16.3 Other Fuzzy Extensions of Fuzzy Logic

cutoff for true insofar as the value 1 counts as true and all values less than 1 do not count as true, and similarly for false. The introduction of fuzzy linguistic truthvalues allows that truth may be a matter of degree rather than black and white. The value 1 may be the only one that counts as true to degree 1, but other values can count as true to high degrees as well. Or perhaps values less than, but close to, 1 may even count as true to degree 1. There’s a lot of flexibility here.

16.3 Other Fuzzy Extensions of Fuzzy Logic

There are other fuzzy ways that fuzzy logic has been extended. Although they are beyond the scope of this text, we’ll mention three examples as further avenues of study for the interested reader.

Quantifiers can be fuzzy as well as crisp. The universal and existential quantifiers are paradigmatic crisp quantifiers, as are quantifiers representing specific cardinalities: two people, fifty-eight heartbeats, . . . They are crisp in the sense that they tell us exactly how many things we are talking about. (This may seem counterintuitive in the case of the existential quantifier, which we read as: at least one. But at least one requires exactly a positive integer.) By contrast, few and many are fuzzy quantifiers: How many people, for example, are many people? In a group of 100 people there are surely many people, but are there many people in a group of 20? Of 10? There’s no specific cutoff point between many and not many, just as there is no specific cutoff point between tall and not tall.

Modalities, studied in modal logic, can also be fuzzy. The standard crisp modalities are necessary and possible (although it can certainly be argued that there are degrees of necessity and possibility). Probable is a fuzzy modality. To be sure, we can give specific probabilities for many things, and there are very precise logics of probability. But probable is a fuzzy modality because there is no specific probability or range of probabilities that counts as being probable.6

Finally, there is yet another issue of vagueness that we haven’t explored in this text, which we’ll now describe. We’ve noted that if we add the fuzzy external assertion operator ∆ to Lukasiewicz, Godel,¨ or product fuzzy logic, we can use the formula ( x)(¬ Tx ¬ ¬Tx) to express the existence of borderline cases for the predicate tall. But now recall that Bertrand Russell posited “higher-order” fringes so that we are not forced to recognize a precise cutoff point between a vague predicate’s extension and its fringe. Objects in the first higher-order fringes for a predicate, for example, are not in the predicate’s extension (or counterextension), nor are they in the fringe. The issue, first raised by Crispin Wright (1987), is finding a way to express the existence of higher-order fringes.

Wright proposed expressing the existence of first-order fringes thus: there is

no pair of heights that differ by 1/ such that one is definitely tall and the other is

8

6The reader interested in exploring these two extensions—fuzzy quantifiers and fuzzy modalities—would do well to start with Hajek´ (1998b).

definitely not tall. Using D for Wright’s definitely operator, we can symbolize this as ¬( x)( y)((DTx Eyx) D¬Ty). The existence of a second-order fringe between the predicate’s extension and first-order fringe would be expressed as ¬( x)( y)

((DDTx Eyx) D¬DTy): there is no pair of heights that differ by 1/ such that one is

8

definitely definitely tall and the other is definitely not definitely tall, and the existence of a third-order fringe between the predicate’s extension and second-order fringe would be expressed as ¬( x)( y)((DDDTx Eyx) D¬DDTy). . . . 7 Because these formulas are supposed to express the existence of distinct fringes, the formulas T, DT, DDT, DDDT, DDDDT, . . . , must all be nonequivalent to one another. And this means that the external assertion operator cannot do the work of the definitely operator, because any formula that prefixes one or more external assertion operators to the formula ∆P is equivalent to ∆P.

Wright didn’t propose a semantics for the definitely operator, but Richard Heck (1993) satisfactorily analyzed it outside fuzzy logic as a modal operator. In response to a concern that fuzzy logic cannot adequately represent higher-order vagueness, Libor Behounek (“A Model of Higher-Order Vagueness”) has begun formally to explore higher-order vagueness in fuzzy logic using such a definitely operator. Behounek doesn’t analyze the operator as a modality but rather uses fuzzy higherorder (higher than first-order) logic in which we can quantify over, and say things about, fuzzy sets.

16.4 Exercises

SECTION 16.1

1We noted one possible exception to our function for the hedge close to: we may want to say that a height that is tall to degree 1 is not close to tall at all. Do you agree with this intuition? If so, define a function for close to that captures the intuition. If not, explain why you believe the intuition is incorrect.

2We claimed that a height may be very tall, very very tall, very very very tall, close to tall, close to close to tall, very close to tall, close to very tall, or somewhat tall, but that it may not be somewhat somewhat tall, very somewhat tall, or somewhat very tall. Can you provide rules governing which combinations of the three hedges very, close to, and somewhat are permissible in English?

3Given the following interpretation:

D: set of heights between 5 and 6 2 by 1/8 increments, inclusive

I(T)(<x>) = (x – 5 )/14

I(very)(n) = n2

I(close-to)(n) = min (1, n + .1)

I(not)(n) = 1 – n

7Wright and subsequent analysts have had a lot more to say about this operator than this brief exposition suggests, but that literature would take us too far afield. For an overview see Keefe and Smith (1997).

I(a) = 6 7

I(b) = 6 1

I(c) = 5 5

I(d) = 5

I(e) = 4 7

determine the truth-value of each of the following formulas:

a.very Ta

b.very Tb

c.very Te

d.close-to Ta

e.close-to Tb

f.close-to Tc

g.close-to Td

h.very very Ta

i.close-to very Ta

j.very very Td

k.very close-to Tb

l.very close-to Tc

m.very very close-to Tb

n.very very close-to Tc

o.close-to very very Tb

p.close-to very very Tc

q.not close-to Te

r.not close-to very Td

4Provide reasonable interpretations for the following hedges:

a.somewhat

b.extremely

c.slightly

d.infinitesimally

e.hardly

f.clearly

SECTION 16.2

5Assuming that Tj has the value .9 and Tk has the value .2, and given the sample definitions in this chapter, determine the truth-value of each of the following formulas:

a.true Tj

b.not true Tj

c.false Tj

d.not false Tj

e.true Tk

f.not true Tk

g.false Tk

h.not false Tk

i.very true Tj

j.very false Tk

k.not very very false Tk

l.not very true and not very false Tk

m.close-to true Tj

n.close-to false Tk

o.not very close-to true Tj

6Using your definition of clearly in Exercise 4, what degrees of truth will be clearly true ? Does this seem correct?