Needs Verification

Non-CYCLIC Form (Dag Property)¶

Definition & Conceptual Goal¶

The Non-Cyclic Form (DAG Property) metric assesses whether the intra-theory specialization and derivation relations strictly form a Directed Acyclic Graph (Balzer et al., 1987, pp. 172–173).

In formal philosophy of science and epistemology, specializations \(\alpha\) and logical derivations must be asymmetric and strictly irreflexive. Circular specialization (\(T_1 \alpha T_2 \alpha \dots \alpha T_1\)) represents a fatal logical fallacy (petitio principii / vicious circle) where a fundamental law claims grounding in a specialization that presupposes that very same fundamental law.

Theoretical Grounding & Model Formulation¶

The specialization relation \(\alpha\) on the set of theory-elements \(T\) is defined as a strict partial order:

Irreflexivity: \(\forall T_i \in T, \neg (T_i \alpha T_i)\)
Asymmetry: \(\forall T_i, T_j \in T, (T_i \alpha T_j \implies \neg (T_j \alpha T_i))\)
Transitivity: \(\forall T_i, T_j, T_k \in T, (T_i \alpha T_j \land T_j \alpha T_k \implies T_i \alpha T_k)\)

Consequently, a valid theory-net graph \(G = (V, E_{spec})\) is mathematically required to be a DAG.

Mathematical Specification & Graph Formulation¶

Let \(G\) be a theory graph as defined in the Formal Graph Schema (TheoryNet). We evaluate acyclicity on the subgraph restricted to hierarchical edge types, i.e., where \(\lambda_e(e) \in \{\mathsf{specializes}, \mathsf{derives\_from}, \mathsf{explains}\}\).

Binary Acyclicity Check¶

\[\text{IsDAG} (G) = \begin{cases} 1 & \text{if } \text{Cycles} (G) = \emptyset \\ 0 & \text{if } |\text{Cycles} (G)| > 0 \end{cases}\]

Cycle Contamination Metric¶

To measure the severity of circularity in extracted candidate graphs:

\[\text{CircularityPenalty} (G) = \frac{|V_{cyclic}|}{|V|}\]

where \(V_{cyclic} = \{v \in V \mid |SCC(v)| > 1 \lor (v, v) \in E\}\) is the set of vertices participating in any structural cycle, derived via Strongly Connected Components (SCCs) and self-loops.

Measurement & Graph Implementation¶

Tarjan's / Kosaraju's Algorithm: Partition graph \(G\) into strongly connected components. If any component has size \(> 1\) or contains a self-loop, cycle violation is detected.
Topological Sort: Compute Kahn's algorithm or DFS-based topological ordering. The presence of back-edges indicates circular dependencies.
Automated Feedback in Pipeline: When a cycle is detected during pipeline execution, the cycle path is extracted and submitted to the conflict resolution module to determine which edge was misclassified or inverted.

Diagnostic & Metascientific Value¶

Measurement	Diagnostic Finding
\(\text{IsDAG} = 1\)	Valid foundational hierarchy. Epistemic justification flows strictly from core axioms to peripheral applications.
\(\text{IsDAG} = 0\)	Logical circularity error. Edge direction inversion during relation extraction or conceptual conflation in source text.

Grounding References¶

[Balzer et al., 1987] Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An Architectonic for Science. Reidel Publishing, pp. 172–173.
[Schurz, 2024] Schurz, G. (2024). Philosophy of Science: A Unified Approach. Routledge, pp. 119–122.