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Kuhnian / Paradigm-Guidance¶

Definition & Conceptual Goal¶

The Kuhnian / Paradigm-Guidance metric evaluates whether the historical evolution of a theory-net represents coherent "normal science" or erratic, ad-hoc shifts (Balzer et al., 1987, pp. 175–176; Stegmüller, 1976, pp. 169–171, 194).

In Kuhn's philosophy of science (as formalized structurally by Stegmüller), normal scientific research proceeds by developing specialized laws in the outer branches of a theory-net that strictly descend from an immutable paradigmatic core (\(K_0\)), while leaving the core laws protected and stable.

Theoretical Grounding & Model Formulation¶

A historical sequence of theory-nets \(\langle N_1, N_2, \dots, N_k \rangle\) is paradigm-guided iff:

Core Invariance: There exists a fundamental paradigm core \(K_0\) present in every net \(N_t\).
Core Specialization: All newly introduced theory-elements \(K_i (t)\) at subsequent times are formal core specializations (\(K_i (t) \alpha K_0\)) of the root paradigm \(K_0\).
Domain Conservation: The fundamental paradigm applications \(I_0\) remain continuously embedded in the intended application set (\(I_0 \subseteq I (t)\)).

Formal Specification¶

The formal framework addresses limitations in basic graph coverage models by explicitly tracking specialization filtering, incremental dynamics, and domain conservation.

A. Refined Paradigm Adherence Ratio (\(PAR\))¶

Let \(G_t = (V (G_t), E (G_t))\) be the directed graph representing the theory-net at revision stage \(t\), where vertices \(V (G_t)\) denote the set of all active theory-elements. Let \(T_0 \in V (G_1)\) denote the foundational paradigm root node established at inception.

To ensure nodes correctly align with the paradigm, we restrict paths to directed specialization edges \(E_\alpha \subseteq V (G_t) \times V (G_t)\), where \((T_i, T_j) \in E_\alpha\) signifies that \(T_j\) is a direct formal specialization of \(T_i\). We denote the existence of a directed specialization path from \(T_0\) to \(T\) via the reflexive-transitive closure over \(E_\alpha\), written as \(T_0 \xrightarrow{\alpha, *} T\).

The Paradigm Adherence Ratio is then defined as:

\[PAR (G_t, T_0) = \frac{\big\lvert \{ T \in V (G_t) \mid T_0 \xrightarrow{\alpha, *} T \} \big\rvert}{\lvert V (G_t) \rvert}\]

Interpretation: \(PAR \in [0.0, 1.0]\). A value of \(1.0\) indicates that every node in \(G_t\) strictly descends from \(T_0\) through a chain of formal specializations.

B. Incremental Paradigm Adherence Ratio (\(\Delta PAR_t\))¶

To prevent historical nodes from masking recent unguided additions, we define the adherence ratio specifically for newly added nodes \(\Delta V_t = V (G_t) \setminus V (G_{t-1})\):

\[\Delta PAR_t (G_t, T_0) = \begin{cases} \frac{\big\lvert \{ T \in \Delta V_t \mid T_0 \xrightarrow{\alpha, *} T \} \big\rvert}{\lvert \Delta V_t \rvert} & \text{if } \lvert \Delta V_t \rvert > 0 \\ 1.0 & \text{if } \lvert \Delta V_t \rvert = 0 \end{cases}\]

C. Domain Conservation Index (\(DCI_t\))¶

Let \(I_0 \subseteq M_{pp}\) be the initial paradigmatic intended applications established by the founders, and let \(I (t) \subseteq M_{pp}\) denote the total set of intended applications claimed by the theory-net at revision \(t\).

To operationalize Kuhn's condition that original paradigm applications (\(I_0\)) must not be discarded during revisions, we define:

\[DCI (I (t), I_0) = \frac{\lvert I_0 \cap I (t) \rvert}{\lvert I_0 \rvert}\]

Interpretation: \(DCI = 1.0\) iff \(I_0 \subseteq I (t)\) (full conservation). If \(DCI < 1.0\), intended paradigm applications are being abandoned, signaling an anomaly or domain shift.

D. Core Stability Predicate (\(\text{CoreStable}\))¶

Let \(K_0 (t) = (M_p, M_{pp}, M, GC, GL)_0\) be the core tuple of \(T_0\) at time step \(t\):

\[\text{CoreStable} (G_1, \dots, G_k) = \begin{cases} 1 & \text{if } \big|\bigcap_{t=1}^k M_0 (t)\big| = \lvert M_0 (1) \rvert \text{ and } \big|\bigcap_{t=1}^k GC_0 (t)\big| = \lvert GC_0 (1) \rvert \\ 0 & \text{otherwise} \end{cases}\]

E. Composite Paradigm Guidance Index (\(PGI_t\))¶

Combining structural core adherence and empirical domain conservation into a single score (\(w_1 + w_2 = 1\)):

\[PGI (G_t, T_0) = \text{CoreStable} (G_1, \dots, G_t) \cdot \left[ w_1 \cdot PAR (G_t, T_0) + w_2 \cdot DCI (I (t), I_0) \right]\]

\(PGI\) Score	Research Dynamic	Metascientific Interpretation
\(PGI = 1.0\)	Pure Normal Science	All theoretical nodes descend from \(T_0\) and all original applications \(I_0\) are preserved.
\(0.5 \le PGI < 1.0\)	Auxiliary Shifts / Drift	Emergence of semi-independent models or minor loss of original application domain.
\(PGI < 0.5\)	Paradigm Crisis	Severe fragmentation; newly introduced models bypass \(T_0\) or fundamental applications fail.

Updated Graph Implementation (Cypher)¶

To match the refined mathematical specification \(T_0 \xrightarrow{\alpha, *} T\), the database query must explicitly specify the relationship type :SPECIALIZES:

// Query to identify unguided nodes (nodes not reachable via specialization from paradigm root)
MATCH (root:TheoryElement {is_paradigm_core: true, theory_id: $theory_id})
MATCH (t:TheoryElement {theory_id: $theory_id})
WHERE t <> root 
  AND NOT (root)-[:SPECIALIZES*]->(t)
RETURN 
  count(t) AS unguided_node_count,
  collect(t.id) AS unguided_node_ids;

Grounding References¶

References & Theoretical Grounding¶

Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An Architectonic for Science. D. Reidel Publishing Company.
Stegmüller, W. (1976). The Structure and Dynamics of Theories. Springer-Verlag.
Schurz, G. (2014). Philosophy of Science: A Unified Approach. Routledge.
Newman, M. (2018). Networks (2nd ed.). Oxford University Press.
Nováček, V. (2015). Formalising Hypothesis Virtues in Knowledge Graphs: A General Theoretical Framework and its Validation in Literature-Based Discovery Experiments. arXiv:1503.09137.