Conservatism & Epistemic Safety¶

Definition & Conceptual Goal¶

The following metric will discuss two core metrics from epistemology and scientific theory.

• Conservatism evaluates the conceptual detour of a claim. A conservative theory introduces incremental semantic steps that stay closely anchored to known paradigms, whereas radical hypotheses make large conceptual leaps across disparate topics (Novacek, 2015).
• Epistemic Safety evaluates the evidential decay of a claim. It measures how much confidence is retained when traversing the chain of reasoning from empirical axioms to a final conclusion.

Topological Preconditions¶

To calculate both metrics, the underlying theory graph (as defined in Layer 3 of the Formal Model) must explicitly support a directed argumentative structure. The metrics require identifiable paths representing the flow of inference, specifically originating from an empirical core and terminating at a theoretical claim.

A standard required path structure takes the form:

flowchart LR
    Source([Empirical Sentences]) --> H[Hypothesis]
    H --> P[Premises]
    P --> AL[Assignment Laws]
    AL --> Target([Conclusion])
    style Source fill: #e1f5fe, stroke: #01579b, stroke-width: 2px
    style Target fill: #fce4ec, stroke: #880e4f, stroke-width: 2px

By defining explicit Source nodes (empirical grounding/axioms) and Target nodes (conclusions), we can meaningfully measure both the semantic distance and the confidence drop-off between them.

Theoretical Grounding & Model Formulation¶

Conservatism (Conceptual Detour)¶

Following Novacek (2015), conservatism is a topological and semantic distance metric. It assesses whether the semantic "distance" between the premises and the conclusions is minimized. Novacek describes the node's conceptual position with the term 'characteristic context vector'; in order to avoid confusion with eigenvectors, we will consistently use " weighted adjacency vector" instead. A hypothesis path is conservative if the direct distance between its start and end is highly correlated with the sum of the individual step distances along the path.

Epistemic Safety (Confidence Propagation)¶

Safety operates strictly on the explicit confidence weights of the relationships (edges). It assumes that every logical step introduces some uncertainty. Therefore, safety models how confidence propagates and decays through the graph.

Mathematical Specification & Graph Formulation¶

Let \(H = (V_H, E_H)\) be a directed hypothesis graph containing a path \(p = (v_1, v_2, \dots, v_{|p|})\) from an empirical source \(v_1\) to a conclusion \(v_{|p|}\).

Conservatism Formula (Novacek, 2015)¶

To measure the overall conservatism of a hypothesis graph \(H\), we compute the arithmetic mean of the conservatism for all shortest paths \(p \in \pi_s (H, \delta)\) within that graph. The formula exactly follows Novacek (2015):

(Note: \(\delta\) is the Euclidean distance between the weighted adjacency vectors of the vertices. A value closer to 1 implies a highly conservative, straight-line conceptual graph, while a value approaching 0 indicates a radical conceptual detour).

Epistemic Safety Formula¶

Safety operates strictly on explicit confidence weights \(w (e) \in [0, 1]\) assigned to edges. For a target node \(v_{target}\) supported by a source node \(v_{source}\) with initial confidence \(c (v_{source})\), safety seeks the Maximum Reliability Path:

Measurement & Graph Implementation¶

1. Path Identification: Query the graph for paths matching the required topology (e.g., Empirical Source \(\to \dots \to\) Conclusion).
2. Measuring Conservatism: Extract the weighted adjacency vectors for nodes along the path. Calculate the pairwise distances and compute the conservatism ratio.
3. Measuring Safety (Implementation Trick): While Safety uses the product of weights, it is typically implemented using standard shortest-path algorithms (like Dijkstra's) by transforming the edge weights: \(\text{cost} (e) = -\log (w (e))\). The shortest path in the transformed graph corresponds exactly to the Maximum Reliability Path.

Diagnostic & Metascientific Value¶

Grounding References¶

• [Novacek, 2015] Novacek, V. (2015). Formalising Hypothesis Virtues in Knowledge Graphs.