From: thomas7243@bellsouth.net   
      
   This is to announce a new result, which some may find interesting. In   
   the process of preparing a revised 2nd edition (forthcoming) of my   
   Fuzziness and Probability (ACG Press, 1995), I revisited the question of   
   the fuzzy connectives (and, or, not), and the semantic laws which they   
   obey. The connectives put forward in the 1st edition were, in effect,   
   linear combinations of the Zadeh (1965) min-max, Lukasiewicz   
   bounded-sum, and probabilistic product-sum rules, connected through a   
   semantic consistency coefficient, eta, that allowed law of excluded   
   middle (LEM) and law of non-contradiction (LC) to be restored. However,   
   associativity, distributivity, and absorption laws were not shown to   
   hold, rather clearly do not hold for arbitrary choices of eta. I have   
   now been able to develop a representation result, supported both by   
   mathematical theorem and computational experiments, that show, under   
   certain conformance requirements, how associativity, distributivity, and   
   absorption laws may also be upheld.   
      
   This seems to me to be an important theoretical result, because it means   
   that a fuzzy-set theory of semantics may obey all the semantic laws   
   familiar from the ordinary bivalent logic, and without doing away with   
   the fuzziness.   
      
   It is also important in that the result is "term-functional", in the   
   sense that the semantic consistency coefficient may be determined   
   entirely from the membership functions sought to be combined. The set-up   
   is as follows:   
      
                 { (1-t)*(a*b) + t*min(a,b) ,          if 0<= t <= 1,   
     a AND b  =  {   
                 { (1-t)*(a*b)  + t*max(0,a+b-1),      if -1<= t < 0.   
      
      
   where t is here eta, the semantic consistency coefficient joining the   
   two membership functions a and b sought to be combined. Similarly,   
      
      
                 { (1-t)*(a+b-a*b) + t*max(a,b) ,    if 0<= t <1,   
     a OR b  =   {   
                 { (1-t)*(a+b-a*b)  + t*min(1,a+b),      if -1<= t < 0.   
      
   Note that when t=1, the result is Zadeh min-max, corresponding to   
   maximal positive semantic consistency. When t= -1, we have the   
   bounded-sum (Lukasiewicz) rules corresponding to maximal negative   
   semantic consistency. And when t=0, we have the probabilistic   
   product-sum rules corresponding to semantic independence. In this way,   
   LEM and LC are easily restored, assuming t= -1 when a and b are complements.   
      
   Associativity, distributivity and absorption are not so easy to arrive   
   at. Take associativity. We begin with three membership functions a, b,   
   and c, and ask whether  (a AND b) AND c = a AND (b AND c), and   
   correspondingly for OR. Here we have four semantic consistency   
   coefficients to establish: t1 for that between a and b, t2 for that   
   between b and c, t3 for that between (a AND b) and c, and t4 for that   
   between a and (b AND c). The problem is to find a consistent set, t1,   
   t2, t3, and t4 for which associativity holds. And likewise for   
   distributivity, where there are five such coefficients that must be   
   found, and absorption, where there are three such coefficients.   
      
   Since Buckley-Siler (1998, 1999), and Chen et al (2000), we have known   
   that such consistent sets exist. In other words, these authors have   
   shown existence results. The present result consists in showing a   
   term-functional form of representation. One first computes the   
   correlation coefficient between the two functions. This scalar quantity   
   is then used as the argument to a function, phi, whose form is fully   
   specified, which then determines the semantic consistency coefficient   
   eta. Thus all of the information necessary to determine the result   
   resides within the membership functions a and b. This is the natural   
   generalization of the concept of truth-functionality, which clearly does   
   not hold, point-wise, when the construct of the semantic consistency   
   coefficient is introduced. It is argued that this notion of   
   term-functionality is fully consistent with the original notion of   
   truth-functionality put forward by Wittgenstein so many decades ago.   
      
   This result may be helpful from a practical perspective. Builders of   
   inference engines, in whatever field of application, have always had to   
   contend with the question, what connectives to use (e.g. Patel, 2003).   
   None of the extreme cases -- Zadeh, Lukasiecwicz, product-sum -- are   
   always appropriate. The present, flexible, connectives, are put forward   
   as a practical, not just theoretical alternative which, it is hoped,   
   will be of value to builders of inference engines.   
      
   The revised 2nd edition should be ready on the market within the next   
   3-6 months. In the meantime, I am making the chapter extract -- 64   
   pages, 24 figures, 7 tables -- available via email in pdf format to   
   those who might be interested. To help cover costs, a paypal donation of   
   $8.99, is kindly requested. Please send to the email address below with   
   your request for a copy.   
      
   Regards,   
   S. F. Thomas   
   sthomas at decancorp.com   
      
   References:   
   1. J. J. Buckley and W. Siler. (1998). "A new t-norm." _Fuzzy Sets and   
   Systems_, v. 100, pp. 283-290.   
   2. J. J. Buckley and W. Siler. (1999). "L-infinity fuzzy logic." _Fuzzy   
   Sets and Systems_, v. 107, pp. 309-322.   
   3. Dan Chen, Huacan He, Yuqin Ji, Hui Wang. (2000). "A new continuous   
   t-norm and its applications in fuzzy control." In _Proceedings of the   
   3rd World Congress on Intelligent Control and Automation_. June 28-July   
   2, 2000. Hefei. China.   
   4. Ambalal V. Patel.(2003). "Analytical structures of fuzzy PI   
   controllers with multifuzzy sets under various t-norms, t-conorms, and   
   inference methods (invited paper." _International Journal of   
   Computational Cognition_ v. 1, No. 3, pp. 93-141, Sept. 2003.   
   5. S. F. Thomas. (1995). _Fuzziness and Probability_. Wichita, KS: ACG   
   Press.   
   6. L.A. Zadeh. (1965).  "Fuzzy Sets."  _Information and Control 8_, pp.   
   338-353.   
      
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