**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.
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