1. IEEE Intelligent Systems ‘2010, London, UK On Intuitionistic Fuzzy Negations and 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. IEEE Intelligent Systems ‘2010, London, UK Introduction Let x be a variable. Then its intuitionistic fuzzy truth-value is represented by the ordered couple V (x) = 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 1/20

3. IEEE Intelligent Systems ‘2010, London, UK Introduction Everywhere below, we shall assume that for the three variables x; y and z equalities hold: V (x) = V (y) = V (z) = 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. IEEE Intelligent Systems ‘2010, London, UK Introduction In some definitions we shall use the functions sg and sg:

5. IEEE Intelligent Systems ‘2010, London, UK Introduction In ordinary intuitionistic fuzzy logic, the negation of variable x is N(x) such that V (N(x)) = For two variables x and y, operations conjunction (&) and disjunction (  ) are defined by: V(x & y) = V(x) & V(y) = V(x  y) = V(x)  V(y) =

6. IEEE Intelligent Systems ‘2010, London, UK List of IF implications

7. IEEE Intelligent Systems ‘2010, London, UK List of IF implications

8. IEEE Intelligent Systems ‘2010, London, UK List of respective IF negations

9. IEEE Intelligent Systems ‘2010, London, UK List of respective IF negations

10. IEEE Intelligent Systems ‘2010, London, UK List of respective IF negations

11. IEEE Intelligent Systems ‘2010, London, UK List of respective IF negations

12. IEEE Intelligent Systems ‘2010, London, UK Modified Law of excluded middle First, we shall give the LEM in the forms:  = (tautology-form) and  = (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:   = (tautology-form) and   = (IFT-form) where 1  p  q  0 and p + q  1.

13. IEEE Intelligent Systems ‘2010, London, UK 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. IEEE Intelligent Systems ‘2010, London, UK Now, on the following table, we will show the behaviour of the separate negations with respect to the special constants: V (true) = V (false) = V (full uncertainty) =

15. IEEE Intelligent Systems ‘2010, London, UK

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 shown on the following table, where b = 1.

19. IEEE Intelligent Systems ‘2010, London, UK 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. IEEE Intelligent Systems ‘2010, London, UK Conclusion Therefore, there are intuitionistic fuzzy negations that lose their properties when they are restricted to 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. IEEE Intelligent Systems ‘2010, London, UK Conclusion Finally, we must note that in the IFS theory there have 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. IEEE Intelligent Systems ‘2010, London, UK 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