IEEE Intelligent Systems ‘2010, London, UK On Intuitionistic Fuzzy Negations and Law of Excluded...
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I E E E I n t e l l i g e n t S y s t e m s ‘ 2 0 1 0 , L o n d o On Intuitionistic Fuzzy On Intuitionistic Fuzzy Negations Negations and Law of Excluded and Law of Excluded Middle Middle Krassimir T. Atanassov Krassimir T. Atanassov Centre of Biophysics and Biomedical Engineering (Centre of Biomedical Engineering) Bulgarian Academy of Sciences e-mail: [email protected]
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Transcript of IEEE Intelligent Systems ‘2010, London, UK On Intuitionistic Fuzzy Negations and Law of Excluded...
IEEE In
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On Intuitionistic Fuzzy On Intuitionistic Fuzzy NegationsNegations
and Law of Excluded and Law of Excluded MiddleMiddleKrassimir T. AtanassovKrassimir T. Atanassov
Centre of Biophysics and Biomedical Engineering(Centre of Biomedical Engineering)
Bulgarian Academy of Sciencese-mail: [email protected]
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Introduction
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
1/20
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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.
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Introduction
In some definitions we shall use the functionssg and sg:
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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)>
IEEE In
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List of IF implications
IEEE In
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List of IF implications
IEEE In
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List of respective IF negations
IEEE In
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List of respective IF negations
IEEE In
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List of respective IF negations
IEEE In
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List of respective IF negations
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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.
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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.
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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>
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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.
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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.
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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.
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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.
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Thank you for your Thank you for your attention!attention!Krassimir T. AtanassovKrassimir T. Atanassov
Centre of Biophysics and Biomedical Engineering(Centre of Biomedical Engineering)
Bulgarian Academy of Sciencese-mail: [email protected]
