Avoiding the Paradoxes

In introducing the columnar view, I said that one of its key attractions is that it avoids the paradoxes of higher-order vagueness, so it’s worth making it explicit how exactly it does this. Bobzien has offered responses to most of the arguments presented in the previous chapter, so I’ll go through replies to each of them, (critically) making use of her responses where possible.

We’ll start with Wright’s 1992 argument. Recall that this used the (DEF) rule (If DA₁, . . . , DA_nentail C, then DA1, . . . , DA_nentail DC) and a definitised gap principle D¬∃xⁿ(DDFxⁿ∧ D¬DFxⁿ⁺¹) to show that if an object was definitely not-definitely red, say, then an object slightly more red is also definitely not-definitely red, meaning that if there are any definitely non-definitely red objects, there are no definite cases of

‘red’ at all.

Bobzien’s reply to this argument on behalf of the columnar view is slightly puzz-ling. She first claims that, besides accepting the conclusion (and so that higher-order vagueness is paradoxical), we could either reject (DEF) as a rule of inference, reject the definitised gap principle, or reject the ‘second’ claim that Wright uses in his argu-ment, that there are some clear borderline cases. The way she presents this suggests that she has in mind here that rejecting this last option is to be done by rejecting that any objects are definitely not-definitely red, and this is her suggested response (2010, 22).⁶

5Bobzien lets the mask slip on this line sometimes — she describes ‘vague’ itself as ‘undoubtedly’

a vague predicate, for example (2010, 3), then later says she can make no judgement on it (2015, 76).

6Here’s what she specifically says. In her formal reconstruction of Wright’s argument, line (2) says ‘Def(¬Def(Fx^′)’ (D¬DFxⁿ⁺¹in our terms) and is labelled ‘2nd assump.’. She then goes on to

Rejecting this claim would make Wright’s argument toothless, since even if the con-clusion of the main argument,∀xⁿ(D¬DFxⁿ⁺¹ → D¬DFxⁿ) were true, if there were no definitely non-definite cases of ‘red’ to begin with, there would be nowhere to start the chain of reasoning showing that some obviously red objects were in fact not def-initely red. But I say that this is puzzling for two reasons. First, because the claim that there are no definitely non-definite cases of ‘red’ is not the same as the claim that there are no definite borderline cases of ‘red’. Saying that there are no definitely non-definite cases of ‘red’ does entail that there are no non-definite borderline cases of ‘red’, but it also entails that there are no definitely definite cases of ‘not red’: a blue shade, say, is definitely definitely not red, and so it’s surely definitely not a definite shade of red.

So if Bobzien takes her preferred approach, she also seems forced to deny that there are definitely definite cases of ‘not red’, which is a strange thing to say, at least.

The second puzzling feature of Bobzien’s response is that she correctly identifies a crucial distinction which would allow her to give a better response to Wright on be-half of the columnar view, and yet does not actually give that response. This crucial distinction is that Wright’s argument uses a definitised, rather than non-definitised, form of a gap principle. This distinction is important because, while the columnar view tries to preserve higher-order vagueness, and so seems bound to accept that vague expressions satisfy gap principles at many orders, it can avoid saying that any of them definitely do.

To see how, we need to think back to the discussion of how to characterise vague-ness presented in chapter one. I suggested that epistemic tolerance — roughly, that

say ‘[The] options we have are: either give up DEF, or give up the definitization of [¬∃xⁿ(DDFxⁿ∧ D¬D¬Fxⁿ⁺¹] (the 1st assumption), or give up definite borderline cases of the first order (the 2nd as-sumption), or admit that higher-order vagueness is paradoxical’ (2010, 22, her emphasis). The ‘first assumption’ she talks about here is labelled as such at line (1) of her reconstruction of Wright’s argu-ment, in the same way that the ‘second assumption’ is labelled. She then says that her implementation of the columnar view ‘rejects the existence of clear (or definite) borderline cases. In other words, it rejects Wright’s 2nd assumption. Thus, even if DEF and [¬∃xⁿ(DDFxⁿ∧ D¬D¬Fxⁿ⁺¹] are admitted, no higher-order vagueness paradox ensues’ (2010, 22).

there are no knowable sharp boundaries — is characteristic of vague expressions, and subsequently presented an argument from Greenough showing that if an expression is epistemically tolerant, it has (epistemic) borderline cases. Now, the claim that an expression is epistemically tolerant is actually just a kind of gap principle — ‘D’ is just interpreted epistemically — so Greenough’s argument can be used in exactly the same way to show that if an expression makes a gap principle true (at some order), it also has borderline cases at the relevant order. A defender of the columnar view could use these resources to show why we should reject the definitised form of any gap principle by helping themselves to Bobzien’s claim that no expressions clearly have borderline cases, and inferring from there that no expressions clearly make any gap principles true. Such an inference would be acceptable (it seems) because if an expression did clearly make a gap principle true, it would also clearly have borderline cases, and accepting the latter would just be to deny Bobzien’s claim. In sum, then, columnar views can avoid Wright’s conclusion by denying the only premise of his argument, the definitised form of a ‘second-order’ gap principle.

Moving on to Fara’s formulation of the paradox, we need to take a slightly differ-ent approach because no use is made of any claims which take some expression to be clearly vague in any sense. Fara’s argument works by applying gap principles success-ively to reach a contradiction from the truth of DF a1 and D¬Fan— that some object is clearly red and some object much less red than it clearly isn’t red, say — using the (DEF*) rule (If A1, . . . , A_nentail C, then A1, . . . , A_nentail DC).

Bobzien’s response to Fara’s version is just to reject (DEF*) as a valid rule of infer-ence, since it is just inconsistent with the columnar view (2010, 23). If (DEF*) were a valid rule then, since BA ⊢ BA is presumably valid, (DEF*) would tell us that BA ⊢ DBA is also valid. But while the columnar view is consistent with there be-ing borderline cases, it is not consistent with there bebe-ing clear borderline cases. So

BA⊢ DBA could not be valid, and so (DEF*) must in turn not be consistent with the columnar view.

Zardini’s argument can be dealt with in much the same way as Wright’s. His ar-gument makes use of D^mF a₁ and D^m¬Fan(for some arbitrarily high m), the closure of ‘D’ under consequence, and a number of gap principles, definitised to different de-grees, to reach a contradiction. To avoid Zardini’s conclusion, then, a defender of the columnar view should reject any one of these definitised gap principles, again point-ing out that the truth of one of these would be inconsistent with the columnar view itself, as it would entail the existence of clear borderline cases.

Finally, both Shapiro’s argument and the argument that I presented can be deflec-ted easily by the columnar view. Shapiro’s makes use of the assumption that border-line categories do not overlap at all, and the argument I presented used the assumption that these categories don’t overlap completely. These assumptions are just false on the columnar view, since it takes all borderline categories to overlap completely.