Why is set intersection intuitively multiplication?
My expectation is that a set multiplication would give me a new set consisting of all pairs consisting of a single element from each set.
My expectation is that a set multiplication would give me a new set consisting of all pairs consisting of a single element from each set.
"Pairs" (and, respectively, cartesian product) is a bit more complex, and not so much related to boolean algebra deduction concept.
edit: I should add that I'm not claiming "you can't build measure theoretic probability from this formulation of booleans" is a strike against the project. Just addressing the math question.
g(y) = ∑ᵢ cᵢ × (y ∈ Aᵢ)
which is useful when you define integrals and expectations:
E g(Y) = ∫ g(y) f(y) dy = ∑ᵢ g(cᵢ) P(y ∈ Aᵢ)
where Y is a random variable with density function f. Any integrable function can be approximated as the limit of step functions, so this is a well-behaved way to get a general theory of integration.
Of course, one could replace (y ∈ Aᵢ) with 1 - (y ∈ Aᵢ) if one wanted to use "0" to represent the event (y ∈ Aᵢ) and "1" to represent its complement and not affect the truth of the math, but then there will be lots of termf floating around just to convert the notation into the terms that you need for the math.