Friday, January 10, 2014

On Logisitic, 5 of 8; Carrollian Identities

5. Carrollian Identities

Here are tables for addition, reduction and Einstein addition:
+     0         {+}  0          ~   0       1/x
0     0          0   0  0        0   0
0              0      0
Therefore +, {+}, ~ and 1/x are equivalent to, respectively, Boolean ‘and’, ‘or’, ‘iff’ and ‘not’, if we identify 0 with True and ∞ with False. Also, M is equivalent to the majority operator.

Consider these identities:

x        =  (x+x){+}(x+x)
=  (x{+}x)+ (x{+}x)
=  (x+x+x){+} (x+x+x){+} (x+x+x)
=  (x{+}x{+}x)+ (x{+}x{+}x) + (x{+}x{+}x)
=
These are just the cancellation laws x = (2/2)x = (3/3)x = …; but consider them translated into logic:

x        equals       (x and x) or (x and x)
x        equals       (x or x) and (x or x)
x        equals       (x and x and x) or (x and x and x) or (x and x and x)
x        equals       (x or x or x) and (x or x or x) and (x or x or x)

I call these identities “Carrollian” because of their Wonderlandish sound. Here are some other Carrollian identities:
(x+y) ~ (x {+} y)    =    (x~y) + ( x {+} 1/x {+} y {+} 1/y )
(x {+} (y~z) )  +  (y {+} (z~x) )  +  (z {+} (x~y) )      =
1/(x{+}x{+}y{+}y{+}z{+}z) {+} (M(x,y,z)+M(x,y,z))

M(x,y,y)   =   ( ( x {+} y ) + y )  {+} (x + x + y )

M(x,y,y)   =   ( ( x + y ) {+} y )  + (x {+} x {+} y )

(x ~ (y+z) )  {+}  (y ~ (z+x) )  {+}  (z ~ (x+y) )    =
(x+y+z){+} (x+y+z){+} (x+y+z){+} 1/M(x,y,z) {+} 1/M(x,y,z)

(x ~ (y{+}z) )  {+}  (y ~ (z{+}x) )  {+}  (z ~ (x{+}y) )    =
1/(x{+}y{+}z) {+} 1/(x{+}y{+}z) {+} 1/(x{+}y{+}z)  {+}
M(x,y,z){+}M(x,y,z))

(1/x ~ (y+z) )  +  (1/y ~ (z+x) )  +  (1/z ~ (x+y) )    =
1/(x+y+z) + 1/(x+y+z) + 1/(x+y+z) + M(x,y,z) + M(x,y,z)

(1/x ~ (y{+}z) ) + (1/y ~ (z{+}x) ) + (1/z ~ (x{+}y) )    =
(x{+}y{+}z) + (x{+}y{+}z) + (x{+}y{+}z) + 1/M(x,y,z) + 1/M(x,y,z))

Behold the last two:
(not x iff (y and z)) and (not y iff (z and x)) and (not z iff (x and y))
equals
not (x and y and z) and not (x and y and z) and not (x and y and z) and  (most of x,y,z)  and (most of x,y,z)
(not x iff (y or z)) and (not y iff (z or x)) and (not z iff (x or y))
equals
(x or y or z) and (x or y or z) and (x or y or z) and not (most of x,y,z) and not (most of x,y,z))