From: mario.drobics@meduniwien.ac.at.nospam   
   Copy: hmhallani@gmail.com   
      
   To clearify things a bit: Dimitry is right concerning fuzzy sets,   
   however, you have to take care when you want to compute with these   
   membership grades. Typically, you will use fuzzy logic to do so. There   
   are different possible interpretations of what a gradual membership   
   value actually are. However, all interpretations have in common that   
   these memberships correspond to gradual truth values (of a certain   
   event!), whereas in probability/possibility theory you are dealing with   
   the uncertainty of events. This has important consequences on how to   
   compute with these values (conjunction/disjunction, etc.). While fuzzy   
   logic is (as the name suggests) a logic, possibility theory for example   
   is not even fully computational!   
      
   For a more detailed discussion, please see:   
   @article{DuboisPrade01,   
      author =	 {D. Dubois and H. Prade},   
      title =	 {Possibility theory, probability theory and multi-valued   
   logic: A clarification},   
      journal =	 {Annals of Mathematics and Artificial Intelligence},   
      year =	 {2001},   
      volume =	 {32},   
      pages =	 {35--66}   
   }   
      
   Dmitry A. Kazakov schrieb:   
   > On 4 Dec 2006 16:22:25 -0800, hmhallani@gmail.com wrote:   
   >   
   >> Someone asked me what was supposed to be an easy question: What does   
   >> the y-axis represents in the Membership Function? I know it is the   
   >> degree to which an input belongs to the appropriate fuzzy set. But the   
   >> person said, i need a more tangible answer.   
   >>   
   >> So please can you give me an answer to this question.   
   >   
   > It is a conditional set measure mA. The set being measured is the singleton   
   > {x}. So y=mA({x}). Any meaning of m is beyond the scope of the fuzzy set   
   > theory. In possibility theory m means the possibility that {x} belongs to   
   > the set A:   
   >   
   >    y=pos(A|{x})   
   >   
   > In probability theory it would the probability of.   
   >   
   > In general it is the theory of uncertainty you take, which assigns a   
   > "meaning" to the measure. However, any such theory would do this in some   
   > opaque terms like "probability" or "possibility," which themselves cannot   
   > be explained and taken for fundamental, intuitively defined. (Actually   
   > there were many attempts to reduce probability to something else. None of   
   > those was any successful. In my view it were just impossible to do.)   
   >   
      
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