Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$ . (By a state we mean a nonnegative bounded $\sigma$ -additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$. (By a state we mean a nonnegative bounded $\sigma$-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

Proceedings of the American Mathematical SocietyJan 1, 19875 citations

Quantum Logic and Related Systems

Akira Ocirc IDE

Annals of the Japan Association for Philosophy of ScienceJan 1, 19831 citation

Let

L

be a quantum logic and let

S(L)

denote the set of all states on

L

. (By a state we mean a nonnegative bounded

\sigma

-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].