Boole didn't work with XOR, as such, and certainly not with F2. Boole worked with polynomials over the integers. Not all such polynomials directly represented

Boole didn't work with XOR, as such, and certainly not with F2. Boole worked with polynomials over the integers. Not all such polynomials directly represented propositions for him. But those which sent 0 or 1 valued inputs to 0 or 1 valued outputs did represent propositional formulae (with 0 as false and 1 as true). Thus, he used a notion of + for which, in the familiar way, 1 + 1 = 2, distinct from 0. And then he was able to interpret the inclusive disjunction of A and B as 1 - (1 - A)(1 - B) = A + B - AB. But most often, he spoke of this in cases where A and B were already known to be disjoint from each other, so that AB = 0 and this simplifies to A + B. That presumed disjointness/exclusiveness of A and B might be confused with this being XOR. But he did not use the principle A + A = 0. He worked firmly within familiar non-modular algebra. (He spoke of all this in a fashion a bit less familiar to modern eyes than the way I'm describing it, or anyone else in this discussion is. He never talks about "the OR operation" or any such thing, for example. He just saw himself as applying ordinary numeric algebra to the laws of thought/logic, not making up some new algebraic structure. But the above is indeed in essence what he did.)