1 Introduction
Preference is an important concept in knowledge representation. Whenever we aim to design a framework that does not depend solely on logical considerations, a possible way to incorporate extralogical information is to treat it as preference. Preference is subjective. Yet, preference is not based on a beyond analysis personal taste. If that was the case, it would have been pointless to seek a logic for preference.
Preference is based on available information, both implicit (facts we learned and believed) and explicit (facts we empirically verified). In many cases, we can assume that two persons who were exposed to similar information have the same preferences. If their preferences diverge, we look for a difference on their background knowledge and motives. What constitutes a basis of preference is beyond the scope of this paper but labeling on the basis of criteria as the above gives preference a social dimension, which in turns makes preference a basis of reasoning.
What is the logic of preference? A simple but crucial first step has been made by Shoham ([22],[23]) with the introduction of preferential models. Preferential models are models equipped with a (nonreflexive, transitive) preference ordering. Models of this sort are not, strictly speaking, new as they can be reduced to Kripke models or some other labeled order or relation. What is original about them is the nature of the preference relation. This relation seeks to maximize some function. To make this point clearer, let us suppose we have some box emitting binary streams, it has emitted until now, and we want to order two binary streams and according to our preference for its future behavior. Our first impulse would be to rank them equally, as both are possible. This is what we would do if we knew nothing about the box. However, some background information might make us choose one over the other, for example
. In both cases, (conditional) probability would prevail. On the other hand, if some profit is to be made by choosing the less probable
then again our ordering would be a biased one. This preference would seek to maximize utility.The above discussion points implicitly to conditional information and therefore to nonmonotonic inference defined through preferential models. Indeed, what Shoham did is, by fixing a preferential model, to define: preferentially entails iff holds on all minimal models of under the preference relation. Preferential entailment is nonmonotonic as minimal models of might differ from those of . The preferential model approach to nonmonotonicity is a semantical oasis in the overridden world of syntactic nonmonotonic formalisms. It should be pointed out, however, that preferential models have their roots in McCarthy’s Circumscription ([18]) as the latter is a syntactic formalism of selecting the minimal models in a relation that prefers predicates with a smaller extension.
The second important step was made by subsequent work of Kraus, Lehmann, and Magidor ([11]) when they showed that preferential entailment on models whose preferential relation satisfies the additional second order property of smoothness or stopperedness is characterized by the the system (see Table 1),
where means preferentially entails . This result made a connection between the preferential model approach and work on (sceptical) nonmonotonic consequence operators introduced by Gabbay ([6]) and studied by Makinson ([16]). System is a simple yet powerful sequentlike consequence relation that has been recognized ([11],[17]) as the strongest basis for nonmonotonic inference. Any system stronger than is bound to be nonHorn and therefore loose some of its prooftheoretic content. However, apart from greatly diverging from the theory of (monotonic) logical consequence, preferential entailment has the additional defect of the inability of expressing credulous nonmonotonic inference, that is, to express extensions.
The purpose of this paper is to introduce a binary relation among sentences, called partial entrenchment, that has the feature of being monotonic and express extensions and show that any class satisfying system can be generated as the intersection of those extensions. The subclass of partial entrenchments consisting of total preorders is Gärdenfors and Makinson’s expectation orderings which characterize expectation inference ([8]) and Lehmann and Magidor’s rational inference ([12]). Restricting the class of expectation orderings with properties parameterized by theories one gets epistemic entrenchment, a well known class of linear preorders of sentences characterizing the AGM postulates for belief revision ([1]). A further generalization of partial entrenchment led to a uniform characterization of all nonmonotonic inference relations ([10]).
The plan of this paper is as follows. In Section 2, we shall introduce partial entrenchment, explain its function and compare its features with other approaches. In Section 3, we define a nonmonotonic consequence relation based on partial entrenchment called maxiconsistent inference and prove some of its properties. Maxiconsistent consequence satisfies the properties of system and, in Section 4, we show that once we restrict the class of partial entrenchment to an appropriate subclass we get a bijective correspondence.
2 Partial Entrenchment
In this paper, we will not give a semantic account of entrenchment relations but a procedural one. We will now proceed with the formal definition of partial entrenchment. We will use a propositional language of atomic variables, denoted by Greek lower case letters , , , etc., and closed under the usual propositional connectives (negation), (disjunction), (conjunction), and (implication). Entrenchment relations assume an underlying logic. We will use classical propositional consequence denoted with . Such a choice is almost dictated by the choice of connectives and the theory we will develop but, in addition, our intention is to build nonclassical reasoning on top of a classical one. This has the advantage of making our choices simpler and clearer. The set of consequences of a set of sentences under , will be denoted by and we will write and for and , respectively.
Definition 1 A binary relation on is called a partial entrenchment when it satisfies the following properties:
1. if and , then ,  (Transitivity) 

2. if , then , and  (Dominance) 
3. if and then .  (Conjunction) 
We write for but .
Partial entrenchment relations can be read as rules for extending theories. The meaning of , where denotes the partial entrenchment is:
can extend our theory provided we first extend it with .
So entrenchment encodes constraints on theory extensions. Therefore, entrenchment is a priority mechanism for building extensions: we shall consider only extensions that satisfy the entrenchment rules. The larger the extension the better. The reader can easily verify that our reading of partial entrenchment satisfies the above properties.
Partial entrenchment can also be expressed as a consequence relation that extends classical logic. The main point here is that partial entrenchment respects neither disjunction nor negation.
We shall now describe informally how entrenchment gives rise to a nonmonotonic consequence relation, i.e., a conditional of the form . This paper is taking a different approach on entrenchment by defining inference directly on an entrenchment preorder. Here, the entrenchment relation becomes the primary basic notion and nonmonotonic inference takes a secondary higherorder place much like any consequence relation given some underlying proof theoretic mechanism. Entrenchment is not a proof mechanism as it lacks truth functionality but is essentially a priority preorder encoding our preferences. Inference can be roughly described as follows:
In order to evaluate a conditional drop all sentences that could imply . What remains are the sentences compatible with . Form all maximal consistent subsets and consider their intersection. Then holds if together with this set implies .
Similar proposals for evaluating conditionals have a long history in the philosophical logic literature going back to Lewis ([14]) (see [4] for a relevant discussion). Entrenchment is the mechanism for keeping track of this compatibility relation. A sentence is compatible with (we use coherent in Definition 3) if it is not less than . This is also the main idea of Gärdenfors and Makinson. The novelty of our work is that we consider partial preorders and show that the same way of evaluating conditionals still applies, giving rise to preferential inference. As partial preorders give a multitude of possible maximal compatible sets we consider their intersection, that is a ’sceptical’ sort of inference.
Consider the following simple example. In Figure 1, a path upwards from to indicates that , where denotes the entrenchment relation. The partial entrenchment of Figure 1 says, for example, that is less entrenched than all formulas, is less entrenched than , and , while is less entrenched than , and .
For instance, let us assume and suppose we want to extend the classical theory of , to a consistent theory. We can add any sentence to it, provided we do not add or any sentence implying . However, our entrenchment example says that apart from and any sentence implying we should exclude any sentence less than in the entrenchment relation. We shall see that the definition of entrenchment will ensure us that and all sentences stronger than are less than in the entrenchment relation. So we can use the entrenchment relation alone and exclude all sentences less than . So we are left with . We can add those to to form the extension .
Now, let us assume nothing but true sentences and see how we can extend . As before, we should only exclude formulas less or equal to . In this case, we cannot consider together all sentences that are not less or equal to , because this set of sentences is inconsistent. However, we can choose consistent subsets from this set. We must only take care that such sets are upper closed so that they obey the entrenchment relation constraints. Further, we want to add as many sentences as possible so these sets must be maximal. There are two such upperclosed maximal consistent sets of sentences: one contains and and the other and . Adding those to , we can form two extensions: and . Therefore, it is possible to have more than one alternative for extending the theory of our assumptions leading to the wellknown phenomenon of multiple extensions.
Considering nontruth functional orderings of sentences while respecting conjunction is rather an old idea, going back to Schackle ([20]), and used in different disguises in works of Levi ([13]), Cohen ([2]), Shafer ([21]), Zadeh ([25]), Spohn ([24]), and Dubois and Prade ([3]). The above authors use an ordering of sentences satisfying the partial entrenchment properties. However, they impose an additional constraint:
for all , either or . (Connectivity)
A partial entrenchment satisfying connectivity will be called connected. The important contribution of Gärdenfors and Makinson was to show that such connected preorders characterize exactly (not only define) expectation inference. Subsequently, the author showed that these orderings characterize also Lehmann and Magidor’s rational inference in [9]. The main contribution of this paper is showing that dropping the connectivity condition, the resulting class of orderings, that is, the class of partial entrenchment defined above, gives rise to preferential inference as a sceptical form of nonmonotonic inference.
Lindström and Rabinowicz ([15]) were the first to propose dropping connectivity from the GärdenforsMakinson connected entrenchment. Their epistemic entrenchment orderings form a subclass of partial entrenchment by satisfying additional postulates related to a fixed theory and were used for describing a relational belief revision system. Their approach is slightly different to ours as they require an overall consistent entrenchment. However, a common central idea of both approaches is that such relations point to more than one extension.
The linear preorder that Gärdenfors and Makinson introduced by the name of epistemic entrenchment had apart from connectivity two other important features: Transitivity and Dominance. Transitivity shows that we deal with a simple notion of transitive preference while Dominance shows that more specific sentences should be prefered over more general ones. These properties are the basic characteristics of entrenchment and form also a part of the definition of partial entrenchment.
There are at least two other previous attempts of characterizing nonmonotonic inference through some ordering of sentences. These are Michael Freund’s preferential orderings ([5]) and Hans Rott’s generalized epistemic entrenchments ([19]). Both have a similar approach giving a correspondence with nonmonotonic consequence relations^{1}^{1}1Strictly speaking, Rott is characterizing weaker than rational nonHorn belief contraction systems.. Both build on a syntactic condition that translates rational consequence relations to preorders. Hans Rott is using the Gärdenfors and Makinson condition on belief contractions while Freund is using the Kraus, Lehman and Magidor condition ( iff ). In order to generate a preferential inference relation they consider a translation of a connected entrenchment: Freund is using the contrapositive (page 236 in [5]) and Rott the complement of the inverse. Then they relax properties of the translated entrenchment. However this approach leads to preorders that if, they are translated back to entrenchment would fail either Dominance in Freund’s case (property , page 237 in [5]) or Transitivity in Rott’s case (SEE1, page 52 in [19]).
This loss of these properties is not however the main difference between the work presented here and those proposals. Those proposals insist on generating consequence relation in a deterministic way given a preferential ordering. In a partial setting, preference gives rise to more than one alternative, that is. a multitude of most preferred possible situations and the process of inferring statements becomes nondeterministic.
3 Maxiconsistent Inference
We shall now proceed in describing nonmonotonic inference through partial entrenchment. In defining inference, we shall make heavy use of negation, or better, of consistency. This is a very important point often overlooked by previous works on entrenchment. This is the only place where entrenchment makes effective use of the underlying logic, in our case, classical logic. Inference, as illustrated in the above example, consists of two steps. First, we exclude all sentences less than the negation of our assumption. Second, we choose maximal, upperclosed, consistent, deductively closed sets of sentences that form our extensions. Adding to those extensions the classical theory of our assumptions and closing under intersection yields the nonmonotonic theory of our assumptions. This procedure only makes sense for a finite set of assumptions, as negation plays a central role in its definition, so the resulting nonmonotonic consequence relation is a subset of .
A partial entrenchment relation is clearly a partial preorder. A subset of will be called upperclosed iff and implies . A subset of will be called closed under conjunction iff implies . An upperclosed, closed under conjunction, proper subset of , is a filter. A filter of the partial entrenchment is also a filter of the BooleanLindenbaum algebra of and, therefore, deductively closed, that is, . The converse is not true. A deductively closed set might fail to be a partial entrenchment filter. However, the upperclosure of is the least filter containing . This fact is a consequence of Dominance and Conjunction. Principal upper closed sets are filters and deductively closed, that is, .
Given a partial entrenchment, we shall denote its set of filters with . The space is itself a complete semilattice with intersection as meet. It has also directed joins because if two filters are included in a third then the intersection of all filters containing their union is again a filter. This kind of partial order is often called a dcpo.
Definition 2 Let be a partial entrenchment. The set of coherent sentences for a sentence is the set
The base of is the set
The maximal base of is the set
The extension set of is the set
The sceptical extension of is the set
and now define
and say that maxiconsistently infers in the partial entrenchment . ^{2}^{2}2Filters have been employed by Lindström and Rabinowicz for defining multiple revision outcomes. In [15], our and are called fallbacks and maximal fallbacks of .
Note that
Unless equals , i.e. the inconsistent ordering, is nonempty. As a corollary of Zorn’s lemma, every filter not containing is included in an element of . Therefore, if is nonempty then is nonempty. On the other hand, can be empty, even though is not inconsistent. This can only happen if , for all . In this case, we have that . In fact we have the following
The following properties of bases will be useful in the subsequent proofs.
Lemma 3
For all we have:

if then ,

,

,

,

if then ,

if and only if .
Proof. We have which implies . This shows that if is a filter and then and we conclude Part (1).
For the right to left direction of Part (2), use Part (1) to show that . For the other direction, observe that if then either or , since is closed under conjunctions. Hence and we conclude Part (2).
For the left to right inclusion of Part (3), assume . Observe that implies , else there exists such that and . We have , since . So , by Part 1, a contradiction.
For the other inclusion, assume
Let with . We have either or , by Part (2). In the first case, we have , as is lowerclosed, and this can only happen if . So and . The other case is similar and, therefore, .
Now, we turn to Part (5). By Part (4), we have . If , we are done. Suppose that , and let . Further, let such that . By our hypothesis, we have . Also, we have , since otherwise . Therefore . Hence .
For Part 6, suppose that then there exists such that . Therefore, and, since , . The other direction is similar.
It is worth noting that from the algebra of sentences we moved to the algebra of theories and onto the algebra of the powerset of theories. The last algebra is of considerable interest as is the algebra pertaining to nonmonotonic inference. For example, we could dispense with maximal filters and study directly the lattice of the powerset of . Our intention, however, is to introduce as little theory overhead as possible.
We now have everything we need for characterizing preferential inference. However, we should first verify our claim that maxiconsistent inference is a preferential one.
Theorem 4
Given a partial entrenchment , the consequence relation satisfies the system rules.
Proof. We verify the following list of rules: Supraclassicality, Left Logical Equivalence, Right Weakening, And, Cut, Cautious Monotony, and Or. ^{3}^{3}3Cut is redundant, see [11].
For Supraclassicality, suppose that then , for all .
For Left Logical Equivalence, suppose that and . By Lemma 3(1) we have that . So, for every filter , if then .
For And, suppose that and , for all . Then .
For Right Weakening, Suppose that for all we have and . Then by (classical) Cut we get .
For Cut, suppose that and . Suppose that then . By Lemma 3(5), we have and, therefore, . By (classical) Cut, we have . Hence .
For Cautious Monotony, suppose that and , and let . By Lemma 3(5), we have . Thus , and therefore . Hence .
For Or, suppose that and , and let . By Lemma 3(4), there are three cases to consider: either (i) , or (ii) with , or (iii) with . In case (i), we have and , so . In case (ii), we have , and therefore . Again , as above. Case (iii) is similar.
Given the above results we can now give a simple translation of the property of Rational Monotonicity.
Corollary 5
Let be a partial entrenchment. Then satisfies
if and only if
and implies .
A very natural subclass of partial entrenchments is the original class of connected entrenchments introduced by Gärdenfors and Makinson ([7],[8]). This class was shown to be equivalent to the class of rational nonmonotonic consequence relations ([9]) under the following translation:
iff either , for all , or there is a such that .
It is easy to see, by Definition 3, that the above way through which a connected entrenchment gives rise to a nonmonotonic consequence relation is a special instance of the definition of maxiconsistent inference, that is, . Now we can give an alternative proof of the fact that connected entrenchments give rise to rational nonmonotonic consequence relations by showing that a connected entrenchment satisfies the property of Lemma 5. In fact, it satisfies a much stronger property as the following lemma shows.
Lemma 6
If is a connected entrenchment then
implies .
Proof. This is immediate because if is connected then is either a singleton or empty for all .
In the next section we shall exhibit a class of nonconnected entrenchment relations that satisfy the property of Lemma 5, and therefore give rise to rational inference relations. We leave open the question whether there is a simple firstorder property of that relaxes connectivity and still implies the property of Lemma 5. The above corollary shows why maxiconsistent inference makes partial entrenchment a generalization of the Gärdenfors and Makinson original notion of entrenchment. It is well known that connected entrenchments not only give rise to rational inference but they are in bijective correspondence as well. Given a rational inference relation one can construct a connected entrenchment with using the translation below
iff either or ,
proposed in [9] which is a slightly modified version of the one proposed by GärdenforsMakinson for expectation inference relations (see [8]). In the case of partial entrenchment relations the above translation no longer works. In the next section, an alternative way to generate entrenchment given a preferential inference relation will be presented.
4 Reducing Preferential Inference to Partial Entrenchment
In this section, we show that every preferential consequence relation can be expressed as a maxiconsistent inference of a partial entrenchment. The class of maxiconsistent inference relations is much wider than that of preferential inference. Maxiconsistent inference expresses sceptical nonmonotonic consequence by an intersection of possible extensions. Therefore, we can construct two different partial entrenchments assigning different sets of extensions for the same assumptions while still agreeing on the intersection of the extensions.
Given a preferential inference relation, we will construct a partial entrenchment with the same maxiconsistent inference. This construction will be canonical, in the sense that one can safely identify a preferential inference relation with the partial entrenchment constructed. The main idea is to construct a partial entrenchment with all possible extensions of the sceptical extension. This way their intersection will also provide the sceptical extension. It turns out that such partial entrenchments can be described syntactically by adding the following rule to Dominance, Transitivity, and Conjunction. For all
if and then . (Weak Disjunction)
A partial entrenchment satisfying Weak Disjunction is called weakly disjunctive. The class of weakly disjunctive partial entrenchment is properly contained in that of partial entrenchments as the following simple counterexample shows.
Example 8 Let , and define an ordering as follows
The preorder is a partial entrenchment. However, it is not weakly disjunctive, for and but .
The main property of weakly disjunctive partial entrenchments is given in the following proposition.
Proposition 9
Let be a weak disjunctive partial entrenchment. Then for all and , either or .
Proof. Fix an and and suppose and , towards a contradiction. As is maximal in , we have . This implies that there exists such that . Similarly, there exists such that . For we have both and . Now observe that is classically equivalent to . So
Also, we have
So, by Conjunction,
Similarly,
Applying Weak Disjunction on the last two, we have
Therefore
Since , we have
a contradiction as .
Corollary 10
Let be a weak disjunctive partial entrenchment. Then
Proof. For the right to left direction, assume and . We have that , for all , and . We have . Choose such that . However, contains and therefore , a contradiction. Note that this direction does not use Weak Disjunction.
For the left to right direction, assume . We must show . Let . By Proposition 9, We have either or . We cannot have as so .
We can go back and forth between a preferential inference relation and a partial entrenchment through a syntactic translation given in the following definition.
Definition 11 Given a partial entrenchment and a nonmonotonic consequence relation , then define a consequence relation and a relation as follows
()  iff  
()  iff  . 
We shall denote and with and , respectively.
Definition is akin to a preorder defined in [11] by (see also Makinson’s comments in [17], page 78). The maps defined in Definition 10 are inverses of each other.
Lemma 12
Let and be a partial entrenchment and a preferential inference relation, respectively. Then

, and

.
Proof. Let . We have iff , where . Now, we have iff , by definition. The latter holds iff iff , by Dominance. Now, implies , by Transitivity, and implies , by Conjunction and Dominance.
Let . We have iff iff iff , by Left Logical Equivalence, iff , by And, Right Weakening and Reflexivity.
Theorem 13
If is a preferential inference relation, then the relation defined by () is a weakly disjunctive partial entrenchment relation such that, for all , in ,
Proof. We must only show that is a weakly disjunctive partial entrenchment.
For Dominance, suppose that . Thus , and so . Hence .
For Transitivity, suppose that and . By the definition above, these translate to and , respectively. Further, the following rule is derivable in the preferential system (Lemma 5.5 in [11])
So we have . Hence .
For Conjunction, suppose that and . We must show that . Our assumption translates to and , respectively. Applying Or and Left Logical Equivalence, we get . So . Hence .
For Weak disjunction, suppose and . These translate to and . By And, we have . The latter translates to as desired.
We can now give a bijective correspondence between the class of rational nonmonotonic consequence relations and weakly disjunctive partial entrenchments. It is enough to translate the property of Rational Monotonicity using :
Weakly disjunctive relations satisfying Splitting will be called rational.
Corollary 14
If is a rational inference relation, then the relation defined by () is a rational weakly disjunctive partial entrenchment relation such that, for all , in ,
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