Logic hyperoperators

[hixidom]

Something that occurred to me recently (maybe it's already been mentioned here somewhere)...

Say a[1]b = a+b
and a[2]b = ab,

Then if "true" and "false" are defined as 1 and 0, respectively, and if a and b are restricted to boolean values, then

H(a[1]b) = a OR b
and H(a[2]b) = a AND b.
where H(x) is Heaviside function with H(0)=0.

So, by analogy, H(a[1.5]b) would be some sort of interpolant of AND and OR.

Alternatively, you could use modulus operator, %:
(a[1]b)%2 = a XOR b
(a[2]b)%2 = a AND b
therefore (a[1.5]b)%2 interpolates AND and XOR.

Somehow I feel that the use of H and % corrupt the validity of this idea, but I'm just curious if anyone else can get further with this than I can.

Thanks for reading.
Andy