## Complementary Logic

Dear peolpe,

I am a poor formalist, but have some ideas, which are based on structural|quantitative point of view on Math language.

They can be found here: http://www.geocities.com/complementa...y/CATpage.html

Maybe you can help me to address these ideas in a rigorous formal way.

By doing it, we can check what idea can survive rigorous definitions.

I think that only then we can move to the next step, which is: to examine its originality.

Thank you,

Yours,

Organic

----------------------------------------------------------------------------
Short overview:

Boolean logic is based on 0 Xor 1.

Fuzzy logic is fading transition between 0 Xor 1.

A non-Boolean logic is based on 0 And 1.

My 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.

This fading transition 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 (no uniqueness)
(((+1)+1)+1*2)=     (((1),1),1,1)
((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)
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) )

Through my point of view, there are connections between structure's symmetry-degree and information's clarity-degree.

High Entropy means maximum level of redundancy and uncertainty, which are based on the highest symmetry-degree of some system.

For example let us say that there is a piano with 3 notes and we call it 3-system :

DO=D , RE=R , MI=M

The highest Entropy level of 3-system is the most left information's-tree,
where each key has no unique value of its own, and vice versa.
Code:

<-Redundancy->
M   M   M  ^<----Uncertainty
R   R   R  |    R   R
D   D   D  |    D   D   M       D   R   M
.   .   .  v    .   .   .       .   .   .
|   |   |       |   |   |       |   |   |
3 = |   |   |       |___|_  |       |___|   |
|   |   |       |       |       |       |
|___|___|_      |_______|       |_______|
|               |               |
An example of 4-notes piano:

DO=D , RE=R , MI=M , FA=F
Code:
------------>>>

F  F  F  F           F  F           F  F
M  M  M  M           M  M           M  M
R  R  R  R     R  R  R  R           R  R     R  R  R  R
D  D  D  D     D  D  D  D     D  R  D  D     D  D  D  D
.  .  .  .     .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
|  |  |  |     |__|_ |  |     |__|  |  |     |__|_ |__|_
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|  |  |  |     |     |  |     |     |  |     |     |
|__|__|__|_    |_____|__|_    |_____|__|_    |_____|____
|              |              |              |

4 =
M  M  M
R  R                     R  R  R        R  R
D  R  D  D      D  R  D  R     D  D  D  F     D  D  M  F
.  .  .  .      .  .  .  .     .  .  .  .     .  .  .  .
|  |  |  |      |  |  |  |     |  |  |  |     |  |  |  |
|__|  |__|_     |__|  |__|     |  |  |  |     |__|_ |  |
|     |         |     |        |  |  |  |     |     |  |
|     |         |     |        |__|__|_ |     |_____|  |
|     |         |     |        |        |     |        |
|_____|____     |_____|____    |________|     |________|
|               |              |              |

D  R  M  F
.  .  .  .
|  |  |  |
|__|  |  |
|     |  |
|_____|  |
|        |
|________|
|
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