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

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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: enric.trillas@sofcomputing.es (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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