# Probabilities in the logic of quantum propositions

## TL;DR

This paper explores whether probabilities in quantum logic are fundamental or emergent, showing that environmental interactions induce irreducible randomness in quantum propositions.

## Contribution

It demonstrates that environmental interactions cause irreducible randomness, providing insight into the origin of probabilities in quantum logic.

## Key findings

- Environmental interaction induces irreducible randomness.
- Probabilities in quantum logic are emergent from system-environment interactions.
- Pure states alone do not determine probability measures.

## Abstract

In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.