On judgment aggregation in abstract argumentation

Abstract

Judgment aggregation is a field in which individuals are required to vote for or against a certain decision (the conclusion) while providing reasons for their choice. The reasons and the conclusion are logically connected propositions. The problem is how a collective judgment on logically interconnected propositions can be defined from individual judgments on the same propositions. It turns out that, despite the fact that the individuals are logically consistent, the aggregation of their judgments may lead to an inconsistent group outcome, where the reasons do not support the conclusion. However, in this paper we claim that collective irrationality should not be the only worry of judgment aggregation. For example, judgment aggregation would not reject a consistent combination of reasons and conclusion that no member voted for. In our view this may not be a desirable solution. This motivates our research about when a social outcome is 'compatible' with the individuals' judgments. The key notion that we want to capture is that any individual member has to be able to defend the collective decision. This is guaranteed when the group outcome is compatible with its members views. Judgment aggregation problems are usually studied using classical propositional logic. However, for our analysis we use an argumentation approach to judgment aggregation problems. Indeed the question of how individual evaluations can be combined into a collective one can also be addressed in abstract argumentation. We introduce three aggregation operators that satisfy the condition above, and we offer two definitions of compatibility. Not only does our proposal satisfy a good number of standard judgment aggregation postulates, but it also avoids the problem of individual members of a group having to become committed to a group judgment that is in conflict with their own individual positions.

Probabilistic Argumentation: An Equational Approach

“…Given S, R and an arbitrary desired assignment E of elements that are in (and consequently also determining elements that are out) for S, this E may not be legitimate in taking into account R, so we need to modify it to get the best proper extension E nearest to E (cf. [2,6]). …”

“…Then P λ (x) gets contributions from both (1) and (2). The only option is that then λ(x) = in, and so all attackers of y i of x are out, so α 0 ¬y i and so P λ ( i ¬y i ) = 1, because it gets contributions from both (1) and (2). (v) Assume P λ (x) = 0.…”

“…In this 3-valued semantics, a valuation would satisfy x ↔ ¬x if and only if it gives the value 1 2 to x. 2 If we consider the equational approach [5], then we can write…”

“…(1) x = 1 means that x = in (all attackers y i = out) (2) x = 1 2 means that x = und (no attacker y i = in and at least (3) one attacker y j = und)…”

Senses of ‘argument’ in instantiated argumentation frameworks

“…Such a system determines the acceptability status of propositions generally in three steps as in Figure 1 (from Caminada & Wu, 2011). Starting with an inconsistent KB comprised of facts and rules, we construct arguments (nodes) and attacks (arcs) from this KB, resulting in an AF (Step 1); evaluate the AF according to a variety of semantics, resulting in extensions (sets) of arguments (Step 2); and extract the conclusions from the arguments, resulting in extensions of conclusions (Step 3).…”