## Thursday, July 12, 2018

### Laws of Triple Form: 4 of 12

Chapter 3 : Kleenean Logic

Isomorphic to Kleenean logic
Differentials
Bochvarian operators

3A. Isomorphic to Kleenean logic

Consider these truth tables for Kleenean logic:

xVy   x   F   I   T
y
F         F   I   T
I         I   I   T
T         T   T   T

~x        T   I   F

x&y   x   F   I   T
y
F         F   F   F
I         F   I   I
T         F   I   T

These tables are isomorphic to bracket forms twice, via S2. In one matching, [xy] corresponds to  (x nor y);  in the other, [xy] corresponds to  (x nand y); the “nor-gate” and “nand-gate” interpretations.

These laws apply:

Swap:         x V y  =  y V x   ;    x & y = y & x
Recall:        x V x     =    x & x    =   x
Identity:     x V F   =   x & T    =   x
Attractor:   x V T  =  T     ;    x & F = F

3B. Differentials

Define the “differentials” this way:

The “lower differential”:   dx   =       x & ~x
The “upper differential”:   Dx  =       x V ~x

These are, respectively, in bracket forms
[x[x]]                    ;                  x[x]
-      in the nor-gate interpretation; reverse order for nand-gate.

They have these tables:

F   I

dx   F   I   F
Dx   T   I   T

These laws apply:

Domination:        (x & dx)   =   (Dx & dx)   =   dx   ;
(x V Dx)   =   (dx V Dx)   =   Dx  ;
(dx V x)    =   (Dx & x)    =    x

Negation:             ~ dx    =  Dx   =  D(~x)  ;
~ Dx   =   dx   =  d(~x)

Leibnitz Laws:    d(x&y)  =   (x&dy)V(y&dx)
d(xVy)  =   (~x&dy)V(~y&dx)
D(x&y)  =   (~xVDy)&(~yVdx)
D(xVy)  =   (xVDy)&(yVDx)

3C. Bochvarian operators

Here are the “Bochvarian” operators, defined in Kleenean logic:

x &B y         =       (x&y) V dx V dy  =      (x&y) V (x&~x) V (y&~y)
x VB y         =       (xVy) & Dx & Dy          =      (xVy) & (xV~x) & (yV~y)

These are, respectively, in bracket forms
[[x][y]] [x[x]] [y[y]]
[ [xy] [x[x]] [y[y]] ]
-      in the nor-gate interpretation; reverse order for nand-gate.

They have these tables:

xVBy   x  F   I   T
y
F         F   I   T
I         I   I   I
T         T   I   T

x&By   x  F   I   T
y
F         F   I   F
I         I   I   I
T         F   I   T

These laws apply:

Swap:                  x VB y  =  y VB x   ;    x &B y = y &B x
Recall:                  x VB x     =    x &B x    =   x
Identity:               x VB F   =   x &B T    =   x
Attractor:            x VB I   =   x &B I    =   I
Differentials:       x VB T   =  Dx      ;         x &B F    =   dx

In Bochvarian logic, I is an infinite value; it dominates any expression it’s in. In Kleenean logic, I is an intermediate value; it can be dominated.