>>> John Boone on Fuzzy Sets 
 WE>  JB> a textbook, "Fuzzy Logic for the Management of Uncertaintiy;"  
 JB> it is "serious" material with truth tables, etc. 
 WE> other book, what's introductory chapters like?  With substance? 
 JB> It depends upon the definiton of "substance."  I have, 
 JB> however, found it usefull as I learned from it, only 
 JB> on page 30 for the past 2 years.  
Substance is explicit definitions, theorems, proofs, formulas.
 
 WE>  JB> In Fuzzy logic, it is my recollection from the book, the 
 WE>  JB> "truth values" of -an item- ranges from 0 to 1.  
 WE> Is this like probability that assigns a probability to 
 WE> statements.  How is it different? 
 JB> Hmm, it didn't look like probability to me.  The author did 
 JB> comment that many say "fuzzy logic" is "probability" in disguise. 
That is true.  x e A (p) is not the same as x e A with probability p.  A half 
apple isn't the same as a random choice from a fruit basket half full of 
apples.
 JB> I agree, but since sets is nothing more than a collection of 
 JB> elements, regardless of what that element is, including sets 
 JB> beings elements of sets.   I ask, question here, not arguing, 
 JB> is there a requirement, that elements be distinct from sets? 
This is true of sets.  Fuzzy sets are different.  They are not a selection of 
elements, they're an assignment of belonging to every element.  The need to 
distinguish elements and sets has historical origin in the Russell paradox.  
All resolutions of this paradox have resulted is some form of recognition of 
this difference with resulting restrictions on what can be a set.
 JB> I am not sure if the "fuzzy set" only includes degrees, 
 JB> or an ordered pair of element, with degree.  I need to read more. 
 JB> Not unexpected, this definitional point is key for the rest 
 JB> of our discussion.  Let me try to read and share some of 
 JB> what I read at a future date, but I need to start to taper 
 JB> my other disussions on this echo.  
A fuzzy set A is a function from the universe U of elements into the closed 
unit interval, x e A (d) could be written A(x) = d.  As a function is a set 
of ordered pairs, the answer is yes.  Note that every fuzzy set contains all 
of the elements x e U to varying degrees.  The fuzzy set U contains all the 
elements with degree 1.  The fuzzy null set not U contains all the elements 
with degree 0.
Note the fuzzy set U is different than the set U.  U is the set of elements, 
for example {x,y,z}, while the fuzzy set U is the function U(x) = 1, U(y) = 
1, U(z) = 1 thus assuring that x e U (1), y e U (1), z e U (1) for all the 
elements in the universe {x,y,z} of my example.
 JB> I remember from Bart Kosko's 
 JB> book, somewhere between page 0 and 30, I believe around page 
 JB> 6 or so, Bart introduced the concept of fuzzy sets, e.g., 
 JB> the set of {apples.}  He asked at which point, in consumption 
 JB> as in eating (at least we are having fun , does the 
 JB> apple become {not apple?} 
This sounds like descriptive stuff.  Does he give any explicit definitions 
anywhere in the book?  The apple is not an apple when it's an apple core.  A 
half eaten apple is half apple.  A slice of an apple is 1/8 apple.  Simple 
enuf.  Now when a apple is rotten, how included in Apples is it?  How about 
apple sauce, cider?  Oh how language can out strip mathematical formalism.
 
 WE> Me neither as I'm not up on extensions of fuzzy set theory 
 WE> beyond the first strata. 
 JB> Thanks! 
It could be as mind twisting as asking to glue two mobius strips together 
edge to edge.
 
 WE> A <= B if for all elements x in U, x e A (a) & x e B (b) implies a <= b.
 JB> Thanks!  It does my heart good to see, a mathematical defintion 
 JB> of subsets and equivalent sets.  It has been YEARS since I have 
 JB> seen such, thanks. 
This is the sort of substance I'm inquiring if Kosho's book contains.  Do my 
notions jib with the actual theory of fuzzy sets?
... Is Occam's razor sharp enuf to split hairs?
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