T-AIF: Trichotomic Argumentation Representation

• T-AIF is a formal framework that encapsulates logos, ethos, and pathos, representing dialogue as weighted graphs with distinct nodes for content, inference, and actors.
• It models inferential structure, trust networks, and actor commitment using explicit weighted edges to quantify support, attack, and endorsement relations.
• The framework advances computational argumentation by enabling nuanced multi-agent debate analysis and integration with existing argument mining systems.

The Trichotomic Argumentation Interchange Format (T-AIF) is a formal framework designed to capture all three classical Aristotelian aspects of argumentation—Logos, Ethos, and Pathos—within a single, structured representation. Building upon previous frameworks such as the Argument Interchange Format (AIF), its extension AIF+, and the Social Argument Interchange Format (S-AIF), T-AIF introduces explicit mechanisms to model not only inferential structure but also speaker credibility and commitment, thereby enabling nuanced characterization of multi-agent reasoning patterns and dialogue (Göttlinger et al., 2018).

1. Formal Structure of T-AIF

T-AIF represents argumentation as a labeled, weighted directed graph $G = (V, E, \tau, \omega)$, where the node set $V$ is partitioned into three disjoint types:

• $V_L$: locution-nodes (L-nodes), each containing a string of dialogue text;
• $V_I$: illocution-nodes (I-nodes), each associated with a logical formula $\phi$ denoting propositional content;
• $V_A$: actor-nodes (E-nodes), each corresponding to a participant or agent.

Edges in $E \subseteq V \times V$ are typed by $\tau_E$ and, where applicable, annotated with a weight $\omega(e) \in [0,1]$:

• reply ($V_L \times V_L$): links replies within dialogues;
• force ($V_L \times V_I$): connects utterances to their illocutionary content;
• supp, att ($V_I \times V_I$): encode support and attack relations (Logos);
• trust ($V_A \times V_A$): weighted, representing ethos as trust between actors;
• commit ($V_A \times V_I$): weighted, representing pathos as the degree of commitment to a proposition.

The full schema is succinctly expressed as: $$G = (V_L, V_I, V_A, E_{reply}, E_{force}, E_{supp}, E_{att}, E_{trust}, E_{commit}, \omega)$$ with $$\omega: E_{trust} \cup E_{commit} \rightarrow [0,1]$$.

2. Representation of Logos, Ethos, and Pathos

Logos: Inference and Conflict

T-AIF inherits from AIF+ a two-layered, scheme-based framework for encoding inferential and conflict relationships. Application instances of argumentation schemes (e.g., Walton schemes) are formalized as binary edges between I-nodes:

• supp (support) edges are annotated by a scheme identifier $s$, encoding premise-conclusion links;
• att (attack) edges model conflicts, also denoted by scheme type.

Each scheme $s$ in sets $S_{sup}$ or $S_{att}$ is defined by:

• premise arity $\alpha(s)$
• exception arity $\epsilon(s)$
• instance set $I_s$ of tuples $$(p_1,\ldots,p_{\alpha(s)}, e_1,\ldots,e_{\epsilon(s)},q)$$

This representation enables the compact encoding of classical argument structures and relevant extensions such as hyperedges collapsed for clarity.

Ethos: Weighted Trust Network

Ethos is rendered explicit via a weighted trust network over actor-nodes. For each pair $(a, b) \in V_A \times V_A$, a directed edge with weight $\omega_t(a,b) \in [0,1]$ expresses the degree to which actor $a$ attributes credibility to actor $b$: $\omega_t: V_A \times V_A \rightarrow [0,1]$ Indirect trust is computed via min-max fuzzy composition, i.e. for actors $x, y$: $$T^{(1)}(x, y) = \max_{x \rightarrow_t z} \min(\omega_t(x, z), \omega_t(z, y))$$ and extended to paths up to length $n$: $$T^{(n)}(x, y) = \max_{x = z_0 \rightarrow_t z_1 \rightarrow_t \cdots \rightarrow_t z_n = y} \min_{i = 0,\dots,n-1} \omega_t(z_i, z_{i+1})$$ The effective trust value $T^*(x, y)$ is the supremum over all finite-length paths.

Pathos: Weighted Commitment Edges

Pathos is captured by per-actor weighted commitment to propositions. That is, $(a, \phi)$ is associated with $\omega_c(a, \phi) \in [0,1]$, interpreted as the degree of endorsement or advocacy: $\omega_c: V_A \times V_I \rightarrow [0,1]$ A weight of 1 denotes full endorsement, 0 indicates minimal or perfunctory mention.

3. Fuzzy Labelling Semantics

T-AIF extends Dung's extension-based semantics into the fuzzy, triadic domain, enabling graded assessment of propositional acceptability.

A fuzzy labelling is a function $\ell: V_I \rightarrow [0,1]$ assigning a real-valued degree of acceptance to each proposition. Support and attack are recursively defined using minima/maxima over the structure of the argumentation schemes:

• Support:

$\Sup_\ell(q) = \max_{s \in S_{sup},\, (p_1,\dots,e_{j},q)\in I_s} \min\left(\min_{i}\ell(p_i), \min_{j}(1-\ell(e_j))\right)$

• Attack:

$\Att_\ell(q) = \max_{s \in S_{att},\, (p_1,\dots,e_j,q)\in I_s} \min\left(\min_{i}\ell(p_i), \min_{j}(1-\ell(e_j))\right)$

Defence, consistency (conflict-freeness), admissibility, completeness, preferredness, stability, and groundedness are all formulated as fuzzy analogues of their classical counterparts, each taking values in $[0,1]$. For instance:

• Admissibility:

$\Al(\ell) = \Co(\ell) \land \min_{\phi \in V_I} \min(\ell(\phi), \Def_\ell(\phi))$

• Groundedness characterizes least complete extensions.

These semantics enable nuanced, trust- and commitment-sensitive evaluations of arguments, reducing to classical binary labellings when all weights are $0$ or $1$.

4. Computational Properties and Translation

The structure of T-AIF is compatible with established computational frameworks such as ASPIC+. Each Rule Application (RA) and Conflict Application (CA) is translatable into (defeasible) rules and contraries, making existing solvers for such formats applicable. The computational complexity of standard reasoning tasks remains analogous to that for classical bipolar argumentation frameworks: credulous acceptance under preferred semantics is NP-complete, skeptical acceptance is coNP-complete, while grounded extension computation is achievable in polynomial time by fixpoint iteration.

Trust and commitment information can be compiled into rule-priorities or actor-specific credibility values, enabling per-actor argumentation theories.

5. Illustrative Example

Consider a T-AIF instance with four actors $A, B, C, D$ and four propositions $p, q, r, s$. The following elements are configured:

• Actor commitments: $\omega_c(A,p)=0.9$, $\omega_c(B,q)=0.8$, $\omega_c(C,r)=0.5$, $\omega_c(D,s)=0.6$
• Trust weights:
  • $\omega_t(A,B)=0.7$, $\omega_t(A,C)=0.4$, $\omega_t(B,C)=0.8$, $\omega_t(C,D)=0.9$, $\omega_t(D,A)=0.3$
• Arguments: $p$ supports $r$, $q$ attacks $r$, $r$ supports $s$

Indirect trust is maximized via the optimal path (e.g., $T^*(A,C) = 0.7$ via $B$, since $\min(0.7,0.8)=0.7 > \omega_t(A,C)=0.4$). Initial fuzzy labellings for each proposition incorporate both direct commitment and aggregated trust: $$\ell_0(p) \approx 0.9 \times \frac{1+0.7+0.7+0.3}{4} = 0.6075$$ After performing grounded extension computation, attack and support interactions yield $$\ell_1(r) = \max(\ell_0(p),0) - \max \ell_0(q) \approx 0$$, and $$\ell_1(s) = \ell_1(r) \times \text{support} \approx 0$$. Despite some commitment to $r$, a stronger attack defeats it; thus, $r$ and $s$ are rejected.

6. Distinctions from AIF, AIF+, and S-AIF

T-AIF advances over AIF, AIF+, and S-AIF in several crucial respects:

• Ethos (trust) is modeled as an explicit, weighted, and aggregatable network, rather than as unsubstantial metadata.
• Pathos (commitment) is formally tracked per speaker and proposition, supporting dynamic modeling of changing speaker commitments.
• Logos retains the full structure of AIF+ schemes in a streamlined graph formalism, integrating support and attack logic directly into the hypergraph.

These enhancements facilitate semantic actor profiling, trust-adjusted acceptability, and trichotomic analyses across both human debate mapping and automated argument mining (Göttlinger et al., 2018).

7. Significance and Applications

The explicit trichotomic modeling in T-AIF provides a foundation for analyzing argumentation in multi-agent dialogues, enabling:

• Rich profiling of participant reasoning patterns through trust and commitment.
• Integration with argument mining pipelines and dialogue systems sensitive to both content and source credibility.
• Quantitative study of debate dynamics and retreat/recovery of commitments over time.

A plausible implication is that T-AIF supports a unified framework for the computational study of persuasive argumentation, aligning classical logical structure with socially situated, agent-sensitive factors.