A puzzle about fuzzy boundaries

10/11/2025

Almost all the topics of ordinary thought and talk are vague: they have fuzzy boundaries. Here, Professor J Robert G Williams presents and plays around with a puzzle by Crispin Wright, showing a clash between two tempting claims about fuzzy boundaries.

I can be uncertain about whether what I’m doing is procrastinating (I’m writing this blog post, but in doing so, I’m neglecting other urgent things on my to-do list). Plausibly, this uncertainty is just a reflection of my own epistemic limitations. Someone could, in principle, know! Generalising: if a psychologist were in ideal conditions (I), and if I were procrastinating (P), then the psychologist could know I’m procrastinating (KP). Call this Epistemic Constraint:

(EC)  I→(P→KP)

My recent history has involved clear cases of procrastination and clear cases of non-procrastination. It also contains cases on the fuzzy boundary, borderline cases of procrastination. Much of the literature on the nature of fuzzy boundaries (vagueness) takes borderlineness to block knowledge: if it’s borderline whether I’m procrastinating, then nobody can know that I’m procrastinating, and nobody can know that I’m not procrastinating. Call this Verdict Exclusion:

(VE) borderline (P)→~KP&~K~P

I take it that if VE and EC are true at all, they’re necessarily true. I also assume that the presence of an ideal psychologist wouldn’t rule out my borderline-procrastinating. But now we’re in trouble, via an argument I’m adapting from Crispin Wright’s work:

1. I & borderline(P)        (something that should be possible)
2. ~KP&~K~P                (from 1, VE)
3. P→KP                       (1, EC)
4. ~P→~KP                   (1, EC)
5. ~P                             (2,3)
6. ~~P                           (2,4) 

We have a contradiction, by watertight reasoning, in a situation assumed possible. It looks like either VE or EC must go.

The appeal to (EC) at line (4) is a place to poke. The original motivation for EC is that an ideal psychologist could know that one is procrastinating whenever one is procrastinating. But line (4) requires a variation of this: that an ideal psychologist could know that one is not procrastinating whenever one is not procrastinating. Perhaps the lesson is that ideal detectability of properties is one thing; ideal detectability of their absence is another. 

Yet even giving this up, a fragment of the reasoning remains. We can argue:

1. I & borderline(P)        (something which should be possible)
2. ~KP&~K~P                (from 1, VE)
3. P→KP                       (1, EC)
4. ~P                             (2,3)

So we can now conclude that (at least when conditions are ideal) whenever P is borderline, ~P. 

To pin down the oddness, consider this pretty plausible claim:

(U) when P is borderline, the correct attitude is to be uncertain about P.

When the correct attitude towards P is uncertainty, then a belief that P is incorrect. Presumably, knowledge requires correct belief. (U) is a possible explanation of why (VE) holds.  

An assumption of the original paradox was that I & borderline(P) is possibly true. If it can be possibly known to be true, if we can know EC and VE, and knowledge is preserved over watertight reasoning, then we can conclude ~P must be known. But in the light of (U), this attitude would be incorrect, and so not knowledge. A paradox is regained. 

So perhaps either positive (EC) fails for procrastination, or else the “ideal” conditions (exhaustive knowledge of context, concepts, and psychology, boundless resources) are so rare that it’s not embarrassing to say that nobody could know they are in them. 

Even conceding this, ideal conditions for other fuzzy properties are easier to achieve. Consider the property looks red to me right now. This has borderline instances (it’s borderline whether my curtains look red to me right now). I’m sober, have plenty of time, and good viewing conditions. I know these normal, ordinary conditions obtain. Those are (I think) enough, in the actual world, for me to be an ideal judge of whether the curtains appear red to me right now. One undebunked instance of paradox is enough to teach us we haven’t yet understood how the machine works.  

Is the lesson that VE (and U) has to go? That’s the moral Wright drew. It’s something I’ve explored in my own work. The idea is that the characteristic syndrome of it being borderline whether I am procrastinating is to prompt some kind of tentative “taking of an opinion” over the question of whether I’m procrastinating. And, in general, we’re in no position to declare such tentative opinions not to be knowledge. 

But again, we need to consider the full generality of the puzzle. Even if we were in no position to assert that knowledge is excluded in borderline cases, perhaps something else is. For example, consider the tentativeness of our opinion, mentioned above. Wright suggests (plausibly) that this consists, in part, of tolerating contrary judgements. If we agree a case is borderline, and you tentatively think it’s procrastination and I tentatively don’t, neither of us should think the other’s view is based on bad evidence or poor processing. There need be no mistake. By contrast, I see my hands typing these words, and I confidently judge that there are hands. Intolerantly, I maintain: if you judge otherwise, you have made a mistake. Much of our knowledge is confident, but the hypothesis is that putative knowledge in borderline cases cannot be. If we use “K*” to stand for confident knowledge, then we have:

(VE*) borderline (P)→~K*P&~K*~P

And now if procrastination (or: looking red to me right now) is something which is ideally confidently detectable in positive and negative instances, we could lay down:

(EC*)  I→(P→K*P)

But now we can rerun the original argument for paradox, simply swapping K* for K throughout. (I leave the reconstruction of my second version as an exercise for the reader). 

So the moral of the puzzle may well be that (EC*) must be given up. That’s weird! Neither I in normal conditions, nor the ultimate omniscient deity looking down at me, could confidently know in all cases that the curtains look red to me, when they look red to me. And this generalises to every fuzzy property---which is to say, pretty much every property beyond pure mathematics. Omniscience and ordinary self-knowledge, even if possible, must operate with a lower grade of knowledge than the confident knowledge we commonly have of ordinary facts.  

J Robert G Williams is a Professor in the School of Philosophy, Religion and History of Science at the University of Leeds. He works in areas throughout theoretical philosophy (including logic, language, formal epistemology, mind, metaphysics, and metaethics). Particular foci of his have included: how theories of rationality need to adapt to accommodate nonclassical logics and indeterminacy; how conditionals (if-statements) work and what they’re for; and how our words and concepts get meaning. In the last decade, he has mostly been occupied running two big projects funded by the European Research Council – the first of which led to his book The Metaphysics of Representation (OUP, 2020) and the second of which explored whether and how collectives (groups, institutions, nations) could believe, desire, or think. He joined the University of Leeds in 2005 after completing my PhD under the supervision of Crispin Wright. 

Recommended texts / further reading

Stanford Encyclopedia of Philosophy Entry on “Vagueness”. General background, other paradoxes, and the main theories of vagueness. 

Rosanna Keefe, Theories of Vagueness, 2000. An accessible book-length introduction to the literature. 

Crispin Wright, The riddle of vagueness, OUP 2021. Especially “Vagueness: a fifth column approach”, “On being in a quandary” and “On the characterization of borderline cases”.

Robert Williams, “Decision making under indeterminacy” Philosopher’s Imprint 2014. My own account of a verdict-exclusion-denying account of borderline cases (and indeterminacy more generally).