The ASPIC⁺ framework for structured argumentation: a tutorial

Abstract

This article gives a tutorial introduction to the ASPIC⁺ framework for structured argumentation. The philosophical and conceptual underpinnings of ASPIC⁺ are discussed, the main definitions are illustrated with examples and several ways are discussed to instantiate the framework and to reconstruct other approaches as special cases of the framework. The ASPIC⁺ framework is based on two ideas: the first is that conflicts between arguments are often resolved with explicit preferences, and the second is that arguments are built with two kinds of inference rules: strict, or deductive rules, whose premises guarantee their conclusion, and defeasible rules, whose premises only create a presumption in favour of their conclusion. Accordingly, arguments can in ASPIC⁺ be attacked in three ways: on their uncertain premises, or on their defeasible inferences, or on the conclusions of their defeasible inferences. ASPIC⁺ is not a system but a framework for specifying systems. A main objective of the study of the ASPIC⁺ framework is to identify conditions under which instantiations of the framework satisfy logical consistency and closure properties.

Notes

1. n is a partial function since you may want to enforce that some defeasible inference steps cannot be attacked.

2. In this paper we use ⊃ to denote the material implication connective of classical logic.

3. In our further examples we will often leave the logical language and the n function implicit, trusting that they will be obvious.

4. Note that in Modgil & Prakken (2013) we motivate the use of the ASPIC⁺ attack relation to define conflict-free sets, and then only use the ASPIC⁺ defeat relation to determine the acceptability of arguments. It turns out that under certain conditions, this way of evaluating the status of arguments is equivalent to Definition 3.1’s use of the defeat relation for both determining conflict freeness and acceptability of arguments.

5. Notice that it suffices to restrict to finite sets since ASPIC⁺ arguments are assumed to be finite (in Definition 3.14) and so their sets of ordinary premises/defeasible rules must be finite.

6. In all examples below, sets that are not specified are assumed to be empty.

7. In the examples that follow we may use terms of the form s_i, d_i or f_i, to identify strict or defeasible inference rules or items from the knowledge base. We will assume that the d_i names are those assigned by the n function of Definition 3.2; sometimes we will attach these names to the ⇒ symbol. Note that the s_i and f_i names have no formal meaning and are for ease of reference only.

8. Although antecedents of rules formally are sequences of formulas, we will sometimes abuse notation and write them as sets.

9. As explained above, this strictly speaking is not a rule but a scheme or rules, with meta variables ranging over .

10. See Chapter 4 of Caminada (2004) for a very readable overview of the discussion.

11. In fact, given that the ASPIC⁺ arguments are restricted to those with consistent premises, satisfaction of the postulates also requires that if for some set of premises S is minimally (under set inclusion) indirectly inconsistent (see Definition 3.4), then , . Modgil and Prakken (2013)show that this property is also satisfied for Tarskian-based ASPIC⁺ ATs.

12. One way to argue why classical simulations may give counter-intuitive results is to recall that a number of researchers provide statistical semantics for defeasible inference rules. These semantics regard a defeasible rule of the form as a qualitative approximation of the statement that the conditional probability of Q, given P, is high. The laws of probability theory then tell us that this does not entail that the conditional probability of ≠g P, given ≠g Q, is high. The problem with the classical-logic approach is then that it conflates this distinction by turning the conditional probability of Q given P into the unconditional probability of P ⊃ Q, which then has to be equal to the unconditional probability of .