Thursday, December 19, 2013

Odd Logic, 4 of 5; 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,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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