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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2601.00118 |
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| _version_ | 1866915966623940608 |
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| author | Grillo, Sergio Daniel |
| author_facet | Grillo, Sergio Daniel |
| contents | 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 $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The universal logic of repeated experiments Grillo, Sergio Daniel Logic Mathematical Physics Rings and Algebras 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 $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$. |
| title | The universal logic of repeated experiments |
| topic | Logic Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2601.00118 |