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System COHERENCE (HARMONY, \(H\))¶

Definition & Conceptual Goal¶

The System Coherence (Harmony, \(H\)) metric measures the global explanatory stability and constraint-satisfaction state of the entire epistemological network (Thagard, 1989, p. 443; Schurz, 2024, sec. 5.3).

It quantifies how well the competing and supporting hypotheses satisfy mutual constraints: maximizing simultaneous activation of mutually supportive (excitatory) propositions while minimizing simultaneous activation of contradictory (inhibitory) claims.

Theoretical Grounding & Model Formulation¶

In Thagard's connectionist architecture (ECHO), explanatory coherence is framed as a parallel constraint satisfaction problem analogous to Hopfield neural networks or Boltzmann machines.

The global state of the network at iteration \(t\) is evaluated by a Harmony Function (\(H (t)\)). As node activations update over successive iterations, the network relaxes toward a state of maximum harmony, resolving conflicts and selecting the globally coherent theory.

Mathematical Specification & Graph Formulation¶

Let \(V = \{1, 2, \dots, N\}\) be the set of all proposition nodes (hypotheses and evidence units). Let \(w_{ij}\) be the symmetric connection weight between node \(i\) and node \(j\):

\(w_{ij} > 0\) for positive explanatory / coherence links (excitatory constraints).
\(w_{ij} < 0\) for contradiction / incompatibility links (inhibitory constraints).
\(a_i (t) \in [-1, 1]\) be the continuous activation of node \(i\) at iteration step \(t\).

Global Harmony Function¶

\[H (t) = \sum_{i=1}^{N} \sum_{j=1}^{N} w_{ij} a_i (t) a_j (t)\]

Normalized System Coherence Score (\(SCS\))¶

\[SCS (G) = \frac{H (t^*)}{\sum_{i=1}^N \sum_{j=1}^N |w_{ij}|}\]

where \(t^*\) is the convergence iteration where \(\Delta H (t^*) < \epsilon_{\text{tol}}\).

Measurement & Graph Implementation¶

Network Initialization: Construct symmetric adjacency matrix \(W\) with excitatory (\(w > 0\)) and inhibitory (\(w < 0\)) weights.
Synchronous Relaxation Loop: Update node activations until \(\|\mathbf{a} (t+1) - \mathbf{a} (t)\| < 10^{-4}\).
Harmony Calculation: Compute quadratic form \(\mathbf{a}^T W \mathbf{a}\) at equilibrium.

Diagnostic & Metascientific Value¶

\(SCS(G)\) Score	Epistemic Coherence State	Metascientific Diagnostic
High \(SCS\) (\(> 0.7\))	Highly Harmonious & Coherent	Hypotheses and evidence mutually reinforce without unresolved contradictions.
Low \(SCS\) (\(< 0.2\))	Fractured / Conflicted Paradigm	Intense internal contradictions and un-reconciled anomalies across competing hypotheses.

Grounding References¶

[Thagard, 1989] Thagard, P. (1989). Explanatory Coherence. Behavioral and Brain Sciences, 12 (3), pp. 435–467.
[Schurz, 2024] Schurz, G. (2024). Philosophy of Science: A Unified Approach. Routledge, sec. 5.3.