> I see your difficulty. You think that if A is a fuzzy term, and its > membership function is denoted simply by a, let's say, then the > one-minus rule of negation gives the membership function of NOT A as > 1-a. Hence the "middle" is included, so to speak, and LEM and LC > should fail, as indeed it obviously does if the min-max rules are then > applied. For we have A AND NOT A being modeled in the meta-language as > min(a,1-a), which gives us the well-known middle with a peak at 0.5 > (assuming of course that a has its max at 1, its min at 0, and there > is gradation in-between).
> Now let's try another rule of conjunction, in particular the > Lukasiewicz bounded-sum rule, for which we have for two membership > functions a and b, and their corresponding terms A and B,
> mu[A AND B] = a AND b = max(0, a+b-1).
> In the particular case where B is NOT A, and b=1-a, we have under this > rule
> a AND b = max(0,a+1-a-1) = 0 everywhere
> and in accordance with the law of contradiction, the term A AND NOT A > is rendered as the comstant absurdity whose membership value is > everywhere 0. LC is upheld.
I am employed at the Silesian Technical University as a university teacher. In my work I have to very often answer to the following question. "Does John know topic x?" or "What is the relation between topic x and Mr John's knowledge?" Sometimes it is very difficult to answer this question. In order to answer to this question I use number between 2 and 5. If John know topic x, then I use number 5. If John don't know topic x, I use number 2. If I am not sure that John know topic x, I use number between 2 and 5. I think that this is a definition of fuzzy set. For example. John belong to the set of people which know topic x with degree 4= = John get 4 at the class test. Let's us consider the following situation? John get 3 at the class test. ( m(John | Topic x)=3) Marry get 4 at the class test. ( m(Marry | Topic x)=4) Do John and Mary know topic x? What is the answer to this question? a) m(John and Marry | Topic x)=min{3, 4}=3 b) m(John and Marry | Topic x)=(3+4)/2 (I think that this is quite good solution.) c) m(John and Marry | Topic x)=max(2, 3+4-5)= 2 (I think this is cruel.) What is the correct answer? Andrzej Pownuk P.S. I belong to the set of people who know English language with degree of membership 3=. I apologise for that. ------------------------------------------------ MSc. Andrzej Pownuk Chair of Theoretical Mechanics Silesian University of Technology E-mail: pownuk@zeus.polsl.gliwice.pl URL: http://zeus.polsl.gliwice.pl/~pownuk ------------------------------------------------