Density conditions for quantum propositions

As has already been pointed out by Birkhoff and von Neumann, quantum logic can be formulated in terms of projective geometry. In three-dimensional Hilbert space, elementary logical propositions are associated with one-dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis $\{{\vec u},{\vec v},{\vec w}\}$, successive application of the binary logical operation $(x,y)\mapsto (x\vee y)^\perp$ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis $\{{\vec u},{\vec v},{\vec w}\}$ is orthogonal to the other ones.

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