**Odd Logic**

1. Odd Sums in Logic

2. Matrix Logic

3. Ultrasums

4. Odd Sums of Objects

5. Odd Sums of Kleenean Logic

**ODD SUMS IN LOGIC**

Define the

**x+y+z of three boolean truth values to be the value which occurs among x, y and z an odd number of times; that is, one or three; so that***odd sum*
T+T+T = T

T+F+T = F

T+F+F = T

F+F+F = F

Then:

x+y+z = y+x+z = x+z+y = etc. symmetry

x+x+y = y cancellation

x+x+x = x recall

We can define higher odd sums:

x+y+z+u+w = (x+y+z)+u+w

x+y+z+u+w+a+b = ((x+y+z)+u+w)+a+b

and so on; and the parentheses associate.

x xor y = x+y+F

x iff y = x+y+T

So we can define even sums, but in two different ways.

If we identify F with 0 and T with 1, then + and xor become
addition modulo 2, 'and' becomes
times mod 2.

If we identify F with 1 and T with 0, then + and iff become
addition modulo 2, 'or' becomes times
mod 2.

Working back from mod-2 arithmetic yields two distributive
laws:

a and (x xor y) = (a and x) xor (a and y)

a or (x iff y) = (a or x) iff (a or y)

"And" does not double-distribute over
"iff", nor "or" over "xor", but they do triple-distribute:

a and (x iff y iff z)
= (a and x) iff (a and y) iff (a
and z)

a or (x xor y xor
z) =
(a or x) xor (a or y) xor (a or z)

This implies:

a and (x iff y) = (a and x) iff (a and y) iff (a)

a or (x xor y) = (a
or x) xor (a or y) xor (a)

Now it turns out that

x+y+z = x xor y xor z
= x iff y iff z

in either identification of {T,F} with {0,1}.

So:

a and (x+y+z) = (a
and x) + (a and y) + (a and z)

a or (x+y+z) = (a or x) + (a or y) + (a or z)

but also:

not (x+y+z) = (not x) + (not y) + (not z)

(x+y+z) --> a
= (x --> a) + (y --> a) +
(z --> a)

a --> (x+y+z)
= (a --> x) + (a --> y) +
(a --> z)

(x+y+z) - a = (x - a) + (y - a) + (z - a)

a xor (x+y+z) = (a xor x) + (a xor y) + (a xor z)

a iff (x+y+z) = (a iff x) + (a iff y) + (a iff z)

and in general:

F(x+y+z) = F(x) + F(y) + F(z)

for any boolean function. This is 'triple distribution' or

'trilinearity'. More general
is 'oddlinearity':

F(x+y+z+u+v)
= F(x)+F(y)+F(z)+F(u)+F(v)

F(x+y+z+u+v+a+b) =
F(x)+F(y)+F(z)+F(u)+F(v)+F(a)+F(b)

and so on.

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