arxiv: 2605.04802 · v1 · submitted 2026-05-06 · 🧮 math.PR

Recognition: unknown

Revisiting the logical independence

Chuanfeng Sun

Pith reviewed 2026-05-08 17:03 UTC · model grok-4.3

classification 🧮 math.PR

keywords logical independencesigma-logical independenceprobability extension theoremlimit theoremssigned measuresfoundations of probabilityrandom variables

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The pith

Logical independence can be defined before probability and still yields the standard extension and limit theorems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces logical independence and its sigma variant to construct a probability extension theorem that reverses the conventional order of definitions. This move addresses the reconciliation of logical and probabilistic notions of independence, the formulation of independence for signed measures and collections of measures, and the persistence of limit theorems when their usual assumptions are not met in practice. A sympathetic reader would care because the result treats independence as prior to measure and equips the logical version with the same computational tools previously reserved for the probabilistic version. The work thereby unifies two historically separate concepts within a single coherent framework.

Core claim

By introducing logical independence and σ-logical independence, the probability extension theorem is established. This result demonstrates that independence ought to be defined before probability, endows logical independence with probabilistic machinery, and thereby renders it computationally tractable in the same manner as probabilistic independence. The paper then shows how independence should be defined when multiple measures are involved and proves that limit theorems remain valid under the two conditions of σ-logical independence and identical range of the random variables.

What carries the argument

The probability extension theorem, which constructs measures from the prior notion of logical independence together with its sigma version.

If this is right

Independence receives a definition that applies directly to signed measures.
Families of probability measures can be equipped with a uniform notion of independence.
Limit theorems continue to hold when only σ-logical independence and identical ranges are assumed.
Logical independence acquires the same calculational status as probabilistic independence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Foundational presentations of probability could begin with logical relations rather than measure axioms.
Computational checks for independence in discrete or logical settings may now be transferred directly into probabilistic calculations.

Load-bearing premise

Logical independence can be defined so that it remains consistent with the axioms and operations of probability measures without generating contradictions.

What would settle it

An explicit example of random variables that satisfy σ-logical independence and identical range yet fail to obey a classical limit theorem such as the law of large numbers.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches a program to put logical independence first and derive probability from it, but the abstract gives no proofs or definitions so the extension theorem and limit theorem relaxations stay untested.

read the letter

The central claim is that logical independence and its sigma version let you build a probability extension theorem, which reverses the usual order and supposedly makes logical independence as usable as the probabilistic kind. The paper then extends the idea to signed measures and families of measures, and shows that classical limit theorems survive under sigma-logical independence plus identical ranges for the variables. That program is coherent on the surface and ties three separate questions together in one framework. It is new at least in the specific way it packages these moves, even if pieces of the independence discussion have appeared before. The abstract does not cite enough prior work to judge overlap, but the stated results are presented as solutions to open issues. The soft spot is obvious: no definitions of logical independence appear, no proof outline for the extension theorem is given, and nothing checks whether the construction stays consistent with standard measure theory or avoids circularity when probability is recovered from the logical notion. The reconciliation of the two independence concepts is the load-bearing step, and it cannot be evaluated from the given text. Readers who care about foundational questions in probability or who work on limit theorems under weakened assumptions might get something from the full version if the proofs hold. For everyone else the paper is still too thin to engage. It should go to peer review so that experts can examine the actual mathematics rather than the abstract outline.

Referee Report

2 major / 1 minor

Summary. The paper introduces the notions of logical independence and σ-logical independence to reconcile logical and probabilistic independence. It claims to prove a probability extension theorem showing that independence must be defined prior to probability, endows logical independence with probabilistic structure for computational tractability, extends the framework to signed measures and families of measures, and establishes that classical limit theorems remain valid under the conditions of σ-logical independence together with identical ranges of the random variables.

Significance. If the probability extension theorem and the limit-theorem results are rigorously established without circularity or inconsistencies in the measure-theoretic setting, the contribution would be substantial: it would address foundational questions on the ordering of probability and independence, provide a coherent definition of independence for signed and multiple measures, and relax the hypotheses of limit theorems to more intuitive conditions. The explicit construction of a probability extension theorem and the recovery of limit theorems would constitute concrete, falsifiable advances.

major comments (2)

[Abstract] The abstract asserts that the probability extension theorem demonstrates independence ought to be defined before probability, yet the provided manuscript text contains no derivation, no statement of the theorem, and no verification that the construction avoids circularity when logical independence is used to generate a probability measure. This is load-bearing for the central claim.
[Abstract] The claim that limit theorems hold under σ-logical independence plus identical ranges is stated without any indication of the proof strategy, the precise statement of the theorems recovered, or the counter-examples to classical assumptions that are now covered. Without these details the result cannot be assessed for correctness.

minor comments (1)

Notation for σ-logical independence is introduced with LaTeX markup in the abstract; ensure the full text defines the concept formally before its first use and maintains consistent typography throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We believe the points raised highlight areas where the presentation can be improved for clarity, and we address them point by point below, indicating the revisions we intend to make.

read point-by-point responses

Referee: [Abstract] The abstract asserts that the probability extension theorem demonstrates independence ought to be defined before probability, yet the provided manuscript text contains no derivation, no statement of the theorem, and no verification that the construction avoids circularity when logical independence is used to generate a probability measure. This is load-bearing for the central claim.

Authors: We agree that the abstract's claim regarding the probability extension theorem is central and requires explicit support in the manuscript to be fully convincing. While the introduction motivates the idea that logical independence is defined set-theoretically prior to introducing measures, and the abstract summarizes the theorem's implications, we acknowledge that a formal statement of the theorem, its derivation, and an explicit argument against circularity are not sufficiently detailed in the current version. To rectify this, we will revise the manuscript by adding a clear statement of the Probability Extension Theorem early in Section 3, including a proof sketch that begins with the definition of logical independence on the algebra of events (without any probabilistic structure), followed by the construction of a finitely additive set function, and then the extension to a probability measure on the sigma-algebra. We will also include a paragraph explaining that this ordering avoids circularity because the independence relation is purely logical and precedes the measure. This revision will make the foundational claim verifiable and address the referee's concern directly. revision: yes
Referee: [Abstract] The claim that limit theorems hold under σ-logical independence plus identical ranges is stated without any indication of the proof strategy, the precise statement of the theorems recovered, or the counter-examples to classical assumptions that are now covered. Without these details the result cannot be assessed for correctness.

Authors: We appreciate this observation, as the limit theorem results are intended to demonstrate the practical utility of our framework. The manuscript asserts that under σ-logical independence and identical ranges, classical limit theorems such as the law of large numbers and the central limit theorem continue to hold, even in cases where standard probabilistic independence may not apply. However, we concur that the current text lacks the necessary details on the proof strategy, exact theorem statements, and illustrative counterexamples. In the revised version, we will expand the section on limit theorems to provide: the precise formulations of the theorems under our conditions; an outline of the proof, which leverages the probabilistic tractability from the extension theorem to apply standard measure-theoretic arguments; and specific examples where random variables are not independent in the classical sense but satisfy σ-logical independence with identical ranges, showing that the conclusions still hold. This will enable a proper assessment of the results' correctness and novelty. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces logical independence and σ-logical independence as primitive notions prior to probability measures, then derives an extension theorem from them. The abstract and stated program show a one-way construction: new independence concepts are defined first, the theorem follows, and probabilistic machinery is then attached. No equations or steps are quoted that reduce a claimed prediction or theorem back to a fitted parameter or self-referential definition. No self-citations appear in the provided text, and the reconciliation with signed measures and limit theorems is presented as a consequence rather than an input. The derivation remains self-contained against external benchmarks with no load-bearing reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on the newly introduced notions of logical independence and σ-logical independence together with the probability extension theorem; none of these are substantiated or derived in the abstract.

axioms (1)

domain assumption Logical independence can be defined independently of probability measures and then extended to them without contradiction
This premise is required for the probability extension theorem and the claim that independence should precede probability.

invented entities (2)

logical independence no independent evidence
purpose: To serve as a primitive notion that reconciles logical and probabilistic independence
Newly introduced concept whose independent evidence is not supplied in the abstract.
σ-logical independence no independent evidence
purpose: To provide a countable version of logical independence sufficient for limit theorems
Strengthened variant of the new concept, again without external validation shown.

pith-pipeline@v0.9.0 · 5441 in / 1409 out tokens · 77747 ms · 2026-05-08T17:03:45.667440+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

[1]

A. N. Kolmogorov.Foundations of the Theory of Probability. Chelsea Publishing Company, 1950

1950
[2]

de Finetti

B. de Finetti. Foresight: Its logical laws, its subjective sources. In S. Kotz and N. L. Johnson, editors, Breakthroughs in Statistics: Volume 1: Foundations and Basic Theory, pages 134–174. Springer, New York, NY, 1992

1992
[3]

D. R. Cox. Some misleading arguments involving conditional independence.Journal of the Royal Sta- tistical Society: Series B (Methodological), 41(2):249–255, 1979

1979
[4]

Fitelson and A

B. Fitelson and A. H´ ajek. You say you want a revolution: two notions of probabilistic independence. Philosophy of Science, 90(3):583–602, 2023

2023
[5]

Pap.Handbook of Measure Theory, volume 1-2

E. Pap.Handbook of Measure Theory, volume 1-2. Elsevier, Amsterdam, 2002. Zbl 0998.28001. 13

work page arXiv 2002