 According to Hailperin(1986), the proof-theoretic side of Boole’s algebra is simply that of non-trivial commutative rings with unit and distinguished idempotent elements, but without non-zero additively or multiplicatively nilpotent elements. His favorite models were rings of signed multi-sets and he used them to  explain why Boole’s theorems are correct for the algebra of  logic of universal propositions. (Hailperin’s analysis did not apply to particular propositions.)