Revisiting the logical independence

TL;DR

This paper introduces logical independence concepts to reconcile probabilistic and logical independence, resolving longstanding issues in probability theory and extending classical limit theorems.

Contribution

It establishes the probability extension theorem using logical independence, showing independence should be defined prior to probability and making logical independence computationally tractable.

Findings

Probability extension theorem demonstrated.

Independence should be defined before probability.

Limit theorems hold under $\sigma$-logical independence and identical ranges.

Abstract

It has been widely acknowledged that probabilistic independence and logical independence cannot be coherently reconciled. By bridging these two notions, this paper addresses three long-standing problems that have puzzled the field of probability theory: Should probability be defined prior to independence, or independence prior to probability? How ought independence to be formulated for signed measures and families of probability measures? Why do the conclusions of classical limit theorems remain valid even when practical scenarios violate their underlying assumptions? By introducing logical independence and $\sigma$-logical independence, we establish the probability extension theorem. This result not only demonstrates that independence ought to be defined before probability, but also endows logical independence with probabilistic machinery, thereby rendering it computationally tractable in the same manner as probabilistic independence. Then, we investigate how independence should be defined when multiple measures are involved. Finally, we prove that limit theorems can hold true under two intuitive conditions: $\sigma$-logical independence and identical range of random variables.