Dear prof. Zadeh, > > Underlying these versions there is a basic problem which is exemplified > by the following. X is a variable ranging over positive integers. What I > know about X is that it is not a small integer, with "small integer" > defined as follows. If n is a positive integer then the degree to which > n fits the description "small integer" is 1/n. What is the probability > that X is 15? More generally, what is the probability that X is n? > > Do existing decision theories provide rational answers to these questions? >
I agree that the problem with "small" integer numbers is not related to probability. Let us consider an apple. If we cut this apple, then we have 50% of apple. Then we can say that this object is a piece of apple or apple which belong to the sets of apples with the degree 50%. Similarly we can create other "fuzzy" sets of apples by taking into account only relative volume of the apple (this is probably not perfect method but this is only an example). Similar problem is with Robert who is half-German, quarter- French and quarter- Italian. The numbers 0.5, 0.25 and 0.25 have very clear interpretation. Now we can now define the "fuzzy" sets of Germans, French and Italians. There are no problems with that. Such "fuzzy" classifications are commonly used in every day life. There is nothing extraordinary with that and in my opinion these problems are not connected with probability. Let us consider a little more interesting example. Probably everybody knows that SCO Company (http://www.sco.com) is trying to prove that some Linux code is a part of UNIX OS. >From mathematical point of view this problem can be described in the following way. Let us consider a set of all UNIX codes and some CodeX that belong to that set. What happened if we change two lines in that code? Is this program still a UNIX code? Probably yes. What happened if we change 1000 lines? Is this code still a UNIX code? Well, maybe in 50%. As we can see the set of UNIX code has no crisp boundary and the problem "How well CodeX fit to the set of UNIX software?" is a million dollar question today (of course I don't know the answer to that question). >From mathematical point of view this question is about degree of membership (or rather degree of ownership ... ). Of course in order to show the problem I simplified the real situation (and I don't know who is right in this dispute between SCO and Linux community). The process of changing the software is more or less continuous, because of that appropriate description of that changes should be also continuous (not only based on 0 and 1 numbers or "Yes" or "No" statements). Additionally I do not see any connection of this problem with probability. As we can see "fuzzy" descriptions are commonly used in everyday life and these are not probabilistic descriptions in my opinion. I don't know what the interpretations of t-norm in these cases are but this is another problem. Regards, Andrzej Pownuk - -------------------------------------- Ph.D., research associate at: Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology URL: http://zeus.polsl.gliwice.pl/~pownuk E-mail: [EMAIL PROTECTED] - --------------------------------------
