The Doctrinal Paradox and Majority Rule

02 Apr 2014

Consider the following hypothetical…

Three judges need to decide on a particular point of law:

  1. the defendant intended to kill the victim
  2. the defendant did in fact kill the victim
  3. the defendant is guilty of murder (only if 1 & 2 are true)

Judge A agrees with 1, 2, and therefore 3;
Judge B agrees with 1 but not 2, and therefore not 3; while
Judge C agrees with 2 but not 1, and therefore not 3.

A paradox arises when we see that there is majority agreement for both 1 and 2 but not 3. Individual reasoning (1 & 2) -> 3 is consistent but the majority reasoning is not. If we collectively reason by aggregating premises first, then we often reach the opposite outcome than if we simply reason individually and aggregate conclusions. If the judges reason step by step - the majority agrees they defendant had intent, and the majority agrees he did kill the victim, then we must find him guilty. But if each judge reasons individually, and we then take the majority view on the conclusion, we get the opposite result.

This is known as the Doctrinal Paradox or Discursive Dilemma and arises whenever there are multi-stage collective decision procedures. Its a paradox that affects any collective inference process that includes aggregation - voting being just one example. On the one hand, it seems pretty straight forward - the intersection of any two non-identical sets is always smaller than their union.

But on the other, there is a deeper problem lurking here. Unless it is a tautology or axiom, any premise in an argument must be the conclusion of another set of premises. So the premise the defendant did in fact kill the victim can only be justified by the following type of argument:

  1. the victim is in fact dead
  2. shooting was the cause of death
  3. a witness saw the defendant shoot the victim
  4. the defendant did in fact kill the victim

Here, 1, 2, & 3 are all necessary to infer 4. If there is any disagreement on these sub-premises, then the discursive dilemma can apply to this premise (the original premise 1). And again, these sub-premises need to be justified by their own arguments, so we have paradoxes all the way down. So depending on what collective inference procedure we use - premise aggregation vs conclusion aggregation - majority decisions (ie voting) can endorse opposite positions even though individual position remain unchanged.

In this regard, the discursive dilemma is analogous to Simpsons’s Paradox, in that the result of our final inference depends on the perspective or level of granularity of the desiderata. And just like Simpson’s Paradox, there are no solid arguments as to which perspective or procedure we should use.

This poses a significant challenge to certain consequentialist accounts of democratic authority. If we should obey the outcomes of a majority voting procedure - ie the outcomes are morally binding - then it seems necessary that those outcomes are not arbitrary. But if they outcomes depend purely on what process is used - ie premise vs conclusion aggregation - and there are no good justifications for one over the other, then that’s exactly what the outcomes seem to be - arbitrary.