1. INTRODUCTION
Which is more impressive – St Paul's Cathedral or the Eiffel Tower? It seems that neither is more impressive than the other. Are they equally impressive? If they were, then a minute improvement to the impressiveness of St Paul's would make it more impressive than the Eiffel Tower. But a minute improvement does not seem sufficient to shift the balance. So, the two cannot be equally impressive. Therefore, none of the trichotomy of comparative relations ‘more impressive than’, ‘less impressive than’ and ‘equally impressive’ apply between the Eiffel Tower and St Paul's. This is the Small Improvement Argument (SIA).Footnote 1
For some SIAs, our failure to have confidence in any of the components of the trichotomy can be explained by what we might call our ‘contingent ignorance’ about the properties of the two options, as when there is some fact of the matter that we just haven't yet found out. Others can be explained by our conceptual incompetence, such as when we haven't thought hard enough about the comparative concept in question. However, there are some ‘hard cases’ which cannot be explained in this manner. The comparison of the Eiffel Tower and St Paul's is arguably one example: even once we have all the information and are sufficiently conceptually competent, we may still be unable to conclude that one is more impressive or that they are equally impressive. We, along with a number of other philosophers, believe that these hard cases are borderline cases of vague comparative predicates (Broome Reference Broome and Chang1997). Just as we cannot come to a definitive answer in these hard comparisons, we cannot come to a definitive answer about when a man is bald, even if we know all the facts about the number of hairs on his head and how they are dispersed, and even if we completely understand the concept of baldness. The reason for this, in our view, is that these are both instances of vagueness.
Ruth Chang (Reference Chang2002) has notably rejected this claim. She argues that hard cases (which she calls ‘superhard cases’) can be distinguished from borderline cases by considering some of the permissible practical responses to the two types of case. We will not engage with that argument here for reasons of space and because our arguments would largely repeat those made elsewhere (Broome Reference Broome and Chang1997; Wasserman Reference Wasserman2004; Gustafsson Reference Gustafsson2013; Williams Reference Williams2016). Instead, we take it for granted here that the hard cases raised in some SIAs exploit vagueness, and our conclusion is accordingly conditioned on that assumption.
The question that follows is, what is the correct theory of vagueness, and what are its implications for the SIA? Almost all vagueness-based accounts of the SIA have thus far assumed the truth of supervaluationism, one leading theory of vagueness. According to supervaluationism, sentences involving borderline cases of vague predicates are neither true nor false. Supervaluationist accounts of the SIA thus say of the above case that it is neither true nor false that the Eiffel Tower is more impressive than St Paul's, nor that it is less impressive, and nor that the two are equally impressive. However, supervaluationism is not universally accepted as a theory of vagueness, and it has some problematic features (Williamson Reference Williamson1994: ch. 5). For example, it implies that a true disjunction can have no true disjuncts. With respect to the SIA set out above, for instance, it says that the disjunction ‘St Paul's is either more impressive than, less impressive than, or as impressive as the Eiffel Tower’ is true, but that none of the individual disjuncts is true.Footnote 2 This seems problematic. Indeed, the natural thing to say in response to such a theory may be that it simply has the wrong account of what ‘or’ means; we need a theory which is consistent with ‘or’, and supervaluationism does not fit the bill. Appeals to semantics of this kind command widespread assent in other domains. For example, the most popular response to the claim that the relation ‘better than’ is intransitive is simply that this must be wrong, as a semantic matter (Huemer Reference Huemer2013). Thus, supervaluationism's hegemony in debates about the SIA is certainly open to question.Footnote 3
Although this of course does not count as decisive criticism of supervaluationism, it does give us prima facie reason to explore viable alternatives to it. One of the leading alternatives to supervaluationism is epistemicism. In contrast to supervaluationism, epistemicism holds that borderline propositions have exactly one truth value – true or false – but that we are incurably ignorant of it. Consider the example of a plump man, Jim, who is a borderline case of ‘is fat’. The epistemicist denies that it is neither true nor false that Jim is fat. Rather, this proposition has exactly one truth value, but we cannot know what that truth value is.
In the wake of persuasive recent defences of epistemicism and its growing philosophical popularity, epistemicism at least deserves a seat at the table in discussions about vagueness (Sorensen Reference Sorensen1988; Williamson Reference Williamson1994).Footnote 4 Whether those defences succeed is, of course, a judgement about which reasonable people will differ, but we think it is clear that epistemicism cannot, in view of these defences, simply be ignored.
Yet in spite of epistemicism's large and growing philosophical popularity, an epistemicist account of the SIA has yet to be fully developed in the literature. Our goal here is to fill this gap. We argue that on an epistemicist vagueness-based account of the SIA, items are comparable in small improvement cases. In other words, if epistemicism is true, in small improvement cases, incomparabilists confuse our ignorance of a ranking with the non-existence of a ranking. (Note that our argument only establishes that items are comparable in small improvement cases, and therefore is compatible with the possibility that items are incomparable in other cases not involving vagueness. For example, it might be argued that any amount of profound artistic contemplation is absolutely incomparable with any amount of base pleasure.) This is potentially important for axiology. If epistemicism is true, then the SIA does not show the betterness ordering to be incomplete in small improvement cases.
Our argument is doubly conditional: if all hard cases raised in SIAs are borderline cases of vague predicates, and if epistemicism is true, then items are comparable in small improvement cases. We make no attempt to defend either part of the antecedent here. Nonetheless, as we have argued above, there is sufficient reason to explore what follows if both parts of the antecedent are true.
The paper is structured as follows. In Section 2, we define comparability, formalize the SIA, and define borderline cases of vague predicates. In Section 3, we introduce in more depth, and discuss the implications of, supervaluationism and epistemicism for the SIA. Since epistemicism is deductively logically consistent with both comparabilism and incomparabilism about small improvement cases, we develop and discuss versions of both, arguing that we ought to accept epistemic comparabilism. Interestingly, one of the arguments we use also shows that even if hard cases are not borderline cases of vague predicates, if epistemicism is true, then items cannot be on a par. Lastly, we consider the treatment of higher order hard cases by supervaluationism, epistemicism (our preferred theory), and Chang's parity view. We argue that, as things stand in the literature, only the epistemicist view provides a compelling account of how to account for such cases.
2. COMPARABILITY AND THE SMALL IMPROVEMENT ARGUMENT
2.1. Comparability
How one defines comparability depends upon certain assumptions, and we use a definition with which some philosophers disagree. We believe that what Ruth Chang (Reference Chang2002) has called the ‘Trichotomy Thesis’ is true, and will take it for granted here. Chang's argument against the Trichotomy Thesis rests in part on the claim that not all SIAs can be explained by contingent ignorance or vagueness. As we have mentioned above, we believe that others have provided decisive criticism of this view.Footnote 5
Trichotomy Thesis: Two items, x and y, are comparable in terms of their Fness, if and only if x is Fer than, less F than, or as F as y.
According to the Trichotomy Thesis, two options are incomparable in terms of their Fness if and only if it is not true that x is Fer than, less F than, or as F as y. Comparabilism can be understood as the claim that for any two items in F’s domain, one of the trichotomy of ‘Fer than’, ‘less F than’, or ‘as F as’ applies between them. Incomparabilism is the denial of comparabilism. For instance, a comparabilist about the creativity of artists would claim that, for all pairs of artists, one is either more creative than, less creative than, or as creative as the other. An incomparabilist about the creativity of artists would deny this.
Note that on our definition, it is sufficient for incomparabilism that all of the components of the trichotomy are not true. On some logics, propositions can be neither true nor false (Broome Reference Broome and Chang1997). We believe that, as a semantic matter, if all of the components of the trichotomy are not true, that is sufficient for incomparability. The claim that it is false that one of the components holds is stronger (given that false implies not true) and so is also sufficient.Footnote 6 If all of the components of the trichotomy are not true between a pair of items, then the items are not ordered in terms of their Fness, and it is the ordering of items in terms of Fness that the concept of comparability ought to capture.
One final thing to note about our definition is that it allows that it might be the case that between two comparable items, we do not or cannot know, in a trivial sense, which of the components of the trichotomy is true. Some such cases are straightforward. Suppose we have to discern which of two carrots is longer, but one of them is in a locked safe to which we do not have access. We might ordinarily say ‘the length of these two carrots is incomparable’. However, it is not, on the present understanding of ‘incomparable’. It is true that one carrot is longer than the other or that they are equally long. That we cannot measure them does not change this fact. They are, then, in the relevant sense, comparable. We should not confuse incomparability with ignorance about the ranking of options.
2.2. The Small Improvement Argument
We can now set out the Small Improvement Argument. A range of SIAs have been presented in the literature. Consider this version, which we believe makes the case particularly strongly. Suppose that we are comparing a painter, Francis, with a musician, Kate, and that it seems as though it is not true that one is more creative than the other. Now consider Francis+, who is slightly more creative than Francis: suppose that one of his frescos manifests slightly more creativity. If Francis and Kate were exactly equally creative, then Francis+ would be more creative than Kate. However, intuitively it is not true that Francis+ is more creative than Kate, since a small improvement to one of Francis's frescoes could not tip the balance in this way. Therefore, Francis and Kate cannot be equally creative. Therefore, it is not true that Francis is more creative than, less creative than, or as creative as Kate. By the Trichotomy Thesis, Francis and Kate are therefore incomparable in terms of their creativity.
The SIA may be generalized and formalized as follows:
Premises
P1. ‘Equally F’ is a transitive relation.
P2. For some F, and some options x, x+, and y in F’s domain:
(a) It is not true that x is Fer than y
(b) It is not true that y is Fer than x.
(c) It is not true that x+ is Fer than y.
(d) x+ is Fer than x.
P3. If it is not true that x is Fer than or less F than y, and it is not true that x and y are equally F, then x and y are incomparable with respect to F.
Argument
[Suppose for reductio that]:
4. x and y are equally F
[From P1, P2 (d) and 4]:
5. x+ is Fer than y.
[But this contradicts P2 (c). So]:
6. It is not true that x and y are equally F.
[From P2 (a) and (b) and 6]:
7. It is not true that x is Fer than or less F than y, and it is not true that they are equally F.
[From P3 and 7]:
C. x and y are incomparable with respect to F.
Our target is P2. We argue that, assuming that epistemicism is true, it is not possible for P2 (a), (b), (c) and (d) to be true at the same time. If epistemicism is true, incomparabilists confuse our ignorance of the truth of P2 (a), (b) and (c) with the truth of all of P2 (a), (b), (c) and (d).Footnote 7
2.3. Hard cases and borderline cases
In our view, many SIAs which have been presented in the literature involve a significant amount of contingent epistemic limitation, not unlike our example of the locked-away carrots. SIAs which only involve contingent ignorance are uncontroversially unsound because they confuse our merely contingent ignorance of a ranking with the non-existence of a ranking. Similarly, other SIAs exploit poorly understood concepts. It may sometimes be the case, for instance, that two soldiers seem incomparable in terms of their bravery, but only because we do not understand the concept of bravery well enough. However, it is widely believed that there remain hard cases in which no amount of further empirical checking or conceptual analysis would allow us to determine which of the components of the trichotomy applies (Messerli and Reuter Reference Messerli and Reuter2016). The creativity SIA set out above seems to present an example of such a hard case.
As we have said, we believe that these hard cases are borderline cases of vague predicates. Before we proceed, it is worth being clear about what it means to say that something is a borderline case of a vague predicate. There is disagreement both about the definition of vagueness and of borderlineness. Sensitivity to sufficiently small changes but sensitivity to larger ones (‘tolerance’), susceptibility to the sorites paradox, and susceptibility to borderline cases have all been defended as definitive of vagueness, and have also been subject to criticism (Sorensen Reference Sorensen1988; Greenough Reference Greenough2003; Bueno and Colyvan Reference Bueno and Colyvan2012). In light of this controversy, we remain neutral on the sufficient conditions for something to be a borderline case of a vague predicate. However, we follow the most popular account of borderlineness by holding that it is a necessary condition of borderline cases of vague predicates that they are resistant to inquiry (Williamson Reference Williamson1994: 2; Keefe and Smith Reference Keefe and Smith1997; Sorensen Reference Sorensen and Zalta2013).Footnote 8 A case is a borderline case only if we cannot determine whether a predicate applies or does not apply by reducing our contingent ignorance or by improving our conceptual competence.
3. VAGUENESS AND IGNORANCE
We will now briefly review the implications of supervaluationism for the SIA before moving on to consider the implications of epistemicism.
3.1. Supervaluationism
Before beginning, we should make clear that what follows is simply a discussion, rather than a critique, of the implications of a supervaluationist account of the SIA for the truth of incomparabilism. Our claim is that the implications of that view on this question are ambiguous, but we do not take this to be a flaw per se of a supervaluationist account of the SIA. We do, however, also happen to believe that the source of this ambiguity – supervaluationism's rejection of classical logic and semantics – is a good reason to disprefer supervaluationism as a theory of vagueness more generally. But establishing that goes beyond the scope of our argument here, which is conditional on the truth of epistemicism.
Supervaluationism is the only theory of vagueness that has been fully developed in relation to comparability and the SIA, though it nonetheless has an ambiguous relationship to comparability.Footnote 9 According to supervaluationism, in borderline cases ‘x is Fer than, less F than or as F as y’ is true, but all of the disjuncts of this disjunction are neither true nor false. This is explained in the same way that supervaluationism explains the law of the excluded middle (Broome Reference Broome and Chang1997: 82). According to the supervaluationist, vague concepts can be admissibly ‘sharpened’ in numerous ways. For instance, different admissible sharpenings of ‘more creative than’ will draw the boundary between ‘more creative than’ and ‘not more creative than’ in different places. For the borderline case ‘Francis is more creative than Kate’, on some admissible sharpenings, Francis is more creative than Kate, whereas on others he is not. Supervaluationism dictates that ‘Francis is more creative than Kate’ is therefore neither true nor false. ‘Francis is more creative than Kate’ is only true (or as it is also said in the literature is ‘supertrue’) when Francis is more creative than Kate on all admissible sharpenings.
However, even in cases when ‘Francis is more creative than Kate’ is neither true nor false, i.e. is not supertrue, the disjunction ‘Francis is more creative than, less creative than, or as creative as Kate’ is true on all sharpenings, i.e. is supertrue, because all sharpenings entail a complete ordering. Therefore, on supervaluationism, the disjunction is true, even though each of its disjuncts is neither true nor false.
The truth status of the disjunction makes supervaluationism seem like a form of comparabilism about small improvement cases, whereas the truth status of the individual disjuncts makes supervaluationism seem like a form of incomparabilism. This puzzle is explained by the fact that the intuitive definition of comparabilism follows the dictates of classical semantics, which does not allow a true disjunction to have no true disjuncts. Since supervaluationism rejects this principle of classical semantics, it is not clear whether supervaluationism is a comparabilist or incomparabilist theory about small improvement cases.Footnote 10
This ambiguity can also be brought out by examining the formalization of the SIA we set out above. In our presentation of the argument, we included truth predicates in the premises themselves, and thus constructed a ‘metalinguistic’ version of the SIA. Here is P2 (a), (b) and (c):
P2. For some F, and some options x, x+, and y in F’s domain:
(a) It is not true that x is Fer than y
(b) It is not true that y is Fer than x.
(c) It is not true that x+ is Fer than y.
We arrive at P2 (a), (b) and (c) in the following way. Consider P2 (a). Because ‘x is Fer than y’ is a borderline case, it is neither true nor false that x is Fer than y, and by consequence it is not true that x is Fer than y. The same reasoning applies for P2 (b) and (c).
On classical logic and semantics (according to which amongst other things, the principle of bivalence holds, and true disjunctions cannot have no true disjuncts) merely appending the truth predicate to a premise cannot affect the soundness of the argument of which the premise is a part. For instance, if my argument depends on the premise
P. S
then revising the premise to instead claim that
P′. ‘S’ is true
would make no difference to the argument's soundness. But this is not so on supervaluationism. According to supervaluationism, P2 (a), (b) and (c) are true, whereas versions of these premises which do not include truth predicates would be neither true nor false. To see this, consider a non-metalinguistic version of P2, P2′, which does not include truth predicates in the premises:
P2′. For some F, and some options x, x+, and y in F’s domain:
(a) x is not Fer than y
(b) y is not Fer than x
(c) x+ is not Fer than y
Now consider P2′ (a). On supervaluationism, since P2′ (a) concerns a borderline case, it is neither true nor false. That is, in such a case, ‘x is not Fer than y’ is neither true nor false. But for the SIA to be sound, its premises must be true. Similar remarks apply to P2′(b) and P2′(c). The premise is therefore not true. Therefore, supervaluationism entails the unsoundness of the non-metalinguistic SIA. By contrast, each component (a–c) of the metalinguistic premise P2 is true according to supervaluationism; and thus supervaluationism supports the soundness of the metalinguistic SIA.
This is another way of bringing out supervaluationism's ambiguous relationship to comparability, and also shows why we use the metalinguistic version of the SIA: since (or so we argue) epistemicism implies comparabilism, the metalinguistic SIA highlights the fact that there is a distinction between the way that supervaluationism and epistemicism treat some versions of the SIA.
3.2. Epistemicist comparabilism or incomparabilism?
We now turn to the implications of epistemicism for comparability. First, we will provide a brief account of epistemicism. As is well known, epistemicism claims that borderline propositions are classically truth-functional but are distinguished by being propositions whose truth value cannot be known. For instance, the epistemicist claims that there is a precise point as we take hairs way at which ‘Tom is bald’ becomes true where before it was false, though we do not and cannot know where this point is (Williamson Reference Williamson1994: 198–201). The principle of bivalence is true: propositions have exactly one truth value – true or false; no proposition can be neither true nor false (Williamson Reference Williamson1994: 187–89). The epistemicist explains the inquiry resistance of borderline cases by appealing to our ignorance: no amount of further empirical checking would enable us to know whether Tom is bald or not – our epistemic deficit is non-contingent and incurable. Contingent ignorance can be resolved by further empirical checking, whereas, according to the epistemicist, the ignorance manifested in borderline cases cannot.
It might appear that epistemicism straightforwardly entails that options are comparable in small improvement cases. This is not so. Epistemicism is (deductively) logically compatible with both comparabilism about small improvement cases and incomparabilism. Consider first what we call epistemicist comparabilism.Footnote 11 Epistemicist comparabilism says that for all x and y in F’s domain, x is either Fer than y, less F than y, or x and y are equally F and that only one of the disjuncts is true. For example, suppose that Francis is slightly less creative than Kate. There is then some slightly more creative version of Francis, Francis+, who is as creative as Kate, and a slightly more creative version again, Francis++, who is more creative than Kate. However, we cannot know the truth value of any of these three comparative propositions because the trichotomy of ‘more creative than’, ‘as creative as’ and ‘less creative than’ are vague. Epistemicist comparabilism posits a single point of precise equal creativity, and says that items either side of this point are either more creative than, or less creative than the items used as the standard of comparison. Hence, the comparative ordering of options is complete (unless there is some other non-small improvement-based argument showing the ordering to be incomplete).
Epistemicist comparabilism can be visually represented by Broome's standard configuration (Broome Reference Broome and Chang1997). Suppose that there is a linear scale of Fness, and that the standard, x, is in the middle of this scale, and that there is a set of options (y 1, y 2. . ., yn) ordered in terms of their Fness. Epistemicist comparabilism posits that only one of the options in the y-set, ye, is exactly equally as F as x. The subset of options that are less F than y are also less F than x, and the subset of options that are Fer than y are also Fer than x. The grey box in Figure 1 indicates the set of borderline cases for which we are incurably ignorant which of the components of the trichotomy holds.
A visual representation of epistemicist comparabilism
However, as Broome (Reference Broome1999: 152) has pointed out, epistemicism is also (deductively) logically compatible with incomparabilism about small improvement cases. Epistemicist incomparabilism posits a zone of incomparability sandwiched by two zones of comparability. More precisely, it posits that there is a subset, S1, of y options which are Fer than x, and a subset, S3, which are less F than x. However, it posits that, on the scale of Fness there is another subset, S2, of y options sandwiched between S1 and S3, which are neither Fer than nor less F than x. Provided the options in S2 and x are not equally F, the options in S2 and x are incomparable with respect to F. There is a sharp transition from the zones of comparability into the zone of incomparability. However, it cannot be known when this transition occurs because each component of the trichotomy is vague. This is incomparabilism with an incurable epistemic deficit.
A first pass at visually representing epistemicist incomparabilism by the standard configuration is presented below. (As we discuss in our criticism of epistemicist incomparabilism below, it is actually subject to a qualifying proviso, which makes visual representation difficult.) Suppose again that there is a linear scale of Fness, and one item, x, is in the middle of this scale, and that there is a set of options (y 1, y 2. . ., yn) ordered in terms of their Fness. The grey box in Figure 2 indicates the set of borderline cases for which we are incurably ignorant of which of the components of the trichotomy holds.
A first pass at a visual representation of epistemicist incomparabilism
Should we accept epistemicist comparabilism or epistemicist incomparabilism? We cannot think of any reasons to favour epistemicist incomparabilism over epistemicist comparabilism. There are, however, two abductive reasons to believe that epistemicist comparabilism is true.
Firstly, epistemicist comparabilism posits a single point of precise equality. The epistemicist incomparabilist treatment of equality, by contrast, is problematic. It appears that there are two possible treatments. The first would be to claim that there is a solitary point within the gap between the application of ‘Fer than’ and ‘less F than’ (that is, within S2) at which options are equally F and are therefore comparable. However, it cannot be true that there is a solitary point of equality and that options are incomparable. If x and ye are equally F, then all other options (apart from ye) in the sequence of items (y 1, y 2,. . . yn) that are ordered in terms of their Fness, must be Fer than or less F than x. The subset of options that are Fer than ye must also be Fer than x because x and ye are equally F; and the subset of options that are less F than ye must also be less F than x because x and ye are equally F. Precise equality crowds out incomparability.
The second way for epistemicist incomparabilism to treat equality would be to claim that no pairs of items can be equally F within the gap between the zones of comparability (that is, within S2). If this were true, however, the question of whether any two items were equally F could never be inquiry resistant, since it would have a determinate answer (‘no’), and therefore there could never be borderline cases of ‘equally F’. As discussed earlier, this has the consequence that ‘equally F’ cannot be vague.Footnote 12 There are reasons to believe that this is a mistake.
Namely, it seems that the question of whether predicates of the form ‘is equally F’ apply sometimes can be inquiry resistant. Suppose again that we are assessing whether Kate and Francis are equally creative. Intuitively, it seems that once we have all the information and are conceptually competent, we might still be unable to settle with complete confidence whether the two are equally creative; we would not be certain that they were not equally creative. But such certainty is what epistemicist incomparabilism requires, since on that view, the fully determinate answer to whether any two items are equally F is ‘no’. Epistemicist comparabilism is, by contrast, compatible with the intuitive lack of certainty about this case. Given that there is no deeper theoretical reason to accept epistemicist incomparabilism's stance on this issue, it is hard to see why we would accept the incomparabilist view. Indeed, all proponents of vagueness-based accounts of the SIA have hitherto assumed that predicates of the form ‘is equally F’ can be vague: according to supervaluationists, in borderline cases, all components of the trichotomy are neither true nor false. This suggests that there is significant intuitive support for the proposition that predicates of the form ‘is equally F’ can be vague.
A second abductive argument in favour of epistemicist comparabilism goes as follows. Epistemicist incomparabilism as we have set it out says that there is a zone of incomparability that is sandwiched by two zones of comparability. However, this characterization should be qualified; the story is still more complicated. This can be demonstrated by what we call the ‘Monadic-Dyadic Argument’.
This argument begins by noting that, according to epistemicism, there is what we will call a precise ‘monadic threshold’ at which the truth value of a statement with a vague monadic predicate switches from true to false. This is, for instance, the threshold at which, as his creativity improves, ‘Francis is creative’ switches from being false to true. If this is true, then in many cases ‘Fer than’ will apply within the gap in which all of the components of the trichotomy are supposed to be false. This can be shown using the following principle:
Monadic-dyadic principle: If x is F and y is not F, then x is Fer than y
We take the monadic-dyadic principle to be uncontroversial for a large class of comparatives (though there may be some exceptions, which we return to below).Footnote 13 If one person is creative, then she is more creative than all people who are not creative. Now imagine a musician, Elton, and a singer, Freddie, who stand just on either side of the monadic threshold of ‘is creative’: Elton is creative but Freddie is not. Therefore, given the monadic-dyadic principle, Elton is more creative than Freddie. However, since Elton and Freddie are very close to one another in terms of creativity and manifest creativity in different ways, ‘Elton is more creative than Freddie’ is also a borderline case, and so we cannot know whether or not it is true. Therefore, ‘Elton is more creative than Freddie’ is true, but we cannot know whether or not it is true. Yet according to epistemic incomparabilism, ‘Elton is more creative than Freddie’ is false because, in virtue of being a borderline comparative case, it lies within a zone of incomparability.
Thus, epistemicist incomparabilism must be qualified. It should say that for some borderline cases of some vague comparatives, two items are incomparable, except when they are either side of the monadic threshold.Footnote 14 In that case, one item is Fer than the other, so the items are comparable. These cases, it is important to note, will be pervasive. There is an infinite number of possible cases in which two items are either side of the monadic threshold and are a borderline case of the comparative form of that monadic predicate. Therefore, there are points of comparability in what we initially said must be a zone of incomparability.
Furthermore, if Elton and Freddie stand only marginally on either side of the monadic threshold, then Elton is only slightly more creative than Freddie; they are extremely close to each other in terms of their creativity. But if this is true, then it seems as though the following must also be true: there is some small improvement to Freddie's creativity which would make him as creative as Elton, where before he was not. But as we argued above, precise equality crowds out incomparability. If this line of argument is sound, then epistemicist incomparabilism collapses entirely. There may be other ways to qualify the theory so that it allows pockets of comparability in the zone of incomparability, but doing so seems ad hoc and lacks deeper theoretical or intuitive justification.
(It is worth noting in passing that, if epistemicism is true, this argument also counts against tetrachotomist views, such as Chang's parity view. Chang would argue that the comparison of Elton and Freddie is a ‘superhard’ case of ‘more creative than’ and that therefore they are on a par with respect to creativity. However, the Monadic-Dyadic Argument shows that in fact one is more creative than the other. Thus, if epistemicism is true, then there are points of comparability in what was initially posited to be a zone of parity. If epistemicism is true, this seems like a fatal flaw of all tetrachotomist theories. Note that this argument does not assume or imply that hard cases of comparison are borderline cases of vague dyadic comparatives. Rather, it moves from an assessment of vague monadic predicates to the claim that items are not on a par in terms of a shared property. This is compatible with it being true that hard cases of comparison are not borderline cases of vague dyadic comparatives.)
As we mentioned above, the monadic-dyadic principle is uncontroversial for a wide range of comparatives. However, one possible response to the Monadic Dyadic Argument is to argue that the monadic-dyadic principle is not true for axiological betterness.Footnote 15 There is a great deal of controversy about whether goodness can be reduced to betterness or vice versa (Gustafsson Reference Gustafsson2014), and some of this controversy may extend to the monadic-dyadic principle. For example, Gustafsson (Reference Gustafsson2017) has defended ‘blank critical range utilitarianism’ in population ethics, according to which the monadic-dyadic principle is false. In brief, on this theory, lives can be good, bad, neutral, or ‘blank’ for the person living them. If a life is blank, it is neither good, bad nor neutral. Due to the possibility of a blank life and the resultant (non-vagueness-based) incompleteness in the ranking of the goodness of lives, it is possible that some good lives are not better than blank lives, even though blank lives are not good (Gustafsson Reference Gustafsson2017: 15).
While it is true that the Monadic-Dyadic Argument fails if Gustafsson's axiology is correct, we would offer two rejoinders. Firstly, if the blank critical range axiology is true, then there are still no reasons to believe that epistemicist incomparabilism rather than comparabilism is true – we have merely lost one reason that favours comparabilism. And if the critical range theory is false and some other plausible axiology that is compatible with the monadic dyadic principle is true, then the Monadic-Dyadic Argument alone is sufficient to refute epistemicist comparabilism. The blank critical range axiology, moreover, is highly controversial, so epistemicist incomparabilism is also for this reason less probable than epistemicist comparabilism. This point generalizes for all controversial axiologies which are putatively incompatible with the monadic-dyadic principle, not just the blank critical range axiology.
Secondly, the monadic-dyadic principle can be amended to avoid this counterexample. The blank critical range theory only implies the falsity of the monadic-dyadic principle as it applies to lives, but not to other items that we can compare in terms of axiological betterness, such as experiences. All that is needed to decisively refute epistemicist incomparabilism is for some version of the Monadic-Dyadic Argument to commit the incomparabilist to some pockets of comparability in the zone of incomparability. In general, it seems unlikely that there could be an argument showing that the monadic-dyadic principle fails to hold for all possible items that we can compare in terms of axiological betterness; it is plausible that it will always be possible to develop some version of the monadic-dyadic principle which is sufficient to refute epistemicist incomparabilism.
In sum, if we believe epistemicism is true, then incomparabilism about small improvement cases is drained of all of its intuitive force. Incomparabilism was initially intuitively plausible because it provided an explanation of the phenomena in small improvement cases. If we can explain those phenomena with appeal to unknowable boundaries and unknowable precise equality, there is no reason to accept the intuitive and theoretical problems that come with a conjunction of unknowable boundaries and incomparabilism. Unknowable boundaries ought to be treated as a substitute for incomparability.Footnote 16 If epistemicism is true, then in small improvement cases incomparabilists confuse our ignorance of a ranking with the lack of a ranking.
3.3. Sharp boundaries and higher-order vagueness
There is one other point about epistemicist comparabilism that is worth making. Many people take the idea of an unknowable sharp boundary to be so counter-intuitive as to render epistemicism as a wider theory of vagueness completely implausible. There must, these people believe, be some more plausible non-epistemicist theory of vagueness, which does not unambiguously entail comparabilism. Sharp boundaries, on this view, are a liability.
In fact, however, this intuition provides no leverage against epistemicism, as things stand in the literature on comparability. This is because all existing treatments of small improvement cases posit sharp boundaries. The only difference is that they move them. Not only that, having moved them, they treat them differently from the first-order boundaries they attempted to avoid in the first place.
According to Broome's (Reference Broome and Chang1997) early form of supervaluationism, there is a precise and knowable answer to the question ‘how many frescoes must Francis paint for ‘Francis is more creative than Kate’ to shift from being neither true nor false to being true?’Footnote 17 Similarly, according to Chang's parity view, for some predicates ‘Fer than’ and ‘as F as’, there is no sharp transition from ‘x is Fer than y’ to ‘x is as F as y’, but there is a sharp transition from ‘x is Fer than y’ to ‘x is on a par with y with respect to F’ which, it also seems, is precise and knowable.Footnote 18 The problem is that it seems to be just as hard to determine where these boundaries are as it does to determine where the first-order boundaries are: hard cases recur at higher orders. If Broome and Chang were to say that these boundaries are sharp but unknowable, then they would be committed to an epistemicist account of higher-order hard cases. But then, it is difficult to see why they would not just accept an epistemicist account of first-order hard cases.
Thus, two of epistemicism's main rivals posit sharp and knowable boundaries.Footnote 19 This seems at least as unbelievable as the sharp and unknowable boundary posited by epistemicist comparabilism. It is the unknowability of the boundary that accounts for the intuitive inquiry resistance of hard cases, hard hard cases and so on. In this respect, existing rivals fail to explain the phenomena. They treat higher-order hard cases differently from the first-order cases raised by the SIA, despite the apparently identical phenomenology of first-order and higher-order cases.
By contrast, epistemicists happily accept higher-order vagueness. For the epistemicist, higher-order vagueness is understood as ignorance of our ignorance at lower orders of vagueness (Williamson Reference Williamson1994: 3). The zone of borderline cases has sharp boundaries, but we cannot know where they are because there are borderline borderline cases. For instance, we cannot know which is the first borderline case of the monadic ‘is bald’ or the dyadic ‘more creative than’. Therefore, it seems to be a dialectical advantage of epistemicist comparabilism that it bites the first ‘sharp boundary bullet’ that comes its way.
Could these rival views adapt to better account for higher-order cases? We find it hard to see how Chang could plausibly alter her account to deal with higher-order hard cases. In accord with the spirit of parity, she could propose that in the hard cases of the application of ‘on a par’, there is a zone of ‘super-parity’. But such a move seems to be prima facie unattractive. Supervaluationism seems to be better placed in this regard since supervaluational accounts of higher-order vagueness have been presented in the vagueness literature (Williamson Reference Williamson1999). This has, though, yet to be done in the literature on comparability. Moreover, even if it is done in the future, supervaluationism has to posit knowable sharp boundaries somewhere (Sorensen Reference Sorensen2001: 82–3). In sum, epistemicist comparabilism is not only worth taking seriously in the debate about the SIA, it has significant advantages over existing rival accounts.
It is true that some philosophers deny the existence of higher-order vagueness (Wright Reference Wright, Dietz and Moruzzi2010). Nonetheless, as things stand in the debate on comparability, epistemicist comparabilism has an advantage on the higher-order vagueness front. The phenomenology of first-order borderline cases and putative higher-order borderline cases does appear to be the same: further inquiry will not enable us to know whether a predicate applies at higher-orders.Footnote 20 Thus, it certainly seems as though an explanation is required from proponents of non-epistemicist theories of comparability, which has not yet been developed. They must either explicitly deny the possibility of higher-order vagueness or explain why they treat it differently from first-order vagueness.
4. CONCLUSION
Many have thought that the hard cases raised in some SIAs simply cannot be cases of ignorance. How can they be, given that all of the relevant information is in, yet the cases remain resistant to inquiry? Since they are not cases of ignorance, it must be the case that each component of the trichotomy fails to hold. Epistemicism about vagueness provides an escape from this train of thought. If these hard cases are borderline cases of vague predicates, and if epistemicism is true, then one of the components of the trichotomy is true but we cannot know which. The unknowable boundaries posited by epistemicism force us towards comparabilism about small improvement cases. Furthermore, even if hard cases are not cases of vagueness, if epistemicism is true, then options cannot be on a par.
We have argued for several further conclusions as well. For one, epistemicist comparabilism has an important advantage over existing rival accounts of the SIA. Epistemicist comparabilism appears to be better equipped than any existing rival theory to account for all hard cases: it can account for hard cases, hard hard cases, and so on. Rival theories, on the other hand, fail to cope with higher-order hard cases. For another, our argument may be practically important for axiology: many have thought that the SIA provides one argument for the incompleteness of the betterness ordering, by showing that some options can be incomparable with respect to goodness. But if our thesis is correct, then the SIA provides no such argument.
ACKNOWLEDGEMENTS
We would like to thank Tim Williamson and John Broome for helpful comments on a much earlier version of this paper. We are grateful to an audience at the White Rose Early Career Ethics Researchers Conference at the University of York for helpful comments and suggestions, and to Roy Sorensen for advice. Johan Gustafsson, Michelle Hutchinson and Marinella Capriati also provided incisive and supererogatory comments. We would especially like to thank Fabienne Peter and two anonymous reviewers at Economics and Philosophy for their penetrating criticisms.
