Let $\mathsf{E}$ be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal $κ$ , which is the event space of the $κ$ -times repeated experiment? In the case of classical experiments, where $\mathsf{E}$ is a (complete) Boolean algebra on some set $S$ , i.e. a classical or distributive logic, the answer is more or less known: the (complete) Boolean algebra on $S^κ$ generated by $\mathsf{E}^κ$ . But, what if $\mathsf{E}$ is not a Boolean algebra? In this paper we give a constructive answer to this question for any $κ$ and in the context of general orthocomplemented complete lattices, i.e. general logics. Concretely, given a general logic $\mathsf{E}$ defining the event space of a given experiment, we construct a logic $\mathsf{U}_κ\left(\mathsf{E}\right)$ representing the event space of the $κ$ -times repeated experiment, in such a way that $\mathsf{U}_κ\left(\mathsf{E}\right)$ and $\mathsf{E}$ are isomorphic if $κ=1$ , and such that $\mathsf{U}_κ\left(\mathsf{E}\right)$ is distributive if and only if so is $\mathsf{E}$ . We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal $κ$ is arbitrary. This gives rise to tensor products $\bigotimes_{α\inκ}\mathsf{E}_α$ of families $\left\{ \mathsf{E}_α\right\} _{α\inκ}$ of orthocomplemented complete lattices, in terms of which $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$ .Let $\mathsf{E}$ be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal $κ$, which is the event space of the $κ$-times repeated experiment? In the case of classical experiments, where $\mathsf{E}$ is a (complete) Boolean algebra on some set $S$, i.e. a classical or distributive logic, the answer is more or less known: the (complete) Boolean algebra on $S^κ$ generated by $\mathsf{E}^κ$. But, what if $\mathsf{E}$ is not a Boolean algebra? In this paper we give a constructive answer to this question for any $κ$ and in the context of general orthocomplemented complete lattices, i.e. general logics. Concretely, given a general logic $\mathsf{E}$ defining the event space of a given experiment, we construct a logic $\mathsf{U}_κ\left(\mathsf{E}\right)$ representing the event space of the $κ$-times repeated experiment, in such a way that $\mathsf{U}_κ\left(\mathsf{E}\right)$ and $\mathsf{E}$ are isomorphic if $κ=1$, and such that $\mathsf{U}_κ\left(\mathsf{E}\right)$ is distributive if and only if so is $\mathsf{E}$. We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal $κ$ is arbitrary. This gives rise to tensor products $\bigotimes_{α\inκ}\mathsf{E}_α$ of families $\left\{ \mathsf{E}_α\right\} _{α\inκ}$ of orthocomplemented complete lattices, in terms of which $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$.

arXiv:2601.00118Dec 31, 20250 citations

Non-Boolean derived logics for classical systems

Michael D. Westmoreland, Benjamin W. Schumacher

Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting ``classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this ``derived logic'' can be non-Boolean even for theories much less radical than quantum theory.Birkhoff and von Neumann [Ann. Math. 37, 823 (1936)] proposed a nonstandard logic to describe quantum mechanics, in which the distributive laws of Boolean logic do not hold. In this paper we develop two algebras of propositions for classical mechanics that, like the quantum logical algebra, are based on a measurement theory. We adopt a simple classical measurement theory that allows the determination of any continuous phase-space function to any finite precision. Surprisingly, the resulting ``classical logics'' are non-Boolean, though the distributive laws hold. It appears that any physical theory with a mathematical space of physical states and an adequate description of measurement naturally yields a logiclike structure of experimental propositions, and that this ``derived logic'' can be non-Boolean even for theories much less radical than quantum theory.

Physical Review AAug 1, 199319 citations

Let

\mathsf{E}

be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal

κ

, which is the event space of the

κ

-times repeated experiment? In the case of classical experiments, where

\mathsf{E}

is a (complete) Boolean algebra on some set

S

, i.e. a classical or distributive logic, the answer is more or less known: the (complete) Boolean algebra on

S^κ

generated by

\mathsf{E}^κ

. But, what if

\mathsf{E}

is not a Boolean algebra? In this paper we give a constructive answer to this question for any

κ

and in the context of general orthocomplemented complete lattices, i.e. general logics. Concretely, given a general logic

\mathsf{E}

defining the event space of a given experiment, we construct a logic

\mathsf{U}_κ\left(\mathsf{E}\right)

representing the event space of the

κ

-times repeated experiment, in such a way that

\mathsf{U}_κ\left(\mathsf{E}\right)

and

\mathsf{E}

are isomorphic if

κ=1

, and such that

\mathsf{U}_κ\left(\mathsf{E}\right)

is distributive if and only if so is

\mathsf{E}

. We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal

κ

is arbitrary. This gives rise to tensor products

\bigotimes_{α\inκ}\mathsf{E}_α

of families

\left\{ \mathsf{E}_α\right\} _{α\inκ}

of orthocomplemented complete lattices, in terms of which

\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}