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Organic
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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 non-boolean 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 non-boolean 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 symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's clarity-degree
(((+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 symmetry-degree,
Maximum information's clarity-degree
(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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