In this section, we give definitions of new generated implications and prove some useful properties of them.
3.1. Fuzzy Implications Generated by One Increasing Function and Two fuzzy negations
Theorem 1. Ifare two fuzzy negations andis an increasing and continuous function with g(0) = 0, then the functiondefined byis a fuzzy implication. Proof. Let be an increasing and continuous function with g(0) = 0 and
If then
is decreasing, i.e., I satisfies (1)
Let . If , then
is increasing, i.e., I satisfies (2)
, i.e., I satisfies (3)
, i.e., I satisfies (4)
, i.e., I satisfies (5)
Therefore, I ∈ FI. □
Proposition 3. Let I be the fuzzy implication of Theorem 1, then the fuzzy implication N-reciprocal of I is Proposition 4. Ifare strong negations, then the fuzzy implication of Theorem 1 satisfies additionally the left neutrality property (14) and the exchange principle (15).
Proof. , y ∈ [0, 1], i.e., I satisfies (14)
Thus, we have
i.e., I satisfies (15). □
Theorem 2. If φ ∈ Φ and I is the fuzzy implication of Theorem 1, then Iφ is a fuzzy implication.
Proof. According to Remark 1, is a fuzzy implication. □
Proposition 5. If = (the least fuzzy negation), then the fuzzy implication of Theorem 1 satisfies the Identity Principle (16).
Proof.
i.e., I satisfies (16). □
Proposition 6. If the fuzzy implication of Theorem 1 satisfies the Identity Principle (16), then it satisfies the Ordering Property (17).
Proof. Let x,y ∈ [0,1] and x ≤ y, then I(x,y) ≥ I(y, y) = 1. Thus, I(x, y) = 1.
If ⇔
⇔
⇔
⇔
x = 0 ≤ y or y = 1≥ x.
Thus, we have x ≤ y. □
Proposition 7. The natural negationof the fuzzy implication of Theorem 1 is Proposition 8. Whenare strong negations, then the fuzzy implication of Theorem 1 is an (S, N)–implication.
Proof. When
are strong negations, according to Theorem 1 and Proposition 4, the fuzzy implication
satisfies (I1) and (EP). Moreover, if
are continuous negations, then
is also a continuous fuzzy negation. We deduce that I is an (S, N) – implication (see [
1] Theorem 2.4.10). □
Example 2. Let .
Example 3. LetThus,Then, it is the Reinchenbach Implication.
The graph of the above surface is plotted in Figure 2. Example 4. . Then,
The graph of the above surface is plotted in Figure 3. Proposition 9. Letbe three fuzzy negations. Let us suppose also thatis a strong fuzzy negation. If g: [0,1] → [0,∞) is an increasing and continuous function with g(0) = 0 and, then the fuzzy implicationdefined in Theorem 1 satisfies the law of contraposition (18) with respect to.
Proof. g: [0,1] → [0,∞) is an increasing and continuous function
□
Lemma 1. (see [1] Proposition 1.5.3).Letbe three fuzzy negations with the propertiesbeing strict ones andadditionally being a strong negation. If g: [0,1] → [0,∞) is an increasing and continuous function with g(0) = 0 and, then the fuzzy implicationdefined by Theorem 1 satisfies the left (L-CP) and the right (R-CP) law of the contraposition. Proof. According to Proposition 1.5.3 [
1], I satisfies the left (19) and the right (20) law of the contraposition. □
Using Definition 14 and Proposition 2 of
Section 2, we prove the following:
Proposition 10. Let I be the fuzzy implication defined by Theorem 1, . Let us suppose that T is a t-norm and T satisfies (32), then:
- i.
.
- ii.
, if is a strong negation.
Proof. As I and T satisfy (TC), then for all .
- i.
Let ⇒ .
- ii.
From Proposition 2, we have ⇒
□
3.2. Fuzzy Implications Generated by Two Increasing Functions and Three Fuzzy Negations
Theorem 3. If are increasing and continuous functions withandare fuzzy negations, then the functiondefined byis a fuzzy implication. Proof. Let ∈ [0,1].
, i.e., I(∙,y) is decreasing, i.e., I satisfies (1).
Let ∈ [0, 1]
If
, then
⇒
i.e.,
is increasing, i.e., I satisfies (2).
□
Proposition 11. Letbe the fuzzy implication of Theorem 3. Ifare strong negations, then the neutrality property (14) and the exchange principle (15) are satisfied.
Proof. , i.e., I satisfies (14)
We conclude that (15) is satisfied. □
Proposition 12. Letbe the fuzzy implication of Theorem 3andbe fuzzy negations. If(the least fuzzy negation), then the identity principle (16) is satisfied.
Proof.
Thus, I satisfies (16). □
Proposition 13. If the fuzzy implication of Theorem 3 satisfies the identity principle (16), then it satisfies the ordering property (17).
Proof. Let x,y ∈ [0,1] and x ≤ y, then I(x,y) ≥ I(y, y) = 1. Thus, I(x, y) = 1.
and (
and or
x = 0 ≤ y or y = 1 ≥ x.
Thus, we have x ≤ y. □
Proposition 14. Letbe four fuzzy negations. Let us suppose also thatis a strong fuzzy negation. If: [0,1] → [0,∞) are increasing and continuous functions withand, and, then the fuzzy implication defined in Theorem 3 satisfies the law of contraposition (16) with respect to.
Proof. Thus, I satisfies (16). □
Proposition 15. The natural negationof the fuzzy implication of Theorem 3 is Example 5. Let
Example 6. Let .
Then, .
The graph of the above surface is plotted in Figure 5. 3.3. Fuzzy Implications Generated by n Increasing Function and n + 1 Fuzzy Negations
Theorem 4. If : [0,1] → [0,∞) are increasing and continuous functions, where (i = 1, 2, …n, n ∈ ℕ) and are fuzzy negations (i = 1,2,…n + 1 , n ∈ ℕ), then the function defined byis a fuzzy implication. Proof. Let ∈ [0, 1].
If ⇒ , then
⇒ , i.e., I(,y) is decreasing, i.e., I satisfies (1)
Let x ∈ [0, 1]. If , then
, i.e., I(x,ˑ) is increasing, i.e., I satisfies (2)
, i.e., I satisfies (3)
, i.e., I satisfies (4)
, i.e., I satisfies (5). □
Proposition 16. Letbe the fuzzy implication of Theorem 4. Ifare strong negations, then the left neutrality property (14) and the exchange principle (15) are satisfied.
Proof. y ∈ [0, 1], i.e., I satisfies (14)
We conclude that (15) is satisfied,
□
Proposition 17. Letthe fuzzy implication defined by Theorem 4. Then, if(the least fuzzy negation), the identity principle (16) is satisfied.
Proof. Thus, I satisfies (16). □
Proposition 18. Letn + 2 be fuzzy negations. Let us suppose also thatis a strong fuzzy negation. If: [0,1] → [0,∞) (i = 1, …, n , n ∈ ℕ) are increasing and continuous functions withand, then the functiondefined by Theorem 4,satisfies the law of contraposition (18) with respect to.
Proof. Thus, I satisfies (16). □
Proposition 19. The natural negationof the fuzzy negation of Theorem 4 is