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By ZF set theory we know that {a,a,a,b,b,b,c,c,c} = {a,b,c}
It means that concepts like redundancy and uncertainy are out of the scpoe of set's concet in its basic form.
When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:
I think that any iprovment in set's concept has to include redundancy and uncertainty as inherent proprties of set's concept.
The above point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and nonboolean logic (0 And 1), for example:
Number 4 is fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.
These fading can be represented as:
Multiplication can be operated only among objects with structural identity .
Also multiplication is noncommutative, for example:
2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )
3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )
More about the above you can find here (the first 9 lines defined by Hurkyl):
http://www.geocities.com/complementarytheory/ET.pdf
More about Complementary logic, you can find here:
http://www.geocities.com/complementarytheory/CompLogic.pdf
http://www.geocities.com/complementarytheory/4BPM.pdf
Organic
It means that concepts like redundancy and uncertainy are out of the scpoe of set's concet in its basic form.
When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:
Code:
<Redundancy>
c c c ^<Uncertainty
b b b  b b
a a a  a a c a b c
. . . v . . . . . .
        
   ____  ___ 
      
_______ _______ _______
  
Where:
c c c
b b b
a a a
. . .
  
   = {a XOR b XOR c, a XOR b XOR c, a XOR b XOR c}
  
_______

b b
a a c
. . .
  
____  = {a XOR b, a XOR b, c}
 
_______

a b c
. . .
  
___  = {a, b, c}
 
_______

The above point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and nonboolean logic (0 And 1), for example:
Number 4 is fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.
These fading can be represented as:
Code:
(1*4) ={1,1,1,1} < Maximum symmetrydegree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's claritydegree
(((+1)+1)+1*2) ={{{1},1},1,1} (no uniqueness)
((1*2)+(1*2)) ={{1,1},{1,1}}
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} < Minimum symmetrydegree,
Maximum information's claritydegree
(uniqueness)
============>>>
Uncertainty
<Redundancy>^
3 3 3 3  3 3 3 3
2 2 2 2  2 2 2 2
1 1 1 1  1 1 1 1 1 1 1 1 1 1
{0, 0, 0, 0} V {0, 0, 0, 0} {0, 1, 0, 0} {0, 0, 0, 0}
. . . . . . . . . . . . . . . .
               
    ___   __   ___ ___
           
           
           
_______ ________ ________ _________
   
(1*4) ((1*2)+1*2) (((+1)+1)+1*2) ((1*2)+(1*2))
4 = 2 2 2
1 1 1 1 1 1 1
{0, 1, 0, 0} {0, 1, 0, 1} {0, 0, 0, 3} {0, 0, 2, 3}
. . . . . . . . . . . . . . . .
               
__ ___ __ __     ___  
          
    _____  _____ 
       
_________ _________ ________ ________
   
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1)) ((1*3)+1) (((1*2)+1)+1)
{0, 1, 2, 3}
. . . .
   
__  
  
_____ 
 
________

((((+1)+1)+1)+1)
Also multiplication is noncommutative, for example:
2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )
3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )
More about the above you can find here (the first 9 lines defined by Hurkyl):
http://www.geocities.com/complementarytheory/ET.pdf
More about Complementary logic, you can find here:
http://www.geocities.com/complementarytheory/CompLogic.pdf
http://www.geocities.com/complementarytheory/4BPM.pdf
Organic
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