Odd Logic, 5 of 5; Odd Sums of Kleenean Logic

ODD SUMS OF KLEENEAN LOGIC

        Linearity isn’t always diagonal. Non-atomic functions are always multi-linear, but needn’t be diagonal-linear.  For instance, consider odd sums on Kleenean logic.

        Kleenean Logic has three values F, I, and T; and atomic functions ‘and’, ‘or’, and ‘not’; all defined by:

        Not T = F ;  not I = I ;  not F = T

        (X and Y) = (Y and X); (X or Y) = (Y or X)   for all X and all Y

        (T and X) = (F or X) = X  for all X

        (F and X) = F  ;  (T or X) = T for all X

        (X and X) = (X or X) = X   for all X.

            not   and Y  F I T    or Y  F I T 

        X

        F    T           F F F          F I T

        I    I           F I I          I I T

        T    F           F I T          T T T

        Now consider the odd sum t+i+f: call it “j”. Then by object cancellation, any three of t,i,j,f sum to the fourth. Also:

        not(j) = not(t)+not(i)+not(f) = f+i+t = j

        T and J = (T&T)+(T&I)+(T&F)    = T+I+F = J

        T or J  = (TorT)+(TorI)+(TorF) = T+T+T = T

        F and J = (F&T)+(F&I)+(F&F)    = F+F+F = F

        F or J  = (ForT)+(ForI)+(ForF) = T+I+F = J

        I and J = (I&T)+(I&I)+(I&F)    = I+I+F = F

        I or J  = (IorT)+(IorI)+(IorF) = T+I+I = T

        J and J = (T&T)+(T&I)+(T&F)+(I&T)+(I&I)+(I&F)+(F&T)+(F&I)+(F&F)

               =  T+I+F+I+I+F+F+F+F  =  T+I+F = J

        J or J  =

         (TorT)+(TorI)+(TorF)+(IorT)+(IorI)+(IorF)+(ForT)+(ForI)+(ForF)

               =  T+T+T+T+I+I+T+I+F  =  T+I+F = J

            not   and Y  F T I J    or Y  F T I J 

        X

        F    T           F F F F          F T I J

        T    F           F T I J          T T T T

        I    I           F I I F          I T I T

        J    J           F I F J          J T T J

        In Kleenean logic, ‘x and y’ is the lesser of the two values, under the order F<I<T; and ‘x or y’ is the greater of the two values; and ‘not’ pivots the three values around the middle value I.

        When you add J, then “and” is the minimum operator, and “or” is the maximum operator, for this diamond-shaped lattice:

                       I

               <              <

        F                              T

               <              <

                       J

        Also, ‘not’ flips this diamond through its I-J axis.

        This is “diamond logic”, a paradox logic including Kleenean logic, and here derived from it by odd sums.

        Now consider the function d(x):

               d(x) = (x and not x)

        d(T) = F ; d(I) = I ;  d(F) = F

        therefore d(T)+d(I)+d(F) = F+I+F = I;

        but d(F+I+T) = (J and not J)  =  (J and J) =  J.

        d is not diagonal-linear; d(t+i+f) needs nine terms:

        d(t+i+f) =  (t+i+f) and not(t+i+f) 

        =       (t and not t) + (t and not i) +(t and not f) +

               (i and not t) + (i and not i) +(i and not f) +

               (f and not t) + (f and not i) +(f and not f)

        =       f + i + t +

               f + i + i +

               f + f + f 

        =       f+i+t  

        =       j

        Here the terms off the lower-left-to-upper-right diagonal cancel out. That counter-diagonal is f+i+t, or j; the main diagonal is f+i+f, or i. So d is multi-linear, but not diagonal-linear.