Formal Graph Schema (TheoryNet)¶

This document establishes the unified formal mathematical architecture of the Theory Graph (TheoryNet) in Grund Episteme. It synthesizes epistemological theory, structuralist philosophy of science, abstract argumentation, and graph-theoretic models into a single, mathematically rigorous framework designed to represent, extract, and evaluate contested scientific literature.

Mathematical Foundations & Graph Representation¶

The Theory Graph is formally defined as a labeled, directed multigraph:

\[ G = (V, E, \lambda_v, \lambda_e) \]

Where:

$V$ is the finite set of vertices (entities, concepts, premises, and claims).
$E \subseteq V \times V$ is the finite set of directed edges (semantic relationships, support, or attack vectors).
$\lambda_v: V \to T_v$ assigns vertex types from an ontology type hierarchy $T_v$.
$\lambda_e: E \to T_e$ assigns edge types from a relationship hierarchy $T_e$.

Isomorphism to the Logical TheoryNet ($TF$)¶

The logical framework of the scientific theory—the TheoryNet—is defined as a relational tuple:

\[ TF = (At, R) \]

where $At$ denotes the set of argumentative atoms and $R$ denotes the set of argumentative relations. To unify computational graph storage (in Neo4j) with model-theoretic argumentation, we enforce a strict isomorphism:

The vertex set $V$ maps directly to the set of argumentation atoms $At$ ($V \cong At$).
The edge set $E$ maps directly to the argumentative relations $R$ ($E \cong R$).

Epistemic Partitioning ($\Delta = (B, A)$)¶

Scientific theories are not flat assertions of fact; they fundamentally differentiate between empirical observations and abstract theoretical hypotheses. We formally partition the vertex set $V$ via the underlying knowledge base $\Delta = (B, A)$:

$B$ (Empirical Observation Base / Beobachtungssätze): The foundation of verifiable, minimally contested empirical data points, localized existential assertions, and raw evidentiary units.
$A$ (Theoretical Antecedents / Hypotheses): Abstract theoretical statements, core axioms, and hypotheses accepted as tenable only under specific inferential conditions.

Consequently, the global vertex set satisfies:

\[ V = B \cup A \quad \text{where} \quad B \cap A = \emptyset \]

Layered Architecture Model (L1, L2, L3)¶

To maintain epistemic purity and prevent semantic drift across processing steps, the theory graph is stratified into three distinct structural layers:

flowchart TD
    subgraph L3 ["Layer 3: Abstract Argumentation / Theory Core (L3)"]
        direction TB
        QBAF["Quantitative Bipolar Argumentation Framework (QBAF)<br/>Λ = ⟨A, R_att, R_sup, τ, ϕ⟩"]
        ATOMS["TheoryAtoms (At) & TheoryRelations (R)"]
        QBAF --- ATOMS
    end

    subgraph L2 ["Layer 2: Deterministic Ontology / Knowledge Base (L2)"]
        direction TB
        ENT["Typed Entities e ∈ V_L2 (Concepts, Persons, Works)"]
        TRIPLES["Factual Relational Triples r ∈ E_L2"]
        ENT --- TRIPLES
    end

    subgraph L1 ["Layer 1: Empirical Manifold / Source Grounding (L1)"]
        direction TB
        CHUNKS["Document Chunks C = (t_1, ..., t_N)"]
        PROV["Textual Envelopes T_n & Provenance Maps"]
        CHUNKS --- PROV
    end

    L2 -->|" MENTIONED_IN / EXTRACTED_FROM "| L1
    L3 -->|" Zuordnungsgesetze Z(a,b) / Epistemic Grounding "| L2
    L3 -.->|" Traversals terminating in B "| L1

Layer 1: Empirical Manifold / Source Grounding ($L1$)¶

Represents raw document chunks, unstructured token sequences $S \in \Sigma^*$, and exact provenance offsets:

\[ L1 = (C, \lambda_c, \pi) \]

Where:

$C = \{c_1, c_2, \dots, c_m\}$ is the set of document chunks.
$\lambda_c: C \to \text{DocumentMetadata}$ assigns structural coordinates (e.g., Table of Contents section, chapter, author, page number).
$\pi: C \to \text{Text}$ provides the verbatim character substring sequence $(t_1, t_2, \dots, t_N)$.

Layer 2: Deterministic Ontology / Knowledge Base ($L2$)¶

Captures entity-centric factual relationships extracted directly from the literature:

\[ L2 = (\mathcal{V}_{\text{L2}}, \mathcal{E}_{\text{L2}}, \gamma, \delta) \]

Where:

$\mathcal{V}_{\text{L2}}$ is the set of typed entity nodes ($e \in \mathcal{V}_{\text{L2}}$), such as Concept, Person, Theory, or Work.
$\mathcal{E}_{\text{L2}} \subseteq \mathcal{V}_{\text{L2}} \times \mathcal{V}_{\text{L2}}$ is the set of directed relationship edges ($r \in \mathcal{E}_{\text{L2}}$), such as INFLUENCED, DEFINES, or ASSOCIATED_WITH.
$\gamma: \mathcal{V}_{\text{L2}} \to [0, 1]$ assigns extraction confidence scores.
$\delta: \mathcal{E}_{\text{L2}} \to [0, 1] \times \mathcal{P} (L1)$ assigns relationship certainty paired with explicit evidentiary text snippets from $L1$.

Layer 3: Abstract Argumentation Framework ($L3$)¶

Encodes argumentative, defeasible, and theoretical dependencies as a Quantitative Bipolar Argumentation Framework (QBAF):

\[ L3 \equiv \Lambda = \langle A, \mathcal{R}_{att}, \mathcal{R}_{sup}, \tau, \phi \rangle \]

Where:

$A$ is the set of argument components / theoretical hypotheses ($a \in \mathcal{V}_{\text{L3}}$), formally categorized as Claims or Premises.
$\mathcal{R}_{att} \subseteq A \times A$ is the class of attacking/undermining relations.
$\mathcal{R}_{sup} \subseteq A \times A$ is the class of supporting/warranting relations.
$\tau: A \to [0, 1]$ is the prior plausibility base score.
$\phi: \mathcal{R}_{att} \cup \mathcal{R}_{sup} \to [0, 1]$ assigns continuous structural relationship weights.
Continuous gradual semantics $\rho: A \to [0, 1]$ subsequently updates and evaluates node acceptability.

Argumentation Framework & Gradual Semantics¶

Scientific reasoning is rarely binary; theoretical statements carry varying degrees of warrant and face conditional objections. We model the logical interaction over theoretical nodes $A$ as a Quantitative Bipolar Argumentation Framework (QBAF).

Super-Role Inclusion (DL-LiteR)¶

Drawing upon Description Logics (specifically DL-LiteR), argumentative interactions are not restricted to monolithic primitives. Instead, "attack" and "support" are formalized as broad super-role classes, denoted as $\mathcal{R}_{att}$ and $\mathcal{R}_{sup}$:

Any fine-grained relational predicate identified in the scientific text (e.g., $r_k \in \{\text{REFUTES}, \text{CRITICIZES}, \text{UNDERMINES}\}$) is treated as a role inclusion inheriting the formal semantics of its parent super-class:

\[ r_k \sqsubseteq \mathcal{R} _{att} \quad \text{or} \quad r_k \sqsubseteq \mathcal{R}_{sup} \]

The complete argumentation space is defined as the tuple:

\[ \Lambda = \langle A, \mathcal{R} _{att}, \mathcal{R}_{sup}, \tau, \phi \rangle \]

Where:

$A$ is the set of theoretical antecedents.
$\mathcal{R}_{att} \subseteq A \times A$ is the class of all attacking/undermining relations.
$\mathcal{R}_{sup} \subseteq A \times A$ is the class of all supporting/warranting relations.
$\tau: A \to [0, 1]$ is the prior plausibility function (vertex labeling $\lambda_v$).
$\phi: \mathcal{R}_{att} \cup \mathcal{R}_{sup} \to [0, 1]$ is the continuous edge weight function (edge labeling $\lambda_e$).

Continuous State Computation & Gradual Semantics¶

The ultimate acceptability of any scientific premise within the Theory Graph is calculated via an update function $\rho: A \to [0, 1]$ that combines its prior plausibility $\tau (a_i)$ with the aggregated strengths of incoming relations.

For any premise $a_i \in A$, the framework calculates an aggregation function $\alpha (a_i)$ summing incoming support and subtractive attack:

\[ \alpha (a_i) = \sum_{ (a_j, a_i) \in \mathcal{R} _{sup}} \rho (a_j) \cdot \phi (a_j, a_i) \; - \sum_{ (a_k, a_i) \in \mathcal{R}_{att}} \rho (a_k) \cdot \phi (a_k, a_i) \]

Using gradual semantics (e.g., Quadratic Energy or Euler-based convergence), this yields the final stability score $\rho (a_i)$, iteratively propagated through the network until convergence.

Dung Abstract Argumentation Mapping¶

In discrete evaluation contexts, the graph can be reduced to Dung's classical abstract argumentation framework:

Arguments: $A \subseteq \mathcal{V}_{\text{L3}}$.
Attack Relations: $\leadsto \; \subseteq A \times A$, where $u \leadsto v \iff (u, v) \in \mathcal{R}_{att}$.
Extensions: Evaluated under standard grounded, preferred, and stable extension semantics.

Zuordnungsgesetze (Mapping Laws) & Epistemic Grounding¶

For an abstract theory core ($A$) to possess empirical creativity rather than remaining purely speculative, it must anchor to the empirical base ($B$). This anchoring is mediated by formal Mapping Laws (Zuordnungsgesetze, Schurz, 2014):

\[ Z: A \times B \to [0, 1] \]

Between an empirical observation $b \in B$ and a theoretical antecedent $a \in A$, the mapping law is governed by two complementary functions:

Correlational Mapping ($Z_1$): [ Z_1: A \times B \to \mathcal{P} ] Models the probabilistic correlation between theoretical constructs and observable indicators, mapped to the continuous $[0, 1]$ interval of the QBAF.
Structural Isomorphism ($Z_2$): [ Z_2: A \times B \to \sigma (\kappa (a), \kappa (b)) ] Measures the structural overlap $\sigma$ between the theoretical model structure $\kappa (a)$ and the empirical domain $\kappa (b)$.

Epistemic Justification Paths ($J (at)$)¶

Rather than storing justifications as static, ungrounded node attributes, the set of epistemic justifications $J (at)$ for any atom $at \in At$ is dynamically inferred via directed graph traversal through supportive and mapping edges terminating in $B$:

\[ J (at) = \{ p = (a_0, e_1, a_1, \dots, e_k, b) \mid a_0 = at, \; b \in B, \; e_i \in \mathcal{R}_{sup} \cup Z \} \]

This ensures that logical inferences remain provably grounded in empirical textual evidence.

Structuralist Model-Theoretic Foundations¶

Complementing abstract argumentation, the pipeline integrates the Structuralist Theory of Science (Sneed, 1971; Stegmüller, 1976; Balzer et al., 1987). Theories are model-theoretic entities rather than syntactic statement lists.

The Model-Theoretic Triad: $M_p$, $M$, and $M_{pp}$¶

A theory-element is defined as a pair $T = \langle K, I \rangle$, where $K$ is the formal theory core and $I$ is the domain of intended applications. The formal core relies on three nested classes of set-theoretic structures:

Potential Models ($M_p$): Defines the mathematical vocabulary and conceptual framework (e.g., coordinate frames, mass functions, force vectors), regardless of whether empirical laws hold.
Actual Models ($M \subseteq M_p$): The subset of potential models satisfying the substantive natural laws or mathematical axioms of the theory.
Partial Potential Models ($M_{pp}$): The empirical substructure obtained by stripping away all $T$-theoretical functions, leaving only non-theoretical, directly observable terms.

Uniform Space $(M_p, \mathcal{U})$ & Admissible Blurs ($\mathcal{A}$)¶

Real-world scientific applications inevitably involve measurement noise and observational tolerances. Rather than requiring absolute mathematical exactness, closeness between empirical observations and theoretical models is formalized via a Uniform Space $(M_p, \mathcal{U})$:

Uniformity ($\mathcal{U}$): A family of entourage relations $u \subseteq M_p \times M_p$ satisfying diagonal containment ($\Delta (M_p) \subseteq u$), filter intersection ($u_1 \cap u_2 \in \mathcal{U}$), and triangular composition ($v \circ v \subseteq u$).
Admissible Blurs ($\mathcal{A} \subseteq \mathcal{U}$): A pragmatically restricted sub-collection of entourages bounded by an upper tolerance limit: [ \text{Bound} (\mathcal{A}) ] If observational discrepancy exceeds $\text{Bound} (\mathcal{A})$, the theory is falsified; conversely, if $\mathcal{A}$ were allowed to be arbitrarily large, the theory would become empirically vacuous.

Tenability and Adequacy¶

These model-theoretic structures define two critical validity conditions:

Tenability (Haltbarkeit): The requirement that intended applications can be formulated in the non-theoretical empirical vocabulary: [ I \subseteq M_{pp} ] If an intended application cannot be expressed within $M_{pp}$, the theory is conceptually untenable for that domain.
Approximative Empirical Adequacy (Adäquatheit): The condition that empirical observations $I$ can be enriched with theoretical functions such that the resulting model lies within an admissible blur $u \in \mathcal{A}$ of an actual model $M$: [ \exists X \in M \quad \text{such that} \quad (I, X) \in u, \quad u \in \mathcal{A} ]

Temporal Evolution & Uncertainty¶

Scientific knowledge is non-static; theories evolve, revise hypotheses, and encounter anomalies.

Temporal Graph Evolution¶

The graph evolves over chronological time $t$:

\[ G (t) = (V (t), E (t), \lambda_v (t), \lambda_e (t)) \]

Description Versioning: When an entity or premise matures into a more specialized conceptualization, earlier descriptions are not overwritten. Instead, the graph retains an immutable historical semantic trace. This ensures that early, less sophisticated terminology remains matchable in vector space during cross-document alignment.
Dynamic Refinement: Relations strengthen, weaken, or undergo modal retraction when confronted with empirical counterexamples.

Probabilistic Semantics¶

To capture extraction and inferential uncertainty, each element carries belief-theoretic scores:

Belief Score: $\beta \in [0, 1]$
Disbelief Score: $\delta \in [0, 1]$
Epistemic Uncertainty: $u = 1 - \beta - \delta \quad (\beta + \delta \le 1)$

For composite argument structures:

Conjunction: $c (A \land B) = \min (c (A), c (B))$
Disjunction: $c (A \lor B) = \max (c (A), c (B))$
Implication: $c (A \to B)$ weighted by evidentiary support and conditional transfer confidence.

Computational Tractability & Search Heuristics¶

Evaluating the global structural coherence of a scientific theory requires determining maximal $B$-consistent subsets $\Sigma \subseteq A$, such that $\Sigma \cup B$ is free of logical contradictions, while any superset $\Sigma' \cup B$ introduces an inconsistency.

Because computing admissible extensions in abstract argumentation and verifying constraint satisfaction over posets is NP-hard, the pipeline avoids exhaustive combinatorial search. Instead, it employs:

Dense Vector Retrieval (Bi-Encoder MIPS): Constrains cross-chunk candidate generation to top-$K$ semantic neighbors.
Cross-Encoder Attention Filtering: Reranks relation candidates using fine-grained multi-head cross-attention.
Machine-Learning Guided Solvers: structural predictors guide search-based solvers to approximate stable extensions in polynomial time.

Quality & Validation Metrics¶

The formalized theory graph is evaluated across multiple structural and empirical quality dimensions:

Metric	Target Layer	Formal Definition	Purpose
Entity Precision & Recall	Layer 2	$\frac{	E_{\text{correct}}
Evidence Precision & Coverage	Layer 1 $\to$ Layer 2	Ratio of accurately localized verbatim text spans grounding entities and triples	Prevents hallucinated relationships and guarantees source auditability.
Semantic Soft Matching	Layer 2 / Layer 3	$\cos(E_{\text{ctx}}(a_{\text{pred}}), E_{\text{ctx}}(a_{\text{gold}})) \ge \tau$	Accommodates paraphrasing and terminological variation in philosophy.
Empirical Content Ratio	Layer 3 $\to$ Layer 1	Ratio of theoretical antecedents grounded via $Z(a, b)$ into $B$	Penalizes ungrounded metaphysical baggage.
Structural Coherence	Layer 3	Parallel constraint satisfaction score ($H(t)$ via ECHO)	Assesses the absence of unmitigated circularities or internal contradictions.

Philosophy of Science: Epistemology & Wissenschaftstheorie
Topologies & Community Structure: Theory-Nets, Posets & Topologies
Dense Alignment & Maturation: Dense Alignment & Grounding
Evaluation Taxonomy: Theory Metrics Subsystem
Software Implementation: Pipeline Architecture
Contracts & Schemas: Schema Reference and Phase Contracts

Formal Graph Schema (TheoryNet)¶

Mathematical Foundations & Graph Representation¶

Isomorphism to the Logical TheoryNet (\(TF\))¶

Epistemic Partitioning (\(\Delta = (B, A)\))¶

Layered Architecture Model (L1, L2, L3)¶

Layer 1: Empirical Manifold / Source Grounding (\(L1\))¶

Layer 2: Deterministic Ontology / Knowledge Base (\(L2\))¶

Layer 3: Abstract Argumentation Framework (\(L3\))¶

Argumentation Framework & Gradual Semantics¶

Super-Role Inclusion (DL-LiteR)¶

Continuous State Computation & Gradual Semantics¶

Dung Abstract Argumentation Mapping¶

Zuordnungsgesetze (Mapping Laws) & Epistemic Grounding¶

Epistemic Justification Paths (\(J (at)\))¶

Structuralist Model-Theoretic Foundations¶

The Model-Theoretic Triad: \(M_p\), \(M\), and \(M_{pp}\)¶

Uniform Space \((M_p, \mathcal{U})\) & Admissible Blurs (\(\mathcal{A}\))¶

Tenability and Adequacy¶

Temporal Evolution & Uncertainty¶

Temporal Graph Evolution¶

Probabilistic Semantics¶

Computational Tractability & Search Heuristics¶

Quality & Validation Metrics¶

Formal Graph Schema (TheoryNet)¶

Mathematical Foundations & Graph Representation¶

Isomorphism to the Logical TheoryNet (\(TF\))¶

Epistemic Partitioning (\(\Delta = (B, A)\))¶

Layered Architecture Model (L1, L2, L3)¶

Layer 1: Empirical Manifold / Source Grounding (\(L1\))¶

Layer 2: Deterministic Ontology / Knowledge Base (\(L2\))¶

Layer 3: Abstract Argumentation Framework (\(L3\))¶

Argumentation Framework & Gradual Semantics¶

Super-Role Inclusion (DL-LiteR)¶

Continuous State Computation & Gradual Semantics¶

Dung Abstract Argumentation Mapping¶

Zuordnungsgesetze (Mapping Laws) & Epistemic Grounding¶

Epistemic Justification Paths (\(J (at)\))¶

Structuralist Model-Theoretic Foundations¶

The Model-Theoretic Triad: \(M_p\), \(M\), and \(M_{pp}\)¶

Uniform Space \((M_p, \mathcal{U})\) & Admissible Blurs (\(\mathcal{A}\))¶

Tenability and Adequacy¶

Temporal Evolution & Uncertainty¶

Temporal Graph Evolution¶

Probabilistic Semantics¶

Computational Tractability & Search Heuristics¶

Quality & Validation Metrics¶

Related Documentation¶