The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.
In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.
Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:
(1) the greatest lower bound of a and Na exists and is 0,
(2) a ≤ b implies Nb ≤ Na,
(3) NNa = a.
Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying
where a ≤ b means a ∧ b = a, as usual.
Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:
An orthomodular lattice gives a Finch implication algebra by defining → by
A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by
The orthocomplementation is unchanged in both cases.