Knowledge Within the Margin for Error
Timothy Williamson
Roy Sorensen s criticism of my use of margin for error principles to explain igno-
rance in borderline cases fails because it misidentifies the relevant margin for error
principles. His alternative explanation in terms of truth-maker gaps is briefly
criticized.
Roy Sorensen does not mention that the margin for error principles he
criticizes in  Knowledge Beyond the Margin for Error (Sorensen 2007)
are not those my account in Vagueness (Williamson 1994) treats as basic
to vagueness. Those he criticizes follow from the more basic margin for
error principles only under simplifying conditions. Those simplifying
conditions obtain for paradigmatic vague predicates, but not for the
more complex predicates he discusses. The margin for error principles
basic to vagueness still hold for Sorensen s examples. He has done no
damage to my account.
On my account, vagueness is distinctive because it involves margin
for error principles that concern minute shifts in the meaning of vague
expressions, or in vague concepts, too minute for speakers or thinkers
to discriminate. For example, let  small* be a vague atomic predicate
with a meaning very like that of  small , except that its cut-off point is
one less than that for  small . Thus, for all natural numbers n:
(1) n is small* if and only if n + 1 is small.
Someone who believes the proposition that n is small under the guise of
a sentence of the form  n is small could very easily have had a belief
they would have expressed by the same (physically individuated) sen-
tence, even if  small had meant what  small* actually means. The
actual belief is reliable enough to constitute knowledge only if the
counterfactual belief is true. Thus (since n is small* in the counterfac-
tual circumstances only if it is actually small*):
(2) If one knows that n is small, then n is small*.
Here and below, read  one knows that n is small as qualified by  under
the guise of a sentence of the form  n is small  . The margin for error
Mind, Vol. 116 . 463 . July 2007 © Williamson 2007
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724 Timothy Williamson
principle (2) is intimately related to the vagueness of  small . Trivially,
given (1), (2) is equivalent to:
(3) If one knows that n is small, then n + 1 is small.
Sorensen states the margin for error principle in the form of (3) rather
than (2). Although the difference does not matter in this case, it is cru-
cial in the more complex cases on which he bases his argument.
Sorensen considers complex vague predicates such as  either small or
equal to ten . The analogue of (3) for that predicate is:
(3 ) If one knows that n is either small or equal to ten, then n + 1 is
either small or equal to ten.
As he points out, we should not endorse (3 ). For all we know, ten is the
cut-off point for  small ; if so, eleven is not small. Nevertheless, we
know that ten is either small or equal to ten. In that case, (3 ) is false.
However, what matters is the analogue of (2). Since competent
speakers are sensitive to the grammatical structure of Sorensen s com-
plex predicate, and  small is its only relevantly vague constituent, the
proper analogue of (2) is:
(2 ) If one knows that n is either small or equal to ten, then n is ei-
ther small* or equal to ten.
Given (1), (2 ) is equivalent to:
(2 ) If one knows that n is either small or equal to ten, then either
n + 1 is small or n is equal to ten.
When n is ten, (2 ) is obviously true. When n is any other natural
number, (2 ) is not interestingly different from (2) for present pur-
poses. Thus Sorensen s complex predicates pose no threat to the appro-
priate margin for error principles.
One main advantage of margin for error principles is that they ena-
ble us to explain ignorance in borderline cases on independently plausi-
ble epistemological grounds: it is only to be expected, given classical
principles of logic and semantics. Principles such as (2) and (2 ) are
simply another manifestation of a more general epistemological phe-
nomenon, inexactness, which occurs even when expressions or con-
cepts are not relevantly vague, as in numerical judgements of quantity
made on the basis of unaided perception. For instance, when judging
height by naked eye at some distance, this principle typically holds:
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Mind, Vol. 116 . 463 . July 2007
Knowledge within the Margin for Error 725
(4) If one knows that a man is at least n millimetres tall, then he is
at least n + 1 millimetres tall.
We can construct disjunctive predicates like Sorensen s in such cases.
When it is unproblematic to recognize John, although our naked eye
applications of predicates of the form  either at least n millimetres tall
or exactly the same height as John are typically inexact, they are not so
in the special case of applying the predicate to John himself. The ana-
logue of (2 ) holds in this case:
(4 ) If one knows that a man is either at least n millimetres tall or
exactly the same height as John, then he is either at least n + 1
millimetres tall or exactly the same height as John.
When the man is John, (4 ) is obviously true. When he is any other
man, (4 ) is not interestingly different from (4) for present purposes.
Thus disjunctive predicates like Sorensen s give rise to relevantly similar
phenomena in cases of inexact knowledge not involving vagueness. He
has inadvertently helped strengthen the theory he was trying to refute,
by extending the comparison between vagueness and inexactness.
Sorensen s own attempt to explain ignorance in borderline cases pos-
tulates truth-maker gaps. According to Sorensen, such truths without
truth-makers are unknowable. I regard truth-maker theory as naive
and ill-motivated metaphysics. Bracket that view for the sake of argu-
ment. If truth does not require a truth-maker, it is unclear why knowl-
edge should. Moreover, granted the apparatus of standard truth-maker
theory, Sorensen s theory gives the wrong results.
Consider a chip very close to the boundary between red and orange.
We cannot know that it is red, nor can we know that it is orange. In fact,
it is just red. However, we can know that it is either red or orange (if
you think that there are a few shades between red and orange, include
 between red and orange as a third disjunct; it will make no difference
to the overall argument). Given that known truths require a truth-
maker, the known truth that it is either red or orange has a truth-
maker. By standard truth-maker theory, a truth-maker for the disjunc-
tion that the chip is either red or orange is either a truth-maker for the
disjunct that it is red or a truth-maker for the disjunct that it is orange.
Since it is not orange, the latter disjunct has no truth-maker. Thus there
is a truth-maker for the proposition that the chip is red. But that is sim-
ply a standard example of something unknowable by reason of vague-
ness. Hence truth-maker gaps are not doing the required work.
Actually, Sorensen (2001, p. 174) emulates the supervaluationist by pos-
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Mind, Vol. 116 . 463 . July 2007
726 Timothy Williamson
tulating truth-makers for disjunctions none of whose disjuncts have a
truth-maker. In doing so, he cuts off his account from one of the most
plausible candidates for a general principle about truth-makers. Truth-
maker theory in Sorensen s hands is so spineless that its explanatory
value is negligible.
A further worry about Sorensen s appeal to truth-maker gaps is that
it is in tension with accounts of truth-bearers that an epistemicist might
well find attractive. For example, suppose that ten is the largest small
natural number. Then, plausibly, the property (of natural numbers) of
being small just is the property of being at most ten. On a Russellian
account of propositions (though not of numbers), the proposition that
ten is small just consists of ten and the property of being small. One can
know that truth under the guise of a relevantly precise sentence such
 Ten is at most ten , or better by a sentence that results from replacing  at
most ten by a relevantly precise atomic predicate for the same property,
but not under the guise of the sentence  Ten is small . Since the truth
can be known under the former guise, it has a truth-maker. Thus the
proposition that ten is small has a truth-maker after all.
No doubt epicycles can be added to Sorensen s theory. It seems more
promising to investigate the epistemology of vagueness by means of
independently testable epistemological principles.1
New College Timothy Williamson
Oxford OX1 3BN
UK
[e-mail]
References
Sorensen, Roy 2001: Vagueness and Contradiction. Oxford: Clarendon
Press.
  2007:  Knowledge Beyond the Margin for Error . Mind 463,
pp. 717 722.
Williamson, Timothy 1994: Vagueness. London: Routledge.
1
This piece was written when I held a Tang Chun-I Visiting Professorship at the Chinese
University of Hong Kong.
© Williamson 2007
Mind, Vol. 116 . 463 . July 2007