On Intuitionistic Fuzzy Negations and Law of Excluded Middle

1. On Intuitionistic Fuzzy Negationsand Law of Excluded Middle Krassimir T. Atanassov Centre of Biophysics and Biomedical Engineering(Centre of Biomedical Engineering) Bulgarian Academy of Sciences e-mail: krat@bas.bg

2. Introduction 1/20 Let x be a variable. Then its intuitionistic fuzzytruth-value is represented by the ordered couple V (x) = <a, b> so that a, b, a + b[0; 1] where a and b are degrees of validity and of non-validity of x. Obviously, when V is an ordinary fuzzy truth-value estimation, for it b = 1 – a

3. Introduction Everywhere below, we shall assume that for the three variables x; y and z equalities hold: V (x) = <a, b> V (y) = <c, d> V (z) = <e, f> where a, b, c, d, e, f, a + b, c + d, e + f [0; 1]. For the needs of the discussion below, we shall define the notion of Intuitionistic Fuzzy Tautology (IFT) by: x is an IFT, if and only if ab, while x will be a tautology iff a = 1 and b = 0.

4. Introduction In some definitions we shall use the functionssg and sg:

5. Introduction In ordinary intuitionistic fuzzy logic, the negation of variable x is N(x) such that V (N(x)) = <b, a> For two variables x and y, operations conjunction (&) and disjunction ( ) are defined by: V(x & y) = V(x) & V(y) = < min(a, c), max(b; d)> V(xy) = V(x)V(y) = < max(a, c), min(b; d)>

6. List of IF implications

7. List of IF implications

8. List of respective IF negations

9. List of respective IF negations

10. List of respective IF negations

11. List of respective IF negations

12. Modified Law of excluded middle First, we shall give the LEM in the forms: <a, b> <a, b> = <1, 0> (tautology-form) and <a, b> <a, b> = <p, q> (IFT-form) where 1 pq 0 and p + q 1. Second, we shall give the Modified Law of Excluded Middle (MLEM) in the forms: <a, b> <a, b> = <1, 0> (tautology-form) and  <a, b> <a, b> = <p, q> (IFT-form) where 1 pq 0 and p + q 1.

13. Theorems Theorem #1: Only negation 13 satisfies the LEM in the tautological form. Theorem #2: Only negations 2, 5, 9, 11, 13, 16 satisfy the MLEM in the tautological form. Theorem #3: Only negations 2, 5, 6, 10 do not satisfy the LEM in the IFT form. Theorem #4: Only negation 10, does not satisfy the MLEM in the IFT form.

14. Now, on the following table, we will show the behaviour of the separate negations with respect to the special constants: V (true) = <1, 0> V (false) = <0, 1> V (full uncertainty) = <0, 0>

18. The above assertions show that a lot of negations exhibit behaviour that is typical of the intuitionistic logic,but not of the classical logic. Now, let us return from the intuitionistic fuzzy negations to ordinary fuzzy negations. The result is shownon the following table, where b = 1.

19. Therefore, from the list of intuitionistic fuzzy negations we can generate a list of fuzzy negations, such that some of them coincide with the standard fuzzy negation 1.

20. Conclusion Therefore, there are intuitionistic fuzzy negationsthat lose their properties when they are restrictedto ordinary fuzzy case. In other words, the construction of the intuitionistic fuzzy estimation <degree of membership/validity, degree of non-membership/non-validity> that is specific for the intuitionistic fuzzy sets, causes the intuitionistic behaviour of these sets.

21. Conclusion Finally, we must note that in the IFS theory therehave already been defined some other types of intuitionistic fuzzy negations different from the discussed above. Their behaviour will be studied in a next author's research.

22. Thank you for your attention! Krassimir T. Atanassov Centre of Biophysics and Biomedical Engineering(Centre of Biomedical Engineering) Bulgarian Academy of Sciences e-mail: krat@bas.bg