Key Features of Columnar Views

In the previous chapter we saw a number of ways in which we can generate paradoxes from the claim that there is higher-order vagueness, and we finished with a paradox which I took to rely on the fewest additional resources, requiring only that borderline categories don’t completely overlap. Columnar views are the result of dropping that particular assumption — they take all borderline categories to overlap completely.¹ It’s worth taking some time at this point to see exactly what follows from this.

One consequence of the claim that all borderline categories completely overlap is that every borderline case is also a borderline borderline case, and vice versa. So if it’s not clear whether some object is red, it’s also not clear whether it’s clear.² And likewise

1There is some middle ground between ‘no borderline categories overlap completely’ and ‘all bor-derline categories overlap completely’ — namely, we could say that some, but not all, borbor-derline cat-egories overlap completely. But I pointed out before that the paradox I presented could be made to work using only the assumption that, for any borderline category that didn’t overlap completely with every category of a lower order, there’s at least one borderline category at a higher order that doesn’t overlap with any borderline category at a lower order than it. Now denying this but saying there are some non-overlapping categories would commit us to saying that, beyond a certain order, any border-line category at all overlaps completely with at least one of the borderborder-line categories ‘below’ it. That’s a different claim from saying that all borderline categories overlap completely — there are some other, more esoteric, structures that are consistent with this claim. But it seems that the most sensible way to make sense of this claim is to say that, beyond some order, borderline categories all start overlapping completely with one another. And if, beyond some order, borderline categories all start overlapping completely with one another, what I have to say about this case applies just as much to the case in which all borderline categories overlap completely — we just think about the columns as starting ‘higher up’

— and so for simplicity I will stick to discussing the view on which all borderline categories completely overlap.

2In this chapter I’m going to use the word ‘clear’ essentially as a synonym for ‘definite’. This is a fairly natural reading anyway, but I use it here in particular to mesh with the account that Bobzien puts forward is stated in terms of ‘clarity’, and she gives a special definition for this term. Bobzien also uses the letter ‘C’ (for ‘clear’) in her system of logic rather than the ‘D’ (for ‘definite’) that we’ve been using so far. There’s nothing important about this difference so I’m going to continue to use ‘D’.

it won’t be clear whether it’s unclear whether such a case is an unclear case of ‘red’, and so on. The crucial thing to note here is that, on this picture, there are no clear borderline cases of any sort because a clear borderline case is exactly a non-borderline borderline case, and this view says that there are no such things.

This may come across as counterintuitive, but perhaps it can be made to look a little more plausible at this point. I’ve made reference to cases previously which I took to be borderline cases — the walls of the IC for ‘blue’, and I thought I might be a borderline case of ‘borderline tall’ — but I’m not confident that everyone would agree with me about either, and so perhaps neither is a good candidate for being a clear borderline case. And yet clear non-borderline cases are easy to come by — someone with no hairs on their head is clearly not a borderline case of ‘bald’, and likewise one grain of sand is clearly not a borderline case of ‘heap’ — so if we’re struggling to identify any clear borderline cases, shouldn’t we be suspicious of their existence?³ On the other hand, we might attribute these difficulties to issues surrounding context. Perhaps it’s difficult to definitively identify borderline cases because this would require them to be borderline cases regardless of the context in which we’re talking about them. We can lend some support to this thought by pointing out that clear cases are actually harder than it seems to identify once and for all, perhaps for this exact reason — Violet Brown is clearly old when we’re talking about humans, but when we’re talking about living things on Earth, she’s relatively young (apparently the oldest tree alone is over 5,000 years old!). I say all of this to put the claim that there are no clear borderline cases into perspective, as it may seem a little outlandish at first; it will receive more attention soon.

The other side of the coin is that, since all borderline categories overlap completely, all non-borderline cases will be clearly non-borderline. To see why this is so, consider a clear case of ‘tall’ (say, someone who is 8ft tall). Such a person is not a borderline case

3If you think you’ve identified one, try asking 5 people if they agree and see what happens!

of ‘tall’. And the claim that all borderline categories completely overlap entails that if an object is not a member of one borderline category (with respect to some vague term), it is not a member of any. So this person must not be a borderline borderline case of ‘tall’. This means they’re either clearly clearly tall, or clearly clearly not tall, and so from the fact that they’re clearly tall we can infer that they’re clearly clearly tall. The same argument can be applied to show that people who are clearly not tall are likewise clearly clearly not tall, and so we can show that any non-borderline case will be clearly non-borderline.

Given these two features of columnar views, we can see where the term ‘columnar’

comes from. Say we have a set of shades going from Volcanic Red to Tangerine Twist in a succession of small changes. On the columnar view, we can divide them neatly into three sorts: shades which are clearly red, clearly clearly red, and so on, shades which are borderline cases of ‘red’, borderline borderline cases of ‘red’, and so on, and finally those which are clearly not red, clearly clearly not red, and so on. Between them they form three ‘columns’, thinking of the ordered shades themselves marking a ‘horizontal’ axis, and the orders of clarity (or borderlineness) marking the ‘vertical’:

D . . . DF D . . . DF D . . . DF B . . . BF B . . . BF D . . . D¬F D . . . D¬F D . . . D¬F

DDDF DDDF DDDF BBBF BBBF DDD¬F DDD¬F DDD¬F

DDF DDF DDF BBF BBF DD¬F DD¬F DD¬F

DF DF DF BF BF D¬F D¬F D¬F

1 2 3 4 5 6 7 8

Contrast this with the alternative picture, on which borderline categories at differ-ent orders can differ in which objects they contain, and likewise for categories mark-ing different levels of clarity. This gives us what we can call, followmark-ing Bobzien (2010), a ‘hierarchical’ view of higher-order vagueness — displaying this structure graphic-ally tends to yield upside-down pyramids, with the borderline cases at the base, then

second-order borderline cases (as we might expect) on the borderlines of that category, then third-order borderline cases on the borderlines of that category, and so on:

DDDF DDDF . . . BBBF DBBF DDBF DBB¬F BBB¬F ... DDD¬F DDD¬F

DDF DDF BBF DBF BB¬F DD¬F DD¬F

DF DF BF D¬F D¬F

1 2 3 4 5 6 7 8 9