We can also express another logically true version of the Law of Excluded Middle:

P ¬ P.

11.11 Exercises

SECTION 11.2

1 Given an interpretation on which V(P) = 0

V(Q) = 0.3

V(R) = 0.8

V(S) = 0.2

V(T) = 0.5

V(W) = 1

what is the fuzzy value assigned to each of the following FuzzyL formulas?

a.P Q

b.Q T

c.P S

d.P → Q

e.Q → P

f.P ¬P

g.Q ¬Q

h.R ¬R

i.R → S

j.S → R

k.W → P

l.W → R

m.W → S

n.S → S

o.P ↔ Q

p.Q ↔ R

q.P & Q

r.Q & S

s.Q & T

t.P S

u.Q ¬R

v.S → (R → Q)

w.S → (Q → R)

x.R → (R & R)

y.(R & R) → R

z.(R → S) (S → R) z . (R ↔ T) (R ↔ ¬T)

SECTION 11.3

2 Is the argument P →¬P

¬P

valid in FuzzyL? Defend your answer. 3 Is the argument

¬ (P ↔ Q)

(P ↔ R) (Q ↔ R)

valid in FuzzyL? Defend your answer.

4We noted that P → (P & P) and (P P) → P are not FuzzyL tautologies although P → (P P) and P → (P P) are both tautologies in classical logic. Are the latter formulas—using weak connectives—tautologies in FuzzyL? Defend your answer.

5Are the converse formulas (P & P) → P and P → (P P) tautologies in FuzzyL? Defend your answer.

6Is the argument

P P P

valid in FuzzyL? Defend your answer.

7Compare L3 and FuzzyL entailment: are all FuzzyL entailments also L3 entailments? Does the converse hold? Prove that you are right.

8Show that the DeMorgan, Double Negation, and Distribution equivalences hold in FuzzyL.

9Prove that every FuzzyL formula can be mechanically converted to an equivalent FuzzyL formula containing only negation and bold disjunction.

10Use the Aguzzoli-Ciabattoni decision procedure to determine whether the formula ¬(P & ¬ P) is a tautology of FuzzyL.

SECTION 11.4

11Prove Result 11.8.

12Prove Result 11.10.

13a. Give an example of a formula containing conjunction that is a classical tautology but that is not a 1-tautology of FuzzyL when bold conjunction is used in place of the classical connective.

b.Give an example of a formula containing disjunction that is a classical tautology but that is not a 1-tautology of FuzzyL when bold disjunction is used in place of the classical connective.

c.Do the DeMorgan, Double Negation, and Distribution equivalences that are used for converting a formula to normal form hold in FuzzyL when bold conjunction and disjunction are used in place of weak conjunction and disjunction? For each equivalence that does hold, prove it. For each equivalence that doesn’t hold, provide a counterexample.

d.Do Results 11.7 and 11.8 hold when we use bold conjunction and disjunction in place of the weak connectives? If they do, prove it. If they don’t, provide counterexamples.

14 Prove that ¬(P (P → ¬P)) is a 1/ -tautology in Fuzzy .

3 L

15 Prove that the argument A

A → B

B → C

C → D

D → E

E → F

F → G

G → H

H → I

I → J J

is exactly .1-degree-valid in FuzzyL. We have already shown that it is at most

.1-degree-valid, so you need to show that we cannot make the gap between the value of the least true premise and that of the conclusion greater than .9. Hint: One way to do this is to consider cases as follows: Case 1: V(A) ≤ .9; Case 2: V(A) > .9 and V(B) ≤ .8; Case 3: V(A) > .9, V(B) > .8, and V(C) ≤ .7; and so on. In each case show that the gap between the value of the least true premise and that of the conclusion cannot be greater than .9.

SECTION 11.5

16Consider the fuzzy set of FuzzyL formulas = {P: .4, ¬P: .4, Q: .6, S → Q: .8, R: 1}, where P: n means that formula P is a member of the set to degree n, and each formula not listed between the set brackets is a member of to degree 0.

a.Describe the consonant fuzzy truth-value assignments for this set by indicating, for each atomic formula of FuzzyL, the least value that that formula can have on a truth-value assignment that is consonant for

b.For each of the following formulas, state its degree of membership in the fuzzy consequence FC( ):

i.P

ii.¬P

iii.Q

iv.S

v.R → P

vi.P ¬P

vii.Q → S

viii.R & Q

ix.S → (Q → P)

x.W

xi.¬P → W

17 Show how to define n-degree-entailment in terms of fuzzy consequence.

SECTION 11.6

18The conditional P ↔ ¬P has the value 1 in FuzzyL when P has the value .5. What value does P ↔K ¬P have when P has the value .5?

19Consider the argument

P

P →K Q

Q

a.Is this argument valid?

b.Is this argument degree-valid?

c.For what value of n is the argument n-degree-valid?

SECTION 11.7

20a. FuzzyBI conjunction and disjunction do not qualify as a t-norm and t- conorm. Which conditions for t-norms and t-conorms are violated by the FuzzyBI operations?

b.FuzzyBE conjunction and disjunction do not qualify as a t-norm and t- conorm. Which conditions for t-norms and t-conorms are violated by the FuzzyBE operations?

21Show that the algebraic product operation meets conditions 1–4 for t-norms and that the algebraic sum operation meets conditions 5–8 for t-conorms.

22Show that the condition

m tn n = 1− ((1−m) tc (1−n)), for all m, n [0. .1]

follows from the duality condition m tc n = 1− ((1 − m) tn (1 − n)), for all m, n [0. .1].

23Show that

a.bold conjunction and bold disjunction meet the condition for being a dual t-norm, t-conorm pair, and

b.that algebraic product and sum do as well.

24Show that Distribution holds for FuzzyL’s weak disjunction and conjunction but not its bold disjunction and conjunction.

25Complete the proof that every residuation operation is normal, that is, prove the cases where n = 0 and p = 1, and where n = 0 and p = 0.

26Show that given a dual t-norm, t-conorm pair tn and tc, the residuation operation adjunct to tn satisfies the following equation:

m n = (1 – m) tc n, for all m, n [0. .1].

27Show how to define bold conjunction using FuzzyL’s negation and conditional.

SECTION 11.8

28Prove that FuzzyG weak conjunction and disjunction are identical, respectively, to FuzzyG bold conjunction and disjunction.

29Using the FuzzyG versions of the connectives, what are the truth-values for the formulas in problem 1 when the atomic formulas have the indicated values?

30a. Prove that V(P Q) = V(((P → Q) → Q) ((Q → P) → P)) in FuzzyL. b. Prove that if we define FuzzyG weak disjunction as

V(P G Q) = V(((P →G Q) →G Q) G ((Q →G P) →G P)) then weak and bold conjunction are identical in FuzzyG.

31Prove that if we were to define P G Q as ¬G(¬GP G ¬GQ), then the value of any formula P G Q would always be either 1 or 0.

32Is P →G Q equivalent to ¬P G Q, analogously to FuzzyL’s equivalence?

33Use the decision procedure presented at the end of Section 11.8 to determine whether ¬G(P & G ¬GP) is a tautology of FuzzyG.

SECTION 11.9

34If P ↔P Q is defined as (P →P Q) P (Q →P P), what are the truth-conditions for

P ↔P Q?

35Using the FuzzyP versions of the connectives, what are the truth-values for the formulas in problem 1 when the atomic formulas have the indicated values?

36a. Prove that the truth-conditions 6 and 7 for FuzzyP weak conjunction and disjunction follow from the definitions

P P Q =def P &P (P →P Q)

and

P P Q =def ((P →P Q) →P Q) P ((Q →P P) →P P).

b.What would the truth-conditions for FuzzyP weak disjunction be if it were defined as

P P Q =def ¬P(¬PP P ¬PQ)?

c.What would the truth-conditions for FuzzyP weak conjunction be if it were defined as

P P Q =def P &P (¬PP &P Q)?

37a. Prove that P →G (P &G P) is a tautology but P →P (P &P P) is not.

b.Prove that ¬P¬PP →P (((Q &P P) →P (R &P P)) →P (Q →P R)) is a tautology but ¬G¬GP →G (((Q &G P) →G (R &G P)) →G (Q →G R)) is not.

c.Find another formula that is a tautology in FuzzyG but not in FuzzyP, and defend your answer.

d.Find another formula that is a tautology in FuzzyP but not in FuzzyG, and defend your answer.

e.Prove that the Modus Ponens inference is degree-valid in FuzzyG, but not in FuzzyP.

SECTION 11.10

38a. Can ∆P be defined in FuzzyL as ∆P =def ¬LP? Defend your answer.

b.Can ∆P be defined in FuzzyG as ∆P =def ¬GP? Defend your answer.

39Prove that the definition ∆P =def ¬G¬LP correctly defines fuzzy external assertion.

40a. What is the value of the formula ¬G P G ¬G ¬GP in FuzzyG when P has any value other than 1 and 0?

b.What is the value of the formula ¬G P G ¬G ¬GP in FuzzyG when P has the value 1? When P has the value 0?

c.What is the value of the formula ¬P P &P ¬P ¬PP in FuzzyP when P has any value other than 1 and 0?

d.What is the value of the formula ¬P P &P ¬P ¬PP in FuzzyP when P has the value 1? When P has the value 0?