Discussion Graph Semantics of First-Order Logic with Equality for Reasoning about Discussion and Argumentation

TL;DR

This paper addresses the lack of a formal framework for reasoning about general discussion graphs by introducing a discussion-graph semantics for first-order logic with equality. It extends Dung-style argumentation to account for node equivalence via an equivalence-equipped model and defines a broad family of extensions under simple and wide conflict-freeness/defence with closures. The authors prove that all generalised extensions and their acceptability semantics are first-order characterisable, and that propositional characterisation suffices for Dung's extensions. These results enable robust, logic-based reasoning about annotated discussion graphs and provide a solid foundation for applying existing FO tooling to argumentative reasoning, while preserving compatibility with established Dung-formalisms via equivalence handling and FO expressiveness.

Abstract

We make three contributions. First, we formulate a discussion-graph semantics for first-order logic with equality, enabling reasoning about discussion and argumentation in AI more generally than before. This addresses the current lack of a formal reasoning framework capable of handling diverse discussion and argumentation models. Second, we generalise Dung's notion of extensions to cases where two or more graph nodes in an argumentation framework are equivalent. Third, we connect these two contributions by showing that the generalised extensions are first-order characterisable within the proposed discussion-graph semantics. Propositional characterisability of all Dung's extensions is an immediate consequence. We furthermore show that the set of all generalised extensions (acceptability semantics), too, are first-order characterisable. Propositional characterisability of all Dung's acceptability semantics is an immediate consequence.

Figures (7)

• Figure 1: Top: An example of a Toulmin's model as an 'annotated graph' comprising a graph and annotations on nodes and edges. Bottom left: The graph part of the annotated graph. Bottom right: Assignment of annotations, partially shown for two nodes $txt_1$ and $txt_2$ and one edge $(txt_1, txt_2)$. $backing$ is assigned to $txt_1$, $warrant$ is assigned to $txt_2$, and no annotation is assigned to $(txt_1, txt_2)$.
• Figure 2: An example of Dung's model as an annotated graph.
• Figure 3: Another example of annotated graph.
• Figure 4: Left: An example of object-level annotated graph. $u_i$ ($1 \leq i \leq 5$) is a member of $ObjStmts$ and $\alpha_j$ ($1 \leq j \leq 6$) is a member of $Annos$. Right: An example of skeleton annotated graph with 3 placeholders.
• Figure 5: Left: a degree-2 skeleton annotated graph. Right: an object-level annotated graph.

Theorems & Definitions (58)

• definition 1: Object-level annotated graphs
• definition 2: Order on object-level annotated graphs
• proposition 1
• proof
• definition 3: Skeleton annotated graphs
• definition 4: Instantiability
• definition 5: Typed discussion graphs
• definition 6: Discussion graph structures
• definition 7: Satisfaction
• definition 8: Simple and wide admissibilities