On 01-15-98 William Elliot wrote to John Boone... 
 
        Hello William, 
  
 WE>  JB> No, it was an introduction into Fuzzy Logic.  I do have  
 WE>  JB> a textbook, "Fuzzy Logic for the Management of Uncertaintiy;"  
 
 WE> Is the intro just descriptive, without any theory or 
 WE> serious material  I'd suggest skipping it.  How about the  
 
  I have not looked at the intro, but I did flip through 
the book when I got it, and it is "serious" material with 
truth tables, etc. 
  
 WE> other book, what's introductory chapters like?  With 
 WE> substance? 
  
  It depends upon the definiton of "substance."  I have, 
however, found it usefull as I learned from it, only 
on page 30 for the past 2 years.  
  
 WE>  JB> my required study material (I have to read about 20 to 30 
 WE>  JB> biological (mostly medical) magazines per month), etc.  
 
 WE> Are there biological applications for fuzziness?  How about 
 
  Yes, many.  I'll begin by using fractals, which IMO, is related 
to fuzziness.  Fractals are "fuzzy" at the edges.  I'll begin by heart 
rate, HR if one were to plot heart rate as a function of time HR(t), 
and plot HR(t), y, HR(t-1), x, and assign a "spread fucntion", one 
finds, such a manipulation is a good predictor of -future- death, 
in particular myocardial infarctions (I done some research in 
this area). 
  Why the big deal?  At present, current technology limits the 
diagnosis of myocardial infarction by EKG changes which in 
as much of 40 to 60%, source, Emergency Room Medicine, November 
96, or American Family Physician, 97 of myocardical infarction 
patients will be normal or serum changes, the newest being, 
troponin.  The problem, a protion of patients, will have NO 
indication by EKG or serum changes of myocardial infarction. 
If -ONE- EKG with a "spread function" determines myocardial 
infarction, treatment could be started sooner and save 
significant money. 
  If one were to look at the structure of circulatory, respiratory, 
and nervous system, such systems have a fractal nature.   It is 
proposed, the reason, DNA being the making "code" could generate 
such diversity, by following a simple -REITERATIVE- "code" 
(going through the process of transcription and translation) 
in the language of mathematics, algorithim. 
  For diagnosis, what one does is take the set of {signs} and 
{symptoms} and assign to a set known as {disease A or not A}. 
Sets of signs and symptoms are often fuzzy.  For example, 
the disease caused by "Sarcoptes Scabii" -usually- presents 
with "severe pruritus".  How does then place what the 
patient said into or not into, perhaps partly into the 
set "severe pruritus"; in addition, "minor pruritis" 
may be "severe pruritis" in another.  
  I hope this explains, why, I think, fuzzy presents itself 
in the biological sciences, from the genetic level, to 
the cellular level and finally to macrosopic level to 
include the systems and even into the field of medicine. 
  
 WE> the text book, can you find any biological examples? 
 
 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? 
   
  Hmm, it didn't look like probability to me.  The author did 
comment that many say "fuzzy logic" is "probability" in disguise. 
  
        [snip] 
  
 WE>  JB> I don't remember the book mentioning this.  The reason for 
 WE>  JB> my uncertainity in making this definitional point, the set  
 WE>  JB> not A is quite broad and would include classes (items not part  
 WE>  JB> A) that would be also part of A, due to their fuzzy nature.    
 
 WE> No, the notion of set is a stratified notion.  First there are 
 WE> elements, then  
 WE> there are sets of elements.  Third level is sets of the first two 
 WE> levels.   
 WE> Etc.  So there is a universe U of elements, distinct from  
 WE> sets.  A fuzzy set is an assignment of degrees of each  
 
  I agree, but since sets is nothing more than a collection of 
elements, regardless of what that element is, including sets 
beings elements of sets.   I ask, question here, not arguing, 
is there a requirement, that elements be distinct from sets? 
  
 WE> element in U to a real number d in the closed unit 
 WE> interval.  So the fuzzy set not A is the assignment of 1 -  
 
  I am not sure if the "fuzzy set" only includes degrees, 
or an ordered pair of element, with degree.  I need to read more. 
Not unexpected, this definitional point is key for the rest 
of our discussion.  Let me try to read and share some of 
what I read at a future date, but I need to start to taper 
my other disussions on this echo.  
  
 WE> d to x when x is assigned d by A.  For simplicity, I happy 
 WE> to stop at the first level.  Now if you want to extend this  
 WE> to higher strata you may do so.  Beware these strata ascend  
 WE> unto infinities upon infinities culminating in paradoxes.   
 WE> In aggregate, it is called the constructed universe as it  
 WE> is constructed from a bunch of elements and sets generated  
 WE> therefrom. 
   
 WE>  JB> I don't remember the book mentioning this.  The reason for 
 
 WE> Fuzzy sets and fuzzy logic are different.  Perhaps that is why. 
   
  Yes, but they seem intertwined.  I remember from Bart Kosko's 
book, somewhere between page 0 and 30, I believe around page 
6 or so, Bart introduced the concept of fuzzy sets, e.g., 
the set of {apples.}  He asked at which point, in consumption 
as in eating (at least we are having fun , does the 
apple become {not apple?} 
  
        [snip] 
  
 WE>  JB> Using the defintion of x e A and B (min(a,b)) iff x e A(a) 
 WE>  JB> and x e B(b) (where the little letters stand for x elements  
 WE>  JB> degree of inclusion into each set), and extending it to the  
 WE>  JB> total set {B, C, D, E, F,....}, we are to assume -all- these  
 WE>  JB> elements degrees taken together -somehow- become 1-a.  At  
 WE>  JB> the present, I am unsure how this is done.  
 
 WE> Me neither as I'm not up on extensions of fuzzy set theory 
 WE> beyond the first strata. 
   
  Thanks! 
  
 WE>  WE> A or B is the fuzzy set assigning to x the degree max(a,b) 
 WE>  WE> where x e A (a) and x e B (b).  
 
 WE>  JB> Seems reasonable. 
 
 WE>  WE> A and B is the fuzzy set assigning to x the degree min(a,b) 
 WE>  WE> where x e A (a) and x e B (b).  
 
 WE>  JB> Seems reasonable. 
 
 WE> A is included in B, A <= B when for each element x in U the degree a 
 WE> of x in  
 WE> A is <= the degree b of x in B.  A <= B if for all elements  
 WE> x in U, x e A (a) & x e B (b) implies a <= b.  Note that A  
 WE> included in B and B included in A implies that A = B, that  
 WE> is for all elements x in U, x e A (a) & x e B (b) implies a  
 WE> = b.  The assignments A and B of degrees to the elements  
 WE> are the same. 
   
  Thanks!  It does my heart good to see, a mathematical defintion 
of subsets and equivalent sets.  It has been YEARS since I have 
seen such, thanks. 
  
        [snip] 
  
 WE> JB> It would be ok, but my time is limited; so, it would have
 WE> JB> to be in piece meal.   
 
 WE> Well let's piece it together. -)  What mathematical stuff 
 WE> is in the books you already have?  I'd like to get a  
 WE> sharper fix on the difference between fuzzy logic and fuzzy  
 WE> sets. 
 
  Me too, but it will have to be piece meal. 
 
Take care, 
John 
 
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