In the past I have mainly thought of the logic of belief (if not connected with probability) as one of the family of normal modal logics. Now I realize that there is a problem which that cannot accommodate, while it seems that neighborhood semantics can.
Think of the agent who believes, for each natural number n, that there are at least n stars, but also believes that the number of stars is finite. Such examples take the simple form: something is F, 1 is not F, 2 is not F, ….
We get a geometric example with intervals: I dropped a point-particle on the interval [0,1] on the real line. My first belief is that it fell in the half-open interval [1/2,1). My other relevant beliefs are that it fell in each of the intervals (1/2, 1], (1/4, 1], …, (1/2^n,1]…. My first belief has a non-empty intersection with each of the other beliefs. But the intersection of those other beliefs is [1], which is excluded by my first belief.
From any finite subset of such a family of beliefs it is not possible to deduce a contradiction but the family as a whole is not satisfiable. Goedel called this omega-inconsistency.
These examples are, to be reader friendly, generated by simple recipes, and hence amenable to an argument by mathematical induction, leading to a straight contradiction. There are examples not of that sort, I just can’t write them down. In any case, the agent may not have mastered mathematical induction.
The Problem. In the normal modal logic approach, the agent in world w believes that A if and only if A is true in all the worlds w’ which bear a certain relation R to w. If propositions are sets of worlds, and [A] is the proposition that A expresses, with B the ‘the agent believes that’ connective, this amounts to:
BA = {w’: wRw’} ⊆ [A].
But in the case of the omega-inconsistent believer {w’: wRw’} would then have to be part of every proposition in his set of beliefs. And there is no world in which all of those are true. Thus, that case, {w’: wRw’}is empty. But that is no different from a believer who believes that A & ~A.
So there is, in normal modal semantics, no way to distinguish the omega-inconsistent believer from the believer who believes a simple self-contradiction.
The Solution. Here I will rely on my previous post, about logic of belief and neighborhood semantics.
Given simply that world w has neighborhood N(w) and p is believed in w iff p is a member of w, that distinction between ‘ordinary’ and omega-inconsistency can be respected. For N(w) is a filter on the algebra of propositions, merely closed under finite intersections and superset formation.
So suppose the agent in w believes p = (something is F), q(1) = (1 is not F), q(2) = (2 is not F), …. and those propositions generate filter N(w). In that case ~p is not in N(w), so ~B~p is true in w. At the same time, there is no world in which all of N(w) is true, so the agent’s beliefs, taken altogether, imply all propositions, including B~p.
This is an adequate representation of omega-inconsistent belief, with the empty set not in N(w). That shows that condition
(cd) Λ is not in N(w)
must not be read as ‘N(w) is consistent’ but only as ‘Each finite subset of N(w) is consistent’.
The Upshot
What this means is that you can be entirely wrong about what the world is like, with ideas that are not realistic under any possible conditions, and still live a useful, productive, and happy life. And your family and friends might never find out.
Bas van Fraassen's philosophy blog 