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Towards the dissolution of the sorites paradox
Enric Trillas a,*, Luis A. Urtubey b
a European Centre for Soft Computing, Mieres, Asturias, Spainb Universidad Nacional de Cordoba, Cordoba, Argentina
Received 26 March 2007; received in revised form 9 January 2008; accepted 15 January 2008
Available online 31 January 2008
Abstract
This paper tries to open a way of discussion on the existence of effective methods to determine on which cases the point of separation between
P and not-P (suggested by Max Black) can be computed.
# 2008 Elsevier B.V. All rights reserved.
Keywords: Sorites and falakros paradoxes; Point of separation; Fuzzy sets
www.elsevier.com/locate/asoc
Available online at www.sciencedirect.com
Applied Soft Computing 11 (2011) 1506–1510
1. Introduction: sorites and falakros
The statement ‘‘This is a heap’’ is actually a perception-
based statement whose assertion implies some familiarity with
the term heap, namely, the capability to recognize that, for
example, a given set of sand’s grains deserves to be called a
heap. Otherwise, ‘‘This is a heap’’ would be a non-sense
statement, since the word heap is used in a particular situation
where the utterer distinguishes between the set of grains
constituting the heap, and the ‘‘heap’’ in itself.
In the philosophical discussions on the sorites paradox what
customarily matters is the number of grains, without any
reference to its shape. However, it is important to distinguish a set
Hk with k grains from a heap hk with the same number of grains.
Consequently, if n is the initial number of grains, we will
search for the existence of a number m < n, such that:
Hn\hn;Hn�1\hn�1; . . . ;Hm\hm;Hm�1 � hm�1;Hm�2
� hm�2; . . .
indicating by the sign ‘‘�’’ the concept ‘‘indistinguishable
from’’.
This number m, provided it exists, can be called the heap’s
threshold or, in Black’s sense in [8], the point of separation, that
separates a concept called P from its negate or one of its
opposites. Nevertheless, and contrarily to the obvious fact
Hk \ Hk � 1 for all k, it seems that hk can be considered as an
* Corresponding author. Tel.: +34 985 456545; fax: +34 985 456699.
E-mail address: [email protected] (E. Trillas).
1568-4946/$ – see front matter # 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.asoc.2008.01.008
imprecise predicate, since it is difficult to appreciate the
difference between hk and hk � 1 (at least before some value of
k). Consequently, in the following treatment of the issue, heap
will be taken as an imprecise predicate.
Discussions about the so-called sorites and falakros paradoxes
have a long-standing tradition in the philosophical study of
vagueness. More specifically, in the setting of fuzzy set theory,
this paradox shows that classical sets cannot constitute the
extension of vague predicates, as Goguen showed in the early
days of fuzzy logic in [4], since predicates of this kind do not
specify a function which just takes the values 1 and 0 like a
classical set. Therefore, a logical calculus involving vague
predicates calls for an appropriate notion of extension that is to be
different from the notion given in terms of classical sets.
Moreover, one wonders whether these paradoxes still remain
when they are considered from the perspective of fuzzy set
theories.
Sorites paradox, the paradox of the heap, comes from the
following reasoning: taking away a grain of sand does not make
a heap a non-heap. Even by losing this heap another grain, it
will remain a heap. Recursively, process goes in like manner
until just one grain of sand is left. Finally, something without
any grain of sand should count as a heap. This is paradoxical;
the empty set is a heap!
The paradox of the bald man (falakros) follows from a
reasoning like this: if one hair is implanted to a bald man, he
will remain bald. Accordingly, at the end of the process a man
with his head full of hairs is also bald.
Reasoning of this kind, involving the use of words such as
‘‘heap’’ and ‘‘bald’’, must be rejected, from the very use of
E. Trillas, L.A. Urtubey / Applied Soft Computing 11 (2011) 1506–1510 1507
these words, because it leads to unacceptable conclusions.
Which is the point of the process when a heap is no longer a
heap or a not-bald man becomes bald? Similarly, being P an
imprecise predicate, when do the moment arrive at which the
statement ‘‘x is P’’ start to be more ‘‘x is not-P’’ than ‘‘x is P’’? Is
there any separation point between P and not-P?
The aim of this paper is to show that there is an algorithm
that allows finding a separation point between P and not-P, at
least when a fuzzy set represents the use of P by means of a
monotonic membership function mP. The separation point also
exists when P is compared with one of its antonyms [1], antP,
instead of not-P.
Effectively, the problem posed by the paradox boils down to
the manner in which P, not-P and antP are used. What does it
mean ‘‘heap’’ and ‘‘bald’’? Addressing this question is crucial
to carry on the reasoning involved in the sorites and falakros,
since both ‘‘heap’’ and ‘‘bald’’ appear at each step in the
reasoning. Words are used in a given context; therefore, they
have got to be fixed from the beginning. To be able to dissolve
the paradox, in the sense of the later Wittgenstein, it will be
necessary to find the very moment when a thing that was
reasonably called a heap does not deserve this label any more.
This requirement can be met, for example, by knowing a
quantity of grains so that a pile of sand under this specific
number of grains no longer counts as a heap.
Given a predicate P on a universe of discourse X, whose
meaning (or use, following Wittgenstein [6]) is described by a
membership function mP: X! [0,1] and once the negation of P,
not-P = P0, is obtained by means of a strong negation
N:(0,1)! [0,1] (see [3]), the reasoning considered in the
preceding sections leads to study the xs which belong to X such
that for them the inequality
mPðxÞ � mP0 ðxÞ ¼ NðmPðxÞÞ ½��
does hold.
Since N can be defined by some order-automorphism
w::[0,1]! [0,1], (see [2]) as N(a) = w�1(1 � w(a)) for all a in
[0,1] inequality [*] is equivalent to
mPðxÞ � ’�1ð1� ’ðmPðxÞÞ
in its turn equivalent to 2w(mP(x)) � 1. Hence, inequality [*] is
equivalent to
mPðxÞ � ’�1
�1
2
�;
and it seems possible, in some cases, to find a threshold or
separation point of P from the set
P ¼�
x2X; mPðxÞ � ’�1
�1
2
��
For example, if X is a closed interval [a,b] of real numbers, it
is possible to consider either Inf P, or Sup P, or some weighted
mean between a and b.
2. A digression on philosophical approaches to the
paradox
The invention of the puzzles, later known as sorites and
falakros, is attributed to the logician Eubulides of Miletus, a
contemporary of Aristotle, although some scholars trace its
origins even earlier. What is clear is that Eubulides was the first
to employ the sorites, perhaps to attack other philosophical
doctrines. However, it was around 300 (B.C.E.), when sorites
arguments became prominent in disputes between two rival
schools of philosophy, the Stoa and the Academy. Most notably,
the skeptics of the Academy used sorites arguments against the
stoic’s claims to knowledge.
Arguments of this kind had a customary presentation
through series of questions, where the right answers to the first
and last questions are obvious and opposite, but troubles to
answer the successive questions appear sooner or later in the
course of the process. In the case of stoic philosophers, since
for them there truly exist cut-off points in a sorites sequence, the
difficulty in answering the questions comes from our ignorance
of what the answers are. Consequently, according to the stoics,
at some point in the interrogation one should fall silent and
withhold ascent, although not even the wise man can locate the
last clear case with perfect accuracy to stop judging. As
Williamson has pointed out in [10], on which these historical
remarks are largely based, the power of the interrogative force
stems from the fact that the questions force one to take up
attitudes for or against the individual propositions.
Beside this presentation through a series of questions, sorites
reasoning can be and was presented as a formal argument
having logical structure. There are two argument forms of the
sorites that became more popular, namely, one that is based on
conditionals and the application of modus ponens, and another
one employing the principle of mathematical induction. In this
case, the quantified premise 8n(Fan! Fan + 1) replaces the set
of conditional premises. Repeated application of simple logical
rules leads to the unpleasant consequence. Now then, some-
thing is gained by presenting the sorites as an argument with
premises and conclusion, only if the premises have positive
support. In this case, the argument form takes primacy, because
the question form leaves too much unsaid. Galen (CE c.
129 � c. 199) seemed to have been well aware of this, when
saying ‘‘I know of nothing worse and more absurd than that the
being and non-being of a heap is determined by a grain of
corn’’. In this case, it is not allowed to suspend judgment – as
stoics used to advice – on the general claim that one grain does
not make a difference between a heap and a non-heap.
More to the point here is what seemed to have been Galen’s
specific interest in the sorites, which had to do with a dispute
between empirical and dogmatic doctors. The empirical doctors
based their medical knowledge in inductive inferences, which
turns out to be reliable just in case they are supported by
‘‘sufficiently many experiences’’. Precisely this concept of
‘‘sufficiently many’’ is attacked by dogmatic doctors by using
sorites arguments. Interestingly, the empirical replied that if
this notion were debarred much of the commonsense reasoning
would meet the same fate. Notably, logic did not seem to matter
E. Trillas, L.A. Urtubey / Applied Soft Computing 11 (2011) 1506–15101508
to the Empiricist, and they did not try to demarcate the
phenomenon in logical terms. Similarly, some modern authors
have pointed out the connection between sorites puzzles and
concepts applied on the basis of observation, as was also
emphasized in the beginning of this paper. As noted by
Williamson it seems that some commentators of Aristotle did
establish a link of this kind, in which they took the paradoxes to
show that empirical concepts must be applied case by case.
General rules do not substitute observation and good judgment.
Surprisingly, where the authors nowadays take observation to
originate the paradoxes, the Aristotelian commentators took it
to solve them.
Afterward, sorites paradoxes seemed to attract no attention
until the late nineteenth century, when formal logic assumed an
outstanding role in philosophy. As with any paradox, currently
four responses appear to be available. One might (a) deny that
logic applies to soritical expressions, (b) deny some premises,
or (c) deny its validity. These last two options follow from the
first one, in case that one might accept that the sorites paradox
constitutes a legitimate argument to which logic applies.
Finally, what seems to be the last resort, (d) accept the paradox
as sound. Different authors have taken each one of these options
in the last decades and they still animate philosophical
discussions on the subject. Notably, most contenders in the
philosophical arena have confronted the paradox from a strictly
‘‘logico-semantic’’ perspective, both in their methods and
presuppositions.
Joseph Goguen’s brilliant work [4] ‘‘The logic of inexact
concepts’’ constitutes a possible exception to the rule. In his
fundamental contribution, Goguen put forth what turned out to
be one of the most perspicuous treatments of the sorites-type
paradoxes. Notably, Goguen remarked in his paper that
‘‘representing concepts by sets and deduction by method of
traditional logic does not yield an adequate model of our
customary use of inexact concepts and deduction’’. Moreover,
Goguen’s main idea was to represent inexact concepts by means
of fuzzy sets. Notwithstanding, the solution proposed to the
paradox also boils down to the formal criteria of rendering the
inference invalid, inasmuch as successive application of modus
ponens makes the inference less and less valid, according to a
measure of its validity. A more sophisticated approach in line
with this point of view has been developed by Hajeck and
Novak in [9].
Incidentally, some authors, like Black in [8], have insisted
on the importance of being aware that sorites-type reasoning
employs observational terms, i.e., it involves the making of
judgments or at least, statements concerning what judgments
have to be made under certain circumstances. In fact, Black
opened the door towards an empirical point of view concerning
the analysis of sorites that is different from those others that,
preponderantly, belong to a formal and purely abstract view.
This paper aims to contribute as well to look at sorites as a
perception-based problem (see [5]). Nonetheless, in all
honesty, given the predicates’ imprecise character, it is to be
expected that the above-mentioned algorithm will give, for the
most part, approximate results. As it will turn out clear in the
following paragraphs, in some cases it would be more
appropriate to take an interval instead of a single separation
value between P and not-P. Moreover, it will be occasionally
more advisable to take a value over or above the point xs given
by the algorithm.
In short, when the use of a predicate P on a closed interval
[a,b] �R is represented by a membership function mP and their
negations P’ by mP’ = NomP, being N a strong negation, it results
that only for x in part P� [a,b] really can make sense to say ‘x
is not-P’, since in P x is less P than not-P. When x is in [a,b] – P,
it has more sense to say that ‘x is -P’.
Consequently, P turns out to be something like a maximal
classical extension of the imprecise predicate P, once its use is
known. Perhaps this fact could contribute, in the line of thought
of Max Black, to dissolve the sorites, if these problems, as
Wittgenstein says in [7], ‘‘are in a real sense dissolved – like a
lump of sugar in water’’. At least from the point of view of those
words whose meaning depend on perceptions.
3. Toward the dissolution of the sorites
Let n be a natural number big enough to guarantee that n
grains of sand make a heap. Let use the word heap in such a way
that up to 2n/3 grains actually make a heap; between 0 and n/3 is
not a heap and between n/3 and 2n/3, the word heap is
predicated with grade 3x/n � 1. It follows that for H = heap and
being x the number of grains,
mHðxÞ ¼0; if x � n=3
3x
n� 1; if n=3 � x � 2n=3
1; if 2n=3 � x � n;
8><>:
Notice that it turns out to be a decreasing function.
Consequently, non-heap = H’, provided that it is defined as
mH’(x) = 1 � mH(x), do has the membership function:
mH0 ðxÞ1; if x� n=32� 3x
n; if n=3 � x � 2n=3
0; if 2n=3 � x � n;
8><>:
Hence, the separation point will be the greatest x0 such that
mH(x0) < mH’(x0), i.e., 3x0/n � 1 < 2 � 3x0/n: x0 < n/2. The
first integer greater than 2, can be taken as the point of
separation. If it were n = 50,000, for example, it would be
x0 = 25,000, and if n = 50,005, x0 = 25,003; and in that case
mH(25,000) = mH’(25,000) = 0.5
Consequently, if the words heap and non-heap are used in the
above-specified form, and starting from a heap of 50,000 grains
of sand, then with less than 25,000 grains the word heap cannot
be properly used.
Even if the word heap is used in a more complex way, by
considering, for example, a three-dimensional or properly
arranged distribution of grains, the paradox will receive an
analogous treatment. However, both the sorites and the
falakros, in their classical versions, refer only to the number
of grains or hairs, respectively.
E. Trillas, L.A. Urtubey / Applied Soft Computing 11 (2011) 1506–1510 1509
4. The case of ‘‘small’’ in the unit interval
Let X = [0,1] be the unit interval of the real line, and let
assume that the imprecise predicate S = small is used in X
according to the function:
msðxÞ ¼0; if 0 � x � 0:31; if 0:7 � x � 17
4� 5x
2; if 0:3 � x � 0:7
8><>:
Obviously, x = 0.3 gives a point of separation between small
and non-small; however this separation turns out to be
excessively rigid, since it overlooks the points lying in the
interval (0.3,0.7), which are small with grade higher than 0 and
less than 1; only the points in (0.7,1) are no longer small in a
definite sense. If mS’(x) = N(mS(x)), being not-P used in a
decomposable form [2] by means of the strong negation
function N(a) = 1 � a/1 + a, it follows that:
ms0 ðxÞ ¼0; if 0 � x � 0:31; if 0:7 � x � 17
4� 5x
2; if 0:3 � x � 0:7
8><>:
Accordingly, mS(x) < mS’(x) if x > 0.534. Namely, the
points x of X starting from x = 0.534, are more properly
considered not-small than small, according to this use of the
predicate. Depending on the granularity, one can say, for
example, that the point of separation is either 0.53 or 0.535. In
that case, mS(0.535) = 0.413 and mS’(0.535) = 0.416.
5. The algorithm
If P is used in X according to the function mP(x): X! [0,1],
where mP(x) is monotonic, increasing or decreasing, then the
following algorithm allows finding the point of separation xS
between P an not-P.
1. Design the function mP.
2. Consider the use of not-P and design mP’.
3. Solve the equation mP(x) = mP’(x), which has solution Ss.
4. If mP is increasing, take the greatest solution xS.
5. If mP is decreasing, then take the least xS.
6. The case with an antonym
The same algorithm can be used, if instead of not-P, the
antonym of P, antP, is given. Accordingly, the equation
mP(x) = mantP(x) must be solved. For example, inX = [0,2], if the
predicate T = tall is used through the following func-
tion,mTðxÞ ¼1; if 1:75 � x � 2
0; if 0 � x � 1:60x
0:15� 1:60
0:15; if 1:60 � x � 1:75
8><>: then, since
mT(x) is a monotonic increasing function, a use of the predicate
S = short can be represented by the function [1]
mS(x) = mT(2 � x), such that:
mSðxÞ ¼1; 0 � x � 0:25
0; 0:40 � x � 2
2x� =0:15� 1:60=0:15; 0:25 � x � 0:40
8<:
Hence, each x in the interval [0.40,1.60] yields a solution to
the equation mT(x) = mS(x). Therefore, being the initial function
increasing, xS = 1.60 can be taken as the separation point
between tall and short.
7. The case with not-P and ant P
The pair (P, not-P) yields a point of separation x0S, and the
pair (P, antP) another x00S. Consequently, the ultimate point of
separation might be located in a weighted average of both the
values x0S and x00S. For example, taking again the case in
paragraph 4, with P = small and Q = big, a use of Q is given by
[1] mQ(x) = mP(1 � x). Hence,
mSðxÞ ¼1; 0:7 � x � 1
0; 0 � x � 0:37=4� 5=2ð1� xÞ ¼ 5x=2� 3=4; 0:3 � x � 0:7
8<:
Because of that, the following average can be taken as the
point of separation:
xs ¼0:534mPð0:543Þ þ 0:5mQð0:5Þ
mPð0:543Þ þ mQð0:5Þ
¼ 0:534 0:415þ 0:5 0:25
0:415þ 0:25¼ 0:53
8. A case with a negation N 6¼ 1 � id
Take X = [0,1] and P = big, with mP(x) = x. Provided not-
big = P’ is obtained by mP’(x) = N(mP(x)) = N(x), with N given
by
NðaÞ ¼ 1� a
1þ a:
It results mP(x) � mP’(x) equivalent to x � (1 � x)/(1 + x) or to
x2 + 2x � 1 � 0. Hence,
fx2 ½0; 1�; x2 þ 2x� 1 � 0g ¼ fx2 ½0; 1�; x � 0:414g;
and xS = Sup {x2[0,1]; mP(x) � mP’(x)} = 0.414, since mP(x)
is a non-decreasing function. For all x � xS = 0.414 it will be
mP(x) � mP’(x).
9. A case with a non-monotonic function
The example in this case is with P = around 4, in X = [0,10],
with the use of P given by the function:
mPðxÞ ¼
0; if 0 � x � 3:5; 5 � x � 10
1; if x � 4
x� 3; if 3 � x � 4
5� x; if 4 � x � 5
8>><>>:
E. Trillas, L.A. Urtubey / Applied Soft Computing 11 (2011) 1506–15101510
In that case, if not-P is used by means of mP’(x) = 1 � mP(x),
it follows that,
mP0 ðxÞ ¼
1; if 0 � x � 3; 5 � x � 10
0; if x ¼ 4
4� x; if 3 � x � 4
x� 4; if 4 � x � 5
8>><>>:
Therefore, in that case there are two solutions to the equation
mP(x) = mP’(x), that come from:
x � 3 = 4 � x, x0 = 3.5
5 � x = 4 � x, x00 = 4.5
Consequently, it is possible to obtain:
xs ¼3:5xmPð3:5Þ þ 4:5xmPð4:5Þ
mPð3:5Þ þ mPð4:5Þ¼ 3:5x0:5þ 4:5x0:5
0:5þ 0:5¼ 4
The point xS = 4 separates ‘‘around 4’’ from ‘‘not-around 4’’.
Obviously, another use of P and not-P would lead to a different
value for xS.
10. Conclusion
It was one of the paper’s main contentions that, in certain
cases, it is possible to determine the sorites separation point
referred by Black in [8]. Such cases are those given by
predicates P that are measurable in the unit interval. That is, that
for all x in the universe of discourse there is a number indicating
the ‘‘degree up to which x is P’’ = mP(x), and that this number
belongs to [0,1].
Effectively, in a certain way it is necessary to interpret what
Black means when saying, ‘‘ I stop whenever I judge that my
words are ceasing to work properly’’, by finding a point beyond
which it is likely to stop. Naturally, what the imprecise
character of the vague terms involved causes, is that this
threshold or separation point can just be approximately fixed;
notwithstanding, as it has been shown in this paper, this so-fixed
separation point allows in many cases pointing out the line of
demarcation beyond which a given predicate P ceases to apply
properly to the objects in the universe of discourse, since from
there on this objects satisfy P in a strictly lesser grade compared
with the grade that they are not-P. As it should turn out clear, in
some cases it seems to be more appropriate to take an interval
instead of a single separation value between P and not-P. In this
way, when it was allowed to work with antonym and negate, the
ultimate point of separation was located in a weighted average
of the interval’s endpoints. Moreover, and according to the
function, eventually it would be better that a value over or above
the xs given by the algorithm was taken.
What is absolutely essential is to acknowledge that it does
not make any sense to talk about the sorites paradox without
being aware of what is meant by the word ‘‘heap’’. The paradox
will vanish as soon as the use of the predicate is known,
provided that ‘‘heap’’, or the soritical predicate, whichever you
choose, be represented by means of the membership function of
a fuzzy set.
Acknowledgments
The authors wish to thank the two anonymous referees
whose comments helped to improve an earlier version of this
paper
The first author has been partially supported by the Spanish
grant TIN 2005-08943-C02-01.
The paper was accomplished by the second author during a
period he spent as visiting researcher in the European Centre for
Soft Computing (ECSC) in Mieres, Asturias, Spain, in June
2007, mainly supported by the ECSC and partially supported by
the FonCyT grant BID 1728/OC-AR-PICT 2002-04-12822 and
SeCyT (U.N.C.) Argentina. The author is deeply indebted to the
ECSC for the hospitality extended to him there.
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