Classical and Quantum Probabilities as Truth Values

TL;DR

This paper presents a unified logical framework using sheaf topoi to interpret probabilities as truth values, applicable to both classical and quantum physics, extending previous quantum topos approaches.

Contribution

It introduces a general scheme that treats probabilities as truth values within sheaf topoi, bridging classical and quantum theories with a logical perspective.

Findings

Probabilities can be represented as truth values in sheaf topoi.

The approach extends existing quantum topos methods.

Provides a logical interpretation of quantum states.

Abstract

We show how probabilities can be treated as truth values in suitable sheaf topoi. The scheme developed in this paper is very general and applies to both classical and quantum physics. On the quantum side, the results are a natural extension of our existing work on a topos approach to quantum theory. Earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective.