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 | Name | Meaning |
|---|---|---|---|
| ⁻ | U+207B | Superscript minus / Contrariness | Relates a formula to the formulas that conflict with it |
| ≼ | U+227C | Precedes or equal to / Preference | One argument is no stronger than another |
| ⇒ | U+21D2 | Rightwards double arrow / Defeasible rule | Presumptive inference, open to undercut |
| → | U+2192 | Rightwards arrow / Strict rule | Deductively valid inference |
| ∪ | U+222A | Union | Knowledge base is the axioms together with the ordinary premises |
| L | — | Language | Formulas of the underlying logic |
| R_s | — | Strict rules | Deductively valid rules; not themselves attackable |
| R_d | — | Defeasible rules | Presumptive rules, named so that they can be undercut |
| K | — | Knowledge base | Premises available for building arguments |
| K_n | — | Axiom premises | Certain premises; cannot be undermined |
| K_p | — | Ordinary premises | Fallible premises; open to undermining |
| rebut | — | Rebutting attack | On the conclusion of a defeasible inference |
| undercut | — | Undercutting attack | On the applicability of a named defeasible rule |
| undermine | — | Undermining attack | On an ordinary premise |
| defeat | — | Defeat | Attack that succeeds once preferences are applied |
Metatheory. Caminada and Amgoud (2007) set the target: an instantiated argumentation formalism should have its justified conclusions closed under the strict rules and both directly and indirectly consistent, and they showed that naive instantiations fail this, yielding inconsistent conclusion sets from consistent premises. ASPIC+ is engineered to the postulates rather than merely tested against them. Modgil and Prakken (2013) proved that closure and consistency hold under complete, grounded, preferred and stable semantics provided the argumentation system is well formed, the strict rules are closed under transposition (or the language under contraposition), the axiom premises K_n are consistent, and the argument ordering is reasonable in a sense they make precise — the standard last-link and weakest-link orderings qualify. Because the framework is a schema, its reductive claims are instantiation results: assumption-based argumentation is recovered by taking assumptions as ordinary premises with no defeasible rules, and classical-logic argumentation by taking the classically valid inferences as the strict rules.
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. The rationality guarantees are conditional rather than automatic: they hold only if the strict rules are closed under transposition, K_n is consistent and the ordering is reasonable, and dropping any of these yields instantiations whose justified conclusions are inconsistent. Closure under transposition is itself a strong demand, since it forces the contrapositive of every strict rule into the system, which misrepresents any rule meant to be read directionally or causally. Central design choices are left open: last-link and weakest-link orderings rank the same arguments differently, and whether rebut should be restricted to the conclusions of defeasible inferences remains disputed, the two variants validating different arguments on the same input. Preferences are an external input, with nothing in the theory saying where the ordering comes from. The constructed framework can be exponentially larger than the knowledge base and, with unbounded chaining, infinite.
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