Respecting Relevance in Belief Change

I recently read a short and interesting article; Respecting Relevance in Belief Change, which is concerned with investigating the extent to which the formal operations of AGM belief change respect criteria of relevance.

Rohit Parikh proposed a criterion for relevance in belief change, which AGM is not stringent enough to meet. An example given is as follows:

Let p,q be two distinct elementary letters, and put K = Cn(p, q). Then there is an AGM maxichoice contraction that puts K \div p [the contraction of K by p ] to be Cn(p \leftrightarrow q), thus eliminating not only p but also q from K.

Subsequent expansion of Cn(p \leftrightarrow q) with ~p or ~q would then result in K = Cn(~p, ~q).

Whilst it is clear to see how some sense of relevance is violated here, I think that such results can be legitimate. Take a Grove sphere modelling of the above example. The possible world where both p and q are true is in the centre. The closest outer shell contains the possible world where both p and q are false. The second outer shell contains the two possible worlds where one is true and one is false. So the ordering is:

world p q
w1 T T
w2 T F
w3 F T
w4 F F
Ordering of worlds

  1. w1
  2. w4
  3. w2, w3

This ordering can be motivated in a very reasonable and natural way. Say p stands for ‘Tom went to the party’ and q stands for ‘Harry went to the party’.

A third person, Dick, believes that both Tom and Harry went to the party. He also believes that it is almost certain that Tom and Harry only go to parties if both of them are going. So his second most plausible possible world is the one in which neither go to the party. Therefore, if he is told that Tom did not go to the party, as well as contracting p, q will also be contracted.

Leave a Reply

Your email address will not be published. Required fields are marked *