Measuring Logic and Linguistics: A Cross-Domain Metrology of Logic on the Structure–Symbol Sample Space

Abstract

This paper proposes and argues a program: treat logic as a measurable physical object, and linguistics as its operational framework. On the structure–symbol sample space (p,τ,P(τ∣p))(p,τ,P(τ∣p)), the three orthogonal dimensions of logic — determinism H (the strength of the structure's constraint on the symbol), directionality ΔS (the irreversibility of generation), and anchoring A (cross-system semantic agreement) — constitute a complete measuring instrument; the anchoring-alignment hypothesis H1 (high anchoring ⇔ low conditional entropy) is the first empirical law yielded by this instrument. Five-domain measurement results. (i) Code (structures = AST paths, symbols = tokens): H1's direction consistent (ρ = −0.45~−1.00); the three-dimensional metrics reach 10−810−8–10−10010−100-magnitude significance on fix-commit defect labels; below-null flags achieve lift = 2.04 (p = 0.022) in temporal hold-out prospective validation. (ii) DNA (structures = regulatory contexts, symbols = sequence patterns): ρ = −0.59~−0.60. (iii) Exoplanets (structures = metallicity × orbit × type, symbols = mass-density classes): non-significant at coarse granularity; fine metallicity partitions (channel alignment) raise ρ to −0.56~−0.71 (p < 0.01) with 20/20 independent half-sample replications; the tilt theorem ρ(A,|θ|) = −0.604 (p = 0.029). (iv) Asteroid families (structures = family membership, i.e., a real collisional generation event; symbols = albedo classes): 48 families give ρ = −0.691 (permutation p = 0.0002); the shuffled null is positive (+0.594) and the observation lies 13.8σ away. (v) AI-generated text: H1 fails and reverses direction (six-source design, PASS-only subset ρ = +0.833, p = 0.005) — the correct prediction of the domain clause: H1 constrains only lawful generation; AI text is statistical sampling, and hallucination is the breaking of the law. Hallucination thereby obtains a three-dimensional decomposition: the determinism loss ΔH is positive in all 11 structures (breaking is universal); ΔS points consistently toward verbosity (binary taxonomy 11/11, p = 0.001); H1's negative direction survives in the unbroken lawful part (A vs H_g: ρ = −0.600, p = 0.051). Cross-sample disagreement, the runtime form of the anchoring dimension, is validated as a domain-agnostic hallucination measure (self-consistency AUC 0.918; cross-model 0.730; numeric domain 0.928; fused detector 0.922). Metrological lessons form the second pillar of the program: structure partitions must align with generative channels (the granularity ladder); structure and symbol must not share an observable (the lesson of the imputation artifact); language groups must be genuine generative environments; every significant result must pass permutation, half-split, shuffling, and variant-robustness tests, and be confirmed by preregistration. The paper concludes with the programmatic statement of a "metrology of logic" and a list of future domains. Keywords: structure–symbol sample space; metrology of logic; determinism; directionality; anchoring; anchoring-alignment hypothesis; domain clause; logical linguistics; channel alignment; preregistration; cross-domain validation 本文提出并论证一个纲领:把逻辑作为可度量的物理对象,把语言学作为其操作化框架。在结构–符号样本空间 (p,τ,P(τ∣p))(p,τ,P(τ∣p)) 上,逻辑的三个正交维度——确定性 H(结构对符号的约束强度)、方向性 ΔS(生成过程的不可逆性)、锚定 A(跨系统语义一致性)——构成一套完整的测量仪器;锚定对齐假说 H1(高锚定 ⇔ 低条件熵)是该仪器给出的第一条经验定律。 五域测量结果:(i) 代码域(结构=AST 路径,符号=token):H1 方向一致(ρ=−0.45~−1.00),三维度指标在 fix-commit 缺陷标注上达到 10−810−8–10−10010−100 量级显著性,below-null 标记在时间留出前瞻验证中给出 lift=2.04(p=0.022);(ii) DNA 域(结构=调控上下文,符号=序列模式):ρ=−0.59~−0.60;(iii) 系外行星域(结构=金属量×轨道×类型,符号=质量-密度类):粗粒度划分不显著,细金属量划分(通道对齐)使 ρ 达 −0.56~−0.71(p<0.01),20/20 独立半样本复现;倾斜定理 ρ(A,|θ|)=−0.604(p=0.029);(iv) 小行星族域(结构=族归属,即真实撞击生成事件;符号=反照率类):48 族 ρ=−0.691(置换 p=0.0002),打乱零分布为正(+0.594)且观测偏离 13.8σ;(v) AI 生成文本域:H1 不成立且方向反转(六源设计 PASS-only 子集 ρ=+0.833,p=0.005)——这是适用域条款的正确预言:H1 仅约束有法则的生成,AI 文本是统计采样,幻觉是法则的破坏。幻觉由此获得三维分解:确定性损失 ΔH 于全部 11 个结构为正(破坏普遍)、ΔS 方向一致(二元分类 11/11,p=0.001,指向冗长化)、H1 负方向在未破坏的合法部分残留(A vs H_g:ρ=−0.600,p=0.051)。基于锚定维的跨采样分歧度被验证为域无关的幻觉度量(自一致性 AUC 0.918、跨模型 0.730、数值域 0.928;融合探测器 0.922)。 测量学教训构成纲领的第二支柱:结构划分须对齐生成通道(粒度阶梯);符号与结构不得共享观测量(插补假象的教训);语言群体必须是真正的生成环境;一切显著结果须经置换、半分裂、打乱与变体稳健性检验,并以预注册确认。本文最后陈述"度量逻辑学"的纲领内容与未来领域清单。 关键词:结构–符号样本空间;逻辑测量学;确定性;方向性;锚定;锚定对齐假说;适用域条款;逻辑语言学;通道对齐;预注册;跨领域验证