10 - Causality and symmetry

Summary

This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell's inequalities and their generalization to more than four observables.

INTRODUCTION

In this paper we are concerned to present a number of theorems about inferences from phenomenological correlations to hidden causes. In other terms, the theorems are mainly theorems about hidden variables. Most of the proofs will not be given but references will be cited where they may be found.

To emphasize conceptual matters and to keep technical simplicity in the forefront, we consider only two-valued random variables taking the values ±1. We shall also assume symmetry for these random variables in that their expectations will be zero and thus they will each have a positive variance of one. For emphasis we state:

GENERAL ASSUMPTIONThe phenomenological random variables X₁, …, X_N have possible values ±1, with means E(X_i) = 0, 1 ≤ i ≤ N.

We also use the notation X, Y, and Z for phenomenological random variables. We use the notation E(XY) for covariance which for these symmetric random variables is also the same as their correlation ρ(X, Y).