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The sorites paradox and fuzzy logicPetr Hájek a & Vilém Novák† ba Institute of Computer Science , Academy of Sciences, 18207, Prague, Czech Republicb Institute for Research and Applications of Fuzzy Modeling, University of Ostrava , 30.dubna 221, 70103, Ostrava, Czech RepublicPublished online: 17 Oct 2011.

To cite this article: Petr Hájek & Vilém Novák† (2003) The sorites paradox and fuzzy logic, International Journal of GeneralSystems, 32:4, 373-383, DOI: 10.1080/0308107031000152522

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THE SORITES PARADOX AND FUZZY LOGIC

PETR HAJEKa,* and VILEM NOVAKb,†

aInstitute of Computer Science, Academy of Sciences, 18207 Prague, Czech Republic; bInstitute forResearch and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 70103 Ostrava 1,

Czech Republic

(Received 1 January 2003; In final form 7 March 2003)

The sorites paradox (interpreted as the paradox of small natural numbers) is analyzed using mathematical fuzzylogic. In the first part, we present an extension of BL-fuzzy logic by a new unary connective At of almost true and thecrisp Peano arithmetic extended by a fuzzy predicate of feasibility. Then we give examples of possible semantics ofAt and examples of semantics of feasible numbers. In the second part, we present an analysis of the sorites paradoxwithin fuzzy logic with evaluated syntax and show that under a very natural assumption we obtain a consistent fuzzytheory. Thus, sorites is not paradoxical at all.

Keywords: Sorites paradox; Basic fuzzy logic; Fuzzy logic in narrow sense; Peano arithmetic

1. INTRODUCTION

One of the most striking paradoxes disturbing logical reasoning for more than two thousand

years is the sorites paradox attributed to Aristotle’s contemporary, Eubulides—the Magarian

philosopher who is the author of several other known paradoxes such as the liar paradox.

The standard form of the sorites paradox is as follows:

One grain does not form a heap. Adding one grain to what is not yet a heap does not make a heap.Consequently, there are no heaps.

The problem lies in the fact that in small steps proceeding little by little, we pass from truth

to falsehood. Note that another form of the same paradox is the falakros (bald man):

losing one hair does not make a non-bald man bald, thus it follows by induction that there are

no bald men. Similarly, we can proceed when reasoning about a lot of other kinds of

properties such as few, small, etc. Apparently, there are “small numbers” but not all numbers

are small. A common feature of all these examples is vagueness of the concept in concern.

This means that when passing along the possible bearers of vague property, we are

in principle unable to distinguish the moment when the given property ceases to exist.

ISSN 0308-1079 print/ISSN 1563-5104 online q 2003 Taylor & Francis Ltd

DOI: 10.1080/0308107031000152522

*Corresponding author. E-mail: hajek@uivt.cas.cz†E-mail: vilem.novak@osu.cz.

International Journal of General Systems, August 2003 Vol. 32 (4), pp. 373–383

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From the point of view of mathematical logic, let us consider a vague predicate “feasible”

(following Parikh, 1971, who investigated the problem in formal arithmetic). We will

write Fe(n) for “n is feasible”. The sorites paradox is clear: postulating Feð0Þ;ð;xÞðFeðxÞ ) Feðx þ 1ÞÞ and ð’xÞ : FeðxÞ; we immediately come to contradiction with

the induction principle.

Parikh (1971) investigated contradictory extensions of Peano arithmetic by the above

axioms strengthened even by a concrete assumption : FeðtÞ when t is a term like 10101010

.

He showed that even if in this theory one can prove contradiction in t steps, no short proof

can give contradiction (short being precisely [crisply] defined in dependence on t).

Another analysis of this paradox is provided by Vopenka (1979) who introduced the concept

of semiset. This is formally a subclass of some set, not necessarily being a set itself. A typical

semiset is that of finite natural numbers Fn containing 0 and being closed under successor,

addition and multiplication, but not containing all natural numbers. This idea gave rise to a

new alternative set theory (AST) which is equiconsistent with Zermelo–Fraenkel set theory.

Induction fails for (some) formulas containing semiset variables on the constant Fn. But for

each concrete natural number n, AST proves �n [ Fn (�n being the nth numeral).

In this paper, we are going to analyze the notion of small natural numbers inside fuzzy

logic. We will demonstrate solution within two fundamental approaches of it, namely the

basic fuzzy logic as developed in Hajek (1998) and fuzzy logic with evaluated syntax (FLn)

(called also Pavelka logic) presented in detail in Novak et al. (1999). The paper is also a

response to some publications which have appeared in recent years and which provide more

or less improper (or ignorant to recent achievements) analysis of the abilities and

contribution of fuzzy logic to the solution of the sorites paradox (cf., e.g. Read, 1995; Keefe,

2000).

The first analysis of the sorites paradox in fuzzy logic was provided by Goguen (1968–

69). In this paper, we will make his considerations more precise. Namely, we offer two

possible solutions of the sorites paradox. The main idea of the first solution is the following:

if x is small then it is almost true that x þ 1 is small (i.e. the truth degree of “x þ 1 is small” is

only a little less than that of “x is small”). In the first case, “almost true” is taken to be a fuzzy

unary connective (hedge); the reader may compare it with the connective “very true” studied

in Hajek (2001). The second solution employs the possibility to consider axioms true in

various degrees and takes the implication ð;xÞðFeðxÞ ) Feðx þ 1ÞÞ as not being fully true.

Our arithmetic will be crisp except for the predicate Fe (see also Hajek, 2000).‡

The reader is assumed to be familiar with both kinds of fuzzy logics. First, the basic

fuzzy predicate logic BL;, and three famous stronger logics Łukasiewicz Ł;, Godel G;,

and product logic P;. In particular, recall that BL has two conjunctions: the strong

conjunction & (whose standard semantics is any continuous t-norm, the residuum ! of

the t-norm being the semantics of the implication ) ) and the min-conjunction ^ , w ^ cbeing defined as w&ðw ) cÞ which gives ^ the semantics of minimum.

Second, we suppose that the reader knows the predicate fuzzy logic with evaluated syntax

described in details in Novak et al. (1999). Recall that its basic truth structure is Łukasiewicz

algebra LŁ ¼ k½0; 1�;_;^;^;!; 0; 1l and it deals with evaluated formulas, fuzzy sets of

axioms and besides many-valued interpretation as above, it employs the concepts of fuzzy

theory and the provability degree of a formula in it. The completeness theorem states that the

provability and truth degrees of each formula in each fuzzy theory coincide.

‡Note that the notion of a large number is discussed in Hajek (1998) 3.3.21 for Łukasiewicz logic and 4.1.27 forproduct logic. Note also the analysis of the liar paradox in Hajek et al. (2000).

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In Section 2, we present an extension of BL by a new unary connective At of almost true

and the theory PAat—crisp Peano arithmetic extended by the fuzzy predicate of feasibility;

we also prove a simple theorem in PAat. Section 3 contains examples of possible semantics of

At and examples of semantics of Fe (feasible numbers). Section 4 contains analysis of the

sorites paradox within fuzzy logic with evaluated syntax.

2. ALMOST TRUE AND FEASIBLE

We extend the basic predicate logic BL; by a new unary connective At (At (w) being read

“it is almost true that w”) and by the following two axiom schemata:

ðat1Þ w ) AtðwÞ;

ðat2Þ ðw ) cÞ ) ðAtðwÞ ) AtðcÞÞ:

In words, if w (is true) then w is almost true; if w implies c and w is almost true then c is

almost true (minor reformulation using (A5)). This logic is denoted BL;at: Examples of truth

functions of at are given in the next section.

Now we define the arithmetic PAat as follows. We extend the language of Peano arithmetic

PA (see Hajek and Pudlak, 1993) by a new unary predicate Fe (Fe (x) is read “x is feasible”)

and work with the variant of BL; with function symbols (see Hajek, 2000). Thus the

language consists of

(i) binary equality predicate ¼ , unary predicate Fe,

(ii) constant �0 (zero), unary function symbol S (successor),

(iii) binary function symbols þ ,· (addition, multiplication).

The axioms are as follows:

(i) x ¼ y _ x – y (crispness axiom for ¼ ),

(ii) all axioms of Peano arithmetic (including the definition of x # y as ð’zÞðz þ x ¼ yÞÞ;(iii) x , y ) ðFeð yÞ ) FeðxÞÞ;(iv) FeðxÞ ) ðAtðFeðSðxÞÞÞ ^ At ðFeðx þ xÞ ^ At ðFeðx·xÞÞ:

Lemma 1 For each formula w of PA (i.e. not containing the predicate Fe), PAat proves

w_ : w (tertium non datur).

Proof For atomic w this is an instance of the crispness axiom of equality; the rest follows by

induction on the complexity of formulas. Let us show the induction step for quantifiers.

Assume r w_ : w: Then r w _ ð’xÞ : w (since r: w ) ð’xÞ : w), thus

r ð;xÞðw _ ð’xÞ : wÞ by generalization and r ð;xÞw _ ð’xÞ : w by the axiom (;3).

From the last provability we get r ð;xÞw_ : ð;xÞw (since r ð’xÞ : w ): ð;xÞw) (see

Hajek, 1998, 5.1.20).

Similarly, from r w_ : w we get r ð’xÞw_ : w; further r ð’xÞw _ ð;xÞ : w; and finally

r ð’xÞw_ : ð’xÞw: A

Theorem 1 A formula not containing Fe is provable in PA over classical logic iff it is

provable in PAat over BL;at:

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Proof If PA r w then PAat r w thanks to the preceding lemma. Conversely, if PAat r w then

inside PA define AtðaÞ , a for each a and FeðxÞ , x ¼ x: This embeds PAat (over BL;at)

non-conservatively into PA (over classical logic) and PA r w: A

Theorem 2 PAat proves

FeðxÞ ^ Feð yÞ ) ðAtðFeðx þ yÞÞ ^ AtðFeðx·yÞÞ:

Proof We work in PAat and have the following chain of implications:

ðx # y&Feð yÞÞ ) ðx þ y # y þ y&AtðFeð y þ yÞÞ ) AtðFeðx þ yÞÞ

(noticing that PAat proves u # v ) At(Fe(v) ) At(Fe(u))). Hence

x # y ) ðFeð yÞ ) AtðFeðx þ yÞÞÞ;

x # y ) ððFeðxÞ ^ Feð yÞÞ ) AtðFeðx þ yÞÞÞ:

Similarly,

y # x ) ðFeðxÞ ^ Feð yÞÞ ) AtðFeðx þ yÞÞ

and thus, FeðxÞ ^ Feð yÞ ) AtðFeðx þ yÞÞ by axiom (A6) of BL.

The proof for x·y is fully analogous. A

Observe that we have not postulated Feð�0Þ (thus the assumption ð;xÞ : FeðxÞ) is consistent

with PAat and makes the theory equivalent to PA).

Corollary 1 For each term tðx1 . . . xnÞ of PA with the variables indicated, there is a

natural number n such that

PAat r Feð�0Þ ^ Feðx1Þ ^ · · · ^ FeðxnÞ ) At nðFeðtðx1; . . .; xnÞÞÞ

(where At nðwÞ is AtðAtð. . .ðAtðwÞÞ. . .ÞÞ; n copies of At).

3. EXAMPLES

In this section, we give various possibilities for how the connective almost true and predicate

feasible could be interpreted. Let us remark that such examples can be produced ad libitum.

3.1 Examples of Almost True

First let us collect trivial examples: AtðwÞ ,` for all w – everything is almost true;

AtðwÞ ,’ – nothing is almost true; AtðwÞ , w–At(w) just says w. Clearly, these are

uninteresting examples.

Second, let p be a truth constant for a non-extremal truth value and define AtðwÞ , ð p ) wÞ:Clearly, this satisfies our axioms; furthermore, the following becomes provable:

Atðw ) cÞ ) ðw ) AtðcÞÞ;

ðAtðwÞ&Atðw ) cÞÞ ) AtðAtðcÞÞ:

Figure 1 shows graphs of this At for Łukasiewicz, Godel and product t-norms.

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Third, extend our logic by the square root connectiveffi

*p

(for the concept of square root see

Hohle, 1995) satisfying w ,ffiffiffiw*

p&

ffiffiffiw*

p: If our conjunction is interpreted by a continuous

t-norm * putffiffiffix*

p¼ max{yjy*y ¼ x}: For Łukasiewicz one gets ðx þ 1Þ=2; for Godel x,

for productffiffiffix

p: And define AtðwÞ ,

ffiffiffiffiw*

p:

Lemma 2 For the *-square root just defined from a continuous t-norm *, the formula

Atðw ) cÞ , ðAtðwÞ ) AtðcÞÞ

is a tautology.

Proof We proveffiffiffiffiffiffiffiffiffiffiffix ! y*

p¼

ffiffiffix*

p!

ffiffiffiy*

p: First,

ffiffiffix*

p!

ffiffiffiy*

p� �2#

ffiffiffix*

p� �2!

ffiffiffiy*

p� �2¼ x ! y

(using the fact that ððp ) qÞ&ðp ) qÞÞ ) ððp&pÞ ) ðq&qÞÞ is a tautology).

Conversely, if z2 # ðx ! yÞ then z2*x # y; z*

ffiffiffix*

p#

ffiffiffiffiffiffiffiffiffiffiz2

*x*p

#ffiffiffiy*

p;hence z #

ffiffiffix*

p!

ffiffiffiy*

p:

Thusffiffiffix*

p!

ffiffiffiy*

pis the maximal z such that z2 # x ! y and hence,

ffiffiffix*

p!

ffiffiffiy*

p¼

ffiffiffiffiffiffiffiffiffiffiffix ! y*

p: A

Fourth, observe that if a : ½0; 1� ) ½0; 1� is a possible semantics of At and c : ½0; 1� )

½0; 1� is any non-decreasing hedge then

a0ðxÞ ¼ maxðaðxÞ; cðxÞÞ

is also a possible semantics of At. To see this, observe the following:

x ! y # atðxÞ! atðyÞ # maxðatðxÞ; cðxÞÞ! maxðatðyÞ; cðxÞÞ

# maxðatðxÞ; cðxÞÞ! maxðatðyÞ; cðyÞÞ

(recall the BL-tautology ðp ) qÞ ) ðp _ rÞ ) ðq _ rÞÞ: For example, c(x) may be a constant

0 , c , 1: Or, e.g. let dðxÞ ¼ 0 for x # 0:7; dðxÞ ¼ 1 for x . 0:7: Combine these with the

example above.

3.2 Examples of Feasible

(1) Let AtðwÞ , p ) w ( p a truth constant, whose truth value, again denoted by p, is close to

1 but different from 1. Extend the standard model N of natural numbers by the following

interpretation be of Fe: Put q1 ¼ 2; qnþ1 ¼ q2n: Put Feð0Þ ¼ Feð1Þ ¼ Feð2Þ ¼ 1; and for

qn , k # qnþ1 put FeðkÞ ¼ p*k ¼ p* . . .*p where * is our favorite continuous t-norm. This

makes (N, Fe) to a * -model of PAat.

Indeed, let us verify the axioms for Fe, observing that it is equivalent to

Feð0Þ ) AtðFeð1ÞÞ; Feð1Þ ) AtðFeð2ÞÞ; x $ 2 ) ðFeðxÞ ) AtðFeðx2ÞÞÞ:

FIGURE 1 .

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Thus assume k . 2; qn , k # qnþ1; then FeðkÞ ¼ pn; qnþ1 , k 2 # qnþ2; thus Feðk 2Þ ¼

pnþ1 and hence, the formula FeðkÞ ) ðp ) Feðk 2Þ (or equivalently ðFeðkÞ&pÞ ) Feðk 2Þ has

the truth value 1. For k ¼ 0; 1; 2 the verification is obvious. If our * is Łukasiewicz then due

to its nilpotence, for some k we shall have FeðkÞ ¼ 0: If * is product then FeðkÞ . 0 for each

k but inf ðFeðkÞÞ ¼ 0: Thus for Łukasiewicz the model verifies ð’xÞ : FeðxÞ: For product it

verifies : ð;xÞFeðxÞ but not ð’xÞ : FeðxÞ:(2) Let atðxÞ ¼

ffiffiffix*

pand for simplicity work in product logic (thus atðxÞ ¼

ffiffiffix

p—the usual

square root). Recall that we have to guarantee Feð0Þ #ffiffiffiffiffiffiffiffiffiffiffiFeð1Þ

p;Feð1Þ #

ffiffiffiffiffiffiffiffiffiffiffiFeð2Þ

pand

FeðnÞ #ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFeðn2Þ

pfor n $ 2:

The simplest thing to do is to put FeðnÞ ¼ 1=n for n $ 2 and Feð1Þ ¼ 1=ffiffiffi2

p; Feð0Þ ¼

1=ffiffiffi24

p(or, e.g. Feð0Þ ¼ 3=4; Feð1Þ ¼ 2=3Þ: If one finds this to decrease too rapidly put

FeðnÞ ¼ 1=n1 (1 . 0 small) for n $ 2; Feð1Þ ¼ffiffiffiffiffiffiffiffiffiffiffiFeð2Þ

p; Feð0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiFeð1Þ

p:

This model has the disadvantage that we have Feð0Þ , 1; which does not fit the intuition.

To have Feð0Þ ¼ 1; we may put atðxÞ ¼ p !ffiffiffix*

pfor some suitable p , 1:

4. SORITES IN FUZZY LOGIC WITH EVALUATED SYNTAX

Recall that FLn works with evaluated formulas. These are couples of the form a/w where w is

a formula and a [ ½0; 1� is its syntactic evaluation. The latter can also be understood as an

initial information about the truth of w. The truth values are assumed to form the

Łukasiewicz MV-algebra (based on [0,1]). The language of FLn is also supposed to contain

truth constants a (truth value constants as special formulas) for all the truth values a [ ½0; 1�

(alternatively, we may confine ourselves only to all rationals from [0,1]—for the details,

see Hajek, 1998; Novak et al., 1999, Novak and Perfilieva, 2000).

A fuzzy theory T is a fuzzy set of formulas determined by a fuzzy set of axioms (i.e.

axioms need not be completely convincing and thus their truth may be smaller). A formula

may be proved using an evaluated proof (a common definition of a proof accompanied by its

value). Then T ra w means that w is provable in T in the degree a where a is the supremum of

the values of all proofs of w. Then for each model V it holds that a # VðwÞ and T oa w means

that w is true in T in the degree a where a is infimum of the values of w in all models. The

completeness theorem states that both degrees are equal.

Note that FLn is a full generalization of the Fregean understanding to inference in logic: the

inference proceeds with truth of facts and not with the facts themselves (cf. Novak, 1996).

To formalize the sorites paradox, we can proceed in two ways in FLn. The first way is

analogous to that provided for BL-fuzzy logic in the preceding section.

4.1 Introducing Connective At

We introduce a new unary connective At, which must be logically fitting, i.e. there is n . 0

such that the formula

ðw , cÞn ) ðAtðwÞ , AtðcÞÞ ð1Þ

is an axiom with the degree 1. This means that its interpretation at : ½0; 1�! ½0; 1� must be a

Lipschitz continuous function (cf. Mesiar and Novak, 1997). In particular, this means that at

preserves arbitrary infima and suprema.

Furthermore, we will assume axioms (at1) and (at2) in the degree 1 as well as the

following special book-keeping axiom:

ðB2Þ AtðaÞ , atðaÞ; a [ ½0; 1�

where atðaÞ denotes the truth constant for the truth value at(a) when a is given.

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Theorem 3 (Completeness) Let J be a language extended by the logically fitting unary

connective At. Then

T ra A iff T oa A

for every fuzzy theory T and a formula A [ FJðTÞ.

Proof This follows from the assumptions and Novak et al. (1999), Corollary 4.6. A

Lemma 3 For every fuzzy theory T and every formula w,

Tat r ð;xÞðAtðwÞÞ , Atðð;xÞwÞ:

Proof This follows from the completeness theorem and the assumption (1). A

Recall that by at n we denote the n-times composite function of at. Let us now define a

(pseudo-) inverse of the function at by

at ð21Þð yÞ ¼^

{ajy # atðaÞ}:

Lemma 4 For each n [ N and a [ ½0; 1Þ;

(i) at ð21ÞðyÞ # y;(ii) at ð2nÞðyÞ ¼

n times

at ð21Þ. . . at ð21ÞðyÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼V

{ajy # at nðaÞ};

(iii) at nð0Þ ¼ 1 iff at ð2nÞð1Þ ¼ 0;(iv) if at nðaÞ ¼ 1 then at (2n)(1) # a,

(v) if at nð0Þ ¼ a then at ð2nÞðaÞ ¼ 0:

Proof (a) follows from the fact that y [ {ajy # atðaÞ}:(b) For n ¼ 2 we prove that

bj^

{ajy # atðaÞ} # atðbÞn o

¼ ajy # at 2ðaÞ� �

using the fact that at preserves arbitrary infima. Then use induction. (c), (d), (e) follow

immediately from (b). A

The main disturbing fact about the sorites paradox is that using the correct way of

reasoning and stemming from apparent truth, we derive an apparently false conclusion. The

main problem, in our opinion, lies in the fact that after adding one stone to something which

may still not be a heap, its form very slightly (imperceptibly) changes. Classical logic has no

means to distinguish such situations and so it must neglect it. The price it pays for such

negligence is disqualification of the inference process in all cases similar to sorites.

We will now construct a fuzzy theory Tat. Our goal is to be able to introduce explicitly

that Feð�0Þ; ð’xÞ : FeðxÞ may be provable in the degree 1 and show that this is not paradoxical

in FLn.

The theory Tat is given by the following fuzzy set of special axioms:

(i) Axioms of the theory PAat in the degree 1,

(ii) Axiom of the path to the sorites:.

(at3) 1=At m0ð’Þ for some m0 . 0:From (at3) we immediately obtain Tat r At nð’Þ for every n $ m0:

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Let us comment on a somewhat surprising axiom (at3). This axiom explicitly states that

the unary connective “almost true” enables our theory to pass from truth to falsity. This can

be justified as follows.

The connective At says that the given truth value a can be understood as “almost true”.

It is clear that 1 is “almost true” without doubts. Lower a may be “almost true” in a degree

lower than 1 but still greater than a. Hence, it seems natural to suppose that after, say, n steps

we may arrive at the conclusion that a is “n-times almost true” with truth value 1.

Similarly, we may state that 0 is “almost true” in some very small positive degree since in

general, we may not be “100%” sure that something is false. But then, after repeating, say, m

steps, we may arrive at the conclusion that 0 is “m-times almost true” with truth value a. But

then we immediately obtain that at nþmð0Þ ¼ 1; this is reflected by axiom (at3).

The following is immediate.

Lemma 5 Let T2at be the fuzzy theory Tat without axiom (at3). If PAat r w then T2

at r w:

Lemma 6

Tat r At nðwÞ ð2Þ

for every formula w and n $ m0:

Proof From (at3) and the fact that r’) w holds for every formula w, we obtain Eq. (2) by

the following evaluated proof:

1= ’) w{provable formula}; 1=Atð’) wÞ {ðat1Þ; tMP}; . . .;

1=At m0ð’Þ ) At m0 ðwÞ{ðat1Þ; ðat2Þ; tMP}; 1=At m0ð’Þ{ðat3Þ};

1=At m0ðwÞ{tMP}; . . .; 1=At nðwÞ{ðat1Þ; tMP}:

A

Theorem 4 Let the fuzzy theory TFe be obtained from Tat by

TFe ¼ Tat < {1=Feð�0Þ; 1=ð;xÞðFeðxÞ ) AtðFeðSðxÞÞÞÞ; 1=ð’xÞ : FeðxÞ}: ð3Þ

Then TFe is a consistent fuzzy theory.

Proof We will construct a model kN;Fel of TFe. Therefore, we put

Feð0Þ ¼ 1; ð4Þ

Feðn þ 1Þ ¼ at ð21ÞðFeðnÞÞ; n [ N: ð5Þ

It follows from Eq. (5) that FeðnÞ ¼ at ð2nÞðFeð0ÞÞ: Then kN;Fel is a model of TFe. Indeed,

put BðnÞ ¼ {ajFeðnÞ # atðaÞ}: Then

FeðnÞ #a[BðnÞ

^atðaÞ ¼ at

a[BðnÞ

^a

0@

1A ¼ atðat ð21ÞðFeðnÞÞÞ ¼ atðFeðn þ 1ÞÞ

because of continuity of at and thus,

n[N

^ðFeðnÞ ) FeðSðnÞÞÞ ¼ 1:

Finally, by axiom (at3) we get at ð2nÞð1Þ ¼ 0 for each n $ m0; and so FeðnÞ ¼

at ð2nÞðFeð0ÞÞ ¼ 0; i.e. : FeðnÞ ¼ 1 by Eqs. (4) and (5) and Lemma (4). A

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A simple example of the definition of at is to put

atðxÞ ¼ 1 ^ ðx þ 1Þ

for some small 1 . 0: Then at nðxÞ ¼ 1 ^ ðx þ n1Þ and we can find a number m0 $ 1=1 such

that at m0 ð0Þ ¼ 1: This connective also works well in PAat (cf. Sections 2 and 3).

4.2 Sorites in General Fuzzy Theory

The solution presented in the previous subsection does not utilize the advantage of FLn,

which is considering evaluation just in the syntax as the initial information about the truth.

Therefore, let us return to the observation that the step FeðxÞ ) FeðSðxÞÞ is not fully

convincing. We cannot take this formula as an axiom with truth value 1 but with a slightly

lower one. The magnitude of decreasing of 1 depends on the concrete situation—how

precisely we see the heap, i.e. where is the level of inperceptibility of changes in its size and

shape?

The following theorem, whose formulation is very close to the formulation provided by

Parikh (1971) formalizes the sorites paradox using evaluated syntax of FLn.

Theorem 5 Let TPA be a fuzzy Peano theory, i.e. its fuzzy set of special axioms consists of

Peano axioms accepted in the degree 1 and are crisp. Furthermore, let 0 , 1 # 1 and

Fe � JðTPAÞ be a new predicate. Then the fuzzy theory

TFe ¼ TPA < {1=Feð�0Þ; 1 2 1=ð;xÞðFeðxÞ ) FeðSðxÞÞÞ; 1=ð’xÞ : FeðxÞ} ð6Þ

is a consistent conservative extension of TPA.

Proof The proof is based on construction of model kN;Fel in which

Feðn þ 1Þ ¼ FeðnÞ^ð1 2 1Þ; n [ N ð7Þ

where ^ is Łukasiewicz conjunction (details can be found in Novak et al., 1999). A

According to this theorem, we can consistently add axioms of the predicate Fe to the fuzzy

theory containing Peano arithmetic.

Corollary 2 For each n [ N;

TFe reðnÞ Feð�nÞ; eðnÞ ¼ 0 _ ð1 2 n1Þ:

Proof Starting from 1=Feð�0Þ; we construct a proof of Feð�nÞ with the value e(n). On the other

hand, by construction of the model V o TFe; we get VðFeð�nÞÞ ¼ eðnÞ: A

It is clear that we may take arbitrary 10 # 1 in Eq. (7). Alternatively, we may also take

some function n such that nðnÞ # 1 for all n [ N:This solution of the sorites paradox seems to fit the intuition well and, moreover, transition

from full truth to falsity without contradiction is well justified. Namely, it can be seen from

Corollary 2 that eð0Þ ¼ 1 and there is a number n0 such that eðn0Þ ¼ 0: The same holds in any

model of TFe—there are numbers m0 and n0, m0 , n0 such that Feðm0Þ ¼ 1 and Feðn0Þ ¼ 0:The number n0 being the “apparently large number”{ is determined by the threshold 1 and it

{Note that by no means can we say that n0 is the first large number; it is only the number which we surely know tobe large.

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actually depends on the context. Thus it can be, say, n0 ¼ 1015 for a computer memory,

n0 ¼ 105 for the human hair, but n0 ¼ 100 for a bus full of people. We argue that this is

precisely in accordance with our intuition and observation.

5. CONCLUSION

Our axioms appear to be reasonable desiderata for the infinite fuzzy notion of a small natural

number and for the underlying concept of “almost true” understood as a hedge (unary

connective). Let us stress that the aim has been just to isolate some important axioms

characterizing the notion and show their consistency by giving concrete examples (models);

in Section 2, no attempt has been made to select any particular semantics of “almost true”

and “feasible” as “recommended”, “best”, etc. On the contrary, since the notion is fuzzy,

it would be unwanted to make it crisp real-valued by fixing one example as the semantics.

One can suggest further, more restricting axioms (one candidate being Fe(0), or

Fe(1000000), etc).

In Section 4, we have used fuzzy logic with evaluated syntax. First, we have shown that the

previous approach works well here, too. On the other hand, evaluated syntax enables us to

provide initial information about transition from truth to falsity inside sorites simply by

doubting the axiom ð;xÞðFeðxÞ ) Feðx þ 1ÞÞ: The doubt is expressed by setting its initial

truth value slightly smaller than 1. This gives us a consistent fuzzy theory which

mathematizes the paradox.

Acknowledgements

Partial support of grant No. A1030004/00 of the Grant Agency of the Academy of Sciences

of the Czech Republic is acknowledged by the first author.

References

Goguen, J.A. (1968–69) “The logic of inexact concepts”, Synthese 19, 325–373.Hajek, P. (1998) Metamathematics of Fuzzy Logic (Kluwer, Dordrecht).Hajek, P. (2000) “Function symbols in fuzzy predicate logic”, Proc. East West Fuzzy Colloquium 2000

(Zittau-Gorlitz), 2–8.Hajek, P. (2001) “On very true”, Fuzzy Sets Syst. 124, 329–333.Hajek, P. and Pudlak, P. (1993) Metamathematics of First-Order Arithmetic (Springer, Heidelberg).Hajek, P., Paris, J. and Shepherdson, J. (2000) “The liar paradox and fuzzy logic”, J. Symb. Logic 65,

339–346.Hohle, U. (1995) “Commutative residuated l-monoids”, In: Hohle, U. and Klement, E.P., eds, Non-Classical Logics

and Their Applications to Fuzzy Subsets. A Handbook of the Mathematical Foundations of Fuzzy Set Theory(Kluwer, Dordrecht).

Keefe, R. (2000) Theories of Vagueness (Cambridge University Press, Cambridge).Mesiar, R. and Novak, V. (1997) “On Fitting Operations”, Proc. of VIIth IFSA World Congress (Academia, Prague),

pp 286–290.Novak, V. (1996) “Paradigm, Formal Properties and Limits of Fuzzy Logic”, Int. J. General Syst. 24,

377–405.Novak, V. and Perfilieva, I., eds (2000) Discovering the World With Fuzzy Logic, Studies in Fuzziness and Soft

Computing, (Springer-Verlag, Heidelberg) Vol. 57.Novak, V., Perfilieva, I. and Mockor, J. (1999) Mathematical Principles of Fuzzy Logic (Kluwer, Dordrecht).Parikh, R. (1971) “On existence and feasibility in arithmetic”, J. Symb. Logic 36, 494–508.Read, S. (1995) Thinking About Logic (Oxford University Press, Oxford).Vopenka, P. (1979) Mathematics in the Alternative Set Theory (Teubner, Leipzig).

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Prof. RNDr. Petr Hajek, DrSc. graduated in mathematics from Charles

University in Prague. He obtained a PhD and DSc. (doctor of sciences) in

mathematical logic in the Czechoslovak Academy of Sciences. In 1998, he

obtained full professorship in mathematics (awarded by the president of the

Czech Republic). He worked in the Mathematical Institute of the

Czechoslovak Academy of Sciences—now the Academy of Sciences of the

Czech Republic—and then became director of the Institute of Computer

Science of AS CR, where he is now a senior scientist. He is author or coauthor

of 6 monographs and more than 200 scientific papers in the field of

mathematical logic. He is a member of the Association for Symbolic Logic

and of the editorial boards of the Archive for Mathematical Logic, Fundamenta Informaticae, Soft

Computing, and the Czechoslovak Mathematical Journal.

Prof. Ing. Vilem Novak, DrSc. graduated from Mining University, Ostrava

(Czech Republic) as a system engineer. He obtained a PhD in mathematical

logic in Charles University, Prague, and DSc. (doctor of sciences) in computer

science at the Polish Academy of Sciences, Warsaw. He obtained a full

professorship at Masaryk University, Brno (awarded by the president of the

Czech Republic). He worked in the Mining Institute of the Czechoslovak

Academy of Sciences and since 1995 he has worked in the University of

Ostrava in the Department of Mathematics and at the same time is the director

of the Institute for Research and Applications of Fuzzy Modeling. His

research interests are fuzzy logic, modeling of natural language semantics,

fuzzy control and modeling. He is author or coauthor of 4 monographs and more than 140 scientific

papers. He is a member of the European Society for Fuzzy Logic and Technology (EUSFLAT) and also

a member of the editorial board of the International Journal of Fuzzy Sets and Systems, the Journal of

Soft Computing and the Journal of Applied Non-Classical Logics.

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