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The universal logic of repeated experiments

Sergio Daniel Grillo

TL;DR

The article develops a universal framework for the event space of repeated experiments beyond classical Boolean logic. It constructs the universal logic $\mathsf{U}_{\kappa}(\mathsf{E})$ from a given complete orthocomplemented lattice $\mathsf{E}$, embedding $\mathsf{E}^{\kappa}$ and closing under a closure operator to obtain a complete, distributive lattice that preserves the original logic’s distributivity. A key result is that $\mathsf{U}_{\kappa}(\mathsf{E})$ is distributive iff $\mathsf{E}$ is distributive, and that there exists a natural epimorphism from $\mathsf{U}_{\kappa}(\mathsf{E})$ onto the Boolean algebra generated by $\mathsf{E}^{\kappa}$, $\langle \mathsf{E}^{\kappa} \rangle$, linking the universal construction to classical multi-shot experiments. The paper also extends the framework to nonuniform repetitions via tensor products $\bigotimes_{\alpha\in\kappa} \mathsf{E}_{\alpha}$ and provides a universal property characterizing these constructions, enabling a broad applicability to quantum and classical logics alike.

Abstract

Let $\mathsf{E}$ be the event space of an experiment that can be indefinitely repeated, in such a way that the result of one repetition does not affect the result of the other. 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 \textit{classical }or\textit{ distributive logic}, the answer is more and 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. \textit{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 $\mathsf{U}_κ\left(\mathsf{E}\right)=\bigotimes_{α\inκ}\mathsf{E}$.

The universal logic of repeated experiments

TL;DR

The article develops a universal framework for the event space of repeated experiments beyond classical Boolean logic. It constructs the universal logic from a given complete orthocomplemented lattice , embedding and closing under a closure operator to obtain a complete, distributive lattice that preserves the original logic’s distributivity. A key result is that is distributive iff is distributive, and that there exists a natural epimorphism from onto the Boolean algebra generated by , , linking the universal construction to classical multi-shot experiments. The paper also extends the framework to nonuniform repetitions via tensor products and provides a universal property characterizing these constructions, enabling a broad applicability to quantum and classical logics alike.

Abstract

Let be the event space of an experiment that can be indefinitely repeated, in such a way that the result of one repetition does not affect the result of the other. A natural question arises: given a countable cardinal , which is the event space of the -times repeated experiment? In the case of classical experiments, where is a (complete) Boolean algebra on some set , i.e. a \textit{classical }or\textit{ distributive logic}, the answer is more and less known: the (complete) Boolean algebra on generated by . But, what if is not a Boolean algebra? In this paper we give a constructive answer to this question for any and in the context of general orthocomplemented complete lattices, i.e. \textit{general logics}. Concretely, given a general logic defining the event space of a given experiment, we construct a logic representing the event space of the -times repeated experiment, in such a way that and are isomorphic if , and such that is distributive if and only if so is . We also extend our construction to the case in which the event space changes from one repetition to another and the cardinal is arbitrary. This gives rise to tensor products of families of orthocomplemented complete lattices, in terms of which .
Paper Structure (27 sections, 21 theorems, 300 equations)

This paper contains 27 sections, 21 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Classical experiments
  3. The event space of the repeated experiment
  4. An expression for the complement
  5. General experiments
  6. The lattices $\mathsf{E}^{\kappa}$
  7. The free terms of $\mathsf{E}^{\kappa}$
  8. Terms and propositions
  9. The completely distributive lattice $\mathsf{D}_{\kappa}\left(\mathsf{E}\right)$
  10. The join operation and the related order
  11. Completeness
  12. An infimum formula for $\mathsf{D}_{\kappa}\left(\mathsf{E}\right)$
  13. Complete distributivity
  14. The universal logic of repeated experiments
  15. The negation operation
  16. ...and 12 more sections

Key Result

Proposition 7

Given ${ \bigsqcup_{i\in I}}A^{i}\in\mathsf{T}\left(\mathsf{E}^{\kappa}\right)$, with $I\neq\emptyset$, and given in addition ${ \bigsqcup_{j\in J}}B^{j}\in\mathsf{T}\left(\mathsf{E}^{\kappa}\right)$, also with $J\neq\emptyset$, we have that if and only if for all $i\in I$ there exists $j\in J$ such that $A^{i}\subseteq B^{j}$.

Theorems & Definitions (58)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Example 5
  • Remark 6
  • Proposition 7
  • proof
  • Remark 8
  • Proposition 9
  • ...and 48 more