The universal logic of repeated experiments

Abstract

Let $\mathsf{E}$ be the event space of an experiment that can be indefinitely repeated. A natural question arises: given a countable cardinal $\kappa$, which is the event space of the $\kappa$-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 $S^{\kappa}$ generated by $\mathsf{E}^{\kappa}$. But, what if $\mathsf{E}$ is not a Boolean algebra? In this paper we give a constructive answer to this question for any $\kappa$ 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 $\mathsf{U}_{\kappa}\left(\mathsf{E}\right)$ representing the event space of the $\kappa$-times repeated experiment, in such a way that $\mathsf{U}_{\kappa}\left(\mathsf{E}\right)$ and $\mathsf{E}$ are isomorphic if $\kappa=1$, and such that $\mathsf{U}_{\kappa}\left(\mathsf{E}\right)$ 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 $\kappa$ is arbitrary. This gives rise to tensor products $\bigotimes_{\alpha\in\kappa}\mathsf{E}_{\alpha}$ of families $\left\{ \mathsf{E}_{\alpha}\right\} _{\alpha\in\kappa}$ of orthocomplemented complete lattices, in terms of which $\mathsf{U}_{\kappa}\left(\mathsf{E}\right)=\bigotimes_{\alpha\in\kappa}\mathsf{E}$.