Needs Verification

Tenability (Haltbarkeit)¶

Implementation: Implemented as a post-processing pass in Theoretical Enrichment & Tenability Evaluation and governed by ADR 0015: Theoretical Enrichment and Tenability Evaluation.

Definition & Conceptual Goal¶

The Tenability (Haltbarkeit) metric evaluates whether a theory's empirical claim can be considered approximately true or highly plausible given a set of empirical observations (Stegmüller, 1976, pp. 120, 270; Schurz, 2024, sec. 2.3). In the non-statement view of scientific theories, a theory is reconstructed not as a system of linguistic statements, but as a system of set-theoretical models (Stegmüller, 1976, pp. 7, 75).

While Empirical Adequacy measures numerical agreement and residual blur tolerances, tenability represents the non-trivial truth of the theory's empirical claim (Stegmüller, 1976, pp. 78, 101): it ensures that the set of intended applications ($I$)—which have the structure of partial potential models ($M_{pp}$) consisting only of non-theoretical, theory-independent concepts (Stegmüller, 1976, pp. 86, 107; Schurz, 2024, sec. 2.3) —can be theoretically enriched by postulating theoretical terms (such as mass or force) (Stegmüller, 1976, pp. 77, 121) in a way that simultaneously satisfies the core physical laws ($M$), the global inter-application constraints ($GC$), and the intertheoretical links ($GL$) of the theory-core (Stegmüller, 1976, pp. 78, 94).

Theoretical Grounding & Model Formulation¶

In structuralist metatheory, a theory-element is represented as $T = \langle K, I \rangle$, where the core is $K = \langle M_p, M, M_{pp}, GC, GL \rangle$ (Stegmüller, 1976, p. 94). The set of intended applications $I$ is a subset of the partial potential models ($I \subseteq M_{pp}$) (Stegmüller, 1976, pp. 86, 94). Within the context of our Theory Graph, the set of intended applications $I$ corresponds to subsets of the empirical base $B$ (Empirical Observation Sentences). An intended application is thus formed by empirical observations $b \in I \subseteq B$.

The idealized empirical claim of the theory asserts that the intended applications can be enriched to full models satisfying all constraints and links (Stegmüller, 1976, pp. 78, 94): $$I \in \mathrm{Cn} (K)$$ Where the content $\mathrm{Cn} (K)$ is the class of sets of partial potential models that are the non-theoretical restrictions of sets of models satisfying the constraints $GC$ and links $GL$ (Stegmüller, 1976, pp. 78, 94).

The Need for Continuous Approximation over Binary Thresholds¶

In real-world scientific texts, NLP-extracted graphs, and empirical data, absolute exactness is an idealization. Therefore, tenability must be modeled utilizing the formal structuralist apparatus of uniformities ($U$) and admissible blurs ($\mathcal{A}$) (Stegmüller, 1976, pp. 75, 110; Balzer et al., 1987, pp. 209–217).

A classical binary threshold (such as defining $\text{Score} = 1 \text{ if error } < \epsilon \text{ else } 0$) introduces a sharp, non-differentiable step-function. In graph-based extraction and empirical scientific domains, this binary discontinuity is deeply problematic:

A theory that misses a hard threshold by an infinitesimal margin (e.g., error is $0.1001$ instead of $0.1000$) would abruptly be classified as completely untenable ($\text{Score} = 0$).
Conversely, all theories meeting the threshold would be treated as equally "perfect" ($\text{Score} = 1$), obscuring meaningful gradations in empirical fit.

Rather than a binary truth value, Tenability is operationalized as a robust, continuous structural compatibility score in the interval $[0, 1]$ (Stegmüller, 1976, p. 101), assessing the degree of approximation required to satisfy both local laws and global constraints (Stegmüller, 1976, pp. 76, 116). This gradualism is philosophically aligned with the concept of truthlikeness (verisimilitude) and approximate truth (Schurz, 2024, sec. 2.3; Stegmüller, 1976, pp. 56, 106): a theory is not simply "true" or "false," but possesses a measurable degree of empirical success and tolerance (Stegmüller, 1976, pp. 76, 101).

The Topology of Uniform Spaces vs. Metric Spaces¶

In Sneed's and Stegmüller's metatheory, approximation is formalized using Bourbaki's uniform spaces rather than metric spaces ([Bourbaki, 1966]; Stegmüller, 1976, pp. 110, 115). This topological approach is crucial because scientific theories routinely deal with heterogeneous physical quantities (e.g., measuring positions in meters, times in seconds, and masses in kilograms) (Stegmüller, 1976, pp. 113, 137). There is no single, natural, universal metric distance function $d (x, y)$ that can combine these different dimensions without introducing arbitrary scaling factors (Stegmüller, 1976, p. 113).

Instead of a single metric distance, a uniformity $U$ specifies a family of nested relational "blurs" $\{u_\delta\}$ (Stegmüller, 1976, pp. 111, 112). Because one cannot simply write "$\text{distance} < x$," the framework utilizes this family of blurs. Topological neighborhood inclusion is thereby translated into a rigorous basis for continuous approximation.

Mathematical Specification & Graph Formulation¶

Let $y \in I \subseteq M_{pp}$ be an empirical data node (representing a partial potential model containing only non-theoretical variables) (Stegmüller, 1976, pp. 86, 107).

The Dual-Enrichment Architecture ($\Phi$)¶

To computationally implement theoretical enrichment in a graph pipeline without assuming a mathematically untenable universal parametric space, the enrichment function $\Phi (y)$ is constructed as a composite function:

\[\Phi (y) = \Phi_{\text{spec}} (\Phi_{\text{gen}} (y))\]

General Enrichment ($\Phi_{\text{gen}}$): A domain-agnostic parser that maps raw data (or text) into an abstract structural space. Rather than physical metrics, its dimensions consist of structural and relational properties:
- Topology: Set-theoretic relations (e.g., is node $A$ a subset of $B$?)
- Causality: Directed acyclic graphs (e.g., does an edge go from $A \to B$?)
- Covariance: Monotonicity and logical correlation (e.g., if $A$ increases, does $B$ increase?)
It yields an abstract graph where baseline structural tenability (e.g., checking for logical contradictions like $A \to B$ and $B \to A$ where strict acyclicity is required) can be evaluated, but lacks empirical metric dimensions.
Specific Enrichment ($\Phi_{\text{spec}}$): An ontological projector that receives the abstract graph and projects it into the bespoke parametric space of the specific theory ($M_p$). By assigning precise dimensions (e.g., "mass" in $\mathbb{R}^+$), it establishes the local metric rules where the blur $\delta$ is actually evaluated.

This composite $\Phi (y)$ maps the partial potential model $y$ to a full potential model $x = \Phi (y) \in M_p$ by adding theory-specific theoretical terms.

Let $M \subseteq M_p$ be the set of actual models satisfying the fundamental laws (Stegmüller, 1976, p. 78). Let $U$ be an empirical uniformity on $M_p$, and let $u_{\delta} \in \mathcal{A}$ denote a scaled admissible blur (inaccuracy neighborhood) associated with the theory (Stegmüller, 1976, pp. 76, 115).

Local Tenability Score ($TS_{\text{local}}$) and Parameter Estimation¶

The local tenability of an empirical application node $y \in I$ measures how closely its theoretical enrichment satisfies the fundamental law $M$, utilizing the supremum over acceptable approximations:

\[TS_{\text{local}} (y, M) = \sup \{ 1 - \delta \mid \exists x^* \in M : (\Phi (y), x^*) \in u_{\delta} \}\]

Rationale for the Supremum ($\sup$) over Blurs¶

In scientific practice, theoretical parameters (e.g., the mass of a planet, a gravitational constant, or the coupling strength of a node) are not directly observed; they are estimated post-factum from raw empirical observations (Stegmüller, 1976, pp. 57, 61). This corresponds to the enrichment function $\Phi (y)$ which postulates values for these theoretical terms (Stegmüller, 1976, p. 121).

We do not merely ask whether some parameter set meets an arbitrary threshold; we actively search for the best possible fit (Stegmüller, 1976, p. 61). Mathematically, finding the best-fitting model is an optimization problem (analogous to the method of least squares (Stegmüller, 1976, p. 61)): $$\text{Minimize } \delta \quad \text{subject to } \Phi (y) \in u_\delta \text{ of some } x^* \in M$$

Minimizing the error $\delta$ is mathematically equivalent to maximizing the structural compatibility score $1 - \delta$ (Stegmüller, 1976, p. 26). The supremum ($\sup$) represents this exact optimization process: it finds the finest, most precise neighborhood $u_\delta$ in our family of blurs that can still successfully reconcile our empirical data with the laws of the theory (Stegmüller, 1976, pp. 115, 119, 147, 148). This elegantly translates pure topological neighborhood inclusion into a continuous, normalized numerical confidence score for graph processing.

Boundary Conditions: If the system satisfies the law exactly, $\delta = 0 \implies TS_{\text{local}} = 1.0$. If no admissible approximation exists within the boundaries of $\mathcal{A}$, the score drops to $0.0$ (Stegmüller, 1976, p. 142).

Edge Tenability Score ($TS_{\text{edge}}$) via Global Constraints¶

Constraints ($GC$) prevent triviality by linking theoretical parameters (e.g., mass, coupling constants) across different applications (Stegmüller, 1976, p. 108). Let $e = (y_a, y_b) \in I \times I$ be a constraint edge representing an inter-application relation. Under a constraint blur $v_{\delta_C} \in \mathcal{A}$, the consistency of theoretical values across the edge is operationalized as:

\[TS_{\text{edge}} (e) = \sup \{ 1 - \delta_C \mid (\Phi (y_a), \Phi (y_b)) \in v_{\delta_C} \}\]

Propagated Tenability¶

If a theoretical value is calculated at node $y_a$ (where $TS_{\text{local}} (y_a)$ is known) and projected onto a new node $y_b$ via a constraint edge $e = (y_a, y_b)$, the predicted tenability at $y_b$ is the product of the node's local tenability and the edge consistency:

\[\text{Tenability}_{\text{pred}} (y_b) = TS_{\text{local}} (y_a, M) \cdot TS_{\text{edge}} (e)\]

Aggregated Tenability of a Theory-Element $T$¶

To evaluate the global tenability of the theory-element $T = \langle K, \mathcal{A}, I \rangle$ over the graph of its applications $I$ and constraint edges $E_{GC}$, we define the weighted average:

\[\text{Tenability} (T) = w_{\text{local}} \cdot \left (\frac{1}{|I|} \sum_{y \in I} TS_{\text{local}} (y, M) \right) + w_{\text{edge}} \cdot \left (\frac{1}{|E_{GC}|} \sum_{e \in E_{GC}} TS_{\text{edge}} (e) \right)\]

Where $w_{\text{local}}, w_{\text{edge}} \in [0,1]$ are weights such that $w_{\text{local}} + w_{\text{edge}} = 1$, representing the balance between local law-adherence and global cross-application consistency.

Measurement & Graph Implementation¶

Schema Typing & Model Verification: Compare extracted empirical entity attributes and observation spans against the formal predicate signatures in the theory's :PotentialModelClass ($M_p$) and non-theoretical base ($M_{pp}$).
Theoretical Enrichment ($\Phi$): Apply $\Phi_{\text{gen}}$ to extract the domain-agnostic relational graph, then apply $\Phi_{\text{spec}}$ to instantiate domain-specific theoretical edges and parameters connecting empirical observation nodes $y \in I$ to candidate model structures $x \in M_p$.
Local Inaccuracy Minimization: Determine the tightest admissible blur $u_\delta \in \mathcal{A}$ reconciling enriched empirical instances with theoretical laws $M$, computing $TS_{\text{local}} (y, M) = 1 - \delta^*$.
Constraint Consistency Evaluation: Evaluate inter-application constraint edges $e = (y_a, y_b) \in E_{GC}$, scoring theoretical parameter consistency across applications under constraint blurs $v_{\delta_C}$.
Graph Inspection & Anomaly Flagging: Application nodes or constraint edges with $TS < 0.5$ are flagged as category errors, severe law deviations, or cross-application parameter incompatibilities.

Diagnostic & Metascientific Value¶

Tenability Score	Conceptual Fit	Metascientific Interpretation
$\text{Tenability} \ge 0.8$	High Tenability	Applications naturally instantiate the theory; theoretical terms reconcile empirical data within tight blurs.
$0.5 \le \text{Tenability} < 0.8$	Approximative Fit	Acceptable empirical fit; minor deviations or tolerable constraint tensions present across applications.
$\text{Tenability} < 0.5$	Category Error / Untenable	Attempt to apply framework to an ontologically mismatched domain or failure to satisfy laws within blurs $\mathcal{A}$.

Grounding References¶

[Stegmüller, 1976] Stegmüller, W. (1976). The Structure and Dynamics of Theories. Springer-Verlag, pp. 7, 26, 56–61, 75–121, 137–148, 270.
[Schurz, 2024] Schurz, G. (2024). Philosophy of Science: A Unified Approach. Routledge, sec. 2.3.
[Balzer et al., 1987] Balzer, W., Moulines, C. U., & Sneed, J. D. (1987). An Architectonic for Science. Reidel Publishing Company, pp. 209–217.
[Bourbaki, 1966] Bourbaki, N. (1966). General Topology: Part 1. Hermann / Addison-Wesley, Chapter II: Uniform Structures.
[Sneed, 1971] Sneed, J. D. (1971). The Logical Structure of Mathematical Physics. Reidel Publishing Company.

Tenability (Haltbarkeit)¶

Definition & Conceptual Goal¶

Theoretical Grounding & Model Formulation¶

The Need for Continuous Approximation over Binary Thresholds¶

The Topology of Uniform Spaces vs. Metric Spaces¶

Mathematical Specification & Graph Formulation¶

The Dual-Enrichment Architecture (\(\Phi\))¶

Local Tenability Score (\(TS_{\text{local}}\)) and Parameter Estimation¶

Rationale for the Supremum (\(\sup\)) over Blurs¶

Edge Tenability Score (\(TS_{\text{edge}}\)) via Global Constraints¶

Propagated Tenability¶

Aggregated Tenability of a Theory-Element \(T\)¶

Measurement & Graph Implementation¶

Diagnostic & Metascientific Value¶

Grounding References¶

Tenability Score	Conceptual Fit	Metascientific Interpretation
\(\text{Tenability} \ge 0.8\)	High Tenability	Applications naturally instantiate the theory; theoretical terms reconcile empirical data within tight blurs.
\(0.5 \le \text{Tenability} < 0.8\)	Approximative Fit	Acceptable empirical fit; minor deviations or tolerable constraint tensions present across applications.
\(\text{Tenability} < 0.5\)	Category Error / Untenable	Attempt to apply framework to an ontologically mismatched domain or failure to satisfy laws within blurs \(\mathcal{A}\).