Needs Verification

Semantic & Axiomatic Homogeneity¶

Definition & Conceptual Goal¶

The Semantic & Axiomatic Homogeneity metric evaluates the logical unity of a theory, ensuring it cannot be factorized into disjoint, non-interacting sub-theories (Schurz, 2024, pp. 11, 73).

It prevents the famous Tacking Paradox (Tacking-Paradoxie) in philosophy of science: the illegitimate conjunctive combination of two completely unrelated theories (\(T_1 \land T_2\), e.g., Newtonian gravitation conjoined with an arbitrary theological dogma) to falsely claim unified empirical confirmation for both.

Theoretical Grounding & Model Formulation¶

In the statement-view / logical axiomatization of theories: Let \(Ax\) be the natural axiomatization of theory \(T\), and let \(E (Ax)\) denote the relevant deductive consequences of \(Ax\).

Schurz's Factorization Criterion¶

An axiomatization \(Ax\) is axiomatically homogeneous (indecomposable) if and only if there exist no non-empty disjoint sub-axiomatizations \(Ax_1, Ax_2 \subset Ax\) (\(Ax_1 \cup Ax_2 = Ax, Ax_1 \cap Ax_2 = \emptyset\)) such that:

\[E (Ax) = \operatorname{Cn} (E (Ax_1) \cup E (Ax_2))\]

If such a decomposition exists, \(T\) is not a genuine single theory, but an arbitrary tacking together of two independent systems.

Mathematical Specification & Graph Formulation¶

Let \(G_{T} = (V_{Ax}, E_{\text{inf}})\) be the logical inference graph between axioms, premises, and derived empirical theorems.

Axiom Interaction Density (\(AID\))¶

Let \(V_{Ax}\) be partitioned into two candidate subsets \(A_1, A_2\). The cross-axiom mutual inference density is:

\[AID (A_1, A_2) = \frac{|\{ (u, v) \in E_{\text{inf}} \mid (u \in A_1 \land v \in A_2) \lor (u \in A_2 \land v \in A_1)\}|}{|A_1| \cdot |A_2|}\]

Homogeneity Factor (\(H_{\text{axiomatic}}\))¶

\[H_{\text{axiomatic}} (T) = \min_{A_1 \cup A_2 = V_{Ax}} AID (A_1, A_2)\]

\(H_{\text{axiomatic}} (T) > 0\): Indivisible, homogeneous theoretical system.
\(H_{\text{axiomatic}} (T) = 0\): Decomposable / tacked-on system.

Measurement & Graph Implementation¶

Bipartition Inference Scan: Check for disconnected inference subgraphs within the asserted :Theory node container.
Shared Intermediate Variables: Validate whether theorems derived by axiom set \(A_1\) share intermediate theoretical state variables with derivations from \(A_2\).
Graph Decomposition Alert: If \(H_{\text{axiomatic}} = 0\), automatically propose splitting the graph into two separate :Theory instances.

Diagnostic & Metascientific Value¶

\(H_{\text{axiomatic}}\)	Logical Structure	Metascientific Diagnostic
\(> 0\)	Unified Theory	Genuine axiomatic unity; all laws contribute to joint derivations.
\(= 0\)	Tacked-On Conjunction	Tacking paradox detected. Arbitrary union of unrelated hypotheses.

Grounding References¶

[Schurz, 2024] Schurz, G. (2024). Philosophy of Science: A Unified Approach. Routledge, pp. 11, 73.
[Balzer et al., 1987] Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An Architectonic for Science. Reidel Publishing, p. 11.