Revisiting the logical independence

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.