The generalization of continuous variable version of boolean logic can be achieved in various ways as far as I know. One way is Cox's theorem [1], which forms the basis of Bayesian probability theory, and will give you consistent logic and reduces to the standard rules of boolean logic in the limit of 0/1.
The way I came up with my attempt was just to notice that when you are looking at the probability of two independent events happening together, you multiply them;
and if you want the probability of A or B or both, you add them and then subtract the probability of (A and B). I guess the reason the multiplication rule doesn’t work for (A && A) is that those two variables are clearly not independent, since they’re the same variable.
[1] https://en.wikipedia.org/wiki/Cox%27s_theorem