Generalization of Boolean OR?

Question

I have been looking at the Boolean OR function and Im trying to find its integral analogue.

What I mean is:

Boolean AND (x, y) where x and y are Boolean Values with 0 = False, 1 = True is equivalent to multiplication:

xANDy = x*y

xORy = ?

It appears to be equivalent x + y - xANDy = x + y - x*y

Is this a correct intuition? My defense of AND and multiplication comes from their relationship in probability where p of A and B = pA * pB if A and B are independent. Likewise probability p of A or B = pA + pB - pA*pB

Answer 1

Using De Morgan's laws, if you have an expression for "not" and "and", you can get an expression for "or."

The probability intuition is good - indeed, the probability rule for "or" is just a version of De Morgan's law.

Answer 2

If you have algebra where $true \gt false$, then the binary operation $AND$ corresponds to the minimum of the operands. And $OR$ corresponds to the maximum of the operands.
I think, the probability you not need to define $AND$ and $OR$ operations.