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Consistency

Definition & Conceptual Goal

The Consistency metric evaluates whether theoretical concepts and relational definitions maintain logical and semantic coherence across documents and argumentative contexts (Balzer et al., 1987, p. xviii; Schurz, 2024, sec. 5.2).

It detects semantic drift and polysemy, ensuring that a theoretical term (e.g., "mass", "entropy", "utility") does not implicitly switch definitions or adopt mutually conflicting axiomatizations within the same reconstructed theory.

Theoretical Grounding & Model Formulation

In structuralist philosophy, a concept's formal meaning is bound by its potential model class \(M_p\) and cross-application constraints \(C\). If term \(\tau\) appears in multiple theory-elements \(T_1, T_2\), its semantic interpretation must satisfy relational compatibility:

\[C (\tau) \subseteq \mathcal{P} (M_p (T_1) \times M_p (T_2))\]

In natural language text, polysemy or equivocation creates an illusion of logical derivation while violating underlying formal consistency.

Mathematical Specification & Graph Formulation

Let \(E_t = \{e_1, e_2, \dots, e_k\}\) be the set of entity mentions / definitions associated with term \(t \in V\) across documents, with vector embeddings \(\mathbf{v} (e_i) \in \mathbb{R}^d\).

Semantic Embedding Coherence (\(S_{\text{embed}}\))

The semantic consistency of \(e_i\)'s underlying concepts, represented by the graph node \(t\), is measured by the average pairwise cosine similarity of its textual contextual embeddings:

\[S_{\text{embed}} (t) = \frac{1}{\binom{k}{2}} \sum_{1 \le i < j \le k} \frac{\mathbf{v} (e_i) \cdot \mathbf{v} (e_j)}{\|\mathbf{v} (e_i)\| \|\mathbf{v} (e_j)\|}\]

Polysemy / Drift Score

If contextual embeddings form multiple distinct clusters (evaluated via silhouette score \(S_{\text{sil}}\) over \(k\) -means), the polysemy score is:

\[\text{PolysemyDrift} (t) = S_{\text{sil}} (t)\]

Measurement & Graph Implementation

  1. Contextual Embedding Extraction: Extract token/sentence embeddings for all entity mentions associated with a given concept node.
  2. Cluster Analysis: Compute pairwise cosine similarity and check for multi-modal embedding distributions.
  3. Graph Consistency Validation: In the property graph, flag nodes with \(S_{\text{embed}} < \theta_{\text{sim}}\) for human review or automated entity splitting.

Diagnostic & Metascientific Value

Consistency Score Semantic State Metascientific Interpretation
\(S_{\text{embed}} \ge 0.85\) Unambiguous / Monosemous Rigorous conceptual precision throughout corpus.
\(S_{\text{embed}} < 0.60\) High Polysemy / Semantic Drift Concept equivocates across contexts; requires splitting into distinct theoretical entities.

Grounding References

  • [Balzer et al., 1987] Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An Architectonic for Science. Reidel Publishing, p. xviii.
  • [Schurz, 2024] Schurz, G. (2024). Philosophy of Science: A Unified Approach. Routledge, sec. 5.2.