Is ~ K ~ KP a luminous condition?

Abstract

One of the most intriguing claims in Sven Rosenkranz’s Justification as Ignorance is that Timothy Williamson’s celebrated anti-luminosity argument can be resisted when it comes to the condition ~ K ~ KP—the condition that one is in no position to know that one is in no position to know P (for some proposition P). In this paper, I critically assess this claim.

1 Introduction

The idea at the heart of Sven Rosenkranz’s Justification as Ignorance is that one has justification for a proposition P iff one is in no position to know that one is in no position to know P. If we let J abbreviate ‘One has justification for…’ and K abbreviate ‘One is in a position to know…’, we can put the idea like this: JP < – >  ~ K ~ KP. Rosenkranz claims a number of benefits for this view—and one of the most striking is that it allows us to defeat Timothy Williamson’s well-known anti-luminosity argument and maintain that justification is a luminous condition. A condition is said to be luminous iff, whenever it obtains, one is in a position to know that it does. Justification will be a luminous condition iff we have the theorem JP – > KJP which, for Rosenkranz, corresponds to the theorem ~ K ~ KP – > K ~ K ~ KP.

Williamson’s argument—outlined in Chap. 4 of Knowledge and Its Limits (2000)—purports to show that there are no non-trivial luminous conditions. According to Williamson, for any condition that obtains in some cases and not others, there will be a possible case in which it obtains but one is in no position to know that it does. A number of conditions have been supposed, by philosophers, to be luminous—phenomenal states, seemings, moral obligations, meanings and, indeed, justification, with epistemic ‘internalists’ claiming that we have special, privileged access to our own justificatory status.¹ If Williamson’s argument is successful—and many think that it is—then a lot of traditional philosophical theorising turns out to be misguided. And if Rosenkranz has genuinely hit upon a way of resisting the argument, at least when it comes to justification, then this too would represent a very significant result.

I am inclined to think that the anti-luminosity argument, as standardly formulated, really does falter when it comes to ~ K ~ KP. As I shall show, however, there is a fix available—the argument can be adapted to get around this problem. My primary aim here is to set out a new variant of the anti-luminosity argument, relying on slightly different assumptions to the original, which will work as effectively against ~ K ~ KP as any other candidate non-trivial luminous condition. This new variant of the argument may be of some independent interest and may be better able to withstand certain extant objections to the original—but I won’t explore this here.

What is it to be in a position to know a proposition? Some take this to imply that one could come to know the proposition without needing to conduct further empirical inquiry—come to know the proposition purely by reflection, introspection, a priori reasoning, etc. No detailed account of being in a position to know will be offered here, but I assume at a minimum that if one is in a position to know P, and one has done everything that one is in a position to do to determine whether P, then one knows P (Williamson, 2000, pp95, Rosenkranz, 2021, pp39). Call this PK₁. This already gives us the result that being in a position to know is factive – KP – > P.

I will table two further related assumptions, both of which are appealed to by Rosenkranz: First, if one has done everything that one is in a position to do to determine whether P, then one has also done everything that one is in a position to do to determine whether ~ P. Second, if one has done everything that one is in a position to do to determine whether one is in a position to know P, then one has also done everything that one is in a position to do to determine whether P. In other words, doing everything that one is in a position to do to determine whether KP involves doing everything that one is in a position to do to determine whether P (plus, potentially, more). The former assumption, which we can call PK₂, might be regarded as trivial. The latter, PK₃, is not a triviality but is I think defensible.² In any case, I won’t further discuss these assumptions here (see, Rosenkranz, 2021, Sect. 3.2, where these assumptions correspond, respectively, to his 2a and 2b).

2 The anti-luminosity argument

There is no single, canonical way of presenting the anti-luminosity argument, with recent presentations departing from Williamson’s original in various respects. The version I will focus on is close to the one used by Rosenkranz (Sect. 4.3), though elaborated in the manner suggested by Srinivasan (2013). Let C be a candidate non-trivial luminous condition. Imagine a sequence of cases α₁, α₂ … linking a case in which C clearly obtains to a case in which ~ C clearly obtains and which proceeds, through time, tracking minute, incremental changes to a relevant dimension, in a such a way that adjacent cases can be made arbitrarily similar to one another.³ The first assumption to be noted (call it AL₁) is that it is, in fact, possible to construct a sequence of this kind for condition C. Suppose that we do have such a sequence and suppose that, throughout the sequence, I am using the optimal available method to detect condition C, and am doing everything that I am in a position to do to determine whether it obtains. As a result, if I am in a position to know that C obtains in any given case in the sequence, then, by PK₁, I do know that C obtains in that case.

The second assumption that drives the argument (call it AL₂) is that if I believe that C obtains in a case α_x, then I could easily believe that C obtains in the adjacent cases α_x-1 and α_x+1. AL₂ is supposed to be motivated by the fact that our powers of discrimination are limited and, since adjacent cases can be made arbitrarily similar, there is some point at which I would simply have no reliable way of telling them apart. The third assumption of the argument (AL₃) is that if I know that C obtains in a case α_x, then I could not easily have falsely believed that C obtains in the next case α_x+1. AL₃ is supposed to be motivated by a kind of safety or margin-for-error requirement on knowledge—in order for one to know a proposition P, it is necessary that one could not easily have believed P falsely, by the same method, in similar cases.

We now have the tools needed to prove that C cannot be a luminous condition. By classical logic, either C or ~ C will obtain in each case in the sequence. As a result, there must be a last case in the sequence—call it α_n—in which C obtains and which is adjacent to the first case—α_n+1—in which ~ C obtains. If I don’t believe that C obtains in α_n, then I don’t know it. But if I do believe that C obtains in α_n, then, by AL₂, I could easily believe that C obtains in α_n+1 from which it follows, by AL₃, that my belief in α_n does not qualify as knowledge. Either way, α_n is a case in which C obtains, even though I fail to know that it does.

Even if I believe that C obtains right up to α_n and then cease to believe it in α_n+1—that is, even if my belief switches at exactly the point that C does—this would just be good fortune. Given the limits on my powers of discrimination, I could easily have continued to believe that C obtains in α_n+1 even if, in actual fact, I don’t. Talk about what could ‘easily happen’ can be usefully understood as existentially quantifying over possibilities that are very similar to actuality. For any case in the sequence α_x, say that a possible case β_x is a near duplicate of α_x iff α_x and β_x are (i) identical with respect to whether C obtains, with respect to the underlying dimension that drives the sequence and with respect to the fact that I’m doing everything I am in a position to do to determine whether C obtains and (ii) are very similar in all other respects. Note that, in any near duplicate of a case in the sequence, just as in the cases themselves, I will know that C obtains whenever I’m in a position to know that it does.⁴ We can now translate talk about easy possibility into talk about near duplicates: what it means to say that I could easily believe that C obtains in a case α_x is that there is a near duplicate of α_x (which could be α_x itself) in which I do believe that C obtains.

Given this translation, AL₂ could be written as follows: if I believe that C obtains in a case α_x then there is a near duplicate β_x+1 of α_x+1 and a near duplicate β_x-1 of α_x-1 such that I believe that C obtains in β_x+1 and β_x-1.⁵ And AL₃ becomes: if I know that C obtains in a case α_x, then I don’t falsely believe that C obtains in any near duplicate of α_x+1. We can now retrace the argument. If I don’t believe that C obtains in α_n, then I don’t know it. If I do believe that C obtains in α_n, then, by AL₂, there is near duplicate β_n+1 of α_n+1 in which I believe that C obtains. By the definition of a near duplicate, ~ C obtains in β_n+1 from which it follows, by AL₃, that my belief that C obtains in α_n does not qualify as knowledge. Either way, α_n is a case in which C obtains even though I fail to know that it does.⁶

Following Williamson, the anti-luminosity argument is often illustrated with the example of feeling cold which, being a phenomenal, introspectable state, might be regarded as a paradigm example of a luminous condition. Suppose I wake at 6am feeling bitterly cold, but I gradually warm up throughout the morning until, at noon, I’m feeling uncomfortably hot. Suppose that, throughout this period, I am introspecting as hard as I can, focussing my attention on the question of whether or not I’m feeling cold. The 6 h from 6am until noon can be divided up into, say, one-millisecond intervals, giving us our sequence of cases. Over the course of a single millisecond, the change in how I’m feeling will be so slight that I would be in no position to reliably detect it. As a result, if I believe that I’m feeling cold at any point throughout the morning, I could easily believe it one millisecond later. Consider then the final millisecond at which I feel cold. If I don’t believe that I’m feeling cold at this point, then I don’t know it. If I do believe that I’m feeling cold at this point, then I could easily believe it, falsely, one millisecond later, in which case my belief does not qualify as knowledge. Either way, there is a point at which I’m feeling cold but do not know it.

3 Rosenkranz on the luminosity of ~ K ~ KP

What happens if we try to run the argument for the condition ~ K ~ KP (for some proposition P)? To begin, we need to imagine a sequence that proceeds, via arbitrarily minute increments, from a case in which ~ K ~ KP clearly obtains to a case in which K ~ KP clearly obtains. Is such a sequence possible? While it is not straightforward to think up a viable example, perhaps the following might work… Suppose I leave my car in a parking lot, and head off to do some shopping. 15 min later, I have good reason to believe (P) that my car is where I parked it, and I’m in no position to know that I’m in no position to know this (~ K ~ KP). Suppose, at this point, I’m standing by the side of a street, when I see a car approaching in the distance. At first, I can’t make out any of the details of the car but, as it gradually draws closer, I notice first the colour, then the make, then the bumper stickers, then the license plate… all of which strike me as very familiar. Two minutes later, the car is driving right past me, and it is obvious to me that this is either my car or a near perfect facsimile. At this point, it’s clear that I am in a position to know that I’m in no position to know that my car is still where I parked it (K ~ KP). This two-minute period can be divided up into arbitrarily small intervals—milliseconds, nanoseconds, etc. The shorter the intervals in question, the smaller the change in my visual impression of the car during each. It’s very plausible that this narrowing will eventually lead to intervals that are witness to so slight a change that I would be in no position to reliably detect it.

In any case, Rosenkranz is willing to accept the relevant instance of AL₁ and grant that we can set up a sequence of the required kind. Suppose, then, that we have our sequence, and suppose that I am, throughout the sequence, using the optimal available method and doing everything that I am in a position to do to determine whether ~ K ~ KP. Let α_n be the last case in which ~ K ~ KP and α_n+1 be the first case in which K ~ KP. If I believe ~ K ~ KP in α_n, does that mean that I could easily believe ~ K ~ KP in α_n+1? According to Rosenkranz, it does not. In α_n+1, like every other case, I am doing everything that I am in a position to do to determine whether ~ K ~ KP. By PK₂ and PK₃, this involves doing everything that I am in a position to do to determine whether ~ KP. But, since K ~ KP holds in α_n+1, this means that I must know ~ KP in α_n+1 by PK₁. Further, it would be irrational for me to believe ~ KP and believe ~ K ~ KP at the same time and, in order for the former belief to qualify as knowledge, I could not simultaneously hold the latter belief.

Given the way the example is described, and the assumptions PK₁, PK₂ and PK₃, I could not believe ~ K ~ KP in α_n+1. And the same is clearly true of any near duplicate of α_n+1. After all, any near duplicate of α_n+1 will also be a case in which K ~ KP and in which I am doing everything that I’m in a position to do to determine whether ~ K ~ KP. In fact, by Rosenkranz’s reasoning, there is no possible case in which these conditions are met and in which I erroneously believe ~ K ~ KP. Assumption AL₂ fails, then, when it comes to the condition ~ K ~ KP—even if I believe ~ K ~ KP in α_n, I could not easily (indeed could not possibly, given the set up) believe ~ K ~ KP in α_n+1. Thus, if I believe ~ K ~ KP in α_n, AL₃ presents no obstacle to this belief qualifying as knowledge.⁷

While it’s straightforward enough to follow Rosenkranz’s reasoning in the abstract, it is somewhat unclear just where it leaves us when it comes to concrete cases. Think again of the above example in which there is a gradual, dawning realisation that an approaching car looks exactly like mine. Are we to think that, at the very nanosecond (indeed the very instant) that K ~ KP becomes true, I am perfectly attuned to this change in such a way that I can’t possibly make the mistake of believing ~ K ~ KP from this point on? Since a nanosecond is too short for me to reliably notice any change in the visual appearances, how am I supposed to manage this? When properly understood, though, Rosenkranz’s argument doesn’t show (of course) that we have some mysterious power to detect K ~ KP from the moment it becomes true. Rather, the set-up of the example ensures that there is a kind of constitutive connection between the obtaining of this condition and our beliefs about it.⁸ Provided that I’m doing everything I am in a position to do to determine whether ~ K ~ KP, my believing ~ K ~ KP would, given PK₁, PK₂ and PK₃, prevent K ~ KP from obtaining.

4 Adapting the argument

One possible way to reinstate the anti-luminosity argument would be to strengthen AL₃ as follows: if I know that C obtains in a case α_x, then there is no near duplicate of α_x+1 in which C is false and in which I invest an at-most-slightly-lower degree of confidence in the proposition that C obtains. This can be motivated by a kind of ‘confidence-safety’ requirement on knowledge—if one knows a proposition P, then there are no similar cases in which P is false and, using the same method, one invests an at-most-slightly-lower degree of confidence in P⁹ (see Williamson, 2000, p101, Berker, 2008, Sect. 4, Srinivasan, 2013, Sect. 5). This may work as a general strategy against those who would seek to undermine the anti-luminosity argument by positing constitutive connections between a given condition and our beliefs about it (Srinivasan, 2013, Sect. 5). In any case, I won’t consider this further here. The new variant of the argument that I will offer doesn’t involve any modification of AL₃ but, rather, a modification of AL₂.

As before, let C be a candidate luminous condition, and suppose we can set up a sequence linking a case in which C clearly obtains to a case in which ~ C clearly obtains and in which adjacent cases can be made arbitrarily similar. Suppose that, throughout the sequence, I am using the optimal method and doing everything that I am in a position to do to determine whether C obtains. As before, we accept an appropriate safety condition on knowledge and endorse AL₃: if I know that C obtains in a case α_x, then I don’t falsely believe that C obtains in any near duplicate of α_x+1.

Instead of AL₂, however, we now introduce a new assumption AL₄: if I do not believe that C obtains in a case α_x, then there is a near duplicate β_x+1 of α_x+1 and a near duplicate β_x-1 of α_x-1 such that I do not believe that C obtains in β_x+1 or β_x-1. AL₄ can, I think, be motivated by the same considerations as AL₂—namely, that our powers of discrimination are limited, while the potential similarity of adjacent cases is not.¹⁰ AL₂ implies that, whenever I stop believing that C obtains, I could easily have stopped a moment later, while AL₄ implies that, whenever I stop believing that C obtains, I could easily have stopped a moment earlier. Both thoughts are plausible for the same reason—the changes from moment to moment are too slight for my beliefs to be perfectly attuned to them.

We now have the tools needed to prove that C cannot be a luminous condition. Let α_n be the last case in the sequence in which C obtains and α_n+1 be the first case in which ~ C obtains. If I believe that C obtains in α_n+1, then, by AL₃, I don’t know that C obtains in α_n. If I don’t believe that C obtains in α_n+1 then, by AL₄, there is a near duplicate β_n of α_n in which I don’t believe that C obtains and, thus, don’t know it. Either way, we have a case (α_n or β_n) in which C obtains, even though I fail to know that it does.

Rosenkranz’s argument against the relevant instance of AL₂ hinged on the fact that, as long I am doing everything that I am in a position to do to determine whether ~ K ~ KP, K ~ KP will be incompatible with my believing ~ K ~ KP. No such argument can be mounted against the relevant instance of AL₄—at least, not on the basis of the assumptions PK₁, PK₂ and PK₃. Perhaps we could motivate other assumptions about being in a position to know that would serve this purpose, but these three, relatively minimal, assumptions are not enough—they are perfectly compatible with ~ K ~ KP being true, and with my failing to believe ~ K ~ KP, even if I am doing everything that I am in a position to do to determine whether ~ K ~ KP.

As noted above, Rosenkranz doesn’t mean to ascribe us any mysterious power to perfectly detect the condition ~ K ~ KP—it is the constitutive connections between ~ K ~ KP and our beliefs about it that enable us to answer the original anti-luminosity argument. But for ~ K ~ KP to be a genuinely luminous condition may, after all, require a mysterious power on our part. Constitutive connections may ensure that my belief in ~ K ~ KP cannot outrun its truth as we progress through the sequence. If I am doing everything I am in a position to do to determine whether ~ K ~ KP, then I cannot make a type II error and believe that ~ K ~ KP obtains when it doesn’t—for the belief is, in a way, self-fulfilling. But it is less clear that there are any constitutive connections which will guarantee that the truth of ~ K ~ KP cannot outrun my belief. Even if I am doing everything I am in a position to do to determine whether ~ K ~ KP, it is less clear whether there are any constitutive connections guarding against the possibility of a type I error.¹¹

Rosenkranz’s attempt to save ~ K ~ KP from the anti-luminosity argument may, in the end, be unsuccessful. But there is, nevertheless, much that we learn by carefully thinking this through. Most obviously, we learn something about the anti-luminosity argument itself and the kinds of assumptions that can be used to drive it. And perhaps we learn something significant about knowledge and justification as well. When it comes to ~ K ~ KP—which is equivalent to JP on Rosenkranz’s theory—it may be that a certain kind of mistake really is impossible. If one is using the optimal method to determine whether this condition holds, then it cannot deliver a false positive.