Repeating this reasoning we eventually end up concluding that even on an assignment v192 such that v192(x) = I(s1) and v192(y) = I(s2), Iv192(Tx), which is Iv192(Ts1), is less than or equal to Iv192(Ts193). But this contradicts the assumption that all of the premises are truer than Ts193. We conclude that the Sorites argument as symbolized in FuzzyG is 1-degree-valid. This reasoning generalizes for longer Sorites chains as well, so the Sorites argument also fails to display decaying validity in FuzzyG .

14.6 Product Fuzzy First-Order Logic

Again, we use the same truth-condition clauses for the quantifiers to define the system FuzzyP :

1.Iv(Pt1 . . . tn) = I (P)(<I*(t1), . . . , I*(tn)>), where I*(ti) is I(ti) if ti is a constant and is v(ti) if ti is a variable

2.Iv(¬PP) = 1 if Iv(P) = 0

0otherwise

3.Iv(P &P Q) = Iv(P) · Iv(Q)

4.Iv(P P Q) = Iv(P) + Iv(Q) – (Iv(P) · Iv(Q))

5.Iv(P →P Q) = 1 if Iv(P) ≤ Iv(Q)

Iv(Q) / Iv(P) otherwise

6.Iv(P ↔P Q): left as an exercise in Chapter 11

7.Iv(( x)P) = glb{Iv (P): v is an x-variant of v}

8.Iv(( x)P) = lub{Iv (P): v is an x-variant of v}

Here bold conjunction is defined to be the algebraic product operation, bold disjunction is defined to be the algebraic sum operation, and negation turns out to be identical to Godel¨ negation. The main semantic results for FuzzyP carry over to FuzzyP . Every formula that is a tautology in FuzzyP is a tautology in classical logic, but not conversely; and every entailment in FuzzyP is a classical entailment, but not conversely. The Sorites argument

Ts1

Es2s1 Es3s2 Es4s3

. . .

Es193s192

( x)( y)((Tx P Eyx) →P Ty)

Ts193

is valid in FuzzyP , as is the version using product bold conjunction in the Principle of Charity premise: ( x)( y)((Tx &P Eyx) →P Ty). But not all of the premises have the value 1 on interpretation SST: in fact, both versions of the Principle of Charity premise have the value 0. When v(x) = 4 71/8 and v(y) = 4 7 , Iv((Tx P Eyx) →P Ty) = 0 because the consequent has the value 0 and the antecedent has the greater

value 1/192, from which it follows that the value of the conditional is the value of the consequent divided by that of the antecedent. Similarly, Iv((Tx &P Eyx) →P Ty) = 0 because the value of the antecedent is also 1/192 in this case. Like the Godel¨ version, the product versions of the Principle of Charity compare poorly with theLukasiewicz

version on interpretation SST given the intuition that the Principle of Charity is close to true. In this connection we note that if we choose a membership function for tall that doesn’t assign 0 as the degree of membership for any height, then the Sorites argument in product fuzzy logic can still have a conclusion whose value is very close to 0 while the Principle of Charity premise has a much higher value.4

We can approximate the n-degree-validity of the bold product conjunction version of the Sorites argument as follows (we approximate in this case because the analysis is considerably more complicated than in the other systems).5 Let m1, m2, . . . , m193 be the values assigned to Ts1, Ts2, . . . , Ts193 on an interpretation I, and let n1, n2, . . . n192 be the values assigned to Es2s1, Es3s2, . . . , Es193s192. We note that when v(x) = I(si) and v(y) = I(si+1) for some i, 1 ≤ i ≤ 192, the value pi of (Tx &P Eyx) →P Ty is 1 if mi · ni ≤ mi+1 and it is mi+1 / (mi · ni) otherwise. In all other cases (i.e., all assignments v that don’t assign such values x and y) we’ll assume that Iv((Tx &P Eyx) →P Ty) = 1 – it can because Iv(Eyx) can be 0 in these cases. (This assumption is insignificant given that we’re trying to find a maximum downward distance.) Thus the value of the Principle of Charity premise is min (p1, . . . , p192). So the maximum possible downward distance between the value of the least true premise and that of the conclusion is then the maximum value that min (m1, n1, . . . , n192, p1, . . . , p192) – m192 can be for some interpretation.

If we restrict our attention to interpretations on which each n1 = n2 = · · · = n192 = m1, then min (m1, n1, . . . , n192, p1, . . . , p192) – m192 reduces to min (m1, p1, . . . , p192) – m192. (Letting the ni values be lower than m1 would decrease the maximum downward distance, so this restriction is also insignificant for our purposes.) If we further restrict our attention to interpretations on which m1 < 1 and mi = m1i for each i > 1, then as i gets larger the values m1i are getting smaller and smaller— and we are trying to make the value assigned to the conclusion as small as we can. Moreover, if mi = m1i for each i > 1 then every pi is 1 because mi · ni = mi · m1 = m1i+1 ≤ m1i+1, and so min (m1, p1, . . . , p192) – m192 now reduces to m1 – m192 or m1 – m1193! (This is why we didn’t want the ni values to be greater than m1.) So we can ask, What is the maximum value that m1 – m1193 can be when m1 is a value in the unit interval [0. .1]? This value is around .97, meaning the argument is at most

.03-degree-valid, clearly a desirable result. The reader will be asked to do a similar

4In fact, the examples in Goguen (1968–1969) use such a membership function—degrees of membership can get close to, but never reach 0. We’ll look at this function in Chapter 17. (Goguen’s membership function does not, however, increase the value of the Godel¨ version of the Principle of Charity.)

We hasten to make one technical point: although the choice of membership function can make a clear difference here, the logical results like validity and n-degree validity do not depend on any particular membership function.

5I am grateful to my colleague Michael Albertson for pointing out that the restrictions used in this analysis yield a simple but good approximation to the n-degree-validity of the Sorites argument.

a.Symbolize this argument using Wx to mean: x dollars is wealth, and letting the domain consist of the set of integers from 10 to 1,000,000.

b.Using the example for the predicate tall in this chapter as a model, find a formula for assigning a fuzzy value in the range [0. .1] to each member of the domain for the predicate W, such that

I(W)(<10>) = 0,

I (W)(<1,000,000>) = 1,

and all other members of the domain are assigned values between 0 and 1 as seems appropriate but subject to the stipulation that I(W)(<x>) ≤ I(W)(<y>) when x < y.

SECTION 14.2

2Consider an interpretation I whose domain D consists of the positive integers {1, 2, 3, . . . } and that makes the following assignments:

I(M)(<x>) = 1/x

I(N)(<x1,x2>) = 1/(x1 + x2) I(a) = 1

I(b) = 5

I(c) = 20

Using the semantic clauses for FuzzyL , what is the (fuzzy) value assigned to each of the following formulas?

a.Ma Mc

b.Mb & Mc

c.( x)Mx

d.( x)¬Mx

e.( x)(Mx & Nxx)

f.( x)(Mx Nxx)

g.( x)(Mx Nxx)

h.( x)(Mx ¬Mx)

i.( x)(Nxx ↔ ¬Nxx)

j.( x)(Nxx → Mx)

k.( x)( y)(Nxy → Mx)

l.( x)( y)(Nxy → (Mx My))

m.( x)( y)(Nxy → (Mx & My))

3What are the values of the formulas i–m in Exercise 2 when the Kleene conditional and biconditional are used in place of the Lukasiewicz connectives?

SECTION 14.3

4For each of the following formulas, state the degree n to which it is an n-tautology in FuzzyL , and explain your answer:

a.( x)Px → ( x)Px

b.( x)(Px → ( x)Px)

c.( x)( y)Lxy →( y)( x)Lxy

d.( x)( y)Lxy → ( y)( x)Lxy

e.( x)Px ( x)¬Px

f.( x)Px ( x)¬Px

g.( x)Px ( x)¬Px

h.( x)Px ( x)¬Px

5For each of the following arguments, state the degree n to which it is n-degree- valid in FuzzyL , and explain your answer:

a.( x)Px ( x)Px

b.( x)Px ( x)Px

c.( x)Px ( x)Rx

( x)(Px Rx)

d.( x)Px ( x)Rx

( x)(Px & Rx)

e.( x)Px Qa → Pa

SECTION 14.4

6Show that the formulas ( x)(¬ Tx ¬ ¬Tx) and ( x)(¬ Tx Tx) are not equivalent in FuzzyL by producing an interpretation on which the formulas have different values.

7We claimed that any variation of SST that assigns to the predicate T in our Sorites argument a noncrisp interpretation in which at least two heights have different degrees of tallness and tallness never increases as height decreases (but that keeps the rest of SST the same) will give the Principle of Charity premise a value less than 1 in FuzzyL . Prove this claim.

8Explain why both Ts1 and the weak conjunction version of the Principle of Charity premise, ( x)( y)((Tx Eyx) → Ty), have the truth-value 192/193 on interpretation SST*.

9Prove that on the interpretation

D:set of heights between 4 7 and 6 7 by 1/8 increments, inclusive I(T)(<x>) = (2 · (x – 4 7 )) / 481/8

I(E)(<x1,x2>) = 384/385 for all x1, x2 I(s1) = 6 7

I(s2) = 6 67/8

. . .

I(s 193) = 4 7

I(Ts1) = I(Es2s1) = I(Es3s2) = · · · = I(Es193s192) = I(( x) ( y) ((Tx & Eyx) → Ty)) =

384/385 in FuzzyL , while I(Ts193) = 0.

10Assume that v is a variable assignment (for an interpretation) such that the values of Tx and Eyx are identical. What is the relation between the values of (Tx Eyx) → Ty and (Tx & Eyx) → Ty on v? Be as specific as you can.

11Prove that .5 is the minimum value that max (191/192 – Iv(Ty), Iv(Ty)) can have for any variable assignment v on interpretation SST.

12a. We claimed that when we use the Kleene conditional in a Sorites argument we don’t get decaying validity as we increase the subscript in the conclusion, but that rather the argument remains .5-degree-valid. Prove this claim. (When we increase the subscript we will automatically add the necessary additional Esi+1Esi premises.)

b.A Kleenean Sorites argument is also .5-degree valid if we decrease the subscript in the conclusion—to a point. What is the smallest subscript for which the Kleenean argument is .5-degree-valid? Prove that you are right.

SECTION 14.5

13 What are the values of the formulas in Exercise 2 when the Lukasiewicz connectives are replaced with Godel¨ connectives?

14What is the value of the Godel¨ conjunctive version of the Principle of Charity, ( x)( y)¬G((Tx &G Eyx) &G ¬GTy), on interpretation SST?

15Show that the Godel¨ conjunctive version of the Sorites argument (using the Principle of Charity ( x)( y)¬G((Tx &G Eyx) &G ¬GTy)) is not valid in FuzzyG .

16Show that if we modify the conclusion of the Godel¨ conjunctive version of the Sorites argument to be ¬G¬GTs193, the resulting argument is valid in FuzzyG .

17Analyze Max Black’s fringe formula ( x)(¬GTx G ¬G¬GTx) in FuzzyG : what is the least value that this formula can have? What is the greatest value?

SECTION 14.6

18 What are the values of the formulas in Exercise 2 when the Lukasiewicz connectives are replaced with product connectives?

19We claimed that if we choose a membership function for tall that doesn’t assign 0 as the degree of membership for any height, then a Sorites argument in FuzzyP can still have a conclusion whose value is very close to 0 while the Principle of Charity premise has a much higher value. Show that this is so by modifying SST’s definition for the fuzzy set denoted for T to be:

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

1/192 if x = 4 7

and then determining the value of the Principle of Charity premise (with either bold or weak conjunction in the antecedent).

20What is the value of the bold product conjunctive version of the Principle of Charity, ( x)( y)¬P((Tx &P Eyx) &P ¬PTy), on interpretation SST?

21Show that the bold product conjunctive version of the Sorites argument (using the Principle of Charity ( x)( y)¬P((Tx &P Eyx) &P ¬PTy)) is not valid in FuzzyP .

22Show that if we modify the conclusion of the bold product conjunctive version of the Sorites argument to be ¬P¬PTs193, the resulting argument is valid in FuzzyP .

23Analyze the n-degree-validity of the (conditional version of the) Sorites argument in FuzzyP when weak product conjunction is used instead of bold product conjunction in the Principle of Charity premise. To simplify, you may restrict your attention to interpretations on which the Esi+1si premises all have the value 1—but be sure to explain why this restriction won’t produce a wildly inaccurate anaysis. Try to think of other simplifications that may help in the analysis.

24Does either product version of the Sorites argument display decaying validity?

25Analyze Max Black’s fringe formula ( x) (¬PTx P ¬P¬PTx) in FuzzyP : what is the least value that this formula can have? What is the greatest value?

propositional Godel¨ and product logics aren’t: they have noncontinuous propositional operations. Here is a table summarizing the situation:

15.2 A Pavelka-Style Derivation System for FuzzyL

In this section we present a Pavelka-style system for FuzzyL , which we call FL PA3. We must add atomic formulas denoting the rational values in the unit interval to the language FuzzyL —we call the expanded system RFuzzyL . The axioms of FL PA include the axiom schemata for the propositional system FLPA:

FL P1. [P → (Q → P), 1]

FL P2. [(P → Q) → ((Q → R) → (P → R)), 1] FL P3. [(¬P → ¬Q) → (Q → P), 1]

FL P4. [((P → Q) → Q) → ((Q → P) → P), 1]

FL P5.1. Includes every graded formula [(m → n) → p, 1] where m, n, and p are atomic formulas denoting rational truth-values m, n, and p in the unit interval such that p = min (1, 1 – m + n)

FL P5.2. Includes every graded formula [p → (m → n), 1] where m, n, and p are as in FL P5.1

FL P6.1. Includes every graded formula [¬m → p, 1] where m and p are atomic formulas denoting rational truth-values m and p such that p = 1 – m

FL P6.2. Includes every graded formula [p → ¬m, 1] where m and p are as in FL P6.1

FL P7. Includes [m, m] for any rational value m in the unit interval, where m is the atomic formula that denotes the value m

FL PA also has two axiom schemata for quantifiers:

FL P8. [( x)(P → Q) → (P → ( x)Q), 1]

where P is a formula in which x does not occur free

FL P9. [( x)P → P(a/x), 1]

where a is any individual constant and the expression P(a/x) means: the result of substituting the constant a for the variable x wherever x occurs free in P.

3Vilem´ Novak´ (1990) generalized Pavelka-syle systems for Lukasiewicz fuzzy propositional logic to first-order systems. FL PA is from Novak,´ Perfilieva, and Mockoˇˇr (1999), which owes partly to Hajek´ (1997).

The rules of FL PA are

MP. From [P, m] and [P → Q, n], infer [Q, p], where p = max (0, m + n – 1) TCI. From [P, m] infer [m → P, 1],

where m is the atomic formula that denotes the value m UG. From [P(a/x), m] infer [( x)P, m]

where x is any individual variable, provided that no assumption contains the constant a and that P does not contain the constant a

The rule UG tells us that if an arbitrary instance of a universal quantification has at least the value m, we may infer that the quantified formula also has at least the value m. This is acceptable because what is true of an arbitrary instance must be true of all instances, so every instance of the quantified formula must have at least the value that an arbitrary instance has.

Using the definition of existential quantification in FuzzyL :

( x)P =def ¬( x)¬P

we have the following derived axiom schemata (we start numbering at 20 because we have all of the derived axioms—as well as the derived rules—from the axiomatic system FLPA):

FL PD20. [( x)(P → Q) → (( x)P → Q), 1]

where Q is a formula in which x does not occur free

Justification. We derive the formula [( x)(P → Q) → (¬( x)¬P → Q), 1], where x does not occur free in Q and a is a constant that does not occur in P or Q:

FL PD21. [P(a/x) → ( x)P, 1]

Justification: Justification is left as an exercise.

We mention these derived axioms because they will appear as explicit axioms in the system for FuzzyG in the next section.

With obvious substitutions we can also produce a derivation for the bold conjunction version of the Sorites argument, using the rule

BCI (Bold Conjunction Introduction). From [P, m] and [Q, n] infer [P & Q, k] where k = max (0, m + n – 1)

which was also derived in FLPA.

It is left as an exercise to produce derivations that show that if the Principle of Charity premise in either version of the Sorites argument has at most the value 191/192 and the other premises have at most the value 1, then the conclusion has at most the value 0. Together these derivations show that if the Principle of Charity premise in the Sorites argument (either version) has exactly the value 191/192 and all of the other premises are true, the conclusion is false.

Recall that our Kleenean version of the Sorites argument is .5-valid. It should therefore be possible to produce derivations in which the premises are all graded with values greater than .5 and the grade of the conclusion is .5 less than the least of these. In fact, in some cases we can infer a higher grade for the conclusion (this doesn’t contradict .5-validity, which gives an upper bound of .5 on the distance

between the values of the premises and that of the conclusion). For example, if the premises Ts1, Es2s1, Es3s2, . . . , Es193s192, and ( x)( y)((Tx K Eyx) →K Ty) all have the value .6, then so must the conclusion. The Kleenean Principle of Charity is equivalent to ( x)( y)(¬(Tx Eyx) Ty) using only Lukasiewicz’s connectives, and if both Ts1 and Es2s1 have the value .6 then ¬(Tx Eyx) has the value .4 and so Ts2 must have the value .6 to give the disjunction the value .6. This reasoning may be repeated over and over again, finally showing that Ts193 must also have the value .6 as well. Here is a derivation showing that if all the premises have at least the value

.6, so must the conclusion (we switch freely between the equivalent forms P Q and (P → Q) → Q, depending on whether we are using the formula as a disjunction or as a conditional). We begin by deriving the formula (Ts2 → 4/10) → 8/10, which contains only Ts2 and truth-value constants:

At this point we know that Ts2 must have at least the value .6, for if it didn’t, the antecedent Ts2 → 4/10 of the formula on line 212 would have a value greater than .8 and the conditional’s value would be less than 1, not 1 as it is graded. Our task now is to derive the formula 6/10 → Ts2 that says that Ts2 has at least the value .6. We will

do this using a strategy that we saw in the derivation of rule CV in Chapter 13, first showing that Ts2 has at least the value .2, then at least .4, then at least .6:

FL P7

FL P5.2

213,214 MP

FL P2, with Ts2 / P, 0 / Q, 4/10 / R

215,216 GMP

212,217 HS

FL PD3, with 8/10 / P, 0 / Q

218,219 HS

220, TRANS

FL P4, with Ts2 / P, 0 / Q

221,222 HS

FL PD19, with Ts2 / P

224, TCI 223,225 GMP

FL P5.2

226,227 HS FL P5.2 213, 229 MP

FL P2, with Ts2 / P, 2/10 / Q, 4/10 / R

230,231 GMP

212,232 HS

FL PD3, with 8/10 / P, 2/10 / Q

233,234 HS

235, TRANS

FL P4, with Ts2 / P, 2/10 / Q

236,237 HS

228,238 GMP

FL P5.2

239,240 HS

FL PD3, with 8/10 / P, 4/10 / Q

212,242 HS

243, TRANS

FL P4, with Ts2 / P, 4/10 / Q

244,245 HS

241,246 GMP

FL P5.2

247,248 HS

. . . . . . {lines 195–202, 204–210, 212, 216–218, 220–226, 228, 231–233, 235–239, 241, 243–247, and 249 repeat 191 times (we don’t need to repeat the formulas that contain only truthvalue constants), with 1 added to the subscripts each time, ending with}

8271 [6/10 → Ts193, 1] 8269,8270 HS 8272 [6/10, .6]

8273 [Ts193, .6]