Quantum logic as an implication algebra

Abstract

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties11. (x → 0) = NX, 12. if x ≤ y then (z → x) ≤ (z → y), 13. if x ≤ y then x ≤ (y ≤ z)= x → z, 14. x ≤ {y → N(y → Nx)}, then the logic is an orthomodular lattice. The lattice operations of join and meet are given by x ∨ y = Nx → N(Nx → Ny) x ∧ y = N(X → N(x → y)) and, in terms of the lattice operations, the material implication is given by (x → y) = (y ∧ x) ∨ NX.Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies 15. (x → y) = (Ny → Nx).