# Discursive dilemma

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Judgment aggregation redirects here.
It is one thing to ask a group to vote on each part of a plan, and another thing to vote on the whole plan itself (because people may have different worries, and a simple vote does not capture those differences)

Discursive dilemma or doctrinal paradox is a paradox in social choice theory. The paradox is that aggregating judgments with majority voting can result in self-contradictory judgments.

Consider a community voting on a plan to fix a road. The community may vote 'Yes' - the roads are important, 'Yes' - the weather is good right now for road repair, and 'Yes' - we have the money to make the repairs. Based on that information, the plan seems to be to fix the roads. And yet, if the community is asked to vote on "Are all the conditions for road repair present?", the community could actually vote 'No', and the plan is now not to fix the roads. This paradox emerges from the complexity of people's opinions on the matter.

Philosopher Philip Pettit believes the discursive dilemma makes it impossible to make simple statements about the beliefs of a collective.

## Overview

Princeton philosopher Philip Pettit says there are hidden challenges of describing the group as though it were a single individual - a metaphorical agent - the way the law sometimes talks about corporations. It is a mistake, he says, to think things can be that simple. In reality, it can be quite difficult to construct a model of the "group mind" by merely asking for a majority opinion. This is because contradictory conceptions of a group can emerge depending on the type of questioning that is chosen.[1]

P Q Do they grant the relation:
C = P & Q
C
Judge 1 yes yes yes yes
Judge 2 no yes yes no
Judge 3 yes no yes no
Majority decision yes yes yes no

To see how, imagine that a three-member court must decide whether someone is liable for a breach of contract. For example, a lawn caretaker is accused of violating a contract not to mow over the land-owner's roses. The judges have to decide which of the following propositions are true:

• P: the defendant did a certain action (i.e. did the caretaker mow over the roses?);
• Q: the defendant had a contractual obligation not to do that action (i.e. was there a contract not to mow over the roses?);
• C: the defendant is liable.

Additionally, all judges accept the proposition $C \equiv P \and Q$. In other words, the judges agree that a defendant should be liable only if P and Q are both true.

Each judge could make consistent (non-contradictory) judgements, and the paradox will still emerge. Most judges could think P is true, and most judges could think Q is true. In this example, that means they would vote that the caretaker probably mowed over the roses, and that the contract did indeed forbid that action. This suggests the caretaker is liable. In contrast, most judges may think that P and Q are not both true. In this example, that means most judges conclude the caretaker not liable. The table above illustrates how majority decisions can contradict (because the judges vote in favor of the premises, and yet reject the conclusion).

This dilemma results because an actual decision-making procedure might be premise-based or conclusion-based. In a premise-based procedure, the judges decide by voting whether the conditions for liability are met. In a conclusion-based procedure, the judges decide directly whether the defendant should be liable. In this formulation, the paradox is that the two procedures don't necessarily lead to the same result; the two procedures can even lead to opposite results.

The discursive dilemma can be seen as a generalization of the Condorcet paradox, as a preference set is just a special kind of proposition set.[clarification needed] Just as the Condorcet paradox can be generalized to Arrow's theorem, List and Pettit argue that the discursive dilemma can be generalized to the List-Pettit theorem, which states that the inconsistency remains for any aggregation method which meets a few natural conditions.

Pettit believes that the lesson of this paradox is that there is no simple way to aggregate individual opinions into a single, coherent "group entity". These ideas are relevant to Sociology, which endeavors to understand and predict group behaviour. Petitt warns that we need to understand groups because they can be very powerful, can effect greater change, and yet the group as a whole may not have a strong conscience (see Diffusion of responsibility). He says we sometimes fail to hold groups (e.g. corporations) responsible because of the difficulties described above, and insists that groups should have limited rights, and various obligations and checks on power.[1]