Complementary Logic

[Organic]

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/complementarytheory/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:

(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.

<-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

------------>>>

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
. . . .
| | | |
|__| | |
| | |
|_____| |
| |
|________|
|