this without embracing a sensitivity requirement, I think. (Depending on exactly how that latter is spelled out, maybe.) I think that’s a pretty substantive assumption. And once we’re thinking about views like E=K, I’m not at all sure there’s much reason to grant it.

  4. Why should we think that we can’t have knowledge based on non-entailing evidence? Given E=K, our evidence will entail what we know. But not all knowledge is self-basing. So I know that it will snow in Ithaca sometime in the next 12 months. But my basis for that is not the fact that it will snow in Ithaca sometime in the next 12 months, but rather, Ithaca’s meteorological history, and general facts about the cyclical nature of the weather.

  5. I think it’s clear that you can have knowledge based on non-entailing evidence, but one worry about E=K that is in the neighborhood of the worry you raise here (but not quite as interesting, unfortunately) arises because of cases like this:

    I see a fox. I see the fox eat blueberries. I’ve read that of the hundreds of observed peckish foxes in the area, only females ate blueberries. It seems I can have inductive knowledge that:

    (1) The fox in my yard is a female fox.

    While I’d like to say that (~1) is consistent with my evidence prior to believing (1) and will remain consistent with my evidence barring any additional observation, what should I say about:

    (2) There is a vixen in my yard.

    Without independent justification for believing (2), I’d like to say that (~2) is consistent with my evidence but cannot say that if (1) is included in my evidence as a consequence of coming to have inductive knowledge that (1) is true. And, since I know (1) iff (2), it seems circular reasoning leads me to conclude that (~1) is indeed inconsistent with my evidence. But, as I said, that’s something we should avoid saying if I don’t acquire new evidence that wasn’t the basis for coming to believe (1) initially.

    Easy fix, of course, is to distinguish processes by which we acquire evidence and processes by which we extend our knowledge via inference. (This is inspired, in part, by Dylan Dodd’s article on Williamson in Phil Quarterly. At the time, I thought the objection rested on the assumption that we cannot have knowledge based on non-entailing evidence if E=K is true but there’s a more complicated way of generating basically the same worry I think led Dodd to question W’s view.)

  6. Clayton, can you say why you need (2) to make your point? Wouldn’t the difficulty arise just with the premise that (~1) “will remain consistent with my evidence barring any additional observation”? For, if you know (1), then (1) is part of your evidence, and so (~1) is not consistent with your evidence.

  7. Hey Jeremy,

    I suppose (2) isn’t necessary, but there’s the response which says (basically) that someone can defend E=K without saying that the evidential basis for things that you know will include those very things, but I take it that there’s the pretty firm intuition that when you know (1) via inductive inference, (~2) will be consistent with your evidence and that given the analytical equivalence between (1) and (2), you wouldn’t want to say that your evidence supports (1) and (2) to different degrees. So, I guess (2) is needed for dramatic emphasis!

    Why is your avatar so much better than mine?

  8. Perhaps one way to accommodate (4) in this framework is by having some evidence that itself concerns the quality of one’s evidence. That is by and large how we actually do get a great deal of the sensitivity that we really have into our belief-forming and -maintaining practices: by learning when & how they can go wrong, and what steps we can do to find out when this is so, & how to fix things when it does.

    In the hypothetical, Jack would likely know full well that if he tries to bootstrap out those first 10 observations for larger and larger values of n, he’d be exposing himself to an ever-greater risk of error. That bit of knowledge is itself part of his evidence set, then. And (4) could thus still hold: it’s what’s in the evidence set that does the work that sensitivity calls for here, to the extent that such work is called for (which is much weaker than a full-blown requirement on knowledge).

    One could try to change the case so that Jack didn’t have any such knowledge, but then I think it is much less likely that Jack can extend his knowledge very far from the initial 10 observations. An agent so clueless about the do’s and don’t‘s of inductive inferences is accordingly not one who we ought to credit with much inferential success.

    Fwiw, I think these issues are connected to some of the ones I was mooting about over here:
    though that is more concerned with abductive than inductive inferences.

  9. Jonathan,

    I think there’s a tricky issue around here concerning how basic inductive reasoning is. It’s very plausible that for particularly basic inferences, we don’t have to know that they are sound in order to soundly make them. The alternative is to fall into the kind of infinite regress that Lewis Carroll satirised.

    Now it’s a very interesting question whether enumerative induction is basic in just that sense, or whether what’s basic is something like “Use rule R when you know rule R is good to use”. If enumerative induction is really one of the basic forms, then I don’t think Jack needs such second order knowledge.

    Also, I think it’s important to be careful what counts as ‘risk of error’ here. By hypothesis, Jack can know R(n). The issue is the inference from R(n) to R(n+1). Or, since we’re not doubting deductive closure, the inference from R(n) to the claim that n+1’th is trustworthy. Intuitively, that seems to get less risky as n gets larger, and Jack could know that. And if that’s right Jack could know that in general enumerative induction on a larger sample is generally less risky than induction on a smaller sample.

    To be sure, Jack does know not just that R(10) is true, but that he’s observed that R(10) is true. Perhaps that second-order knowledge could be what does all the work. That might have interesting consequences for the theory of induction more broadly.

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