From: elias@fitz.TC.Cornell.EDU (Doug Elias)
Subject: Fuzzy Logic and \"George is 'smart'" -- 0.7/
Organization: Software and Consulting Support Group, Cornell Theory Center,
	C.U.
Date: Fri, 22 Jan 1993 15:26:43 GMT


arms@cs.UAlberta.CA (Bill Armstrong) comments that the statement:

                  George is smart -- 0.7

..."express(es) in fuzzy terms something like 70% of people would apply
this *description* to George."

Instead, this statement, if viewed as a proposition in fuzzy logic, is
saying that:
"i-the-speaker feel that this claim, "George is smart", has a truth
value, within the range of 0 [totally false] to 1 [totally true], of
.7" (this is only a simple interpretation of the statement; the more
normally accepted interpretation is described a few paragraphs down);
that means that, with all of the evidence available, the observer
subjectively feels that the claim is more-true-than-not, but isn't
completely true.

The two are very different in interpretation:
*) the former assumes that each observer makes a classical "is/is-not"
   decision, and then you sum over all of those binary 1-or-0's;
*) the latter says that (for a single observer) the strength-of-belief-in,
   or the weight-of-evidence-for the statement is being reported.

Translating this into fuzzy set theory provides a different perspective:
*) classically, the logical claim "George is smart (is true)" becomes the
   set theoretic "George is a member of the set 'smart'";
*) the fuzzy-logical claim "George is smart (is true) to the .7
   membership-grade-level" becomes the fuzzy-set-theoretic "George is
   a member of the fuzzily-defined subset 'smart' (of the overall
   classification-set "intelligence", which consists of
   ('rock-stupid', 'real-dumb', 'dumb', 'dim', 'normal', 'quick',
   'smart', 'real-smart', 'genius')) to the .7 degree of validity".

In the former, "smart" is a characteristic you are considered to
either have or not have; in the latter you can "have" it to varying
degrees over a range of possible values -- for example, you may be
'real-smart' when it comes to things like programming, but
'rock-stupid' when it comes to keeping yourself in shape, and 'dim'
when it comes to finances, etc,: a fuzzy classification scheme allows
you to specify how "well" you represent each category all-at-once, and
your overall "intelligence" classification is the join of your
representativeness in each one of the possible categories.

Getting back to fuzzy logic, "George is smart" would be normally be
applied to the fuzzy set consisting of possible truth-values, say:
    (totally-false, false, slightly-false, neutral, slightly-true, true,
     totally-true)
and the "0.7" would be the membership grade associated with one of
those categories.  This assignment would result, then, in a fuzzy-logical
proposition which could, for example, be combined with other such
claims using fuzzy-logical operators such as "not", "and", "or", and
the logical formulas that can constructed from them.  The "classical"
fuzzy operator-functions "1 - m()" for "negation", "max" for "or", and
"min" for "and" are only single members of essentially an infinite
number of operator-functions that can be used to implement the symbols
"~", "&", and "|".

Armstrong uses the term "linguistic" for what is called "ambiguous",
or "not completely assignable to a single category", within fuzzy set
theory.  "Ambiguous" classifications differ from probabilistic ones
precisely because probability theory requires that a given element of
the universal set be found in one-and-only-one subset of the
classification (there must be no overlap among the possible
alternatives), while fuzzy theory allows an element to maintain
membership in multiple categories at the same time -- i.e.,
probability theory enforces the "Law of the Excluded Middle", while
fuzzy theory relaxs it (another indication of how probability theory
is a proper subset of fuzzy theory, obtained when the "degree-of-relax-
ation of the Law of the Excluded Middle" is set to 0).

Applying these concepts to Armstrong's example propositions of "'George
is smart' == S" and "'George is rich' == R", using the fuzzy operators
~, &, and |, would then result each full fuzzy theorem (e.g., "S & ~R")
being evaluated and that result then being graphed onto the various
"strength-of-truthity" classes described previously.  Let's take this
one case as an example, and set:
    m(S) = .7
    m(R) = .4
Then m(S & ~R) = .7 & .6 = .7, using the classical fuzzy negation and
conjunction operaters.  Now, this simply states that George is a
member of the set of people who are both smart and not rich to the .7
degree, i.e., more-a-member-than-not.  We can now evaluate its various
truthvalues by using the following:

    1	____(At_Least)_Very Weakly True__________________
 M  |  /	                       |    -      -   	-|
 e  |*/**************************** -  |       	      -  |
 m  |/                         -       |           -   - |
 b  ||       (At Least)   -            |        -        |
 e.75|       Weakly True       	       |     - 	     -   |
 r  ||               -         	       |  -              |
 s  ||******************************** -           -     |
 h  ||          -      	            -  |                 |
 i  ||                 	         -     |         -       |
 p .5|      -         	      -	       |                 |
    ||         	   	   -  	       |       -	 |
 G  ||    -    	    True-              |        	 |
 r  ||       	     - 	               |    -   	 |
 a.25|**-*********-********Strongly True-       	 |
 d  ||         -  	        -      |               	/
 e  ||-	    -  	       	-              |               /
    || 	 -       -                     |   Very Strongly True
    || ----------------------------------------------/
    0/0________.25___________.5________|__.75_ __________1
        		               |
       Propositional Truth Value      .7
        			       |
 M  1--------(At-Least)-Very-Weakly-False----------- \
 e  || -    -           	       |	      \
 m  ||    -         -                  |	       \
 b  ||-      -             -           |		\
 e  ||          -        (At Least)    |		 |
 r.75|  	   _    	  -    |		 |
 s  || -	       	       Weakly  |		 |
 h  ||****************-************False		 |
 i  ||             False -             | -      	 |
 p  ||    -                 -  	       |        	 |
   .5|                         _       |        	 |
 G  || Strongly False             _    |       -	 |
 r  ||                  	     _ |                 |
 a  ||*********-********************** -                 |
 d .25  	        	       |  -         -    |
 e  |\  	         -             |     -           |
    | \ 		               |        -      - |
    |Very*************************** -*|	   -   	 |
    | Strongly False    	       |      -       -  |
    |____\_______________________________________________|
    0/0________.25___________.5________|__.75_ __________1


The final result, then, is the determination that
         m((George is smart) & (George is not rich)) == .7
is a member of the fuzzy-set "Truth Value" with membership grades:
   0/Very Strongly True 
 .25/Strongly True
 .70/True
 .85/At Least Weakly True
   1/At Least Very Weakly True
   1/At Least Very Weakly False
 .65/At Least Weakly False
 .30/False
 .25/Strongly False
   0/Very Strongly False

Now, for using this conjunction within a fuzzy logical evaluation, you
first determine what level of "truth value" you want it to have, then
plug in the appropriate membership grade for that level of "truth-ity".


Whew.  "Fuzzy Sets, Uncertainty and Information Theory", by Klir and
Folger (Plenum Press or Prentice Hall, can't remember which), 1988,
covers this in much more detail, and with much better graphics (sic).

doug
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