Compare to probability. Probability of element being in one set AND another distinct set is multiplication of two probabilities (which are, in reality, actually defined by sets).
"Pairs" (and, respectively, cartesian product) is a bit more complex, and not so much related to boolean algebra deduction concept.
x ∈ (A ∩ B) = (x ∈ A) * (x ∈ B) if "x ∈ A" equals 1 when x is in the
set A, 0 when x is not in the set A. Using "indicator functions" like
this also gives you a nice formulation for probability and
integration, etc. that falls apart if you use 1 to represent x ∉ A.
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.
I don't see how you lose boolean algebra, you just need to flip * and + in your equations.
x ∈ (A ∩ B) = (x ∈ A) + (x ∈ B) (and)
x ∈ (A ∪ B) = (x ∈ A) * (x ∈ B) (or)
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.
"Pairs" (and, respectively, cartesian product) is a bit more complex, and not so much related to boolean algebra deduction concept.