**ODD SUMS OF OBJECTS**

To define an odd sum of objects x+y+z, we need to define
predicates and functions on those objects. Let us specify certain predicates
P1, P2, … and functions f1, f2, …; call those the “atomic” predicates and
functions; let other predicates be defined from the atomic predicates plus the
logic operators; and let other functions be defined by composition of atomic
functions. And let all atomic predicates and functions be odd-linear:

P(x+y+z) = P(x)+P(y)+P(z)

f(x+y+z) = f(x)+f(y)+f(z)

Then all predicates defined from logic operators and atomic
predicates will be multi-linear, thus:

If P and Q are atomic predicates, then

(P and Q)(x+y+z)

= P(x+y+z) and
Q(x+y+z)

= (P(x) and Q(x))
+ (P(x) and Q(y)) + (P(x) and Q(z)) +

(P(y) and Q(x)) + (P(y) and Q(y)) + (P(y)
and Q(z)) +

(P(z) and Q(x)) + (P(z) and Q(y)) + (P(z)
and Q(z))

If f, g and h are atomic functions, then

f(g(x+y+z), h(x+y+z))

= f(g(x),h(x)) +
f(g(x),h(y)) + f(g(x),h(z)) +

f(g(y),h(x)) + f(g(y),h(y)) + f(g(y),h(z)) +

f(g(z),h(x)) + f(g(z),h(y)) + f(g(z),h(z))

“And” and f take two inputs, so the odd-sum had 3^2 = 9
terms. For three-imput logic functions on predicates, or three-input functions,
the odd-sum will have 3^3 = 27 terms, and so on.

Of particular importance is the equality predicate, defined
extensionally from the atomic predicates:

(x=y) = conjunction of (P(x) iff P(y))

x+x+y has the same atomic properties as y;

P(x+x+y) = P(x)+P(x)+P(y) = P(y)

Therefore x+x+y = y
; this is

*object cancellation.*
Object cancellation implies that if f is a commuting atomic
function; f(a,b) = f(b,a) ;
then:

f(x+y+z,x+y+z)

= f(x,x) + f(y,x)
+ f(z,x) +

f(x,y) + f(y,y) + f(z,y) +

f(x,z) + f(y,z) + f(z,z) +

= f(x,x) + f(y,y) + f(z,z)

because the off-diagonal terms cancel out. This is

*diagonal linearity.*
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