How to Defeat Higher-Order Regress Arguments for Skepticism
In arguing for radical skepticism about arithmetic knowledge, David Hume uses what I'll call a higher-order regress argument. I was reminded of this style of argument when I read Francois Kammerer's similarly structured (and similarly radical) argument for skepticism about the existence of conscious experiences, forthcoming in Philosophical Studies. In my view, Hume's and Kammerer's arguments fail for similar reasons.
Hume begins by arguing that you should have at least a tiny bit of doubt even about simple addition:
In accompts of any length or importance, Merchants seldom trust to the infallible certainty of numbers for their security.... Now as none will maintain, that our assurance in a long numeration exceeds probability, I may safely affirm, that there scarce is any proposition concerning numbers, of which we can have a fuller security. For 'tis easily possible, by gradually diminishing the numbers, to reduce the longest series of addition to the most simple question, which can be form'd, to an addition of two single numbers.... Besides, if any single addition were certain, every one wou'd be so, and consequently the whole or total sum (Treatise of Human Nature 1740/1978, I.IV.i, p. 181)
In other words, since you can be mistaken in adding long lists of numbers, even when each step is the simple addition of two single-digit numbers, it follows that you can be mistaken in the simple addition of two single-digit numbers. Therefore, you should conclude that you know only with "probability", not with absolute certainty, that, say, 7 + 5 = 12.
I'm not a fan of absolute 100% flat utter certainty about anything, so I'm happy to concede this to Hume. (However, I can imagine someone -- Descartes, maybe -- objecting that contemplating 7 + 5 = 12 patiently outside of the context of a long row of numbers might give you a clear and distinct idea of its truth that we don't normally consistently maintain when adding long rows of numbers.)
So far, what Hume has said is consistent with a justifiable 99.99999999999% degree of confidence in the truth of 7 + 5 = 12, which isn't yet radical skepticism. Radical skepticism comes only via a regress argument.
Here's the first step of the regress:
In every judgment, which we can form concerning probability, as well as concerning knowledge, we ought always to correct that first judgment, deriv'd from the nature of the object, by another judgment, deriv'd from the nature of the understanding. 'Tis certain a man of solid sense and long experience... must be conscious of many errors in the past, and must still dread the like for the future. Here then arises a new species of probability to correct and regulate the first, and fix its just standard and proportion. As demonstration is subject to the controul of probability, so is probability liable to a new correction by a reflex act of the mind, wherein the nature of our understanding, and our reasoning from the first probability become our objects.
Having thus found in every probability, beside the original uncertainty inherent in the subject, a new uncertainty deriv'd from the weakness of that faculty, which judges, and having adjusted these two together, we are oblig'd by our reason to add a new doubt deriv'd from the possibility of error in the estimation we make of the truth and fidelity of our faculties (p. 181-182).
In other words, whatever high probability we assign to 7 + 5 = 12, we should feel some doubt about that probability assessment. That doubt, coupled with our original doubt, produces more doubt, thus justifying a somewhat lower -- but still possibly extremely high! -- probability assessment. Maybe 99.9999999999% instead of 99.99999999999%.
But now we're down the path toward an infinite regress:
But this decision, tho' it shou'd be favourable to our preceeding judgment, being founded only on probability, must weaken still further our first evidence, and must itself be weaken'd by a fourth doubt of the same kind, and so on in infinitum; till at last there remain nothing of the original probability, however great we may suppose it to have been, and however small the diminution by every new uncertainty. No finite object can subsist under a decrease repeated in infinitum; and even the vastest quantity, which can enter into human imagination, must in this manner be reduc'd to nothing (p. 182).
We should doubt, Hume says, our doubt about our doubts, adding still more doubt. And we should then doubt our doubt about our doubt about our doubt, and so on infinitely, until nothing remains but doubt. With each higher-order doubt, we should decrease our confidence that 7 + 5 = 12, until at the end we recognize that the only rational thing to do is shrug our shoulders and admit we are utterly uncertain about the sum of 7 and 5.
If this seems absurd... well, probably it is. I'm sympathetic with skeptical arguments generally, but this seems to be one of the weaker ones, and there's a reason it's not the most famous part of the Treatise.
There are at least three moves available to the anti-skeptic.
First, one can dig in against the regress. Maybe the best place to do so is the third step. One can say that it's reasonable to have a tiny initial doubt, and then it's reasonable to add a bit more doubt on grounds that it's doubtful how much doubt one should have, but maybe third-order doubt is unwarranted unless there's some positive reason for it. Unless something about you or something about the situation seems to demand third-order doubt, maybe it's reasonable to just stick with your assessment.
That kind of move is common in externalist approaches to justification, according to which people can sometimes reasonably believe things if the situation is right and their faculties are working well, even if they can't provide full, explicit justifications for those beliefs.
But this move isn't really in the spirit of Hume, and it's liable to abuse by anti-skeptics, so let's set it aside.
Second, one can follow the infinite regress to a convergent limit. The mathematical structure of this move should be familiar from pre-calculus. It's more readily seen with simpler numbers. Suppose that I'm highly confident of something. My first impulse is to assign 100% credence. But then I add a 5% doubt to it, reducing my credence to 95%. But then I have doubts about my doubt, and this second-order doubt leads me to reduce my credence another 2.5%, to 92.5%. I then have a third-order doubt, reducing my credence by 1.25% to 91.25%. And so on. As long as each higher-order doubt reduces the credence by half as much as the previous lower-order doubt, we will have a convergent sum of doubt. In this case, the limit as we approach infinitely many layers of doubt is 10%, so my rational credence need never fall below 90%.
This response concedes a lot to Hume -- that it's reasonable to regress infinitely upward with doubt, and that each step upward should reduce our confidence by some finite amount -- and yet it avoids the radically skeptical conclusion.
Interestingly, Hume himself arguably could not have availed himself of this move, given his skepticism about the infinitesimal (in I.II.i-ii). We can have no adequate conception of the infinitesimal, Hume says, and space and time cannot be infinitely divided. Therefore, when Hume concludes the quoted passage above by saying "No finite object can subsist under a decrease repeated in infinitum; and even the vastest quantity, which can enter into human imagination, must in this manner be reduc'd to nothing", he is arguably relying on his earlier skepticism about infinite division. For that reason, Hume might be unable to accept the convergent limit solution to his puzzle -- though we ourselves, rightly more tolerant of the infinitesimal, shouldn't be so reluctant.
Third, higher-order doubts can take the form of reversing lower-order doubts. Your third-order thought might be that your second-order doubt was too uncertain, and thus on reflection your confidence might rise again. If my first inclination is 100% credence, and my second thought knocks it down to 95%, my next thought might be that 95% is too low rather than too high. Maybe I kick it back up to 97.5%. My fourth thought might then involve tweaking it up or down from there. Thus, even without accepting convergence toward a limit, we might reasonably suspect that ever-higher orders of reflection will always yield a degree of confidence that bounces around within a manageable range, say 90% to 99%. And even if this is only a surmise rather than something I know for certain, it's a surmise that could be either too high or too low, yielding no reason to conclude that infinite reflection would tend toward low degrees of confidence.
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Well, that was longer than intended on Hume! But I think I can home in quickly on the core idea from Kammerer that precipitated this line of reflection.
Kammerer is a "strong illusionist". He thinks that conscious experiences don't exist. If this sounds like such a radical claim as to be almost unbelievable, then I think you understand why it's worth calling a radically skeptical position.[1]
David Chalmers offers a "Moorean" reply to this claim (similarly, Bryan Frances): It's just obvious that conscious experience exists. It's more obvious that conscious experience exists than any philosophical or scientific argument to the contrary could ever be, so we can reject strong illusionism out of hand, without bothering ourselves about the details of the illusionist arguments. We know in advance that whatever the details are, the argument shouldn't win us over.
Kammerer's reply is to ask whether it's obvious that it's obvious.[2] Sometimes, of course, we think something is obvious, but we're wrong. Some things we think are obvious are not only non-obvious but actually false. Furthermore, the illusionist suspects we can construct a good explanation of why false claims about consciousness might seem obvious despite their falsity. So, according to Kammerer, we shouldn't accept the Moorean reply unless we think it's obvious that it's obvious.
Kammerer acknowledges that the anti-illusionist might reasonably hold that it is obvious that it's obvious that conscious experience exists. But now the argument repeats: The illusionist might anticipate an explanation of why, even if conscious experience doesn't exist, it seems obvious that it's obvious that conscious experience exists. So it looks like the anti-illusionist needs to go third order, holding that it's obvious that it's obvious that it's obvious. The issue repeats again at the fourth level, and so on, up into a regress. At some point high enough up, it will either no longer be obvious that it's obvious that it's [repeat X times] obvious; or if it's never non-obvious at any finite order of inquiry, there will still always be a higher level at which the question can be raised, so that a demand for obviousness all the way up will never be satisfied.
Despite some important differences from Hume's argument -- especially the emphasis on obviousness rather than probability -- versions of the same three types of reply are available.
Dig in against the regress. The anti-illusionist can hold that it's enough that the claim is obvious; or that it's obvious that it's obvious; or that it's obvious that it's obvious that it's obvious -- for some finite order of obviousness. If the claim that conscious experience exists has enough orders of obviousness, and is furthermore also true, and perhaps has some other virtues, perhaps one can be fully justified in believing it even without infinite orders of obviousness all the way up.
Follow the regress to a convergent limit. Obviousness appears to come in degrees. Some things are obvious. Others are extremely obvious. Still others are utterly, jaw-droppingly, head-smackingly, fall-to-your-knees obvious. Maybe, before we engage in higher-order reflection, we reasonably think that the existence of conscious experience is in the last, jaw-dropping category, which we can call obviousness level 1. And maybe, also, it's reasonable, following Kammerer and Hume, to insist on some higher-order reflection: How obvious is it that it's obvious? Well, maybe it's extremely obvious but not utterly, level 1 obvious, and maybe that's enough to reduce our total epistemic assessment to overall obviousness level .95. Reflecting again, we might add still a bit more doubt, reducing the obviousness level to .925, and so on, converging toward obviousness level .9. And obviousness level .9 might be good enough for the Moorean argument. Obviously (?), these are fake numbers, but the idea should be clear enough. The Moorean argument doesn't require that the existence of conscious experience be utterly, jaw-droppingly, head-smackingly, fall-to-your-knees, level 1 obvious. Maybe the existence of consciousness is that obvious. But all the Moorean argument requires is that the existence of consciousness be obvious enough that we reasonably judge in advance that no scientific or philosophical argument against it should justifiably win us over.
Reverse lower-order doubts with some of the higher-order doubts. Overall obviousness might sometimes increase as one proceeds upward to higher orders of reflection. For example, maybe after thinking about whether it's obvious that it's obvious that [eight times] it's obvious, our summary assessment of the total obviousness of the proposition should be higher than our summary assessment after thinking about whether it's obvious that it's obvious that [seven times] it's obvious. There's no guarantee that with each higher level of consideration the total amount of doubt should increase. We might find as we go up that the total amount of obviousness fluctuates around some very high degree of obviousness. We might then reasonably surmise that further higher levels will stay within that range, which might be high enough for the Moorean argument to succeed.
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[1] Actually, I think there's some ambiguity about what strong illusionism amounts to, since what Kammerer denies the existence of is "phenomenal consciousness", and it's unclear whether this really is the radical thesis that it is sometimes held to be or whether it's instead really just the rejection of a philosopher's dubious notion. For present purposes, I'm interpreting Kammerer as holding the radical view. See my discussions here and here.
[2] Kammerer uses "uniquely obvious" here, and "super-Moorean", asking whether it's uniquely obvious that it's uniquely obvious. But I don't think uniqueness is essential to the argument. For example, that I exist might also be obvious with the required strength.