Fuzzy Truth, Consistency and Ontological Vagueness

8/7/2019 Fuzzy Truth, Consistency and Ontological Vagueness

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SOME REMARKS ON TARSKI`S DEFINITION OF TRUTH

Francisco Daz MontillaInstitute of Philosophy and ReligionFaculty of ArtsCharles UniversityEmail: [email protected]

1. Introduction

The discussion about vagueness inevitable becomes a discussion about truth:Which is the truth-value of a vague sentence? Is truth vague? And if so, how can wegive a coherent formulation of such a vague notion?

For us in this paper the question of whether truthis a sharp or vague notion isnot so important. The important point is that truthcan be understood as a sharp notionand as a vague notion as well. Our task, then, will be to characterize truth as a vaguenotion and derive further consequences of this for ontology.

Our strategy will be to generalize the Tarskian definition of truth and so to definethe notion of fuzzy truth for a fuzzy language1. We, then, will consider Haacks critique

of the notion of fuzzy truth and we shall show that such a critique is not conclusive. Insection 4, we will consider the consequences fuzzy truth has for ontology, in particular we shall see that the notion of fuzzy truth does not support the idea of ontologicalvagueness.

2. Fuzzy Truth

Tarski (see [1931]) proposed a definition of truth that is materially adequate andformally correct. Can this definition be generalized or extended to fuzzy languages? Our answer is yes, it is possible.2

1 In what follows we shall assume that the reader is familiarized with the Tarskian definition of truth, so wewill not entry into details.2 Some writers, for instance Haack [1974], [1978] and Woleski [1997], [1998] would reject this point of view. Remember that for them there is a relation between truth and bivalence, the Tarskian definition of truth they say- entails the principle of bivalence (PB).


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For the fuzzy logician truth is a matter of degree, so fuzzy logic can be viewed asa logic with a comparative notion of truth.3 A true sentence in degree 0.7 is less true thananother one that is true in the degree 0.9. Our task, then, is to generalize or to extendTarskian definition to fuzzy languages.

Between the notions that support this comparative conception of truth are thenotion of linguistic variable, the notion of structured linguistic variable and fundamentally- the notion of linguistic hedge or modifier:4

2.1 Definition

A linguistic hedge or a modifier is an operation that modifies the meaning of aterm or, more generally, of a fuzzy set. If A is a fuzzy set, then the modifier m

generates the (composite) term B = (m(A)).

Mathematical models frequently used for modifiers are:

(i) Concentration: con( A)( x) = ( A( x))(ii) Dilatation: dil( A)( x) = A( x)(iii) Contrast intensification:

2( A( x)) for A( x)[0,1] int( A)( x) =

1-2(1- A( x))) otherwise

The following linguistic hedges (modifiers) are associated with (i-iii): if A is a fuzzy term(fuzzy set) then:

Very A = con( A)More ore less A = dil( A)Slightly A = int[plus A and not (very A)] (here and is interpreted possibilistically).

Previous notions give some theoretical elements that allows us formally

characterize the notion of fuzzy truth la Tarski. Lets consider the following:

(1) Mary is old.

3 See Hjek [2001] and [2000].4 SeeZadeh [1973].


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In this sentence the term old admits two readings. In accordance with the firstone, old is a property, while in accordance with the second one, old would be a classor set. In accordance with the first reading it means that Mary has the property of oldness, while in accordance with the second one it means that Mary belongs to the setof old persons. If we used the notion of linguistic variable , we would obtain thefollowing:

M (old) = {(u, old(u)) u [0, 100]}

Where,0 u [0, 50]

old(u) =(1 + u-50 -2 -1 u [50, 100]

5

Thus:

age(Mary) = old = {(u, old( x)) u[0, 100]}Where

0 u [0, 50] old(u) (Mary) =

(1 + u-50 -2 -1 u [50, 100]5

If our basis for uttering (1) were the fact that Mary is 75 years old, then (1) would be truein the degree 0.7.

If oldis or is not a fuzzy set is irrelevant. The important point here is that oldcan be understood or interpreted either as a crisp set in standard sets theory or as a fuzzyset in fuzzy sets theory. If old were understood as a crisp set, it only means thatcharacteristic functions defined in such sets take either 0 or 1 as values, that is: theextreme cases of [0, 1]. Crisp sets, then, can be handled as special cases of fuzzy sets.

In fuzzy sets the membership relation is a matter of degree, so we shall start withthe following: Let A be a fuzzy set. The expression x n A will mean x belongs to A in

the degree n or x is A in the degree n, where n[0, 1]. Thus, n express: (i) the degreeof truth of x is A and (ii) the degree of membership of x to the fuzzy set A. Therefore, to


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say that x is A is true in some degree is to say that x belongs to A in some degree andvice-versa. Supposing that x = Mary and A = old, we would have:

(2) Mary is old is true in the degree n iff old (Mary) = n

If the truth-value were 0.7 as we said above, then we would have:

(2*) Mary is old is true in the degree 0.7 iff old (Mary) = 0.75

Note that the expression in the right side of (2) has the form x is __, where __is the place of any predicate or class. On the other hand, by 2.1 Definition, __ can be

modified bylinguistic hedges, and thus we would obtain: m__, ( x is very __, x is moreor less__ and the like). Furthermore, the application of __ to the object is expressed by:

__( x) or m__( x). We, then, arrive at the following fuzzy T-schema:

(FT): x is __ is true in the degree n iff __( x) = n

Now, as substituting the unbound variable x in x is __ we obtain a sentence we can

write:

(FT*) S is true in the degree n iff S = n

where S = n is the result of applying the characteristic function of the predicate thatappears in S to the object referred to by the subject of S, that is __(x). Thisguarantees the equivalence between the right and left side of (FT*). Therefore, the degree

of truth and the degree of membership are the same.Previous formulation guarantees that the requirement of material adequacy be

satisfied. Now, in the case of formal conditions6 we have to point out the following: infuzzy contexts theusual laws of logic are not always valid. In particular, not always p5 Note that what (2*) says is that the truth-value of Mary is old is the result of applying the characteristicfunction of oldness to the object called Mary.6 See Tarski [1931].


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p = 0 and not always p p =1. It means that neither LEM nor NCP are valid in thisformulation of truth. They only are valid for the extreme cases.

An obvious thing we can see is that T can be formulated as a particular instance of FT*. In particular, if (2*) were a sentence of any classical system and if it took the valuetrue (=1), then we would have:

(2**) Mary is old is true in the degree 1 iff old (Mary) = 17

Therefore, FT* is a generalization of T-schema.If what we have said is correct, then a fuzzy account of truth extending Tarskian

conception of truth can be given. LetL f be a language with variables x1, x2, x3, fuzzy

predicates ( F, G ,...), quantifiers (and), etc. If X, Y has as their range the sequence of objects, and if A , B has as their range the sentences (wffs) of L f , and if xi denotes thei-ththing in any sequence X , then we can define the fuzzy relation of satisfaction as follows:

2.2 Definition Fuzzy relation of satisfaction

i. For one place-predicate:For all i, X: S n8 X, Fxi iff F (Xi) = n 9

ii. For predicates of two-places:For all i, X: S n X, Gxixj iff G Xi, Xj = n

iii. For all X, A : S n X, A iff S 1-n X, A 10

iv. For all X, A, B : S n X, ( A B ) iff S ni X, A and S n j X, B , where n=min{n i , n j }v. For all X, A, B : S n X, ( A B ) iff S ni X, A or S n j X, B , where n = max{n i , n j }

vi. For every X, A , i: S n X, ( xi ) A iff for every sequence Y such that Xj = Yj for all j i S n Y, A .

vii. For every X, A , i: S n X, ( xi ) A iff there is a sequence Y such that Xj = Yj for all j i and S n Y, A

As we can see all closed sentence will be satisfied in some degree either by all or by nosequence. Thus, we obtain:

2.3 Definition7 Or simply: Mary is old is true iff old (Mary) = 18 S n means that the relationS is satisfied in the degree n.9 Note that if n = 1, thenS is satisfied in the classical sense.10 If n = 0.5, then x satisfies both A and A in the same degree.


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True in L f

A closed sentence of L f is true in the degree n iff it is satisfied by all sequences inthe degree n .11

A consequence of this and 2.2 Definition is the following:2.4 Lemma

A closed sentence of L f is true in the degree 1 iff it is satisfied by all sequences inthe degree 1.

That is, Tarskis definition.

3. Haacks criticisms

Susan Haack, nevertheless, rejects a formulation of truth like this. Truth -shesays- does not admit degrees.12

In what follows, we will reconstruct Haacks arguments and then we will makesome comments. In accordance to Haack, although there are adverbs (quite and very)which apply to predicates (tall, old, young, etc.) and which indicate possession of the property in modest or considerable degree, this is not right when such adverbs modify

11 This will be clearer with some examples. Consider the following one-place open sentence:

(3) x is Czech

This sentence will be satisfied in some degree (in which degree will depend on the way be a Czech isunderstood) by all the sequencesMilan Kundera, . In similar way, the two-place open sentence:

(4) x is near to y,

(where x and y represent cities) is satisfied in some degree by the sequencePrague, Podbrady, andthe quantified sentence:

(5) ( x) ( x is Czech)

is satisfied in some degree by all the sequences, , independently of its first or subsequent member, because there is a sequenceMilan Kundera, which differs of any arbitrary sequence at the most in thefirst place and that satisfy in some degree x is Czech. Of course, the above fuzzy formulation of truth for L f is valid only if the vagueness of the sentences of L f were of type-1 vagueness. For a language of type n>1, the formulation will need other theoretical elements.12 Remember that for Zadeh, the claim that truth is a matter of degree rests on the thesis that adverbialmodifiers that apply to true are those that apply to fuzzy predicates.


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true. Quite true and very true she says- do not indicate the possession neither of modest nor of high degree of truth. Adverbs, which typically modify predicates of degree,do not apply to true at all or do not apply such as Zadeh says or thinks.13 For instance,while quite tall contrasts with very tall, quite true can be equated with perfectlytrue, absolutely true, very true or true. When quite modifies a predicate of degree,it cannot be preceded by not (not quite tall is linguistically inadmissible), whereas,when quite modifies an absolute predicate, it can be preceded by not (not quiteready). In conclusion, the behavior of quite with true, so far from supporting thehypothesis that true is a predicate of degree, indicates that it is an absolute predicate(not quite true is fine) (Haack [1974], p. 20).

True, according to Haack, is anachievement predicate . An achievement

predicate is a predicate that, unlike degree and degree/extreme terms, does not takemodifiers of degree and unlike degree terms, does take success modifiers (true does notake extremely but does take absolutely). This, Haack thinks, allows us to explain the behavior of adverbial modifiers which apply to true and which do suggest that true is a predicate of degree. Such locutions, she says, can be better explained by attending morecarefully to the subject of which true is predicated. If p stands for some complexstatements, then:

p is wholly truethe whole of p is true

p is completely true(6) p is partly true part of p is true

p is substantially true most of p is truep is essentially true the essential part of p is truep is approximately true approximately p is true

We, nevertheless, think that Haacks arguments are not conclusive. Haacks says

that adverbial modifiers that apply to true can be better explained by more carefullyattending the subject of which true is a predicate. p is wholly true means: the whole of

13 Rather tall, heavy, intelligent *rather trueFairly tall, heavy, intelligent *fairly trueSomewhat heavy *somewhat intelligentExtremely tall, heavy, intelligent *extremely true


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p is true. However, Haacks proposal is not the only possible. (6) can be reformulatedas:

p is wholly true

the whole of p is wholly truep is completely true(6*)

p is partly true the whole of p is partly true14p is substantially true the whole of p is substantially truep is essentially true the whole p is essentially truep is approximately true the whole p is approximately true

What we mean will be clearer if we considered the following: Let F be a formulacomposed by a chain of sub-formulas jointed by any logical operator- say conjunction or

disjunction-. To every sub-formula a truth-value n[0, 1] is assigned. Lets supposethat F: p1 p2 p3, and that p1 = 1, p2 =0.8and p3= 0.5.We then perfectly obtain:

(7) The whole of F is not wholly trueand(8) The whole of F is partly true because the truth-values of p2, p3 < 1.15 We also obtain:

(9) Part of F is wholly trueand:(10) Part of F is partly true

And so it is clear that truth contrary to Haacks arguments- can be viewed as a fuzzy predicate.

4. Ontological vagueness

There is a relation between ontological (metaphysical) theories and truth. Withoutthe notion of truth we cannot speak about what is. It is sometimes said that a

14 As the whole of p is partly true, it means that part of p is partly true.15 Or why not: the whole of p is not wholly true entails the whole of p is partly true.


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metaphysical theory requires the admission of degrees of truth. Bradley for instancewrote:

There are, one may say, two main views of errors, the absolute and the relative.According to the former view there are perfect truths, and on the other side there aresheer errorsThis absolute view I reject.Ultimately, there are, I am convinced, noabsolute truths, and on the other side there are no mere errorsAll truths and all errorsin my view may be called relative, and the difference in the end between then is a matter of degree.

Bradley ([1914], p. 252.)

For Bradley all truths are in varying degrees erroneous and the whole of the

conditions are not stated, it means that all judgments are less than fully specific, so thattheir true is a matter of degree. It means that one can make an judgment successivelymore specific (and thus approximate to the truth), but never fully specific, because thereare conditions in which the judgment is false as well as conditions in which the judgmentis true. Thus, complete truth is unattainable.

Bradleys arguments are all variations of the theme that truth is a relation between judgments and reality, which can never hold well than imperfectly. So, the imperfection between judgment and reality could be explained by the fact that reality itself is vague.But is this idea coherent?

In [1978], Evans wanted to show that the thesis of ontological vagueness, that is,the claim according to which reality (or parts of reality) is vague is incoherent andtherefore untenable. Provisionally, we shall maintain the following claim:

4.1 Claim

There are no vague objects

To prove that (4.1) is true, we need to prove that the claim that there are vagueobjects is incoherent. If it were the case, then the rejection of such a claim would be justified. The heart of Evans proof is that supposing two termsa and b such that the


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sentence a = b is of indeterminate truth-value, formally:(a = b), then we get a

contradiction:(a = b).16

David Lewis, nevertheless, points out that Evanss proof is false, and if it werefalse, then it seems that we cannot justify 4.1 claim. In accordance to Lewis, the problemwith the proof is that:

It trades on an supposed equivalence between (1) its being vague whether an object a hassuch and such feature (symbolically,(a) and (2) the object being such that it isvague whether it has such a feature (symbolically, x ( x) a). The equivalence breaksdown when we realize thata function non-rigidly in (1) and rigidly in (2).

Evans, according to Lewis, was interested in showing this feature of his proof. Thus, the

correct interpretation of Evanss proof is as follows:

There are vague identity statements, a proof to the contrary cannot be right, and thevagueness-in-describing view affords a diagnosis of the fallacy. His point of view is thatthe vague-objects cannot accept this diagnosisIn fact the vague-objects view does notafford any diagnosis of the fallacy.17

Thus, the thesis would be that if there are vague objects, we must be able to refer tothem using singular terms which functions as precise designators, yet if we do, so we

generate a contradiction. Hence there cannot be vague objects (Maidens, [1998], p. 145).This, however, is not enough for rejecting the idea of vague objects existence.

Maidens points out some interesting facts:1. It might be possible to defend vague objects by suggesting that the problem lies

not with objects themselves but with our assumptions about denotation.2. In the interpretation of quantum formalism, there is a sense in which the

formalism of quantum mechanics is not readily understood as a language withsingular terms and predicate expressions.

3. In substitutions modeled by quantum mechanics the term rigid designator seemsuseless.These three points seem to support the idea that quantum particles like electrons-

are vague objects. Concretely, Maidens arguments point out that quantum universe have16 For details see Evans [1978].17 David Lewis: Vague Identity: Evanss Misunderstood, p. 129, cited by A. Maidens [1998], p. 145.


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such properties that it is not possible to express these properties by means of conceptualoperations that we normally use for referring to macroscopic universe. For this reason,Maidens says that:

Close attention to quantum mechanics suggests that the realm of fundamental particlesmay provide examples of vague objects and help us to see the way in which such objectsresist our attempts to attach name to them. If this is the case, then we cannot rule out theexistence of such objects by the strategy Evans suggests, because we cannot designatethem in the way required to formulate his proof.

Maidens ([1998], p.149).

We agree with Maidens in the sense that Evanss proof does not demonstratewhether there are or there are not vague objects.

An important point in Evanss proof is the idea that the statements which maynot have a determinate truth-value because of their vagueness are identity statements.This idea, nevertheless, is a source of questions: What does mean identity? If identitymeans equality, then Evans argument is structured over a wrong idea and thus, from thefuzzy logic point of view, it cannot proof that the idea of ontological vagueness isuntenable. What we mean will be clearer by means of the following.

4.2 Definition

Let E be a set reference and M = [0-1] its associated membership set. Let A and B be two fuzzy subsets. We will say that A and B are equals if and only if:

x E: A(x) = B(x)

Suppose that E = {a, b, c, d, e, f, g,} and A = {a/0.8, c/0.7, e/0.4, g/0.5}, (where 0.8, 0.7,0.4, 0.5 are the degrees of membership). In accordance with this definition we wouldhave that B = {a/0.8, c/0.7, e/0.4, g/0.5}. Note that if a , belonged to A in the degree 0.9,

then A and B would not be equals, but even so both subsets would be vague or fuzzy.From the fuzzy sets theory point of view, it means that if two sentences or expressionsare vague, this does not entail that they are identical, because they can be vague indifferent degrees. The situation would be different if A and B were synonymousexpressions. In such a case it means that individual concepts18 would be subsumable in

18 See Carnap [1947].


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5. Concluding remarks

The upshots of the present inquiry can be summarized as follows:

(i) Truth as a matter of degree can be formulated in a rigorous way.(ii) The notion of fuzzy truth does not support the idea of ontological

vagueness, because such idea has no genuine logical basis.(iii) Adverbial modifiers modify truth in the same sense that these adverbs

modify fuzzy predicates.

BIBLIOGRAPHY

BALDWIN, J. F. [1979]. A new Approach to Approximate Reasoning Using Fuzzy Logic .FSS 2, pp. 309-325.

BRADLEY, F. H. [1914]. Essays on Truth and Reality . Oxford University Press.

CARNAP, R. [1947].Meaning and Necessity , Chicago University Press, 1956.

DUMMETT, M. [1975]. Wangs Paradox, Synthese 30 (1975), p. 301- 324.

EVANS, G. [1978]. Can There be Vague Objects? Analysis 38, p. 208.

HAACK, S. [1974]. Deviant Logic. Fuzzy logic, The University of Chicago Press.

HAACK, S. [1978]. Philosophy of Logics ( Filosofa de las Lgicas, Traduccin deAmador Antn, Ediciones Ctedra, Madrid, 1991).

HJEK, P. [2000]. The Liar Paradox and Fuzzy Logic. Journal of Symbolic Logic,Vol. 65, Number 1, March 2000, pp.339-346.

HJEK, P. [2001]. On very true , Fuzzy Sets and Systems 124, pp. 329- 333.

MAIDENS, A. [1998]. Vague Objects, Vague Identity and Semantic. In T. Childers(ed.): The Logica Yearbook 1997, pp. 141-151, Filosofia, Prague, 1998.

QUINE, W. V. [1970]. Philosophy of Logic ( Filosofa de la Lgica , Traduccin deManuel Sacristn, Alianza Universidad, Madrid, 1984).


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TARSKI, A. [1931].The Concept of Truth in Formalized Languages . In Tarski: Logic,Semantic and Metamathematics , Oxford, 1956.

WOLESKI, J. [1997]. Semantic Conception of Truth as a Philosophical Theory, in J.Peregrin (ed.)The Nature of Truth (if any), Praha, 1997, pp. 137- 152.

WOLESKI, J. [1998]. Truth and Bivalence, in T. Childers (ed.):The LogicaYearbook 1997, Filosofia, Praha.

ZADEH, L. [1973]. The Concept of a Linguistic Variable and its Application toApproximate Reasoning. Memorandum ERL-M411, Berkeley.

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Fuzzy Truth, Consistency and Ontological Vagueness