Monday, December 16, 2013

Odd Logic, 1 of 5: Odd Sums in Logic

        Odd Logic

1.  Odd Sums in Logic
2.  Matrix Logic
3.  Ultrasums
4.  Odd Sums of Objects
5.  Odd Sums of Kleenean Logic


        Define the odd sum 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

        T+T+T = T
        T+F+T = F
        T+F+F = T
        F+F+F = F

        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}.

        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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