Ordering-based Representations of Rational Inference

07/26/2000
by   Konstantinos Georgatos, et al.
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Rational inference relations were introduced by Lehmann and Magidor as the ideal systems for drawing conclusions from a conditional base. However, there has been no simple characterization of these relations, other than its original representation by preferential models. In this paper, we shall characterize them with a class of total preorders of formulas by improving and extending Gardenfors and Makinson's results for expectation inference relations. A second representation is application-oriented and is obtained by considering a class of consequence operators that grade sets of defaults according to our reliance on them. The finitary fragment of this class of consequence operators has been employed by recent default logic formalisms based on maxiconsistency.

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1 Introduction

A recent breakthrough in nonmonotonic logic is the beginning of study of nonmonotonic consequence through postulates for abstract nonmonotonic consequence relations, using Gentzen-like context-sensitive sequents ([6], [14], [12]). The outcome of this research turns out to be valuable in at least two ways

  • it provides a sufficiently general axiomatic

    framework for comparing and classifying nonmonotonic formalisms, and

  • it gave rise to new, simpler, and better behaved systems for nonmonotonic reasoning, such as cumulative ([6]), preferential ([12]), and rational ([13]) inference relations.

It is unfortunate that these new inference relations enjoy only one, semantical, representation; that of preferential models ([24]). We have that preferential, preferential transitive, and preferential modular or ranked models characterize cumulative, preferential and rational inference relations, respectively ([12], [13]). An additional second-order constraint must be imposed on these models, called stoppering or smoothness. However, this modeling is insufficient because in order to employ the above inference relations, one must be able to generate them. This is crucial when we want to design a system that reasons using the above inference relations. In such a case, one comes up with a set of rules or defaults that one wants to apply, imposes a prioritization on them, and provides a mechanism which ensures that answers are derived according to these inference relations. This is exactly the proof-theoretic approach expressed by default logic. However, no similar proof-theoretic notion is provided in the above framework.

In this paper, we offer two new, alternative representations for rational inference. The first representation is algebraic and obtained through a simple class of orderings of formulas, called rational orderings. The second representation is proof-theoretic and obtained through a class of consequence operators based on the way we handle defaults, called ranked consequence operators. Moreover, a correspondence result between these classes is established.

The first link between nonmonotonic inference relations and a class of orderings of formulas was given by Gärdenfors and Makinson in [7]. However, the nonmonotonic system defined by an ordering of formulas is not one of the previously mentioned systems, but a translation of the well-known belief revision AGM axioms ([1]) into nonmonotonic reasoning, called expectation inference relations. Expectation inference relations are rational inference relations together with a rule called Consistency Preservation. Moreover, Gärdenfors and Makinson’s representation of expectation inference relations with orderings of formulas is not appropriate, in the sense that the correspodence is not bijective. (Two orderings of formulas can generate the same inference relation.) So, two questions remain open. Namely,

  • is there a way to generate one of the independently motivated nonmonotonic inference relations (cumulative, preferential, rational) with a class of orderings of formulas?, and

  • can the correspondence be bijective?

We answer affirmatively both questions for rational inference. Our approach is the following. We study the rule of Consistency Preservation and, by giving it a precise syntactic characterization, show that its role is insignificant in the context of preferential reasoning. Drawing from this intuition, we introduce new defining conditions relating the classes of orderings of formulas and nonmonotonic consequence relations and show that Gärdenfors-Makinson orderings are in bijective correspondence with rational inference. Moreover, we introduce a smaller class of orderings, which, under our translation, is in bijective correspondence with the Gärdenfors-Makinson expectation inference relations. This is how the first representation result for rational inference is obtained. This result adds to a long tradition of defining nonmonotonic logics with orderings of formulas ([4], [19], [7], [20]).

The above representation result is more “constructive” than the semantical completeness of preferential models. However, rational orderings must have a concise, constructive representation. To this end, we encode a natural way of applying defaults into a new class of consequence relations, called ranked consequence operators. Each member of this class generates a rational ordering, and conversely, hence the class of ranked consequence relations coincides with that of rational inference. Also, we show how previous default logic systems in the literature ([17], [18]) reduce to our framework.

The above results pave the way towards a study of nonmonotonicity through orderings of formulas, allow us to translate previous work in belief revision into the context of nonmonotonic reasoning, and provide a framework for designing default systems obeying rational inference.

The plan of this paper as follows. In Section 2, we briefly introduce the relations under study, explain the rule of Consistency Preservation, and provide a characterization for this rule. In Section 3, we introduce the orderings, their translations and our first representation theorem. In Section 4, we introduce ranked consequence operators and our second representation theorem. In Section 5, we show how one can generate a ranked consequence operator given a prioritized family of sets of defaults and, in Section 6, conclude. A preliminary version of the first half of this paper appeared in [10]. Results from the second half were announced in [9].

2 Shifting underlying entailment

Before going to the main result of this section, we shall make a brief introduction to the nonmonotonic consequence relations under study. Assume a language of propositional constants closed under the boolean connectives (disjunction), (conjunction), (negation) and (implication). We shall use greek letters , , , etc. for propositional variables. We shall also use , read as “ normally entails ”, to denote the nonmonotonic consequence relation (). Before we present the first set of rules for , we need a symbol for a classical-like entailment. We shall use . The relation need not be that of classical propositional logic. We require that includes classical propositional logic, satisfies compactness (i.e., if then there exists a finite subset of such that )111We write for ., the deduction theorem (i.e., if and only if ) and disjunction in premises (i.e., if and then ). The reader will notice that these are the only properties we make use of in the subsequent proofs. We shall denote the consequences of with and under and , respectively.

The rules mentioned in the following are presented in Table 1. For a motivation of these rules, see [12] and [15]. (The latter serves as an excellent introduction to nonmonotonic consequence relations.)


(Supraclassicality)

(Left Logical Equivalence)

(Right Weakening)

(And)

(Cut)

(Cautious Monotonicity)

(Or)

(Rational Monotonicity)

(Consistency Preservation)
Table 1: Rules for Preferential, Rational and Expectation Inference

Definition 1  Following ([12], [13], [7]), we shall say that a relation on is an inference relation (based on ) if it satisfies Supraclassicality, Left Logical Equivalence, Right Weakening, and And. We shall call an inference relation preferential if it satisfies, in addition, Cut, Cautious Monotonicity, and Or. We shall call an inference relation rational if it is preferential and satisfies, in addition, Rational Monotonicity. Finally, we shall say that is an expectation inference relation (based on ) if it is a rational and satisfies, in addition, Consistency Preservation.

The most controversial of these rules is Rational Monotonicity, which, moreover, is non-Horn. For a plausible counterexample, see [26].

Expectation inference relations correspond to the so-called AGM postulates for belief revision ([1]), as it was shown in [16], and only differ from rational relations in that they satisfy the following rule, called Consistency Preservation:

where is classical entailment. Consistency Preservation says that a logically not false belief cannot render our set of beliefs inconsistent. This makes a difference between the two classes, in the following sense. Using rational inference, I can rely on an inference such as , where is the statement “I am the Queen of England”. On the other hand, expectation inference would not allow that, since, even if I am certain I am not the Queen of England, one could think a world where I could have been. This becomes more important if, instead of a belief set, one considers a conditional base. For example, consider a database for air-traffic. The statement “two airplanes are scheduled to arrive at the same time and land on the same place” should infer inconsistency on this database, although it is not a falsity. More examples can be drawn from physical laws. This means that rational inference is the logic of “hard constraints”, that is of statements (not necessarily tautologies) I cannot revise without deconstructing the whole inference mechanism. This is not admitted in expectation inference: all statements are allowed to be revised apart from tautologies (or whatever is a consequence of the empty set under the underlying entailment).

In [16] and [15], it was observed that preferential entailment satisfies a weaker form of consistency preservation: there exists a consequence operation with such that satisfies Consistency Preservation with respect to . This was proved by semantical arguments.

In the following theorem, we make this property more precise by expressing it in syntactic terms. We show that the required underlying consequence operation retains the properties of the initial one, as it only differs on the set of assumptions. Therefore, the relation between an expectation and a rational inference relation is that of a logic with its theory.

For the proof of Theorem 3, we shall make use of the following rules (derived in any preferential inference relation).

Lemma 2

In any preferential inference relation, the following rules hold

  1. (Reciprocity)

  2. (S)

Proof. Rules 12 and 3 were introduced and shown to be derived in a preferential relation in [12]. For 4, suppose . Applying S, we get and, by Right weakening, we conclude 4. For 5, suppose . Then, by 3, we get and, by Left Logical Equivalence, we conclude 5.   

Theorem 3

Let be a preferential inference relation based on . Then is a preferential inference relation based on that satisfies the Consistency Preservation rule, where

and

Proof. We must prove that satisfies Supraclassicality, Left Logical Equivalence, Right Weakening and Consistency Preservation with respect to . The rest of the rules are already satisfied since they do not involve an underlying consequence relation.

First notice that Consistency Preservation is immediate by definition of .

For Supraclassicality, suppose then . By compactness of , there exist such that and . By repeated applications of Or, we get . Let , then and . By Supraclassicality of on , we have . By Lemma 2.3, we have , so, by Lemma 2.4, we have . Using Cut, we get , as desired.

For Left Logical Equivalence, suppose , , and , i.e. and . By compactness, there exist such that , , , and . As above, we have and . Therefore, by Lemma 2.1, we get , as desired.

Coming to Right Weakening, suppose and , i.e. there exists such that and . By And, we have , so, using Right Weakening of on , we get , as desired.    

Notice that the result applies to rational inference relations, as well, since the latter are preferential, by definition. We interpret the above result as follows. Once we strengthen the underlying entailment, rational inference will become an expectation inference and, therefore, can be treated as such. It also implies that the logic of hard and soft constraints is basically the same, their only difference being what we consider a consequence of the underlying propositional entailment. Hard constraints are just taking a place in our belief set as “guarded” as that of, say, tautologies. Whatever remains is subject to revision, and hence a soft constraint.

3 Rational inference and orderings

Now that we established the correspondence between rational and expectation inference relations, we shall extend it to a particularly attractive characterization of the latter with orderings of formulas. We shall first review Gärdenfors-Makinson’s results and then present our own.

The intuition behind ordering-based formalisms is common in works on belief revision, possibilistic logic, and decision theory. We order sentences according to our expectations. A relation “” is interpreted as “ is expected more than ”, or “ is more surprising than ”, or “ is more possible than ”. One can treat such an ordering as a primary notion; this is the approach of this paper. However, in case of rational orderings, one can show that such an ordering induces a function from the extensions of formulas to the unit interval. This function induces a possibility measure on the extensions of formulas (see [2]

). A possibility measure is a “weak” probability measure on these extensions. Roughly, it replaces addition with maximisation. Although the connections with probability are not clear yet (see

[3], [25]), probability measures seem especially suited for modeling cases under uncertainty. Further, a possibility measure arises naturally out of a database. Zadeh’s theory for approximate reasoning ([28]) provides a method for turning available information of a certain form (“fuzzy” database) into a possibility measure and, therefore, gives rise to a rational ordering of sentences.

We find that, by a logical point of view, orderings correspond to prioritization. We prefer a proof-theoretic reading, made more explicit in Section 4, “ is more defeasible than ” or “ has lower priority than ”. A notion of proof is developed in Section 4 based on this prioritization and justifies the use of rational orderings without appealing to some probabilistic intuition.

Definition 4 [7] A rational ordering is a relation on which satisfies the following properties:

1. If and , then (Transitivity),
2. If , then (Dominance),
3. or (Conjunctiveness).

The original name of these orderings was expectation orderings. However, we shall see that this name is not justifiable, since expectation inference relations correspond to a smaller class of orderings (see Definition 10).

One can easily derive from the above properties that a rational ordering satisfies

  1. connectivity, i.e. or , and

  2. either , for all , or , for all .

We should mention that the above properties of rational relations are not new. It is not easy to assign credits, but they have appeared in works in belief revision ([7]), possibilistic logic ([3]), [4]), fuzzy logic ([27]), theory of evidence ([23]), and economics ([22]) (see [7] for a historical reference).

Gärdenfors and Makinson define the following maps between the class of expectation inference relations and rational orderings.

Definition 5 [7] Given a rational ordering and an expectation inference relation , then define a consequence relation and an ordering as follows
() iff either or there is a such that and . () iff either or .
We shall also denote and with and , respectively.

Condition () is critical and due to Rott ([21]). Now, one can prove the following.

Theorem 6

[7] Given a rational ordering and an expectation inference relation , then is an expectation inference relation and is a rational ordering. Moreover, we have .

This theorem, although it exhibits the first connection between some class of nonmonotonic consequence relations and orderings of formulas, has two disadvantages. First, the way it achieves consistency preservation is ad hoc. If that was not the case, then the condition () would be inappropriate, since, in the first part, it refers explicitly to the underlying entailment222However, the second part should remain the same since we do not mind having a few more consequences, as long as, the rules which govern the underlying entailment do not change.. Second, it fails to show an isomorphism between the class of expectation inference relations and that of rational orderings, that is . If the second was not the case, then the condition () would use only the expectation inference relation to construct the ordering. Consider the following example.

Example 7  Let and . Now define orderings on as follows

Similarly, for and . We have that , and by Proposition 19, the orderings are rational. However, they generate the same expectation inference relation, using ().

Drawing from the above intuitions and Theorem 3, we define

Definition 8  Given a rational ordering and a rational inference relation , then define a consequence relation and an ordering as follows

() iff either , for all ,
or there is a such that and .
() iff either or .

We shall also denote and with and , respectively.

For Theorem 10, we need the following lemma.

Lemma 9

Let and be a rational ordering and inference relation, respectively. Then

  1. If then , for all , where .

  2. If , for all , then , where .

  3. iff .

Now, everything falls into place.

Theorem 10

Given a rational ordering and a rational inference relation , then is a rational inference relation and is a rational ordering. Moreover, we have and .

Now, if rational orderings are in adjunction with rational inference relations, what is the class of orderings which corresponds to expectation inference relations? For that, observe that by Lemma 9, hard constraints are positioned on the top of rational orderings. So, it is enough to keep exclusively this place for the consequences of the empty set and add this as a condition to rational orderings.

Definition 11  An expectation ordering is a rational ordering which satisfies, in addition, the following property:

If , for all , then .

Now, using the same defining conditions () and (), we can state the improved characterization theorem for expectation inference relations.

Theorem 12

Given an expectation ordering and an expectation inference relation , then is an expectation inference relation and is an expectation ordering. Moreover, we have and .

Theorems 10 and 12 can now be used for giving new straightforward proofs for the characterization of rational inference with ranked preferential models (Theorem 3.12 of [13]) and expectation inference with nice preferential models (Theorems 3.8 and 3.9 of [7]). These proofs will appear elsewhere.

4 Ranked consequence operators

First, a word about the plan of this section. We introduce the notion of ranked consequence operation without referring to an underlying entailment (Definition 4). The reason for such a definition is that we can motivate ranked consequence operators independently of nonmonotonic reasoning. Then we define a smaller class based on an underlying entailment (Definition 4) and show that this class characterize rational inference relations. The same constraints we assumed for a language and an entailment in Section 2 continue to hold here.

Think of a reasoner whose beliefs are ordered accordingly to their defeasibility. Beliefs which are less likely to be defeated come before beliefs which are more likely to be defeated. For instance “Birds fly” will come after “Penguins do not fly” (since the former has more exceptions) and “Mary is married” might come before “Mary is married with children” (since the latter is stronger). There is a natural way to attach a consequence operator to this belief prioritization.

Definition 13  Let be a linear ordering, and be an upward chain of sets of formulas such that iff . Define the following consequence operators (one for each ):

 iff ,
 iff and .

Now let


 iff  either , for some ,
 or , for all .

The consequence operator will be called ranked consequence operator (induced by ).

First, note that ’s are not necessarily deductively closed. Second, notice that, unless we add the last part of the definition of , we do not provide for formulas , where , for all . In order to have , there must either be an such that and , or , for all . This means that if our beliefs can accommodate a context where holds, then we use the part of the ordering that remains consistent after adding . Therefore the ’s which contain both and are irrelevant to the consequence operator.

Indices assign grades of relying on the set of consequences as the next example, formalizing omniscience, shows.

Example 14  Let be the set of natural numbers with the usual order. Now let be the classical consequence relation and let be some set of formulas of propositional logic. Let

and, inductively,

Notice that if is consistent and is the ranked consequence operator defined through then

where is the classical consequence operator of propositional calculus. Note that if is inconsistent, then entails all formulas which are provable from with less steps than , i.e., before we realize inconsistency. Now, if is the set of all tautologies or, better, an axiomatization of them then is exactly the classical consequence operator.

It is clear that the above representation is syntax-based, i.e. depends on the particular representation of ’s. The case where the sets of formulas are closed under consequence is the one we shall deal with in this paper. Doing that is like being logically omniscient; we do not assign any cost to derivations using .

We shall now give a definition of ranked consequence operator using an underlying entailment. In case the ’s are closed under consequence, it coincides with the original definition (by replacing the set belonging relation with proposition entailment ).

Definition 15  A ranked consequence operator based on induced by a chain of sets under inclusion is defined as follows:

We first define a set of consequence operators (one for each ):

 iff and .

Note that we denote with . We can now let

 iff  either , for some ,
 or , for all .

We shall use to denote this operator.

Notice that we can have both and , for all . This translates to the fact that can be true in some possible world but it is unthinkable for us to include it in our beliefs. The above mechanism treats such a case as an instance of a hard constraint: such an implies falsehood.

Again notice that unless , for all , we cannot derive falsity from . The reason is that, in those cases, we are able to form a context based on (a chain of sets of formulas which prove ) which is consistent. Again, the inconsistent ’s are irrelevant to the consequence operator. The following proposition allows us to assume that the ordering is complete, that is it has all meets and joins, and amounts essentially to Lewis’ assumption or smoothness property of preferential models.

Proposition 16

A ranked consequence operator based on is induced by a chain of sets of formulas if and only if it is induced by the closure of this chain under arbitrary unions and intersections.

This result has the following significance: it allows an assignment of a rank to an assertion of the form . Suppose that holds. If for all does not hold, then the set is not empty. Moreover, it is connected. Now, it is easy to see that, in the completion of the chain, this set has a least element (because it is closed under intersection) and a greatest element (because it is closed under unions). Let and be the indices of the least and greatest elements, respectively. The rank of the assertion is and its range . In case , for all , then set the rank of to and its range to , where and are the indices of least and the greatest element of the linear order, respectively. In case of an assertion , observe that its range is of the form , where is the index of the greatest element of the linear ordering.

Finally, notice that a ranked consequence operator is not necessarily monotonic.

Example 17  Let and . We have because and . But we also have that because .

Now, it is interesting to ask what kind of properties a ranked consequence operator satisfies. It turns out that each ranked consequence operator gives rise to a rational inference relation. Although one can show it directly, we define the rational orderings induced by such operators.

Definition 18  Given a ranked consequence operator, let

and call the ordering induced by the ranked consequence operator .

We, now, have the following

Proposition 19

An ordering induced by a ranked consequence operator is rational. Moreover, .

We have immediately the following.

Corollary 20

A ranked consequence operator is a rational inference relation.

The other direction of the above theorem holds, too. We should only show, given a rational ordering, how to generate a total order of sets of formulas. To this end, we shall define a chain of sets which generates a ranked consequence operator equal to . Let be the equivalence relation induced by (a rational ordering is a preorder). The equivalence classes will be denoted by (where ). It is also clear that the set of equivalence classes is linearly ordered. Now, for each , let

Note here that, by Dominance, the sets are closed under consequence. Moreover, we have iff . Now, generate a ranked consequence operator as in Definition 16. This ranked consequence operator turns out to be equal to the one generated by the rational order. So, we have the following.

Theorem 21

A rational inference relation is a ranked consequence operator.

The proof of the above theorem shows that a rational ordering can be defined by a chain of sets which induces a ranked consequence operator and conversely. However, the same rational ordering can be induced by two different ranked consequence operators. This should hardly be surprising, as ranked consequence operators play the role of axiomatizing a nonmonotonic “theory”, that is, a rational inference. Moreover,

  • ranked consequence operators are proof-theoretic in their motivation, and therefore closer to what we want to describe by a rational inference relation, and

  • a ranked consequence operator assigns ranks to assertions as well as to formulas therefore grading the whole process of inference.

We showed that rational and expectation inference relations are exactly the same class of consequence relations if we allow the underlying propositional entailment to “vary”. However fixing , is it possible to tell if a ranked consequence operator satisfies Consistency Preservation? The answer is affirmative, for a formula infers inconsistency () if and only if its negation is a consequence of the first element of the chain which induces the ranked consequence operator (as a corollary of Proposition 16, a first element always exists). To see that, suppose , then, by definition, we must have , for all , and for that it is enough that the first element of the chain implies .

5 Rational default systems

In this section, we shall see how one can design a ranked consequence operator. Suppose we are given a number of sets of (normal, without prerequisites) defaults in a linear well-founded prioritization. Moreover, and this is an important assumption for rational inference, we are asked to, either apply the whole set, or not apply it at all. Let the set of sets of defaults be , where is a well-founded linear strict order, and is preferred from whenever . The are two ways to read this preference.

  • The first way is a strict one: if you cannot add to your set of theorems (that is, you derive inconsistency by adding ) then you cannot add , for all less preferred from .

  • the other is liberal: if you cannot add to your set of theorems, then you can add , where is less preferred from , provided you cannot add , where is more preferred than .

To illustrate this, consider the following example.

Example 22  Let , where , , and . Assume . Following the strict interpretation, we can only infer , from . With the liberal interpretation, we can also infer , since we are allowed to add , and cannot add that leads to a contradiction.

It turns out that those readings are equivalent. Not in the sense that the same set of sets of defaults generate the same consequences, but that a strict extension of a family of sets of defaults can be reduced to a liberal extension of another family of sets of defaults, and conversely. It can be easily shown that strict and liberal extensions of families of sets of defaults are instances of rational consequence operators and, therefore, rational. In particular, Proposition 19 gives us a way to construct the rational orderings of such default systems.

Given , where is well-founded, define the following strict ordering between non-empty subsets of :

iff there exists such that a. but , and
b. for all , iff .

It can be shown that is linear. Now, let .

Definition 23  Let and , where is a total strict order, and , for all . The strict extension of with respect to is defined as follows

where . The liberal extension of with respect to is defined as follows

where .

Thus the liberal extension of is the strict extension of . For the other direction, the strict extension of coincides with the liberal extension of , where .

So, it is enough to construct the ranked consequence operator for the strict extension of . But this is easily achieved. Consider , where .

Thus, strict and liberal extensions of prioritized sets of set of formulas are rational. The above definition, together with Proposition 19, gives us a way to construct the rational orderings of such default systems. Given a prioritized set then the rational ordering of its strict extension is

The rational ordering of its liberal extension is

Assuming a finite language, sets of formulas, intersections, and unions of them correspond to conjunctions, disjunctions, and conjunctions, respectively.

A study of the above default systems under the assumption of finite language, has been carried already in the context of belief revision (therefore, assuming consistency preservation, in addition to finite language), by Nebel ([17], [18]). Our strict and liberal extensions are called prioritized and linear base revision, respectively. Also, Nebel showed in [18] that deciding if a certain formula is contained in the strict or in a liberal extension (that is, deciding ) is and , respectively. We expect that these results carry over to our framework.

6 Conclusion

We summarize our results in the following

Theorem 24

Let be a binary relation on . Then the following are equivalent:

  1. is a rational inference relation, i.e. it satisfies Supraclassicality, Left Logical Equivalence, Right Weakening, And, Cut, Cautious Monotonicity, Or, and Rational Monotonicity.

  2. is characterized by some rational relation on using condition ().

  3. is defined by a ranked consequence operator.

Since rational orderings are in one-to-one correspondence with rational inference, our first representation result has many ramifications. Results in belief revision can be translated in a nonmonotonic framework and vice versa. For instance, selection functions and preferential models can be used for the modeling of both. Proofs of this results are straightforward through our defining conditions for rational and expectation orderings. Work that has already been done on expectation inference relations (e.g. the study of generating expectation inference relations through incomplete rational orderings—see [5]) can be lifted smoothly to rational inference.

Our second representation result reveals the working mechanism of rational inference. It shows that, in order to attain rational inference, we must prioritize defaults in a particular way. We showed how default logic formalisms can fit this pattern. It enables us to assign grades to all components of the reasoning system (formulas and rules). Therefore, it is a particular attractive way to use it as an inference mechanism for nonmonotonic reasoning.

The above characterization results reveal another notion of consequence paradigm hidden behind nonmonotonicity. However, apart from Dubois and Prade’s work on possibility logic, this paradigm has been passed largely unrecognized by logicians as an appropriate method for a treatment of vagueness and uncertainty. Yet, this paradigm arose independently from various studies on different fields and appeared before nonmonotonic logic. In addition, it is applicable. Now, an important question arises: how far this paradigm extends. In other words, is it possible to reduce a nonmonotonic consequence relation to some relation expressing prioritization? The answer is positive and uniform. The important case of preferential consequence relations is treated separately in [8], while the general case (which includes cumulative consequence relations) appears in [11].

Acknowledgements: I would like to thank Gianni Amati and R. Ramanujam for their helpful comments on a preliminary version of this paper.

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Appendix A Proofs

For the proof of Theorem 3, we shall make use of the following rules (derived in any preferential inference relation).

Lemma 2 In any preferential inference relation, the following rules hold

  1. (Reciprocity)

  2. (S)

Proof. Rules 12 and 3 were introduced and showed to be derived in a preferential relation in [12]. For 4, suppose . Applying S, we get , and, by Right weakening, we conclude 4. For 5, suppose . Then, by 3, we get , and, by Left Logical Equivalence, we conclude 5.  

Theorem 3 Let be a preferential inference relation based on . Then is a preferential inference relation based on that satisfies the Consistency Preservation rule, where

and

Proof. We must prove that satisfies Supraclassicality, Left Logical Equivalence, Right Weakening and Consistency Preservation with respect to . The rest of the properties are already satisfied since is a rational inference relation.

First notice that Consistency Preservation is immediate by definition of .

For Supraclassicality, suppose then . By compactness of , there exists such that and . By repeated applications of Or, we get . Let , then and . By Supraclassicality of on we have . By Lemma 2.3, we have , so, by Lemma 2.4, we have . Using Cut, we get .

For Left Logical Equivalence, suppose , and , i.e. and . By compactness there exist such that , , and . As above we have and . Therefore by Lemma 2.1 we get .

Coming to Right Weakening, suppose and , i.e. there exists such that and . By And we have , so using Right Weakening of on we get .

 

For Theorem 10, we shall need the following lemma.

Lemma 9  Let and be a rational ordering and inference relation, respectively. Then

  1. If then , for all , where .

  2. If , for all , then , where .

  3. iff .

  4. If then , for all , where .

  5. If , for all , then , where .

Proof. Part 2 is immediate from defining condition (). For Part 1, suppose that . We must show . Since , we have , by Right Weakening. Applying Or, we get . By hypothesis and And, we get .

The left to right direction of Part 3 is straightforward. For the right to left direction, suppose . Then, by Compactness, there exist such that , for all , and . By Conjunctiveness, we have . So, by Dominance, we have . Hence, by Transitivity, , as desired.

For Part 4, we have, by definition of that , for all or there is such that and . If the former holds then the result follows immediately. If the latter holds then that contradicts , by Dominance. margin: Do Part 5  

Lemma 25

Let and be a rational ordering and inference relation, respectively. Then

  1. implies , where .

  2. implies , where .

  3. If satisfies Bounded Disjunction then implies , where .

Proof. For Part 1, if we get immediately , by Definition (O). If not, that is , then, by And and Right Monotonicity, we have . Again, by Definition (O), we have .

For Part 2, assume . If . Then we have , so , and hence, by Dominance, , as desired. If not then there must be such that and . Therefore . Now, suppose towards a contradiction. By Connectivity, we have and so, by Conjunctiveness, . Hence , a contradiction to Reflexivity.

For Part 3, if then we immediately have , by Definition (C). If not, that is , then applying Bounded Disjunction, we have . The latter implies , so, by Dominance , for all . Hence , by Definition (C).  

Theorem 10 Given a rational ordering and a rational inference relation , then is a rational inference relation and is a rational ordering. Moreover, we have and .

Proof. We shall try not to overlap with the proof of Gärdenfors and Makinson proof of Theorem 6 (see proof of Theorem 3.3 in [7]). Therefore we do not cover the case where the second half of condition (R) applies. The list of rules we verify is Supraclassicality, Left Logical Equivalence, And, Cut, Cautious Monotony, Or and Rational Monotony. Right Weakening follows from the above list.

We shall first show that is a rational inference relation.

For Supraclassicality, suppose that but not for all . So there exists such that . But then and therefore .

For Left Logical Equivalence, suppose that , and for all . Since we have . By Dominance we get and by Transitivity for all . Therefore .

For And, suppose that and . In case for all we have immediately .

Turning to Or, suppose that and . If for all and for all , then by Conjunctiveness we have either or . In either case for all by Transitivity. Therefore . In the mixed case, say for all and there exists such that and , we have . Now suppose that . By Conjunctiveness we must have , a contradiction. Thus . Therefore .

For Cut, suppose that and . If for all then, by definition, . If not, there exists such that and . Now suppose that for all . Observe that . We moreover have that . Therefore .

For Rational Monotonicity, suppose that and . If for all , then we get a contradiction because .

For Cautious Monotony, suppose that and . Observe that in case then the result follows by an application of Rational Monotony. If not, i.e. , then by applying And we have . If for all , then, since , we have but therefore . Otherwise there exists such that and . But then we have that and therefore which is a contradiction to our hypothesis.

Definition () is identical to Gärdenfors and Makinson’s one in the second disjunct. Therefore we shall only treat the first disjunct.

For Dominance, suppose and . We have and . By Or, we get . By Supraclassicality, we have . Applying And, we get .

For Conjunctiveness, suppose and . These imply and , by Left Logical Equivalence. Applying And, we get . By reflexivity of and And, we have . By Left Logical Equivalence again, we have , and so , as desired.

For Transitivity, let and . Suppose . By Lemma 2.5, we have . Lemma 2.3 gives . By S, we have . So, we have . Using the initial hypothesis and Or, we get . By Lemma 2.5, we have , i.e. . Now, suppose and . Then, by Lemma 2.5, we have . Since , we have . Therefore if , then And gives .

We shall now show that the initial rational inference relation and the induced one by the expectation ordering with () are the same.

We show first that . Let . We must show that . If , for all , then it clearly holds. If not, let then . So . Also, . If then , for all (using Lemma 9.1), so, by our hypothesis, . Observe that , and . Right Weakening gives . So and therefore . Hence .

For the other direction, i.e. , let . Suppose first that for all . Therefore . This gives either or . Since obviously the latter does not hold we must have that and hence, by Right Weakening, . Now suppose that there exists with and , i.e.