The Sorites paradox and fuzzy logic

PETR HAJEK(a) and VILEM NOVAK(b)

(a)Institute of Computer Science, Academy of Sciences182 07 Prague, Czech Republic

e-mail: hajek@uivt.cas.cz

(b) University of OstravaInstitute for Research and Applications of Fuzzy Modeling

30. dubna 22, 701 03 Ostrava 1, Czech Republice-mail: Vilem.Novak@osu.cz

Abstract

The sorites paradox (interpreted as the paradox of small natural num-bers) is analyzed using mathematical fuzzy logic. In the first part, wepresent an extension of BL-fuzzy logic by a new unary connective At ofalmost true and the crisp Peano arithmetic extended by a fuzzy predicateof feasibility. Then we give examples of possible semantics of At and ex-amples of semantics of feasible numbers. In the second part, we presentan analysis of the sorites paradox within fuzzy logic with evaluated syntaxand show that under a very natural assumption we obtain a consistentfuzzy theory. Thus, sorites is not paradoxical at all.

1 INTRODUCTION

One of the most striking paradoxes disturbing logical reasoning for more thantwo thousand years is the sorites (heap) paradox attributed to the Aristotle’scontemporary, Eubulides — the Magarian philosopher who is the author ofseveral other known paradoxes such as the Liar one. The standard form ofsorites paradox is as follows:

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

The problem lays in the fact that in small steps proceeding little by little, wepass from truth to falsehood. Note that other form of the same paradox is thefalakros (bald man): loosing one hair does not make a non-bald man bald, thusit follows by induction that there are no bald men. Similarly, we can proceedwhen reasoning about a lot of other kinds of properties such as few, small,

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etc. Apparently, there are “small numbers” but not all numbers are small. Acommon feature of all these examples is vagueness of the concept in concern.This means that when passing along the possible bearers of vague property, weare, in principle, unable to distinguish the moment when the given propertyceases to exist.

From the point of view of mathematical logic, let us consider a vague predi-cate “feasible” (following Parikh (1971) who investigated the problem in formalarithmetic.). We will write Fe(n) for “n is feasible”. The the sorites para-dox is clear: postulating Fe(0), (∀x)(Fe(x) ⇒⇒⇒ Fe(x+ 1)) and (∃x)¬¬¬Fe(x), weimmediately come to contradiction with the induction principle.

Parikh (1971) investigated contradictory extensions of Peano arithmetic bythe above axioms strengthened even by a concrete assumption ¬¬¬Fe(t) when t

is a term like 10101010

. He showed that even if in this theory one can provecontradiction in t steps, no short proof can give contradiction (short beingprecisely [crisply] defined in dependence on t).

Another analysis of this paradox is provided by Vopenka (1979) who intro-duced the concept of semiset. This is formally a subclass of some set, not beingnecessarily a set itself. A typical semiset is that of finite natural numbers Fncontaining 0 and being closed under successor, addition and multiplication, butnot containing all natural numbers. This idea gave rise to a new — alternativeset theory (AST) which is equiconsistent with Zermelo-Fraenkel set theory. In-duction fails for (some) formulas containing semiset variables on the constantFn. But for each concrete natural number n, AST proves n ∈ Fn (n being then-th numeral).

In this paper, we are going to analyze the notion of small natural numbersinside fuzzy logic. We will demonstrate solution within two fundamental ap-proaches of it, namely the basic fuzzy logic as developed in Hajek (1998) andfuzzy logic with evaluated syntax (FLn) (called also Pavelka logic) presented indetail in Novak et al. (1999). The paper is also a response to some publicationswhich appeared in the recent years and which provide more or less improper (orignorant to the recent achievements of) analysis of the abilities and contributionof fuzzy logic to the solution of the sorites paradox (cf., e.g. Keefe (2000); Read(1995)).

The first analysis of the sorites paradox in fuzzy logic has been provided byGoguen (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 ofthe first solution is the following: if x is small then it is almost true that x+1 issmall (i.e. the truth degree of “x+1 is small” is only a little less than that of “x issmall”). 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 inHajek (2001). The second solution employs the possibility to consider axiomstrue in various degrees and takes the implication (∀x)(Fe(x) ⇒⇒⇒ Fe(x + 1)) asnot being fully true. Our arithmetic will be crisp except for the predicate Fe

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(see also Hajek et al. (2000))∗).The reader is assumed to be familiar with both kinds of fuzzy logics: First,

the basic fuzzy predicate logicBL∀, and three famous stronger logics �Lukasiewicz�L∀, Godel G∀, and product logic Π∀. In particular, recall that BL has two con-junctions: the strong conjunction &&& (whose standard semantics is any continu-ous t-norm, the residuum of the t-norm being the semantics of the implication⇒⇒⇒) and the min-conjunction ∧∧∧, ϕ∧∧∧ψ being defined as ϕ&&&(ϕ⇒⇒⇒ ψ) which gives∧∧∧ the semantics of minimum.

Second, we suppose that the reader knows the predicate fuzzy logic withevaluated syntax described in details in Novak et al. (1999). Recall that itsbasic truth structure is �Lukasiewicz algebra L�L = 〈[0, 1],∨,∧,⊗,→, 0, 1〉 and itdeals with evaluated formulas, fuzzy sets of axioms and besides many-valuedinterpretation as above, it employs the concepts of fuzzy theory and provabilitydegree of a formula in it. The completeness theorem states that the provabilityand truth degrees of each formula in each fuzzy theory coincide.

In Section 2 we present an extension of BL by a new unary connective Atof almost true and the theory PAat – crisp Peano arithmetic extended by thefuzzy predicate of feasibility; we also prove a simple theorem in PAat. Section 3contains examples of possible semantics of At and examples of semantics of Fe(feasible numbers). Section 4 contains analysis of the sorites paradox withinfuzzy logic with evaluated syntax.

Acknowledgement. Partial support of the grant No. A1030004/00 of theGrant Agency of the Academy of Sciences of the Czech Republic is acknowledgedby the first author.

2 ALMOST TRUE AND FEASIBLE

We extend the basic predicate logic BL∀ by a new unary connective At (At(ϕ)being read “it is almost true that ϕ”) and by the following two axiom schemata.

(at1) ϕ⇒⇒⇒ At(ϕ),

(at2) (ϕ⇒⇒⇒ ψ) ⇒⇒⇒ (At(ϕ) ⇒⇒⇒ At(ψ)).

In words, if ϕ (is true) then ϕ is almost true; if ϕ implies ψ and ϕ is almost truethen ψ is almost true (minor reformulation using (A5)). This logic is denotedBL∀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 ofPeano arithmetic PA (see Hajek and Pudlak (1993)) by a new unary predicateFe (Fe(x) is read “x is feasible”) and work with the variant of BL∀ with functionsymbols (see Hajek (2000)). Thus the language consists of

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

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

∗)Note that the notion of a large number is discussed in Hajek (1998) 3.3.21 for �Lukasiewiczlogic and 4.1.27 for product logic.

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

The axioms are as follows:

(i) x = y ∨∨∨ x �= y (crispness axiom for =),

(ii) all axioms of Peano arithmetic (including the definition of x ≤ y as(∃z)(z + x = y)),

(iii) x < y⇒⇒⇒ (Fe(y) ⇒⇒⇒ Fe(x)),

(iv) Fe(x) ⇒⇒⇒ (At(Fe(S(x))∧∧∧ At(Fe(x+ x))∧∧∧ At(Fe(x · x)).

Lemma 1For each formula ϕ of PA (i.e. not containing the predicate Fe), PAat provesϕ∨∨∨¬¬¬ϕ (tertium non datur).

proof: For atomic ϕ this is an instance of the crispness axiom of equality;the rest follows by induction on the complexity of formulas. Let us show theinduction step for quantifiers. Assume � ϕ∨∨∨¬¬¬ϕ. Then � ϕ∨∨∨ (∃x)¬¬¬ϕ (since �¬¬¬ϕ⇒⇒⇒ (∃x)¬¬¬ϕ), thus � (∀x)(ϕ∨∨∨(∃x)¬¬¬ϕ) by generalization and � (∀x)ϕ∨∨∨(∃x)¬¬¬ϕby the axiom (∀3). From the last provability we get � (∀x)ϕ ∨∨∨ ¬¬¬(∀x)ϕ (since� (∃x)¬¬¬ϕ⇒⇒⇒¬¬¬(∀x)ϕ) (see Hajek (1998), 5.1.20).

Similarly, from � ϕ ∨∨∨ ¬¬¬ϕ we get � (∃x)ϕ ∨∨∨ ¬¬¬ϕ, further � (∃x)ϕ ∨∨∨ (∀x)¬¬¬ϕ,and finally � (∃x)ϕ∨∨∨¬¬¬(∃x)ϕ. �

Theorem 1A formula not containing Fe is provable in PA over classical logic iff it is provablein PAat over BL∀at.

proof: If PA � ϕ then PAat � ϕ thanks to the preceding lemma. Conversely,if PAat � ϕ then inside PA define At(α) ⇔⇔⇔ α for each α and Fe(x) ⇔⇔⇔ x = x.This embeds PAat (over BL∀at) non-conservatively into PA (over classical logic)and PA � ϕ. �

Theorem 2PAat proves

Fe(x)∧∧∧ Fe(y) ⇒⇒⇒ (At(Fe(x+ y))∧∧∧ At(Fe(x · y)).

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

(x ≤ y&&& Fe(y)) ⇒⇒⇒ (x+ y ≤ y + y&&& At(Fe(y + y)) ⇒⇒⇒ At(Fe(x+ y))

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

x ≤ y⇒⇒⇒ (Fe(y) ⇒⇒⇒ At(Fe(x+ y))),x ≤ y⇒⇒⇒ ((Fe(x)∧∧∧ Fe(y)) ⇒⇒⇒ At(Fe(x+ y))).

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Figure 1:

Similarly,

y ≤ x⇒⇒⇒ (Fe(x)∧∧∧ Fe(y)) ⇒⇒⇒ At(Fe(x+ y))

and thus, Fe(x)∧∧∧ Fe(y) ⇒⇒⇒ At(Fe(x+ y)) by axiom (A6) of BL.The proof for x · y is fully analogous. �

Observe that we have not postulated Fe(0) (thus the assumption (∀x)¬¬¬Fe(x))is consistent with PAat (and makes the theory equivalent to PA).

Corollary 1For each term t(x1 . . . xn) of PA with the variables indicated, there is a naturalnumber n such that

PAat � Fe(0)∧∧∧ Fe(x1)∧∧∧ · · · ∧∧∧ Fe(xn) ⇒⇒⇒ Atn(Fe(t(x1, . . . , xn)))

(where Atn(ϕ) is At(At(. . . (At(ϕ)) . . . )), n copies of At).

3 EXAMPLES

In this section, we give few examples of various possibilities how the connectivealmost true and predicate feasible could be interpreted. Let us remark that suchexamples can be produced ad libitum.

3.1 Examples of almost true

First let us collect trivial examples: At(ϕ) ⇔⇔⇔ � for all ϕ — everything is almosttrue; At(ϕ) ⇔⇔⇔ ⊥ — nothing is almost true; At(ϕ) ⇔⇔⇔ ϕ — At(ϕ) just says ϕ.Clearly, these are uninteresting examples.

Second, let p be a truth constant for a non-extremal truth value and defineAt(ϕ) ⇔⇔⇔ (p⇒⇒⇒ ϕ). Clearly, this satisfies our axioms; furthermore, the followingbecomes provable:

At(ϕ⇒⇒⇒ ψ) ⇒⇒⇒ (ϕ⇒⇒⇒ At(ψ)),(At(ϕ)&&&At(ϕ⇒⇒⇒ ψ)) ⇒⇒⇒ At(At(ψ)).

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

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Third, extend our logic by the square root connective ∗√ (for the conceptof square root see Hohle (1995)) satisfying ϕ⇔⇔⇔ ∗√ϕ&&& ∗√ϕ. If our conjunctionis interpreted by a continuous t-norm ∗ put ∗√x = max{y | y ∗ y = x}. For�Lukasiewicz one gets (x+1)/2, for Godel x, for product

√x. And define At(ϕ) ⇔⇔⇔

∗√ϕ.

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

At(ϕ⇒⇒⇒ ψ) ⇔⇔⇔ (At(ϕ) ⇒⇒⇒ At(ψ))

is a tautology.

proof: We prove ∗√x→ y = ∗√x→ ∗√y. First,

∗√x→ ∗√y)2 ≤ ( ∗√x)2 →( ∗√y)2 = x→ y

(using the fact that ((p⇒⇒⇒ q)&&&(p⇒⇒⇒ q)) ⇒⇒⇒ ((p&&& p) ⇒⇒⇒ (q&&& q)) is a tautology).Conversely, if z2 ≤ (x→ y) then z2 ∗ x ≤ y, z ∗ ∗√x ≤ ∗√

z2 ∗ x ≤ ∗√y, hencez ≤ ∗√x→ ∗√y. Thus ∗√x→ ∗√y is the maximal z such that z2 ≤ x→ y andhence, ∗√x→ ∗√y = ∗√x→ y. �

Fourth, observe that if a : [0, 1] ⇒⇒⇒ [0, 1] is a possible semantics of At andc : [0, 1] ⇒⇒⇒ [0, 1] is any non-decreasing hedge then

a′(x) = max(a(x), c(x))

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

x→ y ≤ at(x)→ at(y) ≤ max(at(x), c(x))→max(at(y), c(x)) ≤max(at(x), c(x))→max(at(y), c(y))

(recall the BL-tautology (p⇒⇒⇒ q) ⇒⇒⇒ (p∨∨∨ r) ⇒⇒⇒ (q ∨∨∨ r)). For example, c(x) maybe a constant 0 < c < 1. Or e.g. let d(x) = 0 for x ≤ 0.7, d(x) = 1 for x > 0.7.Combine these with the example above.

3.2 Examples of feasible

(1) Let At(ϕ) ⇔⇔⇔ p⇒⇒⇒ ϕ (p a truth constant, whose truth value, again denotedby p, is close to 1 but different from 1. Extend the standard model N of naturalnumbers by the following interpretation be of Fe: Put q1 = 2, qn+1 = q2

n. PutFe(0) = Fe(1) = Fe(2) = 1, and for qn < k ≤ qn+1 put Fe(k) = p∗k = p ∗ · · · ∗ pwhere ∗ is our favorite continuous t-norm. This makes (N,Fe) to a ∗-model ofPAat.

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

Fe(0) ⇒⇒⇒ At(Fe(1)), Fe(1) ⇒⇒⇒ At(Fe(2)), x ≥ 2 ⇒⇒⇒ (Fe(x) ⇒⇒⇒ At(Fe(x2))).

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Thus assume k > 2, qn < k ≤ qn+1, then Fe(k) = pn, qn+1 < k2 ≤ qn+2, thusFe(k2) = pn+1 and hence, the formula Fe(k) ⇒⇒⇒ (p ⇒⇒⇒ Fe(k2)) (or equivalently(Fe(k)&&& p) ⇒⇒⇒ Fe(k2)) has the truth value 1. For k = 0, 1, 2 the verification isobvious. If our ∗ is �Lukasiewicz then due to its nilpotence, for some k we shallhave Fe(k) = 0. If ∗ is product then Fe(k) > 0 for each k but inf(Fe(k)) = 0.Thus for �Lukasiewicz the model verifies (∃x)¬¬¬Fe(x). For product it verifies¬¬¬(∀x) Fe(x) but not (∃x)¬¬¬Fe(x).

(2) Let at(x) = ∗√x and for simplicity work in product logic (thus at(x) =√x — the usual square root). Recall that we have to guarantee Fe(0) ≤√Fe(1),Fe(1) ≤ √

Fe(2) and Fe(n) ≤ √Fe(n2) for n ≥ 2.

The simplest thing to do is to put Fe(n) = 1n for n ≥ 2 and Fe(1) = 1/

√2,

Fe(0) = 1/4√

2 (or e.g. Fe(0) = 34 ,Fe(1) = 2

3 ). If one finds this to decrease toorapidly put Fe(n) = 1/nε (ε > 0 small) for n ≥ 2, Fe(1) =

√Fe(2),Fe(0) =√

Fe(1).This model has disadvantage that we have Fe(0) < 1, which does not fit the

intuition. To have Fe(0) = 1, we may put at(x) = p → ∗√x for some suitablep < 1.

4 SORITES IN FUZZY LOGICWITH EVALUATED SYNTAX

Recall that FLn works with evaluated formulas. These are couples of the forma/ϕ where ϕ is a formula and a ∈ [0, 1] is its syntactic evaluation. The latter

can also be understood as an initial information about the truth of ϕ. Thetruth values are assumed to form the �Lukasiewicz MV-algebra (based on [0, 1]).The language of FLn is also supposed to contain truth constants aaa (truth valueconstants as special formulas) for all the truth values a ∈ [0, 1] (alternatively,we may confine ourselves only to all rationals from [0, 1] – for the details, seeHajek (1998); Novak et al. (1999)).

A fuzzy theory T is a fuzzy set of formulas determined by a fuzzy set ofaxioms (i.e. axioms need not be completely convincing and thus, their truthmay be smaller). A formula may be proved using an evaluated proof (a commondefinition of a proof accompanied by its value). Then T �a ϕ means that ϕ isprovable in T in the degree a where a is supremum of the values of all proofs ofϕ. Then for each model V it holds that a ≤ V(ϕ) and T |=a ϕ means that ϕ istrue in T in the degree a where a is infimum of the values of ϕ in all models.The completeness theorem states that both degrees are equal.

Note that FLn is a full generalization of the Fregean understanding to infer-ence in logic: the inference proceeds with truth of facts and not with the factsthemselves.

To formalize the sorites paradox, we can proceed in two ways in FLn. Thefirst way is analogous to that provided for BL-fuzzy logic in the preceding sec-tion.

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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

(ϕ⇔⇔⇔ ψ)n ⇒⇒⇒ (At(ϕ) ⇔⇔⇔ At(ψ)) (1)

is an axiom with the degree 1. This means that its interpretation at : [0, 1] −→[0, 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 wellas the following special book-keeping axiom:

(B2) At(aaa) ⇔⇔⇔ at(a), a ∈ [0, 1]

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

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

T �a A iff T |=a A.

for every fuzzy theory T and a formula A ∈ FJ(T ).

proof: This follows from the assumptions and Novak et al. (1999), Corol-lary 4.6. �

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

Tat � (∀x)(At(ϕ)) ⇔⇔⇔ At((∀x)ϕ)

proof: This follows from the completeness theorem and the assumption (1).�

Recall that by atn we denote the n-times composite function of at. Let usnow define an (pseudo-)inverse of the function at by

at(−1)(y) =∧

{a | y ≤ at(a)}.

Lemma 4For each n ∈ N and a ∈ [0, 1),

(a) at(−1)(y) ≤ y.

(b) at(−n)(y) = at(−1) · · · at(−1)(y)︸ ︷︷ ︸n−times)

=∧{a | y ≤ atn(a)}.

(c) atn(0) = 1 iff at(−n)(1) = 0.

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(d) If atn(a) = 1 then at(−n)(1) ≤ a.

(e) If atn(0) = a then at(−n)(a) = 0.

proof: (a) follows from the fact that y ∈ {a | y ≤ at(a)}.(b) For n = 2 we prove that

{b |∧

{a | y ≤ at(a)} ≤ at(b)} = {a | y ≤ at2(a)}

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

�

The main disturbing fact about sorites paradox is that using correct wayof reasoning and stemming from apparent truth, we derive an apparently falseconclusion. The main problem, in our opinion, lays in the fact that after addingone stone to something which may still not be a heap, its form very slightly (im-perceptibly) changes. Classical logic has no means to distinguish such situationand so, it must neglect it. The price it pays for such negligence is disqualificationof the inference process in all cases similar to sorites.

We will now construct a fuzzy theory Tat. Our goal is to enable to introduceexplicitly that Fe(0), (∃x)¬¬¬Fe(x) may be provable in the degree 1 and showthat 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/

Atm0(⊥⊥⊥) for some m0 > 0.

From (at3) we immediately obtain Tat � Atn(⊥) for every n ≥ m0.Let us comment on a somewhat surprising axiom (at3). This axiom explicitly

states that the unary connective “almost true” enables our theory to pass fromtruth 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 maybe “almost true” in a degree lower than 1 but still greater than a. Hence, itseems natural to suppose that after, say n steps we may arrive at the conclusionthat a is “n-times almost true” with truth value 1.

Similarly, we may state that 0 is “almost true” in some very small positivedegree since in general, we may not be “100%” sure that something is reallyfalse. But then, after repeating, say, m steps, we may arrive at the conclusionthat 0 is ‘m-times almost true” with truth value a. But then we immediatelyobtain that atn+m(0) = 1; this is reflected by axiom (at3).

The following is immediate.

Lemma 5Let T−

at be the fuzzy theory Tat without axiom (at3). If PAat � ϕ then T−at � ϕ.

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Lemma 6

Tat � Atn(ϕ) (2)

for every formula ϕ and n ≥ m0.

proof: From (at3) and the fact that � ⊥⇒⇒⇒ ϕ holds for every formula ϕ, weobtain (2) by the following evaluated proof:

1/⊥⇒⇒⇒ ϕ {provable formula}, 1

/At(⊥⇒⇒⇒ ϕ) {(at1), rMP }, . . . ,

1/

Atm0(⊥) ⇒⇒⇒ Atm0(ϕ) {(at1), (at2), rMP }, 1/

Atm0(⊥⊥⊥) {(at3)},1/

Atm0(ϕ) {rMP }, . . . , 1/

Atn(ϕ) {(at1), rMP }.�

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

TFe = Tat ∪{

1/Fe(0), 1

/(∀x)(Fe(x) ⇒⇒⇒ At(Fe(S(x)))), 1

/(∃x)¬¬¬Fe(x)

}. (3)

Then TFe is a consistent fuzzy theory.

proof: We will construct a model 〈N,Fe〉 of TFe. Therefore, we put

Fe(0) = 1, (4)

Fe(n+ 1) = at(−1)(Fe(n)), n ∈ N. (5)

It follows from (5) that Fe(n) = at(−n)(Fe(0)). Then 〈N,Fe〉 is a model of TFe.Indeed, put B(n) = {a | Fe(n) ≤ at(a)}. Then

Fe(n) ≤∧

a∈B(n)

at(a) = at

⎛⎝ ∧

a∈B(n)

a

⎞⎠ =

= at(at(−1)(Fe(n))) = at(Fe(n+ 1))

because of continuity of at and thus,∧n∈N

(Fe(n) ⇒⇒⇒ Fe(S(n))) = 1.

Finally, by axiom (at3) we get at(−n)(1) = 0 for each n ≥ m0, and soFe(n) = at(−n)(Fe(0)) = 0, i.e. ¬Fe(n) = 1 by (4), (5) and Lemma 4.

�

A simple example of the definition of at is to put

at(x) = 1 ∧ (x+ ε)

for some small ε > 0. Then atn(x) = 1 ∧ (x + nε) and we can find a numberm0 ≥ 1

ε such that atm0(0) = 1. This connective works well also in PAat (cf.Sections 2 and 3).

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4.2 Sorites in general fuzzy theory

The solution presented in the previous subsection does not utilize the advantageof FLn, which is considering evaluation just in the syntax as the initial infor-mation about the truth. Therefore, let us return to the observation, that thestep Fe(x) ⇒⇒⇒ Fe(S(x)) is not fully convincing. We cannot take this formulaas an axiom with truth value 1 but with slightly lower one. The magnitude ofdecreasing of 1 depends on the concrete situation — how precisely we see theheap, 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 formulationprovided by Parikh (1971) formalizes the sorites paradox using evaluated syntaxof FLn.

Theorem 5Let TPA be a fuzzy Peano theory, i.e. its fuzzy set of special axioms consists ofPeano axioms accepted in the degree 1 and are crisp. Furthermore, let 0 < ε ≤ 1and Fe �∈ J(TPA) be a new predicate. Then the fuzzy theory

TFe = TPA ∪ {1/Fe(0), 1 − ε

/(∀x)(Fe(x) ⇒⇒⇒ Fe(S(x))), 1

/(∃x)¬¬¬Fe(x)

}(6)

is a consistent conservative extension of TPA.

proof: The proof is based on construction of model 〈N,Fe〉 in which

Fe(n+ 1) = Fe(n) ⊗ (1 − ε), n ∈ N (7)

where ⊗ is �Lukasiewicz conjunction (details can be found in Novak et al.(1999)). �

According to this theorem, we can consistently add axioms of the predicate Feto the fuzzy theory containing Peano arithmetic.

Corollary 2For each n ∈ N,

TFe �e(n) Fe(n), e(n) = 0 ∨ (1 − nε).

proof: Starting from 1/Fe(0), we construct a proof of Fe(n) with the value

e(n). On the other hand, by construction of the model V |= TFe, we getV(Fe(n)) = e(n). �

It is clear that we may take arbitrary ε′ ≤ ε in (7). Alternatively, we may alsotake some function ν such that ν(n) ≤ ε for all n ∈ N.

This solution of the Sorites paradox seems to fit well the intuition and, more-over, transition from full truth to falsity without contradiction is well justified.Namely, it can be seen from Corollary 2 that e(0) = 1 and there is a number n0

such that e(n0) = 0. The same holds in any model of TFe — there are numbersm0 and n0, m0 < n0 such that Fe(m0) = 1 and Fe(n0) = 0. The number n0

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being the “apparently large number”†) is determined by the threshold ε and itactually depends on the context. Thus, it can be, say n0 = 1015 for a computermemory, n0 = 105 for the human hair, but n0 = 100 for a bus full of people. Weargue that this is precisely in accordance with our intuition and observation.

5 CONCLUSION

Our axioms appear to be reasonable desiderata for the infinite fuzzy notionof 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 beenjust to isolate some important axioms characterizing the notion and show theirconsistency by giving concrete examples (models); in Section 2, no attempt hasbeen made to select any particular semantics of “almost true” and “feasible” as“recommended”, “best” etc. On the contrary, since the notion is fuzzy, it wouldbe 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 haveshown that the previous approach works well here, too. On the other hand,evaluated syntax enables us to provide an initial information about transitionfrom truth to falsity inside sorites simply by doubting the axiom (∀x)(Fe(x) ⇒⇒⇒Fe(x + 1)). The doubt is expressed by setting its initial truth value slightlysmaller than 1. This gives us a consistent fuzzy theory which mathematizes theparadox.

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 WestFuzzy Colloquium 2000 (Zittau-Gorlitz), 2–8.

Hajek, P. (2001) “On very true”, Fuzzy Sets and Systems 124, 329–333.

Hajek, P., Paris, J., Shepherdson, J. (2000) “The liar paradox and fuzzy logic”,Journ. Symb. Logic 65, 339–346.

Hajek, P. and Pudlak, P. (1993) Metamathematics of first-order arithmetic(Springer, Heidelberg).

Hohle, U. (1995) “Commutative residuated l-monoids”, in Hohle, U. and Kle-ment, E. P., Non-Classical Logics and Their Applications to Fuzzy Subsets. A

†)Note that by no means we can say that n0 is the first large number; it is only the numberwhich we surely know to be large.

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Handbook of the Mathematical Foundations of Fuzzy Set Theory (Dordrecht:Kluwer).

Keefe, R. (2000) Theories of Vagueness (Cambridge University Press, Cam-bridge).

Mesiar, R. and Novak, V. (1997), “On Fitting Operations”, Proc. of VIIth IFSAWorld Congress (Academia, Prague), 286–290.

Novak, V. (1996) “Paradigm, Formal Properties and Limits of Fuzzy Logic”,Int. J. of General Systems 24, 377–405.

Novak, V. and Perfilieva, I. (eds.) (2000) Discovering the World With FuzzyLogic (Springer-Verlag, Heidelberg (Studies in Fuzziness and Soft Computing,Vol. 57)).

Novak, V., Perfilieva I. and Mockor, J. (1999) Mathematical Principles of FuzzyLogic (Kluwer, Boston/Dordrecht).

Parikh, R. (1971), “On existence and feasibility in arithmetic”, Journ. 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. is a senior scientist in the Instituteof Informatics of the Academy of Sciences of the Czech Republic. He is au-thor of coauthor of .... monographs and ..... scientific papers in the field ofmathematical logic.

Prof. Ing. Vilem Novak, DrSc. is the director of the Institute forResearch and Applications of Fuzzy Modeling of the University of Ostrava inthe Czech Republic. He is author or coauthor of 4 monographs and more than140 scientific papers in the field of fuzzy logic and fuzzy modeling.

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