16Extensions of Fuzziness

16.1Fuzzy Qualifiers: Hedges

So far we have studied fuzziness in connection with vague predicates, with our main concern the Sorites paradoxes and other logical puzzles. In this chapter we present two extensions of fuzziness. The formula apparatus that we present can be used to augment any of the fuzzy systems we’ve studied: Lukasiewicz, Godel,¨ or product.

We’ll begin with fuzzy qualifiers, known as hedges.1 Consider the adverb very. This adverb combines with vague predicates to form new vague predicates. So, for example, very tall is a vague predicate, as are very bald and very big. The fact that very combines with a given predicate implies that there are degrees of membership in the predicate’s extension: we do not, for example, talk of natural numbers that are very prime; a number either is prime or is not.

Not only does very require that the predicates it qualifies be vague—it systematically produces new vague concepts by raising the threshold for membership in fuzzy sets. Recall the fuzzy membership function for tall in interpretation SST in Chapter 15, where the domain consists of heights between 4 7 and 6 7 :

I (T)(<x>) = (x – 4 7 ) / 24

So I(T)(<6 7 >) = 1; I(T)(<6 5 >) = (roughly) .92; and I(T)(<5 8 >) = .54. When we say that very raises the threshold for membership in this fuzzy set we mean that in general the degree of membership of a height in the fuzzy set very tall will be less than that height’s degree of membership in the fuzzy set tall; that is, it’s harder to have a high degree of membership in the fuzzy set very tall. So, for example, if 5 8 is tall to degree .54, then it is very tall to a lesser degree, perhaps to degree .25. Now, the threshold need not be lowered in every case. For example, 6 7 may be not only tall to degree 1 but also very tall to degree 1. Moreover, the lowering need not be a linear function in those cases where it does occur. In our example, 5 8 is a member of the fuzzy set very tall to a degree that is .29 less than its degree of membership in tall, but the degrees to which 6 5 is a member of the fuzzy sets tall and very tall need not differ by the same amount – 6 5 may be very tall to a higher degree than

.63 (= .92 − .29), perhaps to degree .8 or .9.

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have taken the obvious liberty of using the English expressions rather than Greek letters.)

We must also add a semantic basis clause defining the fuzzy sets corresponding to predicates formed with qualifiers:

I(αPn)(<x1, . . . , xn>) = I(α)(I(Pn)(<x1, . . . , xn>)).

That is, the degree to which <x1, . . . , xn> is αPn is the result of applying the function α to <x1, . . . , xn>’s degree of membership in the fuzzy set Pn. Thus, where T is interpreted as in SST, we have

I(very T)(<x>) = I(very)(I(T)(<x>)) = I(very)((x – 4 7 ) / 24 ) = ((x – 4 7 )/24 )2 I(very very T)(<x>) = I(very)(I (very T)(<x>)) = I(very)(((x – 4 7 ) / 24 )2) =

(((x – 4 7 ) / 24 )2)2, or ((x – 4 7 )/24 )4

I(close-to T)(<x>) = I(close-to)(I(T)(<x>)) = I(close-to)((x – 4 7 )/24 ) = min (1, ((x – 4 7 )/24 ) + .1)

I(close-to close-to T)(<x>) = I(close-to)(I(close-to T)(<x>)) = I(close-to)(min (1, ((x – 4 7 )/24 ) +.1)) = min (1, min (1, (x – 4 7 )/24 + .1) + .1), which is min (1, (x – 4 7 )/24 + .2)

I(close-to very T)(<x>) = I(close-to)(I(very T)(<x>)) = I(close-to)(((x – 4 7 ) / 24 )2) = min (1, ((x – 4 7 ) / 24 )2 + .1)

I(very close-to T)(<x> = I(very)(I(close-to T)(<x>)) = I(very)(min (1, (x – 4 7 ) / 24 + .1)) = (min (1, (x – 4 7 )/24 ) + .1)2

(For perspicuity in our illustration we’ve used English in place of the Greek letters.) The rest of the semantic clauses remain the same, since qualified predicates work the same way as simple predicates when embedded in formulas.

Note that we can also treat not as a predicate qualifier to form expressions like not tall and not very tall. A reasonable interpretation for not in this context is

I(not)(n) = 1 – n

which produces

I(not T)(<x>) = I(not)(I(T)(<x>)) = I(not)((x – 4 7 )/24 ) = 1 – ((x – 4 7 )/24 )

and

I(not very T)(<x>) = I(not)(I(very T)(<x>)) = I(not)((x – 4 7 )/24 )2 = (1 – ((x – 4 7 )/24 ).2

Note that not is modifying very T rather than very; that is, not very T is not (very T), not (not very) T—the latter would require hedges that modify hedges. Obviously, with this interpretation of not, not Tx will be equivalent to ¬Tx in FuzzyL , but not in FuzzyG or FuzzyP .4

16.2 Fuzzy “Linguistic” Truth-Values

In addition to combining with ordinary predicates in English, hedges can modify truth-value attributions. For example, we’ve said often that the Principle of Charity premise in the Sorites paradox is close to true, and we may also say that a particular statement is very true, very close to true, not very true, somewhat true, and so on, or that it is very false, close to false, and so forth. In this book we have also talked about statements that are clearly true, clearly not true, and clearly false rather than just true or false. Lotfi Zadeh (1975) first explored these “linguistic truth-values” (natural language truth-value attributions) in 1975,5 somewhat informally; more recently theoreticians have begun to incorporate linguistic truth-values into formal axiomatic systems (e.g., Hajek´ [2001].

Following Zadeh, we will interpret linguistic truth-values as fuzzy sets of truthvalues. For example, the interpretation of true might be the fuzzy set (over the unit interval [0. .1]) defined by the function

I(true)(n) = 0 if n ≤ .8 2((n − .8)(.2)2 if .8 <n ≤ .9 1 − 2((n − 1)/.2)2 if n > .9

(this example is Zadeh’s). Note that this function makes 1 true to degree 1 and 0 true to degree 0, a minimal requirement we might impose on such a function. The value

.81 is true to degree .05, .9 is true to degree .5, .91 is true to degree .595, and .99 is true to degree .995. The linguistic truth-value true is a propositional operator that combines with formulas just as negation does:

If Q is a formula, so is true Q

and the semantic clause

I(true Q) = I(true)(I(Q))

gives the truth-conditions for formulas formed with the operator true. If Tj symbolizes John is tall, then true Tj symbolizes It is true that John is tall. If Tj has the value

.91, then by the function we have assigned to true, true Tj has the value .595. More generally, interpretations will now include

An assignment to each primitive truth-value t of a function I(t) mapping [0. .1] to [0. .1]:

I(t) [0. .1][0. .1]

to assign fuzzy sets to all primitive truth-values, and the semantic truth-condition clauses will include

I(tQ) = I(t)(I(Q))

to determine the values of formulas formed with linguistic truth-values.

5 Zadeh (1975).