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

Three-valued derived logics for classical phase spaces

Michael D. Westmoreland, Benjamin W. Schumacher, Steven C. Bailey

International Journal of Theoretical PhysicsFeb 1, 19970 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}