From: ponwuk@poczta.onet.pl   
      
   Dear Dmitry,   
      
   > >*******************************************************   
   > >Dear Andrzej   
   > >   
   > >Your examples illustrate the point I made in my message(7-l6-03),   
   > >namely, that standard probability theory, PT, does not address problems   
   > >in which, as in your examples, we encounter partiality of truth and/or   
   > >partiality of possibility. Thus, in the proposition, "Robert is half-   
   > >German, quarter- French and quarter- Italian," the numbers 0.5, 0.25 and   
   > >0.25 are not probabilities but grades of membership or, equivalently,   
   > >truth values.   
   > >   
   > >Cordially yours,   
   > >   
   > >Lotfi   
   > >*******************************************************   
   >   
   > It is difficult to comment. So let me give another example:   
   >   
   > The house I see from my window is 0.1 tall as Empire State Building,   
   > 50.0 wide as my car, its color is 200 red and 80 percent covered with   
   > graffiti by some bastards. Definitely these numbers are not   
   > probabilities etc.   
      
   I agree with that.   
   This is not probability.   
      
   Probability theory is based on "yes" or "no" questions   
   because of that this "fuzzy" description is beyond   
   of the definition of the probability theory.   
   That was my final conclusion in the discussion with Bayesian specialists   
   and they do not give me any argument that my way of thinking is wrong.   
   Of course they don't agree with that :)   
      
   > Note that Prof. Zadeh's carefully avoids to call these arbitrary   
   > numbers possibilities. They are neither possibilities nor   
   > probabilities, but they could be under some conditions. So Prof. Zadeh   
   > is right, but you overstretch his arguments to get more from that.   
   >   
      
   Well, maybe .   
      
   >   
   > You are trying to define some additive measure which will be treated   
   > as the values of a membership function. So far so good.   
   >   
      
   I am not sure that these measures are always so additive.   
   In a real life we always use some measures, levels etc.   
   I am not sure that these descriptions are always additive.   
   ***********************************   
   There is only one requirement.   
   The decryption should be compatible   
   with the phenomena which is just described.   
   ***********************************   
   How to do that?   
   It depends on the application.   
      
      
   That is why I am so skeptical about   
   t-norms and existing fuzzy logic.   
   In existing fuzzy logic we can do the same operations   
   no matter what is going on.   
   Well if the reality is like that, then no problems.   
   But I am not sure that it is so simple.   
      
      
   > The first objection is. How it differs from what the probability   
   > theory does. It also defines an additive measure, and this is where   
   > the problems of the probability theory start.   
      
   If we consider the example of Robert (who know 50% of answers) the question   
   "Does Robert know passwords?" is ambiguous.   
   It is not convenient to answer to that question "yes" or "no".   
   We need another answer.   
   However in traditional sense there is no another answer.   
   Maybe I am wrong   
   but I think that because of that it is not possible   
   to find the answer to that question by using probability theory.   
   The answer "Robert know 50% of question" is precise   
   but this is not the answer in "classical" sense   
   (because the answer in classical sense doesn't exists).   
      
   This is the main difference between the "fuzzy" theory   
   and the probability theory.   
      
   Probability theory cannot find the answers to the ambiguous questions.   
   Probability theory need "yes" or "no" answers.   
      
   > Note that the probability theory is a well established one. Its   
   > additive measure is additive because there is a set of elementary   
   > outcomes which are independent and incompatible. Which is not a fact,   
   > but a premise on which the whole building stands. Remove that premise   
   > and it would collapse.   
      
   OK.   
   My description is not related to crisp problems.   
   In probability theory there is no such things the "half events"   
   of "quarter events".   
   My way of thinking is in completely other direction.   
   There are no events in my examples. No randomness etc.   
      
   > On the contrary, you just claim that something   
   > of unknown origin can be added and *is* a membership grade.   
      
   Well, I don't know what this thing really is.   
      
   > The question is why?   
      
   As far as I know the definition of membership function is completely   
   subjective.   
   We don't know how works our brain then the definition   
   of the membership function actually doesn't exists .   
      
   > Is anything that can be added a membership grade?   
      
   The meaning of grade of something depends on the problem.   
   Additivity is completely not necessary to that problem.   
   What is necessary?   
   Difficult to say in general.   
   Everything depends on the problem.   
   The decryption should be compatible   
      
   with the phenomena which is just described.   
   There is only one condition that have to be satisfied.   
      
      
   Grade of temperature grade at school, grade of height.   
      
   I have example of grade which is probably not additive measure.   
   This example is a little drastic   
   (I am not going to hurt anybody).   
   During the second word war in Germany there were grade of  "Germans".   
   Each person in Germany has some "grade".   
   Some persons are full Germans other has only different degree of   
   "Germnannes".   
   I cannot you give much details how these degree were calculated but   
   as far as I know they have precise meaning   
   (and they are not related to probability).   
      
   > How to mix grades of different origin not knowing what they are? Isn't it   
   > apples and oranges hidden under a cover of nice words?   
      
   Excellent questions.   
   Tell me that.   
      
   The only thing that I know at this moment   
   is that we use different grade in everyday life very often.   
   There is nothing extraordinary in that phenomenon.   
   How to crate general theory of grades?   
   I don't know unfortunately.   
      
   Fuzzy set theory is trying to do that.   
   However I see some weakness in this theory and this is   
   the reason why I wrote this e-mail.   
      
   > Then observe that the probability theory does not claim that the   
   > probability of, for instance, hitting a mark *is* a relation of the   
   > areas of the mark and the rest. It can be *numerically* equal to the   
   > relation under *definite* circumstances. Probability is neither an   
   > area nor a relation of areas. Why do you think that fuzzy membership   
   > grade should be one?   
   >   
      
   What fuzzy membership should be?   
   According to prof. Zadeh:   
   ""approximately a" given u, where u is a real number. In the fuzzy set   
   interpretation, the grade of membership of u in the fuzzy set would be an   
   answer to the question:  On the scale  from 0 to l, what is the degree to   
   which u fits your perception of "approximately a."   
      
   The problem is that this is a "magic" definition.   
   In my opinion in some cases we can train our brain   
   in order to accept some ideas.   
   We don't have to use some undefined statement like   
   "subjective answer to the question on the scale from 0 to 1".   
      
   In some cases this scale is very crisp.   
   There is no need to use subjective answers.   
   The number of passwords is very precise.   
   However the question "Does Robert know passwords?" is ambiguous.   
      
      
   [continued in next message]   
      
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