Knowledge Beyond the Margin for Error
Roy A. Sorensen
Epistemicists say there is a last positive instance in a sorites sequence we just can-
not know which is the last. Timothy Williamson explains that knowledge requires a
margin for error and this ensures that the last heap will not be knowable as a heap.
However, there is a class of disjunctive predicates for which knowledge at the
thresholds is possible. They generate sorites paradoxes that cannot be diagnosed
with the margin for error principle.
No one can know the threshold of a vague predicate. For instance, given
that ten is the last small number, no one can know that ten fits the
description the last small number .1 Knowledge that ten is the last small
number requires knowledge of the weaker proposition that ten is a
small number. Knowledge of thresholds entails knowledge at thresh-
olds. Timothy Williamson ingeniously argues that this weaker knowl-
edge is always impossible. On that basis he claims to have explained our
ignorance of the thresholds of vague predicates.
My general objection is that knowledge at the threshold is often pos-
sible (and actual). Indeed, given any vague predicate for which this
knowledge is impossible, there is a necessarily co-extensive predicate
for which this knowledge is possible. This co-extensive predicate is
vague and epistemically similar to the original. Thus it can generate a
sorites paradox that cannot be diagnosed with a margin for error prin-
ciple.
The epistemicist solution to the sorites paradox is that the induction
step is false:
Base step: One is small.
Induction step: If n is small, then n + 1 is small.
Conclusion: Therefore, a billion is small.
The induction step entails absurdities such as If one is small, then a bil-
lion is small . So epistemicists, such as Williamson and me, claim to
know that the induction step is false. We deduce that there is a threshold
for small ; a number n such that n is small and n + 1 is not small. We
1
For present purposes, number is restricted to the positive integers: 1, 2, 3, and so on.
Mind, Vol. 116 . 463 . July 2007 © Sorensen 2007
doi:10.1093/mind/fzm717
718 Roy A. Sorensen
grant no one can know what the value for n is. Many object that if this
threshold exists, then it is possible that someone can learn what it is.
Williamson replies that ignorance should be expected if there is a
threshold. For knowledge requires a margin for safety:
M: If n can be known to be small, then n + 1 is small.
A known truth must differ significantly from a neighboring falsehood
(even though a truth need not differ significantly from a neighboring
falsehood). The epistemic difference across the threshold of a vague
predicate is insignificant. So it is impossible for there to be a number n
that is known to be small when its neighbour, n + 1, is not small. For
instance, if ten were the last small number, then someone who said Ten
is small would have been lucky he did not choose to make his remark
about eleven.
Williamson reinforces the margin for error principle with reliabilist
epistemology.2 (He delves more deeply than the appeal to luck.) But
enough has been said to set the stage for my specific objection. Con-
sider a sorites paradox with a disjunctive predicate:
(1) One is either small or equal to ten.
(2) If n is small or equal to ten, then n + 1 is either small or equal to
ten.
(3) Therefore, a billion is either small or equal to ten.
Although any specific value for n is unlikely to be a counterexample to
the induction step, ten is a relatively promising candidate:
(C) Ten is either small or equal to ten, but eleven is not small and
eleven is not equal to ten.
Since the first conjunct (italicized) and last conjunct (also italicized) are
certainly true, the whole statement (C) has the same probability as its
remaining conjunct Eleven is not small . (C) is true as long as the last
small number is less than eleven.
The corresponding margin for error principle is:
(MS) If n can be known to be either small or equal to ten, then n + 1
is either small or equal to ten.
Ten can be known to be either small or equal to ten. Therefore, by uni-
versal instantiation from (MS) and then modus ponens, eleven is also
small or equal to ten. If ten is the last number that is either small or
2
In chapter eight of his 1994 and chapter five of his 2000.
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Knowledge Beyond the Margin for Error 719
equal to ten, then (MS) is false. Consequently, (MS) implies that the
threshold of small or equal to ten cannot be ten, which means ten is
not last small number. This is a reckless implication. If an epistemicist
asserted Ten is not the last small number , then his brother epistemicist
would challenge him: How could you know? We epistemicists agree
that ten might be the last small number.
Explanations must be based on principles that are knowable. (MS) is
unknowable because it has an epistemic possibility of being false a
possibility that cannot be eliminated by further inquiry.
Other margin for error principles are false (not just unknowable).
Let x be the last small number. Then the margin for error principle for
either small or equal to x is false. Call that disjunctive predicate
Predicate X .
Predicate X is a vague predicate: it has borderline cases, can generate
a sorites, and passes all other tests for being a vague predicate. Thus it
triggers Williamson s margin for error meta-principle: [W]here knowl-
edge is inexact, some margin for error principle applies. 3 However,
there is no true margin for error principle for Predicate X.
Williamson cannot just restrict his principle to simple predicates.
This would leave his solution to the sorites paradox incomplete.
Disjunctive sorites arguments are not just curiosities. In my 1985, I
used a sequence of disjunctive predicates to argue that vague is vague.
The n-th predicate in the sequence is defined as either small or less
than n . The early members of this sequence are definitely vague but
eventually we reach borderline cases of vague predicate .
Williamson does anticipate disjunctive vague predicates such as
either small or equal to a billion . He plausibly assumes that they can be
tamed by adding the platitude that a predicate can have both definite
boundaries and indefinite boundaries. (Indeed, small number has a
definite lower boundary.) Since the definite boundaries and the
indefinite boundaries have distinct locations, Williamson can restrict
the margin for error principle to the areas near the indefinite bounda-
ries.
However, the upper boundary of small is identical to the boundary
of Predicate X. If x = 10, then the following three predicates have exactly
the same boundaries: small , small or equal to ten , ten or below .
Given x = 10, the three predicates are necessarily co-extensive (like
equilateral triangle and equiangular triangle ). However, the first two
predicates are vague while the third is precise. ( Ten or below has no
borderline cases and cannot generate a sorites.) According to epistemi-
3
2000, p. 227.
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Mind, Vol. 116 . 463 . July 2007
720 Roy A. Sorensen
cism, possession of a definite boundary is an epistemic feature. We can
know the boundaries of Ten or below . We cannot know the upper
boundary of small and cannot know the upper boundary of small or
equal to ten .
Continuing with the supposition that x = 10, the difference between
small or small or equal to ten is that the last small or equal to ten
number is known to be small or equal to ten (because we know ten is
small or equal to ten). We just cannot know it under the stronger
description the last number that is small or equal to ten .
The datum Williamson seeks to explain is that we cannot know the
threshold of a vague predicate. That is, we cannot know a true state-
ment of the form n is F but n + 1 is not F . His margin for error diagno-
sis is that we cannot know the first conjunct of a true threshold
statement. But Predicate X shows that we sometimes know the first
conjunct.
Williamson assumes that the applications of a vague predicate to
individuals nearer and nearer the threshold must decline in plausibility.
This assumption is plausible for predicates that only have a single
source of warrant for their application. But a predicate with two types
of warrant can have a spike in plausibility (into clear knowability) right
at the threshold.
Simple predicates commonly have multiple sources of warrant. So it
is probable that some counterexamples to the margin for error meta-
principle are not disjunctive predicates. They are just harder to specify.
Epistemicism and the margin for error principle are logically inde-
pendent. The supervaluationist Timothy Williamson first espoused the
margin for error principle in Identity and Discrimination. In this first
book he compromises: the vagueness of some predicates may be a mat-
ter of ignorance but the vagueness of other predicates (such as deliber-
ately underdefined terms) involves truth-value gaps. Only later did
Williamson renounce supervaluationism and become an epistemicist.
Thus the history of philosophy shows that one can espouse the margin
for error principle without being an epistemicist.
Since epistemicism pre-dates the margin for error principle, the his-
tory of philosophy further shows that epistemicism can be espoused
without the margin for error principle.
If the margin for error principle were correct, it would play a sup-
porting role in epistemicism just as plate tectonics plays a supporting
role in the theory of continental drift. In 1915, Alfred Wegener suggested
that continents wandered from an original, consolidated landmass.
This explains why the continents fit together like pieces of a jigsaw
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Knowledge Beyond the Margin for Error 721
puzzle and why congruent pieces share fossils, species, and geological
features. The vast majority of geologists regarded Wegener s hypothesis
as incredible. They knew of no mechanism by which huge continents
could move. The continents certainly cannot plow through the earth
like an icebreaking ship (as Wegener initially suggested); the continents
would be deformed by the immense collisions.
In the mid-1960s, J. Tuzo Wilson s plate tectonics filled the explana-
tory gap. Wilson s theory says continents are perched on immense crus-
tal plates. The plates move, some converging, some diverging, some
slipping under other plates. Thanks to plate tectonics, Wegener s long
ridiculed theory is now textbook geology.
In his 1994 Williamson followed the path of Wilson. Williamson
pitched the margin for error principle as an explanation of how we
could be ignorant of boundaries for vague predicates even though these
boundaries exist right under our noses. Just as plate tectonics should
not be identified with continental drift theory, the margin for error
principle should not be identified with epistemicism.
A refutation of the margin for error principle would damage epis-
temicism in the way a refutation of plate tectonics would damage conti-
nental drift theory. The amount of damage would depend on whether
Williamson is correct in conceding that the epistemicist is obliged to
explain ignorance of the boundaries and depend on whether there are
other prospects of plugging the explanatory gap.
In 1875, William Stanley Jevons denied that the correlation between
the sunspot cycle and business cycle was accidental; he conjectured that
there is a causal connection between sunspots and the weather. Some
ridiculed Jevons because he did not know how sunspots could be rele-
vant to weather (either as a cause or through a common cause). There
is still no known mechanism. Nevertheless, most meteorologists agree
with Jevons s meteorological conjecture (though economists disagree
with many of Jevons s assertions about the business cycle).
In Blindspots (1988) I followed the path of Jevons. I dismissed the call
for explanation as above and beyond the call of duty. I hoped someone
would eventually dispell the appearance of word magic. But I did not
wait for resolution of this issue; I accepted epistemicism as it stood,
naked of an explanatory mechanism.
I have offered guesses. In Vagueness and Contradiction (2001) I emu-
late the supervaluationist by postulating truth-maker gaps. Like the
supervaluationist s truth-value gaps, truth-maker gaps explain why we
have trouble conceiving of anything that would make it true that the
© Sorensen 2007
Mind, Vol. 116 . 463 . July 2007
722 Roy A. Sorensen
threshold is at one location rather than another. Both types of gaps
explain why there are absolute borderline cases.
The margin for error principle is relative to a cognizer. So William-
son s borderline cases are always relative to human cognizers. I have
always regarded relative borderline cases as being too weak to constitute
the vagueness that drives the sorites paradox. Consequently, I have
never been able to employ the margin for error principle.
However, the underlying relations of epistemic similarity may play a
role in spreading ignorance from truth-maker gaps to neighboring
propositions that do have truth-makers. Unlike a supervaluationist, I
do not think each unknowable has its own a gap.
At the methodological level, I still deny that an explanation of the
ignorance is a prerequisite for acceptance of epistemicism. Epistemi-
cism follows from obvious premisses by means of classical logic. That is
enough proof.
Nevertheless, an explanation of the ignorance would be welcome.
This supererogatory service would side-step ill-founded suspicion of
the simple but interesting proof that vague predicates have boundaries
that only differ epistemically from the boundaries of precise
predicates.4
Department of Philosophy Roy A. Sorensen
Dartmouth College
Hanover
NH 03755
USA
roy.sorensen@dartmouth.edu
References
Sorensen, Roy 1985: An argument for the vagueness of vague .
Analysis 27/3, pp. 134 7.
1988: Blindspots. Oxford: Clarendon Press.
2001: Vagueness and Contradiction. Oxford: Clarendon Press.
Williamson, Timothy 1990: Identity and Discrimination. Oxford:
Blackwell.
1994: Vagueness. London, Routledge.
2000: Knowledge and its Limits. Oxford: Oxford University Press.
4
This paper was presented at the Workshop on Vagueness in Ovronnaz, Switzerland, on
26 June 2006. I thank my commentator Pablo Rychter and the other participants for their com-
ments and suggestions.
© Sorensen 2007
Mind, Vol. 116 . 463 . July 2007
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