Ways Things Can't Be

Possible worlds semantics has been very useful in modelling not only the intensionality of necessity and possibility, future and past. It has also found its place in modelling the intentionality of propositional attitudes like belief and knowledge. There is something fruitful in analysing a belief as a set of possible worlds. The belief is the set of possible worlds in which the belief is true. The belief is true i the actual world is in the corresponding set of propositions. The possible worlds in the set corresponding to the belief represent how the agent perceives the world to be. If the belief is false, then the world isn't how the agent sees the world to be, and so the actual world isn't in the set of worlds corresponding to the belief [4, 9]. The same can be said of whole belief states just as much as it can be said of individual beliefs. My belief state is the set of worlds consistent with what I believe. This view has been very fruitful, not least because the set-theoretic structure of sets of possible worlds correspnods nicely with the logical structure of entailment relations among propositions and the behaviour of propositional connectives like conjunction, disjunction and negation. However, the story does not deal well with inconsistent belief. Inconsistent beliefs are true in no possible worlds, so they are each modelled by the same set of worlds | the empty set. My beliefs are often inconsistent, and so are those of many of you. For example, people have laboured under the misconception that there are integers x; y; z and n > 2 such that xn + yn = zn. As we now know, this is impossible. Frege thought for years that na ve set theory was consistent. Russell showed us that this theory is inconsistent. David Lewis gives a more homely example