ASPIC+

Origin. Henry Prakken (2010), generalizing the ASPIC framework of Amgoud et al.; developed with Sanjay Modgil (2013, 2018). A general framework for structured argumentation that builds Dung argumentation frameworks from an underlying logical language, rules, and preferences, with rationality guarantees.

Models. Arguments as structured, defeasible proofs. Rather than take arguments as abstract atoms (Dung), ASPIC+ constructs them from a knowledge base and inference rules, defines attacks from the structure (on conclusions, premises, or defeasible steps), resolves them with preferences, and then evaluates the resulting abstract framework by Dung semantics. It is a schema instantiated by many concrete logics.

Formalism.

Argumentation system: A language L with a contrariness relation (⁻), strict rules R_s (deductively valid) and defeasible rules R_d (presumptive), and a naming function for defeasible rules. Knowledge base K = K_n (axioms, certain) ∪ K_p (ordinary premises, fallible).

Arguments: Built by chaining rules from K: an argument has premises, sub-arguments, a conclusion, and its last/defeasible rules. Strict continuation of an argument if only R_s is used.

Attacks: Rebutting: on the conclusion of a defeasible inference (contrary conclusions). Undercutting: on the applicability of a defeasible rule (attacking its name). Undermining: on an ordinary premise in K_p.

Defeat via preferences: An ordering ≼ on arguments (from premise/rule preferences, e.g., last-link or weakest-link). Attack + preference ⇒ defeat: a rebut/undermine succeeds only if the attacker is not weaker; undercuts always defeat (preference-independent).

Evaluation: The arguments and defeats form a Dung abstract argumentation framework; apply grounded, preferred, stable, or complete semantics to get justified conclusions.

Symbols.

Symbol	Unicode	Meaning
R_s, R_d	—	strict, defeasible rules
K_n, K_p	—	axiom, ordinary premises
⁻	U+207B	contrariness / contradictory
≼	U+227C	preference order on arguments
⇒	U+21D2	defeasible inference (vs → strict)
A defeats B	—	successful attack under preference

Metatheory. ASPIC+ is designed to satisfy Caminada–Amgoud rationality postulates — closure under strict rules, direct and indirect consistency — provided the rule set is well-behaved (e.g., closed under transposition or with a suitable contrapositive treatment) and preferences are "reasonable." It generalizes many systems: assumption-based argumentation, classical-logic argumentation, and defeasible logics arise as instances. Its two-level design (structured construction + abstract Dung evaluation) cleanly separates argument formation from acceptability.

Applies to. Structured/defeasible reasoning and nonmonotonic inference. Legal and normative argumentation. Multi-agent dialogue and persuasion. Instantiating and comparing argumentation logics under one schema.

Limitations. Rationality guarantees depend on structural assumptions (transposition/closure, preference "reasonableness") that can be intricate to ensure. Multiple design choices (last-link vs weakest-link ordering, rebut on strict vs defeasible conclusions) yield different behaviors and ongoing debate. Computationally inherits the complexity of Dung semantics on top of argument construction. Modeling a domain requires careful engineering of the rule/preference base.