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THE SMALL IMPROVEMENT ARGUMENT, EPISTEMICISM, AND INCOMPARABILITY
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Edmund Tweedy Flanigan; John Halstead
2018
[This is the penultimate draft of an article that appears in
Economics & Philosophy, Vol 34, iss. 2, pp. 199-219. Please cite
that version:
http://doi.org/10.1017/S0266267118000019.]
## 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).[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 1997). 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 (2002) 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 1997;
Wasserman 2004; Gustafsson 2013; Williams 2016). 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 1994, chap. 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.[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 2013). Thus,
supervaluationism's hegemony in debates about the SIA is certainly
open to question.[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 1988;
Williamson 1994).[4] Whether those defences succeed is, of course,
a judgment 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, formalise 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.
## Comparability and the Small Improvement Argument
### 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 (2002) 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.[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
1997). 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.[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.
### 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 generalised and formalised 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).[7]
### 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 believedthat 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. 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 1988; Greenough 2003; Bueno and Colyvan 2012). 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
1994, 2; Keefe and Smith 1997; Sorensen 2015).[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.
## Vagueness and Ignorance
We will now briefly review the implications of supervaluationism
for the SIA before moving on to consider the implications of
epistemicism.
### 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.[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 the supervaluationism explains the law
of the excluded middle (Broome 1997, 82). According to the
supervaluationist, vague concepts can be admissibly 'sharpened' in
numerous ways. For instance, some 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.[10]
This ambiguity can also be brought out by examining the
formalisation 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, 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 false. 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.
### 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
1994, 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 1994, 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.[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 1997). 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...,
y_(n)) ordered in terms of their Fness. Epistemicist comparabilism
posits that only one of the options in the y-set, y_(e) 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 indicates the set of borderline
cases for which we are incurably ignorant which of the components
of the trichotomy holds:
However, as Broome (1999, 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, S_1, of y options
which are Fer than x, and a subset, S_3, which are less F than x.
However, it posits that, on the scale of Fness there is another
subset, S_2, of y options sandwiched between S_1 and S_3, which are
neither Fer than nor less F than x. Provided the options in S_2 and
x are not equally F, the options in S_2 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..., y_(n)) ordered in terms of their
Fness. The grey box indicates the set of borderline cases for which
we are incurably ignorant of which of the components of the
trichotomy holds:
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 S_2) 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 y_(e) are equally F, then all other options
(apart from y_(e)) in the sequence of items (y_1, y_2,... y_(n))
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 y_(e) must also
be Fer than x because x and y_(e) are equally F; and the subset of
options that are less F than y_(e) must also be less F than x
because x and y_(e) 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 S_2).
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.[12] There are
reasons to believe that this is a mistake.
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.
Secondly, 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 characterisation 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).[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.[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. 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.[15] There is a great deal of controversy about
whether goodness can be reduced to betterness or vice versa
(Gustafsson 2014), and some of this controversy may extend to
the monadic-dyadic principle. For example, Gustafsson (2017) 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 2017, 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 generalises 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.[16] If epistemicism is true, then in small
improvement cases incomparabilists confuse our ignorance of a
ranking with the lack of a ranking.
### 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 (1997) 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?'[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.[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.[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
1994, 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 1999). 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 2001, 82-83). 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 2010). 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.[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.
## 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
fails on this front.
## Acknowledgments
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.
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Biographical Information
Edmund Tweedy Flanigan is a PhD Candidate in Political Theory at
Harvard University. His doctoral research focuses on political and
legal obligation and authority. He also holds an M.Phil. in
Political Theory from the University of Oxford.
John Halstead is a researcher at Founders Pledge in London,. He
holds a DPhil in Political Philosophy from Oxford, and previously
taught at the Blavatnik School of Government.
## Notes
[1] The argument was initially introduced in (de Sousa 1974). For a
classic discussion and overview of the SIA, see (Chang 1997).
[2] We explain why this is in section 3.
[3] This is but one of many criticisms of supervaluationism. For
classic texts both critical and in favor of supervaluationism, see
(Williamson 1994; Keefe and Smith 1997; Sorensen 2001).
[4] A number of other prominent philosophers endorse epistemicism,
including John Hawthorne (Hawthorne and McGonigal 2008), Nicholas
Rescher (2009), and Patrick Greenough (2003). Dunaway (2016) has
explored the implications of epistemicism for practical reasoning.
[5] Elson (2014) provides strong criticism of another crucial part
of Chang's argument against the Trichotomy Thesis. In section 3, we
show that even if hard cases are not cases of vagueness, if
epistemicism is true, then there are very strong reasons to doubt
the possibility of parity, and that parity struggles to deal with
higher-order hard cases.
[6] Some philosophers argue that borderline propositions are both
true and false (Hyde 1997). We bracket these accounts here.
[7] Note that in order to highlight the contrast between
epistemicism and supervaluationism, we include the truth predicates
in the premises themselves. We explain why we do this in the next
section.
[8] One of the virtues of this necessary condition is that it is
ecumenical between different theories of vagueness. The condition
is not sufficient because there appear to be other propositions
that are resistant to inquiry, but which are not borderline cases
of vagueness. For example, Goldbach's Conjecture might be resistant
to inquiry, but this has nothing to do with vagueness (Williamson
1997, 926).
[9] Broome (1997) first developed supervaluationism with respect
to comparability. His discussion at pp. 88-89 reflects the
ambiguity we discuss here. Some philosophers have argued that
supervaluationism is inconsistent with incomparability (Gustafsson
2013; Espinoza 2008).
[10] In his later work, Broome develops a version of
supervaluationism which is about assertability rather than the
truth. He argues that the new theory is incompatible with
incomparabilism (Broome 2004, chap. 12 and 14). However, since the
new theory is about assertability, and the theory severs the
logical connection between assertability and truth, it has no
implications for the SIA or incomparabilism (or the sorites
paradox).
[11] Note again that this theory only implies that items are
comparable in small improvement cases, and is compatible with there
being incomparability for other reasons.
[12] This is intuitive, but the explanation for it is somewhat
cumbersome. For all pairs of items and all comparative predicates,
there are two possibilities. Firstly, there are incomplete
orderings, characterised by vagueness, for which there is one item
x, and a subset of items, S_2 (y_(k), y_(k+1)..., y_(n)), ordered
in terms of their Fness, and all members of this subset are neither
Fer than nor less F than x. For each comparison between x and an
item in S_2, we know that equally F never applies. Thus, these
cases are not inquiry resistant. Secondly, there are complete
orderings that are not characterised by vagueness. For all pairs of
items in these orderings, if we had perfect knowledge and were
conceptually competent, then reducing contingent ignorance and
improving conceptual competence would enable us to know whether or
not 'equally F' applies.
[13] For example, does the fact that Nigel is dead and Brian is not
dead entail that Nigel is deader than Brian? Our intuition is that
this comparative is nonsensical, but others may disagree. We
discuss more difficult cases below, but we claim nonetheless that
the principle holds for a very wide range of comparatives.
[14] Of course, not all borderline cases of comparatives are either
side of the monadic threshold. For example, Mozart and Michelangelo
were both creative. But this is a point about the structure of the
epistemicist incomparabilist view.
[15] We thank a reviewer for pushing us on this point. That
reviewer also suggested the following counterexample: Assume a
hedonic axiology and imagine two states of affairs, S[Jane is happy
to degree 10] and S*[Jane is happy to degree 20 or unhappy to
degree 10]. S, let us say, is a good state; and S*, plausibly, is
not. But because of the uncertainty of S*, it is not obvious that
S is better than S*. Indeed they may seem incomparable. Our
response to this is chiefly that we take relations of axiological
betterness to hold between states of affairs, rather than between
uncertain prospects, and therefore that this cannot show that the
monadic-dyadic principle fails to hold, as a principle about
axiological betterness. (This is only one of several possible
rejoinders; if S* is instead understood as a disjunctive state of
affairs, we would reply that we hesitate to accept the existence of
such states.) Nonetheless, as we discuss in the main text, we
recognise that on some plausible axiologies, the monadic-dyadic
principle may be false.
[16] Moreover, if epistemicism is true, then our argument here
shows that 'comparable to' is not vague. If epistemicism is true,
we know that one of the components of the trichotomy applies. So,
'comparable to' has no borderline cases and is therefore not vague.
Elson (2014) discusses the possibility that 'comparable to' is
vague.
[17] In his later work, Broome acknowledges that there is
higher-order vagueness, but says he excludes it from his theory for
the sake of simplicity (Broome 2004, 180-81). It is not clear that
this is justifiable. After all, we could rule our first-order
borderline cases for the sake of simplicity as well, but we do not
do so because we want to account for the phenomena.
[18] Chang argues that incomparabilism is implausible because
it says that there is a precise transition from a zone of
comparability to one of incomparability (Chang 2002, 673-79).
Chang's theory has an exactly analogous feature. For criticism of
Chang's argument see (Elson 2014).
[19] Raz is not explicit about this, but the same seems to apply to
his arguments for hard incomparability and for incomparability
grounded in semantic indeterminacy (Raz 1986, chap. 13).
[20] Indeed, as we mentioned above, Broome already accepts that
there is higher-order vagueness, and Chang argues that there cannot
be a sharp transition from a zone of comparability to a zone of
incomparability. Thus, she accepts something very close to the
intuition underlying higher-order vagueness.
