Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A countable Boolean algebra that is Reichenbach's common cause complete

Dominika Burešová

TL;DR

The paper addresses whether common cause completeness (CCC) can hold in a small, countable Boolean algebra. It constructs a countable interval-algebra $B$ of finite unions of rational-endpoint sub-intervals of $(0,1]$ with $p$ as the Lebesgue measure restricted to $B$, and shows that for any $a,b$ with $ ext{cov}_p(a,b)>0$ there exists a common cause $c$ with $p(c)=v$ where $v= rac{p(a\land b)-p(a)p(b)}{1+p(a\land b)-p(a)-p(b)}>0$, with $c\le a\lor b$, thereby establishing non-trivial CCC. The same construction idea yields an uncountable CCC example, and the authors discuss extensions to quantum logics via horizontal-sum constructions. A key takeaway is that CCC can arise in countable Boolean algebras without requiring $\sigma$-completeness or a non-atomic measure, providing a simple, explicit counterpoint to prior examples.

Abstract

The common cause completeness (CCC) is a philosophical principle that asserts that if we consider two positively correlated events then it evokes a common cause. The principle is due to H. Reichenbach and has been largely studied in Boolean algebras and elsewhere.The results published so far bring about a question whether there is a small (countable) Boolean algebra with CCC. In this note we construct such an example.

A countable Boolean algebra that is Reichenbach's common cause complete

TL;DR

The paper addresses whether common cause completeness (CCC) can hold in a small, countable Boolean algebra. It constructs a countable interval-algebra of finite unions of rational-endpoint sub-intervals of with as the Lebesgue measure restricted to , and shows that for any with there exists a common cause with where , with , thereby establishing non-trivial CCC. The same construction idea yields an uncountable CCC example, and the authors discuss extensions to quantum logics via horizontal-sum constructions. A key takeaway is that CCC can arise in countable Boolean algebras without requiring -completeness or a non-atomic measure, providing a simple, explicit counterpoint to prior examples.

Abstract

The common cause completeness (CCC) is a philosophical principle that asserts that if we consider two positively correlated events then it evokes a common cause. The principle is due to H. Reichenbach and has been largely studied in Boolean algebras and elsewhere.The results published so far bring about a question whether there is a small (countable) Boolean algebra with CCC. In this note we construct such an example.
Paper Structure (3 sections, 4 theorems, 10 equations)

This paper contains 3 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Results

Key Result

Proposition 3.1

Let $B$ be a Boolean algebra and $p$ be a finitely additive probability measure on $B$. Let $a,b,c \in B$, $c \leq a \land b$, $\operatorname{cov}_p(a,b) > 0$, and $p(c) > 0$. Then $a,b,c$ fulfill the conditions eqn:RCCP1, eqn:RCCP3 and eqn:RCCP4.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5