Belief Revision and Rational Inference

The (extended) AGM postulates for belief revision seem to deal with the revision of a given theory K by an arbitrary formula, but not to constrain the revisions of two different theories by the same formula. A new postulate is proposed and compared with other similar postulates that have been proposed in the literature. The AGM revisions that satisfy this new postulate stand in one-to-one correspondence with the rational, consistency-preserving relations. This correspondence is described explicitly. Two viewpoints on iterative revisions are distinguished and discussed.

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1 Introduction

1.1 Belief Revision

Belief revision is the study of the way an agent revises or should revise its beliefs when acquiring new information. A popular framework for this study has been initiated by Alchourón, Makinson and Gärdenfors in  [1, 2, 3], in which a set of rationality postulates for theory revision were put forward. It assumes beliefs are sets of formulas closed under logical consequence (i.e., theories) and that new information is a formula. The present work fits squarely in this framework. An up-to-date description of this very active research area may be found in [10]. The basic assumptions underlying AGM’s viewpoint may be summarized as follows.

  • The agent holds beliefs: those beliefs constitute some logical theory. There is no additional structure to the agent’s theory. No beliefs are stronger than others.

  • The agent’s knowledge is changing but the world is not. Keller and Winslett [15] noticed that updating an agent’s knowledge about a changing world is a process that is altogether different from theory revision. Katsuno and Mendelzon [14] proposed different postulates for belief update.

  • When a new piece of information is presented to an agent that holds a theory with which is inconsistent, the agent will give precedence to the information , over the theory . The theory is considered as less important or less reliable than the new information with which it conflicts: the agent revises a weakly held theory with a piece of more reliable information. This policy is obviously not the right one in every situation, but AGM assume the agent must have some good reason to prefer to . As noticed just above, this reason is not that describes some change in a world the previous state of which was described by .

  • When revising a theory with some formula , the agent will try keep as much as possible of its previously held beliefs: . In particular, if is consistent with , the agent will keep all of in its new belief set. This assumption will be called the maximal retention assumption.

AGM did not describe any specific method to revise theories that would be the right

way of revising under the assumptions above. They thought that there is probably no unique best way of revising. Instead, they proposed a set of rationality postulates they claimed any reasonable method for belief revision should satisfy. The original postulates were extended in 

[11]. This work accepts, under the assumptions above, the extended AGM postulates. These postulates are meant to express formally the basic assumptions described above. These postulates will be presented now and their adequacy in expressing those basic assumptions will be discussed in section 1.3.

1.2 The AGM postulates

The AGM postulates for belief revision are listed below:  denotes logical consequence and, for an arbitrary theory (i.e., is a set of formulas closed under logical consequence), denotes the result of revising by . The inconsistent theory, i.e., the full language, is denoted by . It is a legal argument for . This paper could have been developed in the more demanding framework, where the first argument of a revision, , must be a consistent theory: the results are essentially the same.

For a justification of these postulates, in addition to the papers cited above, the reader may also consider [18, 7, 19, 12, 21]. Since the notion of a revision is considered by philosophers as not being primitive, but derived from that of theory contraction, the justification of the postulates below is generally given by justifying corresponding postulates for contractions and translating back and forth with the help of the Levi and Gardenförs identities. Not all specialists are in complete agreement concerning the philosophical underpinnings of theory revision, the status of the Levi and Gardenförs identities, or of the postulates below. The present work does not attack those deep questions directly, but presents technical results concerning the postulates that, in turn, help understand what are the processes that the postulates may reasonably be thought of as modelling and what are those they do not model satisfactorily. For any theory and formulas and :

The reader may notice that, in each of the postulates in which the revision symbol () appears more than once (, and ), the theory that appears on the left of the different occurrences of the revision symbol is the same. We take this as an indication that those postulates do not say much on the way the revised theory depends on . We shall come back to this point in section 1.3. We shall call a revision a mild revision, when and a severe revision, if .

The AGM postulates have been found to be intimately related to rational consequence relations (see [17]). The translation is precisely described by the following Theorem.

Theorem 1

If is a revision operation that satisfies the AGM postulates, , and is a theory, the relation defined by

(1)

is rational and consistency-preserving.

Proof: We just remind the reader that consistency preservation is the following property: if , then is a logical contradiction. The definition of all the other properties mentioned in this proof may be found in [16]. Suppose is a theory and satisfies the AGM postulates. The consequence relation satisfies Reflexivity by , Left Logical Equivalence by , Right Weakening and And by , Conditionalization (i.e., (S)) by , and Rational Monotonicity by . It is consistency-preserving by . Any such relation is rational.    

Notice that postulates and are not used in the proof of Theorem 1. This point will be discussed after we have proved Theorem 2.

One may expect the converse to Theorem 1 to hold: if  is an arbitrary, rational and consistency-preserving relation, then, there is a theory and a revision satisfying such that iff . This result holds. It is not difficult to present a direct proof at this stage. An indirect proof will be preferred and presented in Theorem 4.

1.3 Critique of the AGM postulates

The AGM postulates have been discussed in the literature, in particular in [13], [14] and [6]. Some authors reject some of the postulates, in particular , the same and others propose to extend the AGM postulates by additional postulates. The main line of critique of the AGM postulates seems to be that they do not constrain the revisions enough: some revisions allowed by the postulates do not seem reasonable. There are three points, in particular, on which it seems that the postulates do not say enough, or, more precisely, one could feel that additional acceptable postulates may be proposed.

  • The postulates do not seem to enforce the principle of maximal retention when the formula is inconsistent with the theory .

  • They do not seem to impose enough constraints on the revision of two different theories by the same formula, or by related formulas.

  • They do not seem to enforce enough constraints on iterated revisions, e.g., concerning the relation between , , and .

The proposals to drop some of the AGM postulates seem to stem from the desire to add some other postulates with which they are inconsistent, more than from a direct critique of the postulates in question. The first point raised above has been the reason for the interest in maxichoice contractions manifested by AGM, but no suitable additional postulate has been proposed to deal with it. The second point has been very aptly discussed in [17]. The authors write: “Revision is an operation of two arguments, forming out of theory and proposition . On the other hand, nonmonotonic inference conceived as an operation defined as is a function of only one argument . For this reason the logic of theory change is potentially more general than the logic of nonmonotonic inference, in that it allows the possibility of variation in the other argument . Potentially, because this possibility has hardly been explored. The postulates for revision presented in [9] all concern the case where the theory is held constant”. We shall show, in the sequel, that the third point is intimately related to the second one. This paper will present an additional postulate constraining the way one may revise different theories by the same formula. Its main result is that this additional postulate precisely reduces the generality of revisions in a way that makes revision isomorphic to nonmonotonic inference. It will be shown that some of the additional postulates proposed in the literature are incompatible with the AGM axioms and that those that are not, are implied by our postulate.

1.4 Plan of this paper

In section 2, the additional postulates that were previously proposed in the literature, by Katsuno and Mendelzon on one hand, and by Darwiche and Pearl on the other hand, are described and discussed. In section 3 the minimal influence postulate, , is presented and justified on intuitive grounds. Some first consequences of that concern iterated revisions are then proven. In section 3.3 and 3.4, its relations with the postulates of sections 2.1 and 2.2, respectively, are analyzed. In section 4.1, we prove the main result of this paper: revisions that satisfy to stand in one-to-one correspondence with rational, consistency-preserving relations. In section 4.2 a conservative extension result, a model-theoretic description of revision, and the converse to Theorem 1 are proven. We discuss, then, in section 4.3, the meaning of this result for the ontology of theory revision. Section 5 discusses the existence of two conflicting views on iterated revisions, concludes and describes some open questions.

2 Additional postulates previously proposed

2.1 The Katsuno and Mendelzon postulate

In [14], Katsuno and Mendelzon considered the following postulate.

Reasonable as it seems, this postulate is nevertheless inconsistent with the AGM postulates. This is probably the reason why Katsuno and Mendelzon dropped or weakened some of the AGM postulates in the final version of their paper. The postulate that will be proposed in this paper is closely related to , therefore we shall analyze it in some detail. Our first remark is that any revision that satisfies the AGM postulates also satisfies a special case (the mild case) of .

Proposition 1

If satisfies and , then, for any such that , .

The proof is obvious. Our second remark is that may be broken into two halves.

Proposition 2

The postulate is equivalent to the following two properties:

The proof is obvious. Property has been named the postulate of Addition Monotonicity and considered by Gärdenfors, Makinson and Segerberg in [8, 22, 20]. Despite its intuitive appeal (shouldn’t the revised theory depend monotonically on the revised theory?), it has been shown to be inconsistent with the AGM postulates.

Proposition 3

There is no revision operation that satisfies , and .

Proof: Suppose satisfies and . Let be an arbitrary formula that is not a logical contradiction. By , we have:

But, since is not a logical contradiction, and, by , we have . We conclude that , and, therefore, . But the set of all formulas that are not a logical contradiction is inconsistent and is inconsistent, contradicting .    

It will be shown in section 3.3 that the additional postulate we propose, , implies and a special case of .

2.2 The Darwiche and Pearl postulates

In [6], Darwiche and Pearl proposed four additional postulates. Notice that means .

It will be shown in section 3.4 that the postulates (C1), (C3) and (C4) are implied by our additional postulate . The justifications given by Darwiche and Pearl for those postulates are therefore indirect justifications for . The postulate (C2), contrary to the claims of Darwiche and Pearl, is inconsistent with the AGM axioms.

Proposition 4

There is no revision that satisfies and (C2).

Proof: Suppose satisfies and (C2). Let be any tautology, say true and be any logical contradiction, say false. We have . The postulate (C2) therefore implies that . But, for any theory , by and . Therefore, for any , . But, by and , for any consistent theory , . Since does not depend on , we conclude that all consistent theories are equal. A contradiction.    

3 The minimal influence postulate

3.1 The postulate

The postulate we propose to add to the AGM postulates is the following.

The postulate is obviously equivalent to: if , then . Its meaning is that, if is an element of , i.e., the revision of by is a severe revision, then the result of revising by does not depend on . Any revision that satisfies , and , satisfies:

(2)

Such a revision is therefore determined by its restriction to . characterizes those revision systems in which the revision depends mainly on and only minimally on (in the case of a mild revision). Remark also that, for any revision that satisfies and , the new postulate is equivalent to: .

The postulate , at first sight, does not seem to be what one is looking for. First, it seems in danger of being inconsistent with the AGM postulates. This concern will be addressed in full in section 4. But, also, one probably feels that the revisions of should depend on in a major way, even when one considers a severe revision. The feeling that some postulate should be added is widespread and this paper will present formal arguments as to why is the right postulate to add. Those reasons will not completely dissipate a first negative reaction to . The source of this clash between formal arguments and intuitive reaction probably lies in the philosophical underpinnings of theory revision and the general framework chosen by AGM to study theory revision. This paper claims that, in the framework chosen by AGM, is the right postulate to be added. The AGM framework does not seem to be the one in which to study iterated revisions.

We shall now present direct justification for . The postulate is a special case of Darwiche and Pearl’s postulate (C2) described in section 2.2, assuming and . Take, there, to be a logical contradiction. Since , one obtains . Anybody convinced by their defense of (C2) will endorse . Our defense of will be presented now. Notice that is equivalent to the following two properties: they will be justified separately.

The first, is relatively easy to justify. It is a special case of , i.e., the postulate of Addition Monotonicity. This postulate has been favorably considered by many: it is very natural to hope that the more is believed before a revision, the more is believed after. None of the many articles cited above that discuss this postulate rejects it on the grounds its meaning seems unwanted, and the only problem found with it has been that it is inconsistent with the AGM postulates. The special case proposed here, , does not fall prey to this criticism, it is consistent with the AGM postulates, and should be adopted.

The second, , is more difficult to justify. A formally weaker postulate, , will be presented and justified in a direct way. It will then been shown that, in the presence of other AGM postulates, it implies . The consideration of a similar property was suggested by Isaac Levi.

Our justification for is the following. If , then the principle of maximal retention implies we should try to keep in . If , we know that may be kept in (in particular does not contradict ), says that, in this case, we should have in . It is easy to see that implies , in the presence of . It will now be shown that implies .

Proposition 5

Any revision that satisfies , and satisfies .

Proof: Let satisfy , and . Suppose , and . We must show that . But, and . By , we conclude that . By and , then, .    

The rest of this paper is devoted to proving consequences of : the intuitive appeal of those consequences provides indirect justification for it. In particular, in sections 3.3 and 3.4, it will be shown that implies most of the additional postulates that have been previously proposed in the literature. All the arguments in favor of those postulates, in particular those developed in [6] for (C1), (C3) and (C4), provide indirect support for . Our representation result, Theorem 2, and the analysis of section 4.3 provide, both, a soundness result that shows that many revisions satisfy , and an ontology for those revisions. This provides additional indirect justification for .

It seems that contradicts, in a certain degree, the assumption of maximal retention. If , and but , then a revision that satisfies will have to let go of when revising by or to let go of when revising by . The meaning of this remark is that one cannot retain maximally always, one must compromise with the assumption of maximal retention in certain cases to be able to apply it in other cases, as, e.g., in the justification of .

3.2 First consequences: iterated revisions

Our understanding of the AGM framework, in which revisions are operations of two arguments, a theory and a formula, does not require any special mention of iterated revisions: the result of revising first by and then by is the result of revising by the theory that is the result of revising by . Some authors, for example [5], [4] and [24]

, take a different view and prefer to treat revisions as operating on a fixed theory and treat iterated revisions as a special case of varying this theory. This attitude is methodologically at odds with the AGM point of view, and we shall argue, in section 

5.1, that it tries to answer a different question. Discussion about this approach and its relations to the present work is postponed to section 5.1.

It is interesting to consider the meaning of our postulate for iterated revisions. There are two fundamental cases to consider: the case where revising by and then by is equivalent to revising by the conjunction , and the case where it is equivalent to revising by the second formula alone, . Our result concerning the first case does not use : if the second revision is mild then the iterated revision is equivalent to the direct revision by the conjunction.

Proposition 6

Let be a revision operation that satisfies . If , then .

Proof: Assume . By and , . By and , .    

For the second case, we have three different results. First, alone implies that if one has and . Secondly, we have the following.

Proposition 7

Let be a revision operation that satisfies . If , then .

Proof: Suppose . We are going to show that is an element of both and and we shall conclude by . To show that , we consider two cases. First, if , by and , we have

Secondly, if , since and are also elements of (by ), we conclude that is inconsistent. By , is a logical contradiction and is a tautology and, by , an element of . It is left to us to show that . But, by ,

We conclude that .    

The third result of this family is perhaps the most striking. It generalizes the postulates (C1) and (C3) of Darwiche and Pearl.

Proposition 8

Let be a revision operation that satisfies . If , then .

The meaning of Proposition 8 is very natural: if revising by would convince the agent that is true, then revising , first by , and then by amounts to revising, first by , and then by some information, , that conditionally implies the first information. This boils down to revising first by some partial information and then by the full information. This is indeed expected to be equivalent to revising directly by the full information.

Proof: Suppose . Two cases will be considered. Suppose, first, that . By Proposition 6, . But , and, by Theorem 1 and the properties of rational relations, implies that, for any , iff . Therefore . We conclude that .

Suppose, now, that . By , and . But , and, similarly, . We conclude that . By , then, .    

In the next sections, we shall examine in detail the relation between and the postulates previously proposed by, both, Katsuno and Mendelzon, and Darwiche and Pearl.

3.3 and

Our first result is that any revision that satisfies satisfies a special case of .

Proposition 9

Let be any revision operation that satisfies . If , then .

The proof is obvious. Our second result deals with .

Proposition 10

Any revision operation that satisfies , and satisfies .

Proof: For any revision that satisfies the assumptions, we must show that . Suppose, first, that . By , . By ,

Suppose, now, that . We conclude easily by Proposition 9.    

Our conclusion is that implies all the cases of that are consistent with the AGM postulates, the case that is inconsistent with the AGM postulates being, as shown in Proposition 3, the case in which is in but not in . In this case the revision of the intersection may be larger than the intersection of the revisions.

3.4 and the Darwiche and Pearl postulates

Let us, now, consider the postulates proposed by Darwiche and Pearl. We have seen, in Proposition 4, that no revision may satisfy (C2), but the following special case of (C2) is a consequence of .

Proposition 11

Any revision operation that satisfies , and satisfies:

Proof: Suppose and . By , it is enough to prove that , but this is the case since .    

We shall now show that the other postulates considered by Darwiche and Pearl are consequences of .

Proposition 12

Any revision operation that satisfies satisfies (C1), (C3) and (C4).

Proof: For (C1), suppose . By and , and, by Propostion 8, we conclude that .

For (C3), assume that . By Propostion 8, . We conclude that .

For (C4), assume that . We must show that . Assume, first, that . In this case, by Proposition 6, . But and therefore, by , and is not a logical contradiction. By , then, is consistent and . We conclude that . Assume, then, that . By, , . We shall show that and conclude, by , that . To show that , it is enough, by , to show that . We shall show that , i.e., . Since , it is enough to show that , which holds by hypothesis.    

As a corollary, we realize that (C1), (C3) and (C4) follow from (C2’).

Corollary 1

Any revision operation that satisfies and (C2’), satisfies (C1), (C3) and (C4).

Proof: We shall show that satisfies and conclude by Proposition 12. Suppose . Since, , by (C2’), we have . But, by and , .    

We have brought additional indirect justification for : it implies all the postulates proposed by Darwiche an Pearl, except (C2), that has been proved inconsistent. Some special case of (C2) is also implied by .

4 Representation of revisions by rational relations

4.1 Representation result

We shall now prove the main result of this paper: revision operations that satisfy may be represented by rational, consistency-preserving relations. There is, even, a bijection between the former and the latter. This result is an improvement on the results of [17], that described a bijection between AGM revisions of a fixed theory and rational, consistency-presrving relations  such that . This correspondence shows that the postulate is consistent with the AGM postulates: there are revisions that satisfy , as many as rational, consistency-preserving relations.

Theorem 2

There exists a bijection between the set of revisions that satisfy the postulates and the set of consistency-preserving, rational relations. This map associates to every such revision the relation defined by:

(3)

The inverse map associates to any rational, consistency-preserving relation  the revision defined by:

(4)

Note that the relation defined in (3) is none other than the relation defined in Theorem 1. Note also that the revision defined in (4) from the relation  has a very natural meaning. It says that, in the case of a mild revision, revise as mandated by and , and in the case of a severe revision, disregard (since it is contradicted by the more solid information ) and replace it by together with all the formulas that the agent thinks are usually true when is.

Proof: Suppose first that is a revision that satisfies to , and let us show that the relation defined in (3) is rational and consistency-preserving. This follows from Theorem 1 applied to and , since is . Note that , and are not used in this part of the proof.

Suppose now that  is a consistency-preserving, rational relation and let us show that the revision defined in (4) satisfies . The cases of (by right weakening and “and”), (by reflexivity), (if , ), (by definition), (by left logical equivalence), and (by definition) are easily dealt with. Let us show that satisfies . Suppose is inconsistent. Then, either or and is inconsistent. The latter is impossible, therefore and we conclude by consistency-preservation. For , we must show that . We shall consider two cases. Suppose first that . Then too. Suppose . Then, . By conditionalization (i.e., rule S), we have and, since , . Therefore, . Suppose now that . Then, , and . But we have already shown that satisfies , therefore

Finally, for , suppose . We have to show that

or, equivalently that if , then, . We consider again two cases. If, first, , then and also , therefore, by rational monotonicity, and, by “and” and right weakening, . But and therefore and we conclude that . The second case we must consider is: . Then and, by hypothesis, is not a member of this set. This implies that . Therefore, . Therefore .

To complete the proof of the representation theorem, we have to show that the maps and are inverse maps, i.e., that and that . For the first equality, we have iff , iff since . For the second equality, we have

Therefore,

But, if , by , , and if , by and , . Notice that the postulates used now, , and are those that were not used in the first part of this proof.    

In section 1.2, in the discussion following the proof of Theorem 1, we noticed that any theory and any revision operation that satisfies , and to defines a rational, consistency-preserving relation, the relation defined in formula (1). We have shown, in Theorem 2, that any such relation is defined in this way by a specific theory and some revision operation that satisfies not only , and , but also , and . It does not mean that , and are redundant, i.e., derivable from the other postulates, but only that the mapping is not injective. In fact, the reader will quickly realize that no postulate is redundant, though may be weakened to in the presence of and . We take this to mean that postulates , and say nothing about the nonmonotonic inference aspect of revision: they are orthogonal to it.

4.2 Consequences of the representation theorem

The first consequence that we shall draw from Theorem 2 is that the postulate does not constrain the ways one may revise a fixed theory more than the AGM postulates already did: it only constrains the ways one may revise different theories by related formulas. If we consider an AGM revision and a fixed theory , there is a revision that satisfies and that revises the theory , for any formula , exactly as does .

Theorem 3

Let a revision that satisfies the AGM postulates to . Then, for any theory , there exists a revision that satisfies and such that, for any , .

Proof: Let a revision that satisfies the AGM postulates to and an arbitrary theory. By Theorem 1, the relation , defined in equation (1) is rational and consistency-preserving. For simplifying notations, we shall denote by . By Theorem 2, the revision , defined in equation 3, satisfies . But, for any ,

Therefore, by equation (1),