# Argumentation Prerequisites

Under maintenance !!

*What is argumentation *

It’s a verbal, social and rational activity aimed at convincing a reasonable critic of the acceptability of a standpoint by putting forward a constellation of propositions justifying or refuting the proposition expressed in the standpoint.

Argumentation theory allows us to deal with incomplete, uncertain, inconsistent knowledge, conflicting opinions - typical of the real world. It is inherently suited to providing explanations and supports interactions between humans and machines.

*Approaches to argumentation *

As a dialogue mechanism.

By considering the structure of arguments and how it affects the attach relation between them.

From an abstract point of view.

**Abstract Argumentation**

** **

Disregards the internal structure of arguments and focusses instead on acceptability conditions that allow certain sets of arguments to co-exist in a rational manner.

*Abstract Argumentation framework *

Is a tuple <S,R> where S is a set of arguments and R ⊆ S x S is an attack relation. (a,b) —> argument ‘a’ attacks argument ‘b’.

**Argumentation semantics **

An argumentation framework <S,R> can be given semantics in different ways. The idea is to analyse what sets of arguments one can reasonably accept given the attack relation.

Via *extensions *( subsets of S with special properties )
Via *labels *( labelling functions on S with special properties )
Via *equations *( solutions to a system of equations describing the interactions in the argumentation framework <S,R>)

**Extension-based Semantics **

A subset of the set of arguments with special properties is defined: An extension is a set of arguments that are jointly “Acceptable” Arguments are justified according to their statuses in these extensions.

**Conflict-free sets **

A set T ⊆ S is conflict-free if and only if there are no arguments in T that attacks each other.

**Argument Defence **

A set T ⊆ S defends an argument x ∈ S if and only if for every y ∈ S such that y attacks x there is an element z∈T such that z attacks y.

**Admissible sets **

A set A ⊆ S is admissible if and only if A is conflict-free and A defends each argument that is a member of A.

**Complete Extensions **

A complete extension is an admissible set that includes all arguments it defends.

**Maximal Subset **

Take 𝐶 ⊆ 2𝑆, a maximal subset of 𝑆 in 𝐶 is a set 𝑇 ∈ 𝐶 such that no other set in 𝐶 strictly includes 𝑇. Formally, 𝑇 is a maximal subset of 𝑆 in 𝐶 if and only if ∄𝑇′ ∈ 𝐶 such that 𝑇 ⊊ 𝑇′. Take 𝑆 = {𝑎1, 𝑎2, 𝑎3} and let 𝐶 = {𝑎1 , 𝑎2 , 𝑎1, 𝑎2 , 𝑎2, 𝑎3}.

The maximal subsets of 𝐶 in 𝑆 are {𝑎1, 𝑎2} and {𝑎2, 𝑎3} .

**Minimal Subset **

Take 𝐶 ⊆ 2s, a minimal subset of 𝑆 in 𝐶 is a set 𝑇 ∈ 𝐶 such that no other set in 𝐶 is strictly included in 𝑇.
Formally, 𝑇 is a minimal subset of 𝑆 in 𝐶 if and only if ∄𝑇′ ∈ 𝐶 such that 𝑇′ ⊊ 𝑇.
*Example *

Take 𝑆 = {𝑎1, 𝑎2, 𝑎3} and let 𝐶 = {𝑎1 , 𝑎2 , 𝑎1, 𝑎2 , 𝑎2, 𝑎3} . The minimal subsets of 𝐶 in 𝑆 are {𝑎1} and {𝑎2} .

**Grounded Extension **

The Grounded extension is the minimal complete extension with respect to set inclusion. (ie, the minimal subset of the set of complete extensions ).

The grounded extension always exists and is unique. The grounded extension can be empty.

**Preferred semantics **

The preferred semantics tries to maximise the acceptance of arguments. Accept as much as you can defend. A Preferred extension is a complete extension that is maximal with respect to set inclusion (ie, maximal subset of the set of all complete extensions).

**Stable extension **

A stable extension of an argumentation framework <S,R> is a preferred extension E such that for all 𝑦 ∈ 𝑆 ∖ 𝐸 there exists 𝑥 ∈ 𝐸 such that (𝑥, 𝑦) ∈ 𝑅 (in other words, for every argument 𝑦 that isn’t part of 𝐸, there is an argument in 𝐸 that attacks 𝑦). Stable extensions do not always exist.

**Credulous Acceptance **

An argument is credulous accepted by an argumentation framework under a particular semantics if and only if it is part of at least on of the extensions generated by those semantics.

**Skeptical Acceptance **

An argument is skeptically accepted by an argumentation framework under a particular semantics if and only if it is part of all the extensions generated by those semantics.