Closure and Consistency In Logic-Associated Argumentation

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Properties like logical closure and consistency are important properties in any logical reasoning system. Caminada and Amgoud showed that not every logic-based argument system satisfies these relevant properties. But under conditions like closure under contraposition or transposition of the monotonic part of the underlying logic, ASPIC-like systems satisfy these properties. In contrast, the logical closure and consistency properties are not well-understood for other well-known and widely applied systems like logic programming or assumption based argumentation. Though conditions like closure under contraposition or transposition seem intuitive in ASPIC-like systems, they rule out many sensible ASPIC-like systems that satisfy both properties of closure and consistency.

We present a new condition referred to as the self-contradiction axiom that guarantees the consistency property in both ASPIC-like and assumption-based systems and is implied by both properties of closure under contraposition or transposition. We develop a logic-associated abstract argumentation framework, by associating abstract argumentation with abstract logics to represent the conclusions of arguments. We show that logic-associated abstract argumentation frameworks capture ASPIC-like systems (without preferences) and assumption-based argumentation. We present two simple and natural properties of compactness and cohesion in logic-associated abstract argumentation frameworks and show that they capture the logical closure and consistency properties. We demonstrate that in both assumption-based argumentation and ASPIC-like systems, cohesion follows naturally from the self-contradiction axiom. We further give a translation from ASPIC-like systems (without preferences) into equivalent assumption-based systems that keeps the self-contradiction axiom invariant.

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