9.5-9.6, Persuasion in Practical Argument Using Value-based Argumentation Frameworks

Notes taken from ‘Persuasion in Practical Argument Using Value-based Argumentation Frameworks’ by Trevor J. M. Bench-Capon (2003)

5, Value-based argumentation framework

5.1, A value-based argumentation framework (VAF) is a 5-tuple: VAF = (AR, attacks, V, val, P) where AR is a finite set of arguments, attacks is an irreflexive binary relation on AR, V is a non-empty set of values, val is a function which maps from elements of AR to elements of V and P is a set of possible audiences…

… Audiences are individuated by their preferences between values… We can therefore see the elements of P as being names for the possible ordering on V.

5.2, An audience-specific value-based argumentation framework (AVAF) is a 5-tuple VAFa = (AR, attacks, V, val, Valprefa) where AR, attacks, V and val are as for VAF, a is an audience, and Valprefa is a preference relation (transitive, irreflexive and asymmetric)…

… (we) distinguish between one argument attacking another, and that attack succeeding, so that the attacked argument is defeated. We therefore define the notion of defeat for an audience:

5.3, An argument A defeatsa an argument B for audience a iff both attacks(A, b) and not valpref(val(B), val(A)).

… In practice we expect the number of values to be small relative to the number of arguments. Many disputes can be naturally modelled using only two values…

5.4-5.7, Modified definitions of acceptable arguments (acceptablea), conflict-free sets of arguments, admissible sets of arguments and preferred extensions (preferreda) for a particular audience in light of the definition of defeatsa.

Now for a given choice of value preferences valprefa we are able to construct an AF equivalent to the AVAF, by removing from attacks those attacks which fail because faced with a superior value…

6, Acceptance in value-based argument frameworks

We term arguments which are acceptable irrespective of choice of value preferences, i.e. acceptable to every audience, objectively acceptable. Arguments which are acceptable to some audiences are termed subjectively acceptable. Note also that sceptical acceptance in the framework considered as an AF is not only not sufficient for objective acceptance, but is also not necessary…

Note that objective acceptance of an attacked argument requires that the number of values be smaller than the number of arguments: otherwise it is always possible to prefer the value of the attacker, and that value to any of its attackers. A VAF is most useful when the number of values is small, since a single choice of preference between values is then able to determine whether a number of attacks succeed or fail.

6.1, Objective acceptance. Given a VAF, (AR, attacks, V, val, P), an argument A (of AR) is objectively acceptable iff for p (of P), A is in every preferredp.

6.2, Subjective acceptance. Given a VAF, (AR, attacks, V, val, P), an argument A (of AR) is subjectively acceptable iff for p (of P), A is in some preferredp.

An argument which is neither objectively acceptable (such as one attacked by an objectively acceptable argument with the same value) is said to be indefensible.

… While there is a natural requirement for even cycles in a standard AF (e.g. a two-cycle is an obvious way to deal with uncertain and incomplete information), and Dung argues strongly that an interpretation of an AF with an odd cycle is plausible, we believe that they should be avoided in VAFs. An odd cycle means that nothing can be believed: it is akin to a paradox, and paradoxes are best avoided. Even cycles represent dilemmas, and require that a choice between alternatives be made. While such dilemmas have their place in cases of uncertainty, we believe that they should be resolved before practical arguments giving rise to them are advanced. The presence of a single-valued cycle in a VAF is a sure indication that the reasoning which gives rise to it is flawed.

6.3, An argument chain in a VAF, C is a set of n arguments {a1… aN} such that (i) all arguments have the same value; (ii) a1 has no attacker in C; (iii) for all ai in C if i>1, then ai is attacked and the sole attacker of ai is ai-1.

… if a1 is accepted, then every odd argument of C is also accepted and every even argument of C is defeated…

6.4, Every AVAF with no single-valued cycles has a unique, nonempty preferred extension.

6.5, A line of argument for an argument arg, l, comprises a set of n argument chains {C1… Cn}, each relating to distinct values, such that arg is the last argument of C1 and the last argument of Cn attacks the first argument of Cn-1, and the first argument of Cn has no attacker with a value not already present in l.

6.6, For a VAF containing no cycles relating to a single value, in which every argument as at most one attacker, the status of any argument is determined by the parity of the chains in its line of argument.
(a) An argument is objectively acceptable iff it is odd numbered in C1 and the line of argument contains no subsequent odd chain.
(b) An argument is indefensible iff it is even numbered in C1 and the line of argument does not contain a subsequent odd chain.
(c) An argument is subjectively acceptable iff the line of argument contains an odd chain, Cn, where n > 1.

6.7, The preferred extension of a cycle with only two values comprises:
(i) the odd numbered arguments of all chains preceded by an even chain;
(ii) the odd numbered arguments of chains with the preferred value;
(iii) the even numbered arguments of all other chains.
Note that those included under (i) are objectively acceptable and those included under (ii) and (iii) are subjectively acceptable. The even numbered arguments of a chain preceded by an even chain are indefensible.