Yet another method for determining a membership function, the vertical or reverse rating method, asks users of the language to identify, for some selected values vi in [0. .1], the range of heights that are tall to degree vi. The membership function is then constructed from these responses. Pedrycz and Gomide (1998) and Turksen (1991) contain excellent discussions of these and other methods of gathering data from which membership functions can be constructed.

17.3 Logical Relevance?

Does it matter, logically speaking, what shape our membership functions have, or what data we use to construct them? Not really. Logic is concerned with concepts like tautology and entailment, and fuzzy logic is additionally concerned with concepts like n-tautology and n-degree entailment. These logical concepts, fuzzy or not, are defined with respect to all possible interpretations, not just the specific ones we have chosen. From a logical point of view it doesn’t matter whether a specific membership function arbitrarily assigns degrees of membership, just as in classical logic it doesn’t matter how a specific interpretation distributes truth-values. This is our final response to the concern that fuzzy logic relies on assuming that, for any vague predicate, there is a clear cutoff point between cases to which the predicate clearly applies (to degree 1) and the borderline cases (degrees strictly between 1 and 0). N-tautology and n-degree entailment give us measures of tautologousness and entailment that don’t require setting any particular clear cutoff point. So, for example, the results in the table in Section 14.7 of Chapter 14 characterizing the validity, n-degree-validity, and decaying validity of our Sorites argument in the three fuzzy systems are independent of the way the membership function for tall is defined. And that’s the way it should be in logic.

17.4 Exercises

SECTION 17.1

1Using Goguen’s membership function for short, a domain consisting of heights between 3 4 and 8 4 inclusive, by increments of 1/8 , x recording a height in half-foot units, and the obvious interpretation of the predicate G,

a.what is the value of the Principle of Charity for shortness when the FuzzyL conditional and bold conjunction are substituted for the product conditional and bold conjunction?

b.what is the value of the Principle of Charity for shortness when the FuzzyG conditional and bold conjunction are substituted for the product conditional and bold conjunction?

2Can you find a plausible nonlinear membership function for short that will give the Principle of Charity for shortness a high degree of truth in FuzzyG ?

3Suggest other plausible membership functions for short.

4Find a vague noun whose applicability depends on several criteria. What are these criteria, and how would you weight each criterion when defining a membership function for the fuzzy set corresponding to the noun? Defend your answer.

SECTION 17.2

5Of the methods for empirical construction of membership functions presented in Section 17.2, which do you prefer and why?

6Propose another plausible method for empirical consruction of membership functions, and explain why it is plausible.

510

612

The totality of integers, positive, negative, and 0, is also countable. We will list the integers “emanating” from 0, alternating the positive ones with the negative ones:

10

21

3−1

42

5−2

63

7−3

Here each positive integer n is paired with n/2 if it is even, (1 − n)/2 if it is odd. Conversely, each integer m gets paired with 2m if m is positive, and 2(1 − m) − 1 if m is 0 or negative.

Although we have produced formulas showing how the numbers are paired in each of our examples, it’s sufficient in establishing countability to describe how to generate a sequence of the members of the set (our right-hand columns) such that each member of the set must occur exactly once in the sequence. The position of a member in the sequence then gives us the positive integer with which it is paired.

A set is uncountable (has uncountably many members) if it is not countable. Because all finite sets are countable, an uncountable set must at a minimum be infinite. The set of real numbers, as well as the subset of real numbers in the unit interval, are both uncountable. We will prove the latter using an ingenious type of argument developed by the German mathematician Georg Cantor and known as Cantor’s diagonal argument. More specifically, we’ll focus on the real numbers that lie strictly between 0 and 1 (we call this set the open unit interval), and at the end will introduce 0 and 1 into the picture. Every real number in the open unit interval can be written as an infinitely long decimal expression 0.d1d2d3d4d5 . . . , where each di is a single decimal digit. (Note that at some point the trailing digits may all be 0, as in .23000000 . . . – the 0’s allow an elegant presentation of the proof.) Now assume, contrary to what we want to show, that there is a 1-1 correspondence

between the positive integers and the open unit interval, and that a sequence of decimal expressions displaying the correspondence begins as follows:

0.d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8d1,9 . . .

0.d2,1d2,2d2,3d2,4d2,5d2,6d2,7d2,8d2,9 . . .

0.d3,1d3,2d3,3d3,4d3,5d3,6d3,7d3,8d3,9 . . .

0.d4,1d4,2d4,3d4,4d4,5d4,6d4,7d4,8d4,9 . . .

0.d5,1d5,2d5,3d5,4d5,5d5,6d5,7d5,8d5,9 . . .

0.d6,1d6,2d6,3d6,4d6,5d6,6d6,7d6,8d6,9 . . .

0.d7,1d7,2d7,3d7,4d7,5d7,6d7,7d7,8d7,9 . . .

0.d8,1d8,2d8,3d8,4d8,5d8,6d8,7d8,8d8,9 . . .

0.d9,1d9,2d9,3d9,4d9,5d9,6d9,7d9,8d9,9 . . .

We will now show how to find a number 0.e1e2e3e4e5 . . . in the open unit interval that appears nowhere in this sequence. We are going to do this by looking at the digits on the diagonal

0.d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8d1,9 . . .

0.d2,1d2,2d2,3d2,4d2,5d2,6d2,7d2,8d2,9 . . .

0.d3,1d3,2d3,3d3,4d3,5d3,6d3,7d3,8d3,9 . . .

0.d4,1d4,2d4,3d4,4d4,5d4,6d4,7d4,8d4,9 . . .

0.d5,1d5,2d5,3d5,4d5,5d5,6d5,7d5,8d5,9 . . .

0.d6,1d6,2d6,3d6,4d6,5d6,6d6,7d6,8d6,9 . . .

0.d7,1d7,2d7,3d7,4d7,5d7,6d7,7d7,8d7,9 . . .

0.d8,1d8,2d8,3d8,4d8,5d8,6d8,7d8,8d8,9 . . .

0.d9,1d9,2d9,3d9,4d9,5d9,6d9,7d9,8d9,9 . . .

If di,i is 0 we define ei to be 1; otherwise we define ei to be 0. The number 0.e1e2e3e4e5 . . . so defined is different from the first number 0.d1,1d1,2d1,3d1,4d1,5d1,6 d1,7d1,8d1,9 . . . in the list, because we have defined the first decimal digit e1 to be different from d1,1. Similar reasoning shows that 0.e1e2e3e4e5 . . . must be different from every other number in the sequence. So on the basis of the assumption that there is a way to sequence members of the open unit interval we can define a member of the open unit interval that doesn’t appear in the sequence. This is sufficient to show that there is no way to construct a sequence that includes every real number in the open unit interval. No matter how we try to order them in a sequence, we can always define a real number that’s different from every number in the sequence. We conclude that there are uncountably many members of the open unit interval.

It is a simple matter to show that the closed unit interval including 0 and 1 is also uncountable. If there were a 1-1 correspondence between the positive integers and the members of the closed unit interval, then there would be a way to sequence members of the closed interval such that every member occurs exactly once, and removing 0 and 1 from this sequence would leave a sequence of the members of

the open unit interval. But we have just shown that there is no such sequence, so we conclude that the closed unit interval is also uncountable. Generalizing this argument, we may conclude that any set that includes the open (or closed) unit interval must be uncountable—so the set of all real numbers is also uncountable.

The set of rational real numbers in the unit interval is, however, countable. Here’s the beginning of a 1-1 correspondence (that we will describe later) between the positive integers and those rational numbers, expressed as fractions:

10/1

21/1

31/2

41/3

52/3

62/4

73/4

83/5

92/5

103/5

114/5

121/6

135/6

141/7

In the right-hand column we list the beginning of a sequence of the rational numbers in the unit interval, generated as follows: We begin with the denominator 1 and list all fractions representing values in the unit interval that have this denominator— said fractions being ordered by increasing value of the numerator: 0/1, 1/1. Then we do the same for the numerator 2, except that we skip those fractions that equal values already listed—so we skip 0/2 and 2/2. Then we do the same for the numerator 3, and so on. Clearly every rational number in the unit interval will be represented by some fraction in this sequence, and skipping the indicated fraction ensures that no rational number is represented more than once in the sequence.

Finally, we can show that a language contains a countable number of formulas by explaining how to generate a sequence in which each of the formulas of the language occurs exactly once. We’ll first do this for the language FuzzyL, which is defined as:

1.Every uppercase roman letter, with or without an integer subscript, is a formula.

2.If P is a formula, so is ¬P.

3.If P and Q are formulas, so are (P Q), (P Q), (P → Q), (P ↔ Q), (P & Q), and

(P Q).

We (arbitrarily) impose the following alphabetical order on all of the symbols used in formulas of FuzzyL:

A

B

C

.

.

.

Z 0 1 2

.

.

.

9

(

)

¬

→

↔

&

The formulas may now be ordered in a sequence that is ordered overall by their length, the shortest formulas first, and within a single length by their alphabetical ordering. Thus the sequence begins with the twenty-six (nonsubscripted) uppercase roman letters in alphabetical order, these being the shortest formulas, followed by the following sequence:

A1

A2

. . .

A9

B1

B2

. . .

B9

C1

. . .

. . .

Z9

¬A ¬B

. . .

¬Z

—these being all of the formulas that contain exactly two symbols, and so on. Clearly each formula of FuzzyL will appear in this sequence, so the language contains a countable number of formulas.1

The language RFuzzyL is also countable—this can be shown quite simply by adding the symbol / to our alphabetical list of symbols, then defining the sequence as we did in the case of FuzzyL. However, we cannot do the same for RealFuzzyL— precisely because the totality of real numbers in the unit interval is uncountable. If we could produce such a sequence for RealFuzzyL, containing the names of all values in the unit interval, then we could also produce such a sequence consisting of just those real values, by removing all the formulas except the constant atomic formulas that denote real values.

1We need to note an important point here. We have to be careful about how we specify the sequence. We know that every formula will occur in the sequence we’ve described because every formula has a finite length, and we have a finite alphabet, so only finitely many formulas can occur before a given formula P. If we had instead tried to generate a sequence that begins with all formulas—of any length—that begin with the letter A; then all formulas that begin with the letter B, . . . ; then all formulas that begin with the negation operator; then all formulas that begin with a left parenthesis; . . . ; we’d be in trouble. There are infinitely many formulas that begin with the negation operator, and so the sequence would go on forever with these formulas and would never get to the formulas that begin with a left parenthesis!