Are Non-Boolean Event Structures the Precedence or Consequence of Quantum Probability?

TL;DR

This paper explores whether non-Boolean event structures are fundamental to quantum probability, comparing Pitowsky's and QBism's approaches, and aims to clarify their conceptual and mathematical differences.

Contribution

It analyzes Pitowsky's non-Boolean algebra approach and its relation to QBism, proposing a unified perspective on the foundations of quantum probability.

Findings

Pitowsky's theorem derives the Born Rule from non-Boolean algebra.

QBism seeks to treat the Born Rule as a primary empirical postulate.

The paper clarifies the conceptual distinctions between the two approaches.

Abstract

In the last five years of his life Itamar Pitowsky developed the idea that the formal structure of quantum theory should be thought of as a Bayesian probability theory adapted to the empirical situation that Nature's events just so happen to conform to a non-Boolean algebra. QBism too takes a Bayesian stance on the probabilities of quantum theory, but its probabilities are the personal degrees of belief a sufficiently-schooled agent holds for the consequences of her actions on the external world. Thus QBism has two levels of the personal where the Pitowskyan view has one. The differences go further. Most important for the technical side of both views is the quantum mechanical Born Rule, but in the Pitowskyan development it is a theorem, not a postulate, arising in the way of Gleason from the primary empirical assumption of a non-Boolean algebra. QBism on the other hand strives to develop a way to think of the Born Rule in a pre-algebraic setting, so that it itself may be taken as the primary empirical statement of the theory. In other words, the hope in QBism is that, suitably understood, the Born Rule is quantum theory's most fundamental postulate, with the Hilbert space formalism (along with its perceived connection to a non-Boolean event structure) arising only secondarily. This paper will avail of Pitowsky's program, along with its extensions in the work of Jeffrey Bub and William Demopoulos, to better explicate QBism's aims and goals.