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 S_2. 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   T 

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 V_B 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:

xV_By   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 V_B y  =  y V_B x   ;    x &_B y = y &_B x

          Recall:                  x V_B x     =    x &_B x    =   x    

Identity:               x V_B F   =   x &_B T    =   x

Attractor:            x V_B I   =   x &_B I    =   I

Differentials:       x V_B 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.