(a ∧ b) ∨ (a ∧ c) corresponds to states with | p 1 |↾ ≥1 = | p 2 |↾ ≥1 = 0 (this holds even if we defined p differently so as to make such states possible; also, a ∧ b corresponds to | p 1 |↾ ≥1 =0 and p 2 =0). Meanwhile, a corresponds to states with | p |↾ ≥1 = 0. As an operator, p = p 1 + p 2, and nonzero | p 1 |↾ ≥1 and | p 2 |↾ ≥1 might interfere to produce zero | p |↾ ≥1. Such interference is key to the richness of quantum logic and quantum mechanics.