From Boole to Leggett-Garg: Epistemology of Bell-type Inequalities

TL;DR

This paper explores the epistemological foundations of Bell-type inequalities, especially Leggett-Garg inequalities, arguing that their violation does not necessarily imply quantum paradoxes but may stem from assumptions about probability measures.

Contribution

It provides a philosophical analysis connecting Boolean inequalities to quantum inequalities, challenging the interpretation of violations as evidence of quantum paradoxes.

Findings

Leggett-Garg inequality cannot be violated if the assumptions about probability measures are correct.

Violations suggest that probability measures may differ from classical assumptions when data is not directly accessible.

The analysis applies to other quantum puzzles like the two-slit experiment, offering a new perspective.

Abstract

In 1862, George Boole derived an inequality for variables, now known as Boolean variables, that in his opinion represents a demarcation line between possible and impossible experience. This inequality forms an important milestone in the epistemology of probability theory and probability measures. In 1985 Leggett and Garg derived a physics related inequality, mathematically identical to Boole's, that according to them represents a demarcation between macroscopic realism and quantum mechanics. Their formalism, constructed for the magnetic flux of SQUIDS, includes general features and applies also to many other quantum experiments. We show that a wide gulf, a wide divide, separates the "sense impressions" and corresponding data, as well as the postulates of macroscopic realism, from the mathematical abstractions that are used to derive the inequality of Leggett-Garg. If the gulf can be bridged, one may indeed derive the said inequality, which is then clearly a demarcation between possible and impossible experience: it cannot be violated and is not violated by quantum theory. We deduce from this fact that a violation of the Leggett-Garg inequality does not mean that the SQUID-flux is not there when nobody looks, as Leggett-Garg suggest, but instead that the probability measures may not be what Leggett-Garg have assumed them to be, when no data can be secured that directly relate to them. We show that similar considerations apply to other well known quantum interpretation-puzzles including that of the two-slit experiment.