From: mailbox@dmitry-kazakov.de   
      
   Hi William,   
      
   On 20 Oct 2004 16:53:22 -0700, William Siler wrote:   
      
   > "Dmitry A. Kazakov"  wrote in message   
   news:...   
   >>(Snip)   
   >>   
   >> - Intuitionistic fuzzy sets with the operations on them;   
   >> - Fuzzy logic based on the intuitionistic fuzzy sets and the possibility   
   >> theory;   
   >   
   > Dmitry, you have mentioned several times the term "Intuitionistic   
   > fuzzy sets". It is clear that a member of an intuitionistic fuzzy set   
   > has two grades of membership; one paper denotes them by mu and nu. I   
   > had the impression that mu corresponded to necessity, and nu to   
   > possibility.   
      
   it depends on the interpretation of the truth levels for singletons. I am   
   using mu(x)=Pos(x in the set), nu(x)=Pos(x in the complement set). Then mu   
   is possibility, nu is 1-necessity; mu is the upper set, 1-nu is the lower   
   set.   
      
   > For possibility, I have used the basic definitions by Dubois and Prade   
   > that necessity is a measure in [0, 1] of the extent to which the   
   > evidence supports the truth of a proposition, and that possilibility   
   > is a measure of the extent to which the evidence fails to refute a   
   > proposition. In the lack of any evidence, necessity is zero and   
   > possibility is 1. In general,   
   >   
   > Nec(A) <= Pos(A).   
      
   ... provided that A is not contradictory.   
      
   > Consider proposition A, whose current necessity is Nec1(A), and whose   
   > current possibility is Pos0(A). Now we have proposition P which   
   > supports A. Then   
   >   
   > Nec1(A) = max(Nec0(A), Nec(P))   
   > Pos1(A) = Pos0(A)   
      
   Current estimation of Pos(A) and Nec(A) is an evidence as well. Let us   
   denote it as R. Then Pos0(A)=Pos(A|R), Nec0(A)=Nec(A|R). Same with P, it   
   tells something about A provided that P is true, i.e. it is also a   
   conditional: Pos(A|P), Nec(A|P). Now both evidences R and P get combined.   
   We can do it by two ways:   
      
   1. optimistically RVP (here V stands for union, or, max) then   
      
      Pos(A|RVP)=Pos(A|R)VPos(A|P)   
      Nec(A|RVP)=Nec(A|R)&Nec(A|P)   
      
   2. pessimistically R&P (here & is intersection, and, min) then   
      
      Pos(A|R&P)<=Pos(A|R)&Pos(A|P)   
      Nec(A|R&P)>=Nec(A|R)VNec(A|P)   
      
   In your case, assuming that combination is pessimistic and   
   1="Pos(P)"[=Pos(A|P)], we will have exactly the result you are using:   
      
   Pos1(A)<=Pos0(A)   
   Nec1(A)>=Nec0(A)VNec(P)   
      
   Here "Pos1(A)"=Pos(A|R&P), "Nec1(A)"=Nec(A|R&P), "Nec(P)"=Nec(A|P). They   
   need not to be exact values, but also estimations, possibility from above,   
   necessity from below.   
      
   > Now consider propostion Q, that refutes A. Then   
   >   
   > Pos2(A) <= Nec(NOTQ) = 1 - Nec(Q)   
   >   
   > and   
   >   
   > Nec2(A) <= Pos1(A) AND Pos2(A)   
      
   Refuting A, "intuitionistically" is an evidence for ~A (~ is   
   complement/inversion, not, 1-x) so we have: Pos(~A|Q)=~Nec(A|Q) and   
   Nec(~A|Q)=~Pos(A|Q). Then as above:   
      
   1. optimistically SVQ:   
      
      Pos(A|SVQ)=Pos(A|S)VPos(A|Q)=Pos(A|S)V~Nec(~A|Q)   
      Nec(A|SVQ)=Nec(A|S)&Nec(A|Q)=Nec(A|S)&~Pos(~A|Q)   
      
   2. pessimistically S&Q:   
      
      Pos(A|S&Q)<=Pos(A|S)&~Nec(~A|Q)   
      Nec(A|S&Q)>=Nec(A|S)V~Pos(~A|Q)   
      
   Now assuming as you do, that S is the previous combination of R and P, and   
   going pessimistically:   
      
   Pos2(A)<=Pos1(A)&~Nec(Q)<=~Nec(Q)   
      
   So your result is correct, but it is cruder than it might be, because you   
   drop Pos1(A).   
      
   As for Nec2(A), a pessimistic estimation gives:   
      
   Nec2(A)>=Nec1(A)V~Pos(Q)   
      
   Provided as with supporting evidences that 1="Pos(Q)"[=~Nec(~A|Q)]. It   
   becomes:   
      
   Nec2(A)>=Nec1(A)   
      
   Then assuming that everything shall be non-contradictory, you cut it at the   
   level of Pos2(A), which gives your result again.   
      
   > The above is perfectly straightforward, and we have used it for many   
   > years in resolving the contradictions that sometimes appear in fuzzy   
   > reasoning. (Most of our work has been in fuzzy classifications,   
   > including online real-time detection of alarm conditions in a hospital   
   > intensive care unit, preliminary psychiatric diagnosis and   
   > classification of minerals in a rock sample from its X-ray diffraction   
   > spectrum.) It does, however, disagree with the treatment of   
   > possibility in Klir and Yuan, Chapter 7; in their treatment, the   
   > Dubois-Prade definitions of possibility are not used at all. It also   
   > disagrees with the paper by J. G. Sustal on Measures of Conflict for   
   > Intuitionistic Fuzzy Sets and for Fuzzy Classifications.   
      
   I see no open breaches. Except than you take some additional assumptions,   
   which could be well valid in your case, and that some equations are in fact   
   estimations. Not a big deal.   
      
   I remotely recall that I read Sustal's paper some time ago. I will try to   
   find it in papers. Just in case you have electronic references to the   
   papers, let me know.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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