sions to deal with conditionals, together with quantifiers, and other deductive machinery. There are indications of how we would like see things developed [5, 6] but here we rest content with the ^; _; ?; >; ? fragment of the logic.
In what follows we will be using this account of entailment to define belief sets, and we will show how the standard representations of belief revision fare in this new wider context, and that the new account of belief sets enable us to do things with belief sets that were heretofore impossible.
2 Contraction and Revision
From now on we assume that we have at hand a notion Cn of consequence which at least includes first degree entailment. Given that notion of consequence, we define a belief set to be a set K of formulae such that K = Cn(K). Note that any belief set includes >, and any other formula A where > ` A, but that belief sets need not include A_?A, and they can include B ^?B without including ?. The only belief set which includes ? is the trivial belief set K?, which is the set of all propositions in our language.
The simplest operation on belief sets is that of adding another proposition, and closing under logical consequence. We define K+A , the result of adding A to K to be Cn(K [ fAg).
The contraction and revision operations are more interesting. We will start with contraction, the operation of removing a proposition A from a belief set. G?ardenfors' eight original postulates for a contraction of a belief set K are as follows.
(K?1) K?A is closed. (K?2) K?A ? K. (K?3) If A 62 K then K?A = K. (K?4) If > 6` A then A 62 K?A . (K?5) If A 2 K then K ? (K?A )+A. (K?6) If A a ` B then K?A = K?B . (K?7) K?A K?B ? K?A^B. (K?8) If A 62 K?A^B then K?A^B ? K?A .
The most problematic of these eight requirements is the condition (K?5) of recovery. The idea behind recovery is that by removing A from K you only make a minimal change to K, So minimal that by adding A to the result, and closing under consequence you get all of K back. This postulate has been widely criticised, because many intuitive operations of contraction simply do not validate it. This concurs with our approach, because the representations considered in the next section will not
validate recovery. For our purposes, a `contraction operator' will be a function K? satisfiying the conditions K?f1; 2; 3; 4; 6; 7; 8g.
Given a contraction operator K?, we can define a revision operator K? in the usual way. To revise your belief set K by A, you first retract your original belief that ?A (if you have one) and then add A, closing under consequence. So, you can define K?A to be (K??A)+A. This is the Levi Identity. Then we would get a revision operator K? satisfying the following postulates.
(K?1) K?A is closed. (K?2) A 2 K?A. (K?3) K?A ? K+A . (K?4) If ?A 62 K then K+A ? K?A. (K?5) K?A = K? if and only if A ` ? (K?6) If A a ` B then K?A = K?B. (K?7) K?A^B ? (K?A)+B. (K?8) If ?B 62 K?A then (K?A)+B ? K?A^B.
We leave the verification of these postulates to the reader. The reasoning is not any more difficult than the classical case.
Traditionally, we can also define contraction functions from revision functions, by way of the Harper Identity: K?A = K K??A. However, in our non-classical environment, this definition does not always define a contraction function, for the following reason. We need not have A 62 K?A , because A might be both in K, and in K??A. The new theory K??A may be inconsistent about A. This does seem odd, because we would not expect K??A to be inconsistent about A, because we have asked it to revise with respect to ?A. It ought to remove A and then add ?A. However, it could well be that adding ?A might bring with it A. ?A might entail its own negation (as it would if it were of the form B ^ ?B). In that case, adding ?A would bring with it A, no matter how much we would like to avoid this. Classically, this means that ?A ` ?, and hence, > ` A, so A 2 K?A is no problem, since you cannot contract away theorems. However, in our context we need not have ?A ` ?. In other words, A could entail its own negation without A being trivialising. As a result, the Harper identity fails.
That is enough of what does not work in the nonclassical environment. There is a lot of good news about what does work. In this section we will see that each of the standard representation results,