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 properties 11. ( 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 ).