General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations

Transcription

1 General Patterns for Nonmonotonic Reasoning: From Basic Entailments to Plausible Relations OFER ARIELI AND ARNON AVRON, Department of Computer Science, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Abstract This paper has two goals. First, we develop frameworks for logical systems which are able to reflect not only nonmonotonic patterns of reasoning, but also paraconsistent reasoning. Our second goal is to have a better understanding of the conditions that a useful relation for nonmonotonic reasoning should satisfy. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. These generalizations allow the use of monotonic nonclassical logics as the underlying logic upon which nonmonotonic reasoning may be based. Our sequence of frameworks culminates in what we call (following Lehmann) plausible, nonmonotonic, multiple-conclusion consequence relations (which are based on a given monotonic one). Our study yields intuitive justifications for conditions that have been proposed in previous frameworks and also clarifies the connections among some of these systems. In addition, we present a general method for constructing plausible nonmonotonic relations. This method is based on a multiple-valued semantics, and on Shoham s idea of preferential models. 1 1 Introduction Nonmonotonicity is generally considered as a desirable property in commonsense reasoning; Many approaches to basic problems in artificial intelligence such as belief revision, database updating, and action planning, rely in one way or another on some form of nonmonotonic reasoning. This led to a wide study of general patterns of nonmonotonic reasoning (see, e.g., [16, 18, 19, 24, 25, 26, 27, 28, 29, 39]). The basic idea behind most of these works is to classify nonmonotonic formalisms and to recognize several logical properties that nonmonotonic systems should satisfy. The logic behind most of the systems which were proposed so far is supraclassical, i.e.: every first-degree inference rule that is classically sound remains valid in the resulting logics. As a result, the consequence relations introduced in these works are not paraconsistent [11], that is: they are not capable of drawing conclusions from inconsistent theories in a nontrivial way. Moreover, the basic idea behind most of the nonmonotonic approaches is significantly different from the idea of paraconsistent reasoning: While the usual approaches to nonmonotonic reasoning rule out contradictions when a new data arrives in order to maintain the consistency of a knowledge-base, the paraconsistent approach to reasoning accepts knowledge-bases as they are, and tolerates contradictions in them, if such exist. Our goal in this paper is twofold. First, we want to develop frameworks for logical systems which will be able to reflect not only nonmonotonic patterns of reasoning, but also paraconsistent reasoning. Such systems will be useful also for reasoning with uncertainty, conflicts, 1 A preliminary version of this paper appeared in [4]. L. J. of the IGPL, Vol. 8 No. 2, pp c Oxford University Press

2 General Patterns for Nonmonotonic Reasoning 120 and contradictions. Our second goal is to have a better understanding of the conditions that a plausible relation for nonmonotonic reasoning should satisfy. The choice of the various conditions that have been proposed in previous works seem to us to be a little bit ad-hoc, making one wonder why certain conditions were adopted while others (that might seem not less plausible) have been rejected. We would like to remedy this. To achieve these goals, we consider a sequence of generalizations of the pioneering works of Gabbay [18], Kraus, Lehmann, Magidor [24], and Makinson [28]. These generalizations are based on the following ideas: Each nonmonotonic logical system is based on some underlying monotonic one. The underlying monotonic logic should not necessarily be classical logic, but should be chosen according to the intended application. If, for example, inconsistent data is not to be totally rejected, then an underlying paraconsistent logic might be a better choice than classical logic. The more significant logical properties of the main connectives of the underlying monotonic logic, especially conjunction and disjunction (which have crucial roles in monotonic consequence relations), should be preserved as far as possible. On the other hand, the conditions that define a certain class of nonmonotonic systems should not assume anything concerning the language of the system (in particular, the existence of appropriate conjunction or disjunction should not be assumed). The rest of this work is divided into two main sections. Section 2, the major one, is a study of nonmonotonic reasoning on the syntactical level. First we review the basic theory introduced in [24] (Section 2.1), which is based on a classical entailment relation and assumes the classical language. Then we consider nonmonotonic relations that are based on arbitrary entailment relations (Section 2.3). The next generalization (Section 2.4) uses Tarskian consequence relations [44] instead of just entailment relations. Finally, we consider multipleconclusion relations that are based on Scott consequence relations [37, 38] (Section 2.5). For defining the latter relations we indeed need not assume the availability of any specific connective in the underlying language. However, the hierarchy of relations which we consider is based first of all on the question: What properties of the conjunction and disjunction of the underlying monotonic logic are preserved in the nonmonotonic logic which is based on it. Our sequence of frameworks culminates in what we call (following [25]) plausible nonmonotonic consequence relations. We believe that this notion captures the intuitive idea of correct nonmonotonic reasoning. Section 3 provides a general semantical method for constructing plausible nonmonotonic consequence relations. This method is based on a combination of a lattice-valued semantics 2 with Shoham s idea of using only certain preferential models for drawing conclusions ([40, 41]). We show that some well-known plausible nonmonotonic logics can be constructed using this method. Most of these logics are paraconsistent as well (these include some logics that we have considered in previous works [2, 3, 5]). 2 This is a common method for dealing with inconsistent theories see, e.g., [13, 14, 15, 20, 21, 23, 34, 35, 39, 42, 43].

3 General Patterns for Nonmonotonic Reasoning Preferential systems from an abstract point of view In this section we investigate preferential reasoning from an abstract point of view. First we briefly review the original treatments of Makinson [28] and Kraus, Lehmann, and Magidor [24]. Then we consider several generalizations of this framework. 2.1 The standard basic theory A general overview The language that is considered in [24, 28] is based on the standard propositional one. Here, denotes the material implication (i.e., ) and denotes the corresponding equivalence operator (i.e., µ µ). The classical propositional language, with the connectives,,,,, and with a propositional constant Ø, is denoted here by Ð. An arbitrary language is denoted by. Given a set of formulae in a language, we denote by µ the set of the atomic formulae that occur in, and by Ä µ the corresponding set of literals. Definition 2.1 [24] Let Ð be the classical consequence relation. A binary relation 3 ¼ between formulae in Ð is called cumulative if it is closed under the following inference rules: reflexivity: ¼. cautious monotonicity: if ¼ and ¼, then ¼. cautious cut: if ¼ and ¼, then ¼. left logical equivalence: if Ð and ¼, then ¼. right weakening: if Ð and ¼, then ¼. Definition 2.2 [24] A cumulative relation ¼ is called preferential if it is closed under the following rule: -introduction (Or): if ¼ and ¼, then ¼. Note In order to distinguish between the rules of Definitions 2.1, 2.2, and their generalized versions that will be considered in the sequel, the condition above will usually be preceded by the string KLM. Also, a relation that satisfies the rules of Definition 2.1 [Definition 2.2] will sometimes be called KLM-cumulative [KLM-preferential]. The conditions above might look a little-bit ad-hoc. For example, one might ask why is used on the right, while the stronger is on the left. A discussion and some justification appears in [24, 27]. 4 A stronger intuitive justification will be given below, using more general frameworks. 2.2 Generalizations In the sequel we will consider several generalizations of the basic theory presented above: 1. In their formulation, [23, 24, 28, 29] consider the classical setting, i.e.: the basic language is that of the classical propositional calculus ( Ð ), and the basic entailment relation is the 3 A conditional assertion in terms of [24]. 4 Systems that satisfy the conditions of Definitions 2.1, 2.2, as well as other related systems, are also considered in [16, 26, 29, 39].

4 General Patterns for Nonmonotonic Reasoning 122 classical one ( Ð ). Our first generalization concerns with an abstraction of the syntactic components and the entailment relations involved: Instead of using the classical entailment relation Ð as the basis for definitions of cumulative nonmonotonic entailment relations, we allow the use of any entailment relation which satisfies certain minimal conditions. 2. The next generalization is to use Tarskian consequence relations instead of entailment relations (i.e. we consider the use of a set of premises rather than a single one). These consequence relations should satisfy some minimal conditions concerning the availability of certain connectives in their language. Accordingly, we consider cumulative and preferential nonmonotonic consequence relations that are based on those Tarskian consequence relations. 3. We further extend the class of Tarskian consequence relations on which nonmonotonic relations can be based by removing almost all the conditions on the language. The definition of the corresponding notions of a cumulative and a preferential nonmonotonic consequence relation is generalized accordingly. 4. Our final generalization is to allow relations with multiple conclusions rather than the single conclusion ones. Within this framework all the conditions on the language can be removed. 2.3 Entailment relations and cautious entailment relations In what follows denote arbitrary formulae in a language, and denote finite sets of formulae in. Definition 2.3 A basic entailment is a binary relation ½ between formulae, that satisfies the following conditions: R 1-reflexivity: ½. 1C 1-cut: if ½ and ½ then ½. Next we generalize the propositional connectives used in the original systems: Definition 2.4 Let ½ be some basic entailment. A connective is called a combining conjunction (w.r.t. ½ ) if the following condition is satisfied: ½ iff ½ and ½. A connective is called a combining disjunction (w.r.t. ½ ) if the following condition is satisfied: ½ iff ½ and ½. From now on, unless otherwise stated, we assume that ½ is a basic entailment, and is a combining conjunction w.r.t. ½. Definition The 1 means that exactly one formula should appear on both sides of this relation. 6 It could have been convenient to assume also that ½ is closed under substitutions of equivalents, but here we allow cases in which this is not the case. 7 These conditions mean, actually, that basic entailment induces a category in which the objects are formulae.

5 General Patterns for Nonmonotonic Reasoning 123 A connective is called a -combining disjunction (w.r.t. ½ ) if it is a combining disjunction and: µ ½ iff ½ and ½. A connective is called a -internal implication (w.r.t. ½ ) if the following condition is satisfied: ½ iff ½. A constant Ø is called a -internal truth (w.r.t. ½ ) if the following condition is satisfied: Ø ½ iff ½. Definition 2.6 a) A formula is a conjunct of a formula if, or if ½ ¾ and is a conjunct of either ½ or ¾. b) For every ½ Ò is called a semiconjunction of ½ Ò ; If ¼ and ¼¼ are semiconjunctions of ½ Ò then so is ¼ ¼¼. c) A conjunction of ½ Ò is a semiconjunction of ½ Ò in which every appears at least once as a conjunct. Lemma 2.7 (Basic properties of ½ and ) a) ½ is monotonic: If ½ then ½ and ½. b) If is a conjunct of then ½. c) If is a conjunction of ½ Ò and ¼ is a semiconjunction of ½ Ò then ½ ¼. d) If and ¼ are conjunctions of ½ Ò then and ¼ are equivalent: ½ ¼ and ¼ ½. e) If ½ and ½ then ½. PROOF. For part (a), suppose that ½. By 1-reflexivity, ½. Since is a combining conjunction, ½. A 1-cut with ½ yields ½. The case of is similar. We leave the other parts to the reader. Notation 2.8 Let ½ Ò. Then and ½ Ò will both denote any conjunction of all the formulae in. Note Because of Lemma 2.7 (especially part (d)), there will be no importance to the order according to which the conjunction of elements of is taken in those cases below in which we use Notation 2.8. Notation 2.9 µ µ Lemma 2.10 (Basic properties of ½ and, Ø) Let be a -internal implication w.r.t. ½ and let Ø be a -internal truth w.r.t. ½. Then: a) If Ø ½ then ½. b) ½ Ø for every formula. c) ½ iff ½.

6 General Patterns for Nonmonotonic Reasoning 124 d) ½ iff Ø ½. Also, ½ and ½ iff Ø ½. e) If ½ then Ø ½ µ µ; If ½ then Ø ½ µ µ. f) If ½ ¾ are conjunctions of the same set of formulae then Ø ½ ½ ¾. g) If ½ and ½ then ½. PROOF. All the parts of the lemma are easily verified. We only give a proof of the first claim of part (e): If ½, then ½. By Lemma 2.7(a), ½. Thus ½ (combining conjunction), and so Ø ½ µ µ by part (d). Lemma 2.11 Let be a combining disjunction w.r.t. ½. a) is a -combining disjunction iff the following distributive law obtains: ½ ¾ µ ½ ½ µ ¾ µ b) If ½ has a -internal implication then is a -combining disjunction. PROOF. Part (a) is based on the facts that ½, ½, ½, and ½ (see the proof of Lemma 2.7(a)). We leave the details to the reader. Part (b) follows from (a), since it is easy to see that if ½ has a -internal implication then the above distributive law holds. Note It is easy to see that the converse of the distributive law above, i.e. that ½ µ ¾ µ ½ ½ ¾ µ is true whenever and are, respectively, a combining conjunction and a combining disjunction w.r.t. ½. Definition 2.12 Suppose that a language of a basic entailment ½ contains a combining conjunction, a -internal implication, and a -internal truth Ø. A binary relation ½ between formulae in is called Ø ½ -cumulative if it satisfies the following conditions: ½. if ½ and ½, then ½. if ½ and ½, then ½. if Ø ½ and ½, then ½. if Ø ½ and ½, then ½. Note In our notations, a KLM-cumulative relation (Definition 2.1) is Ø Ð -cumulative. Lemma 2.10(d) allows us to further generalize the notion of a cumulative relation so that only the availability of a combining conjunction is assumed:

7 General Patterns for Nonmonotonic Reasoning 125 Definition 2.13 A binary relation ½ between formulae is called ½ -cumulative if it satisfies the following conditions: 1R 1-reflexivity:. ½ 1CM 1-cautious monotonicity: if ½ and, ½ then. ½ 1CC 1-cautious cut: if ½ and, ½ then. ½ 1LLE 1-left logical equivalence: if ½ and ½ and ½, then. ½ 1RW 1-right weakening: if ½ and, ½ then. ½ If, in addition, is a -combining disjunction w.r.t. ½, and ½ satisfies the following rule: 1Or 1- introduction: if ½ and ½, then ½ then ½ is called ½ -preferential. Proposition 2.14 Let be a -internal implication w.r.t. ½ and let Ø be a -internal truth w.r.t. ½. Then a relation is Ø ½ -cumulative iff it is ½ -cumulative. PROOF. Follows easily from Lemma Note From the note after Definition 2.12 and the last proposition it follows that in a language containing Ð, ½ is a KLM-preferential relation (Definition 2.2) iff it is Ø Ð - preferential. Proposition 2.15 Every ½ -cumulative relation ½ is an extension of its corresponding basic entailment: If ½ then ½. PROOF. By 1RW of ½ and ½. Proposition 2.16 Let ½ be a ½ -cumulative relation. Then: a) is a combining conjunction also w.r.t. ½ : ½ iff ½ and ½. b) If Ø is a -internal truth w.r.t. ½ then it is also a -internal truth w.r.t. ½ : Ø ½ iff ½. PROOF. a) µ: Suppose that ½ and ½. Then by 1CM, [1]: ½. On the other hand, by Lemma 2.7(c), ½, and so by Proposition 2.15, [2]: ½. A 1CC, of [1] and [2] yields. Another 1CC with ½ yields that ½. µµ: Suppose that ½. By Lemma 2.7(c), µ ½. By Proposition 2.15 µ ½. A 1CC with ½ yields that ½. Similarly, if ½ then ½. b) By Lemma 2.10(b) and Proposition 2.15, ½ Ø. Now, suppose that ½. A 1CM with

8 General Patterns for Nonmonotonic Reasoning 126 ½ Ø yields Ø ½. For the converse, assume that Ø ½. A 1CC with ½ Ø yields ½. Note Unlike and Ø, in general and do not always remain a -internal implication and a combining disjunction w.r.t ½. Counter-examples will be given in Section 3 (see Proposition 3.24 and the note that follows it). It is possible to strengthen the conditions in Definition 2.13 as follows: s-1r strong 1R: if is a conjunct of then ½. s-1rw strong 1RW: if ½ and ½, then ½. Our next goal is to show that these stronger versions are really valid for any ½ - cumulative relation. Moreover, each property is in fact equivalent to the corresponding property under certain conditions, which are specified below. Proposition 2.17 a) 1RW and s-1rw are equivalent in the presence of 1R and 1CC. b) 1RW and s-1r are equivalent in the presence of 1R, 1CC, and 1LLE. PROOF. a) The fact that s-1rw implies 1RW follows from Lemma 2.7(a). For the converse assume that ½. By Proposition 2.15 (the proof of which uses only 1R and 1RW), ½. A 1CC with ½ yields ½. b) Suppose that ½ and ½. From Lemma 2.7 it easily follows that the first assumption entails that ½ and ½. By s-1r, ½. A 1LLE of the last three sequents yields ½. Finally, by 1CC with ½ we get ½. In the other direction s-1r is obtained from 1RW as follows: Let be a conjunct of. By Lemma 2.7(b) ½. A 1RW with ½ yields that ½. Corollary 2.18 a) s-1r and s-1rw are equivalent in the presence of 1R, 1CC, and 1LLE. b) A relation is ½ -cumulative if it satisfies s-1r, 1LLE, 1CM, and 1CC. PROOF. Immediate from Proposition 2.17 and the fact that s-1r entails 1R. 2.4 Tarskian consequence relations and Tarskian cautious consequence relations The next step in our generalizations is to allow several premises on the l.h.s. of the consequence relations. Definition 2.19

9 General Patterns for Nonmonotonic Reasoning 127 a) A (ordinary) Tarskian consequence relation [44] (tcr, for short) is a binary relation between sets of formulae and formulae, that satisfies the following conditions: 8 s-tr strong T-reflexivity: for every ¾. TM T-monotonicity: if and ¼ then ¼. TC T-cut: if ½ and ¾ then ½ ¾. b) A Tarskian cautious consequence relation (tccr, for short) is a binary relation between sets of formulae and formulae in a language, that satisfies the following conditions: 9 s-tr strong T-reflexivity: for every ¾. TCM T-cautious monotonicity: if and, then. TCC T-cautious cut: if and, then. Proposition 2.20 Any tccr is closed under the following rules for every Ò: TCM Ò if ½ Òµ then ½ Ò ½ Ò. TCC Ò if ½ Òµ and ½ Ò, then. PROOF. We show closure under TCM Ò by induction on Ò. The case Ò ½ is trivial, and TCM ¾ is simply TCM. Now, assume that TCC Ò is valid and for ½ Ò ½. By induction hypothesis ½ Ò ½ Ò and ½ Ò ½ Ò ½. Hence ½ Ò Ò ½ by TCM. The proof of TCC Ò is also by induction on Ò. TCC ½ is just TCC. Assume now that ½ Ò ½µ and ½ Ò Ò ½. By TCM Ò ½ ½ Ò Ò ½. A TCC of the last two sequents gives ½ Ò. Hence by induction hypothesis. The following definition is the multiple-assumptions analogue of Definition 2.4: Definition 2.21 Let be a relation between a set of formulae and a formula in a language. A connective is called combining conjunction (w.r.t. ) if the following condition is satisfied: iff and. A connective is called internal conjunction (w.r.t. ) if the following condition is satisfied: iff. A connective is called combining disjunction (w.r.t. ) if the following condition is satisfied: iff and. In what follows we assume that is a tcr and is a combining conjunction with respect to. Lemma 2.22 (Basic properties of and ) a) If then. b) If then. c) If is a conjunction of ½ Ò and ¼ is a semiconjunction of ½ Ò then ¼. d) If and ¼ are conjunctions of ½ Ò then and ¼ are equivalent: ¼ and ¼. 8 The prefix T denotes that these are Tarskian rules. 9 A set of conditions which is similar to the one below was first proposed in [19], except that instead of cautious cut Gabbay uses cut.

10 General Patterns for Nonmonotonic Reasoning 128 e) If then iff. f) is an internal conjunction w.r.t.. PROOF. Similar to that of Lemma 2.7. Our next goal is to generalize the notion of cumulative entailment relation (Definition 2.13). We shall first do it for consequence relations that have a combining conjunction. Definition 2.23 A tccr is called -cumulative if it satisfies the following conditions: w-tlle weak T-left logical equivalence: if and and, then. w-trw weak T-right weakening: if and, then. TICR T-internal conjunction reduction: for every, iff. If, in addition, has a combining disjunction, and satisfies TOr T--introduction: if and, then then is called -preferential. Notes 1. Because of Proposition 2.22 and w-tlle, it again does not matter what conjunction of is used in TICR. 2. Condition TICR is obviously equivalent to the requirement that is an internal conjunction w.r.t. (see Definition 2.21). Proposition 2.24 In the definition of -cumulative tccr one can replace condition s-tr with the following weaker condition: TR T-reflexivity:. PROOF. Let ¾. A w-t-rw of and yields. By TICR,. We now show that the concept of a -cumulative tccr is equivalent to the notion of ½ -cumulative relation: Definition 2.25 Let ½ be a basic entailment with a combining conjunction. Let ½ be a ½ -cumulative relation. Define two binary relations ½ µ ¼ and ½ µ ¼ between sets of formulae and formulae in a language as follows: a) ½ µ ¼ iff either and ½, or and ½ for every. b) ½ µ ¼ iff and ½. 10 Definition 2.26 Let be a tcr with a combining conjunction. Suppose that is a -cumulative tccr. Define two binary relations µ and µ between formulae in as follows: 10 Since ½ is ½ -cumulative, it satisfies, in particular, 1LLE. Hence, the order in which the conjunction of is taken has no importance (see Lemma 2.7d). Thus ½ µ ¼ is well-defined.

11 General Patterns for Nonmonotonic Reasoning 129 a) µ iff. b) µ iff. Proposition 2.27 Let ½, ½,, and be as in the last two definitions. Then: a) ½ µ ¼ is a tcr for which is a combining conjunction. b) ½ µ ¼ is a ½ µ ¼ -cumulative tccr. c) µ is a basic entailment for which is a combining conjunction. d) µ is a µ -cumulative entailment. e) ½ µ ¼ µ ½. f) ½ µ ¼ µ ½. g) If is a normal tcr (i.e., if then ), then µ µ ¼. h) If then µ µ ¼ iff. i) If is a -combining disjunction w.r.t. ½ and ½ satisfies 1-Or, then ½ µ ¼ is preferential. j) If is a combining disjunction w.r.t. and satisfies T-Or, then µ is ½ - preferential. PROOF. All the parts of the claim are easily verified. We show parts (h) and (i) as examples: (h): Suppose that. Then µ µ ¼ iff µ iff, iff (by TICR). (i): By (b) we only need to show that ½ µ ¼ satisfies TOr. So assume that ½ ¾ Ò ½ µ ¼ and ½ ¾ Ò ½ µ ¼. Then ÎÒ ½ µ ½ and ÎÒ ½ µ ½. By 1-Or, ÎÒ ½ µ µ ÎÒ ½ µ µ ½. By Lemma 2.11, the note that follows it, and 1-LLE, ÎÒ ½ µ µ ½. Thus, ½ ¾ Ò ½ µ ¼. Corollary 2.28 Suppose that is Ð -cumulative [ Ð -preferential]. Define ½ iff. Then w.r.t. Ð, ½ is cumulative [preferential] in the sense of [24] (Definitions 2.1 and 2.2). We next generalize the definition of a cumulative tccr to make it independent of the existence of any specific connective in the language. In particular, we do not want to assume anymore that a combining conjunction is available. Proposition 2.29 Let be a tcr, and let be a tccr in the same language. The following connections between and are equivalent: TCum T-cumulativity: for every, if then. TLLE T-left logical equivalence: if and and, then. TRW T-right weakening: if and, then. TMiC T-mixed cut: for every, if and, then. PROOF. We show that each property is equivalent to TCum: TCum µ TLLE: Suppose that and. By TCum we have that and. A T-cautious monotonicity of the first sequent with yields,

12 General Patterns for Nonmonotonic Reasoning 130 and by T-cautious cut with we are done. TLLE µ TCum: Let ¾, and suppose that. This entails that. Also, by s-r,. Since then by TLLE we have that. But ¾, so. TCum µ TRW: Suppose that. By TCum. TCC with yields. TRW µ TCum: Suppose that and. Then there exists some ¾, and so. By s-tr,, and by TRW. TCum µ TMiC: If is a nonempty set of assertions s.t., then by TCum,. A T-cautious cut of this sequent and gives. TMiC µ TCum: Suppose that, and by TMiC,. Notes is a nonempty set of assertions and. By T-reflexivity, 1. If there is a formula s.t., then one can remove the requirement from the definition of TCum. Indeed, suppose that. If then. Since the l.h.s. of the last entailment is nonempty, then by the original version of Cum,, and by TCC with we have. The other direction is, however, not true: Let, for instance, be some tcr for which there exists ¼ s.t. ¼. Define if and. It is easy to verify that all the conditions of Definition 2.19 as well as TCum are valid for this, but ¼. 2. Being the complement of TMiC, one might consider TRW as another kind of mixed cut. Definition 2.30 Let be a tcr. A tccr in the same language is called -cumulative if it satisfies any of the conditions of Proposition If, in addition, has a combining disjunction, and satisfies TOr, then is called -preferential. Note Since for every ¾, TCum implies s-tr, and so a binary relation that satisfies TCum, TCM, and TCC is a -cumulative tccr. Proposition 2.31 Suppose that is a tcr with a combining conjunction. A tccr is a -cumulative iff it is -cumulative. If has also a combining disjunction, then is -preferential iff it is -preferential. For proving Proposition 2.31 we first show the following lemmas: Lemma 2.32 Suppose that is a tcr with a combining conjunction, and let be a - cumulative tccr. Then Î Ò ½ iff ½ ¾ Ò. PROOF. For the proof we need two simple claims: Î Ò Claim 2.32-A: ½ ¾ Ò ½. Proof: Clearly, ½ ¾ Î Ò ½ Î Î Ò Î Ò ½ Ò and ½ ¾ Ò ½ ½ Î Ò. Now, Ò since ½ ¾ Ò ½ ½ Ò ½ Ò Î, then by TLLE, Î ½ ¾ Ò ½. Ò Claim 2.32-B: Let ½ Ò. Then ½ Ò iff ½. Proof: µµ Î Follows Î Î Î Î Î Ò by applying TLLE on ½ Ò ½ Ò, and ½ Ò ½ Ò, and ½ Ò ½. µ Î Î Ò By applying TLLE on Î Î ½ Ò Î Î Î ½ Ò and ½ Î Ò ½, and, Ò, ½ Ò, we get that, ½ Ò, ½ Ò. Thus, ½.

13 General Patterns for Nonmonotonic Reasoning 131 Lemma 2.32 now easily follows Î Î Ò from the above claims: If ½ then by repeated applications of Claim 2.32-B, Ò ½ ½ ¾ Ò. A T-cautious cut with the property of Claim 2.32-A yields ½ ¾ Ò. For the converse suppose that ½ ¾ Ò. By T-cautious monotonicity with the property Î Î Ò of Claim 2.32-A, ½ ½ ¾ Ò Ò, and by Claim 2.32-B (applied Ò times), ½. Lemma 2.33 Let be a -cumulative relation. Then satisfies TRW. PROOF. Suppose that. By Lemma 2.22(e) µ. Since (s-r), then by w-trw we have that µ. By TICR,, and a TCC with yields that. Note In fact, we have proved a stronger claim, since in the course of the proof we haven t used CM and w-tlle. Now we can show Proposition 2.31: PROOF. of Proposition 2.31 µ Suppose that is a -cumulative tccr. It obviously satisfies w-tlle and w-trw (take and, respectively). Lemma 2.32 shows that also satisfies TICR. Thus is a -cumulative tccr. µµ Suppose that is a -cumulative tccr. By Lemma 2.33 it satisfies TRW, and so it is -cumulative. We leave the second part concerning to the reader. Corollary 2.34 Let be a -cumulative relation, and let be a combining conjunction w.r.t.. Then is a combining conjunction w.r.t. as well. PROOF. For a -cumulative relation the proof is similar to that of Proposition 2.16(a). Hence the claim follows from Proposition Another characterization of -cumulative tccr which resembles more that of a cumulative entailment (Definition 2.13) is given in the following proposition: Proposition 2.35 A relation is a -cumulative tccr iff it satisfies TR, TCM, TCC, TLLE and TRW. PROOF. If is a -cumulative tccr then by Proposition 2.29 and the fact that s-tr implies TR, it obviously has all the above properties. The converse follows from the fact that TRW and s-tr are equivalent in the presence of TR, TCC, and TLLE. The proof of this fact is similar to that of Proposition Scott consequence relations and Scott cautious consequence relations The last generalization that we consider in this section concerns with consequence relations in which both the premises and the conclusions may contain more than one formula. Definition 2.36 a) A Scott consequence relation [37, 38] (scr, for short) is a binary relation between sets of formulae, that satisfies the following conditions: s-r strong reflexivity: if then. M monotonicity: if and ¼, ¼ then ¼ ¼. C cut: if ½ ½ and ¾ ¾ then ½ ¾ ½ ¾.

14 General Patterns for Nonmonotonic Reasoning 132 b) A Scott cautious consequence relation (sccr, for short) is a binary relation between nonempty 11 sets of formulae, that satisfies the following conditions: s-r strong reflexivity: if then. CM cautious monotonicity: if and then. CC ½ cautious 1-cut: if and then. The following definition is a natural analogue for the multiple-conclusion case of Definition 2.21: 12 Definition 2.37 Let be a relation between sets of formulae. A connective is called combining conjunction (w.r.t. ) if the following condition is satisfied: iff and. A connective is called internal conjunction (w.r.t. ) if the following condition is satisfied: iff. A connective is called combining disjunction (w.r.t. ) if the following condition is satisfied: iff and. A connective is called internal disjunction (w.r.t. ) if the following condition is satisfied: iff. Note Again, it can be easily seen that if is an scr then is an internal conjunction iff it is a combining conjunction, and similarly for. This, however, is not true in general. A natural requirement from a Scott cumulative consequence relation is that its singleconclusion counterpart will be a Tarskian cumulative consequence relation. Such a relation should also use disjunction on the r.h.s. like it uses conjunction on the l.h.s. The following definition formalizes these requirements. Definition 2.38 Let be an scr with a combining disjunction. A relation between nonempty finite sets of formulae is called -cumulative sccr if it is an sccr that satisfies the following two conditions: a) Let Ì and Ì be, respectively, the single-conclusion counterparts of and (i.e., Ì iff and Ì iff ). Then Ì is a tcr and Ì is a Ì -cumulative tccr. b) For ½ Ò, denote by (or by ½ Ò ) any disjunction of all the formulae in. 13 Then for every, satisfies the following property: 14 IDR internal disjunction reduction: iff. Following the line of what we have done in the previous section, we next specify conditions that are equivalent to those of Definition 2.38, but are independent of the existence of any specific connective in the language. In particular, we do not want to assume anymore that a combining disjunction is available: 11 The condition of non-emptiness is just technically convenient here. It is possible to remove it with the expense of complicating somewhat the definitions and propositions. It is preferable instead to employ (whenever necessary) the propositional constants Ø and to represent the empty l.h.s. and the empty r.h.s., respectively. 12 This definition is taken from [7]. Definitions 2.4 and 2.21 are obvious adaption of it. 13 It easily follows from (a) above and from the properties of in that the order according to which is taken has no importance here. 14 This property is dual to the property of internal conjunction reduction (TICR, see Definition 2.23) of a -cumulative tccr.

15 General Patterns for Nonmonotonic Reasoning 133 Definition 2.39 Let be an scr. An sccr in the same language is called weakly - cumulative if it satisfies the following conditions: Notes Cum cumulativity: if and, then. RW ½ right weakening: if and then. RM right monotonicity: if then. 1. Since, Cum implies s-r, and so a binary relation that satisfies Cum, CM, CC ½, RW ½, and RM, is a weakly -cumulative sccr. 2. Any weakly -cumulative relation satisfies the following condition: LLE left logical equiv.: if and and then Indeed, by Cum on we have that, and CM with yields. Also, since then by Cum. A CC ½ with yields. Proposition 2.40 Let and be as in Definition A relation is a -cumulative sccr iff it is a weakly -cumulative sccr. PROOF. µ Since is an scr, Ì is obviously a tcr. Also, since is a weakly -cumulative sccr, it satisfies s-r, CM, CC ½, and Cum, thus Ì obviously satisfies s-tr, TCM, TCC and TCum, therefore Ì is a Ì -cumulative tccr. It remains to show that satisfies IDR: Suppose first that for. Since, then by Cum,. A CC ½ with yields. For the converse, we first show that if then. Indeed, RW ½ of and yields. Another RW ½ with yields. Thus,. Now, by an induction on the number of formulae in it follows that if and, then. µµ Let be a -cumulative sccr. Suppose that and. Then. Hence Ì, and since Ì is a Ì -cumulative tccr, Ì. Thus, and by IDR,. This shows that satisfies Cum. For RW ½, assume that and. Since is an scr and is a combining disjunction for it, the first assumption implies that µ µ. By IDR the second assumption implies that µ. Hence µ Ì µ and Ì µ. By TRW (see Proposition 2.29) applied to Ì we get Ì µ. Hence µ. By IDR again,. It remains to show that satisfies RM. Suppose then that and let Æ ¾. Then Æ, and RW ½ with Æ Æ yields Æ. Using IDR it easily follows that Æ, and since Æ ¾ we have that. Note A careful inspection of the proof of Proposition 2.40 shows that if a combining disjunction is available for, then RM follows from the other conditions for a weakly -cumulative sccr. It follows that in this case Cum, CM, CC ½, and RW ½ suffice for defining a weakly -cumulative sccr. The last proposition and its proof show, in particular, the following claim: Corollary 2.41 Let be an scr with a combining disjunction, and let be a weakly - cumulative sccr. Then is an internal disjunction w.r.t..

16 General Patterns for Nonmonotonic Reasoning 134 Part (a) of the following proposition shows that a similar claim about conjunction also holds: Proposition 2.42 Let be an scr with a combining conjunction, and let be a weakly -cumulative sccr. Then: a) is an internal conjunction w.r.t.. I.e., satisfies the following property: ICR internal conjunction reduction: for every, iff b) is a half combining conjunction w.r.t.. I.e, the following rules are valid for : 15 PROOF. a) The proof is similar to that of in the Tarskian case (see Lemma 2.32 and Note 2 after Definition 2.39), using instead of. b) is obtained by applying RW ½ to and. Similarly for. Note Clearly, the condition ICR in part (a) of Proposition 2.42 is equivalent to the following conditions: Á Definition 2.43 Suppose that an scr has a combining conjunction. A weakly -cumulative sccr is called -cumulative if it satisfies the following condition: Á Corollary 2.44 If is an scr with a combining conjunction and is a -cumulative sccr, then is a combining conjunction w.r.t. as well. PROOF. Follows from Proposition 2.42(b). As usual, we provide an equivalent notion in which one does not have to assume that a combining conjunction is available: Definition 2.45 A weakly -cumulative sccr is called -cumulative if for every finite Ò the following condition is satisfied: RW Ò if ½ Òµ and ½ Ò then. Proposition 2.46 Let be a combining conjunction for. An sccr is -cumulative iff it is -cumulative. PROOF. We have to show that if is a combining conjunction w.r.t., then RW Ò is equivalent to Á. Suppose first that satisfies Á. From ½ Òµ it follows, by Á, that ½ Ò. From ½ Ò it follows that ½ Ò. By a RW ½ on these two sequents,. For the converse, assume that and. Since, RW ¾ yields that. 15 The subscripts I and E in the following rules stand for Introduction and Elimination, respectively.

17 General Patterns for Nonmonotonic Reasoning 135 Corollary 2.47 If is an scr with a combining conjunction and is a -cumulative sccr, then is a combining conjunction and an internal conjunction w.r.t.. PROOF. By Proposition 2.42(a), Corollary 2.44, and Proposition Next we consider the dual property, i.e.: conditions for assuring that a combining disjunction w.r.t. an scr will remain a combining disjunction w.r.t. a weakly -cumulative sccr. Our first observation is that one direction of the combining disjunction property for of yields monotonicity of : Lemma 2.48 Suppose that is a combining disjunction for and is a weakly -cumulative sccr. Suppose also that satisfies the following condition: Then is (left) monotonic. PROOF. Suppose that, and let ¾. Then. Since we have also. Hence, by CM,. By this implies that and so. It follows that requiring from a weakly -cumulative sccr is too strong. It is reasonable, however, to require the other direction of the combining disjunction property: Definition 2.49 A weakly -cumulative sccr is called weakly -preferential if it satisfies the following condition, (also denoted by Á ): Or left -introduction: if and, then. Unlike in the Tarskian case, this time we are able to provide an equivalent condition in which one does not have to assume that a combining disjunction is available: Definition 2.50 Let be an scr. A weakly -cumulative sccr is called weakly -preferential if it satisfies the following rule: CC cautious cut: if and then. Proposition 2.51 Let be an scr and let be a weakly -cumulative sccr. Then is a weakly -preferential sccr iff for every finite Ò it satisfies cautious Ò-cut: CC Ò if ½ Òµ and ½ Ò then. PROOF. µ We have to show that satisfies CC. Suppose that Æ ½ Æ for some ½. Since for every ½ we have that Æ and since by assumption, a cautious ½µ-cut of these ½ sequents with yields that. µµ Suppose that satisfies CC. We show the following stronger condition by induction on Ò: If ½ Ò ¼ and ½ Òµ then ¼ ½ Ò

18 General Patterns for Nonmonotonic Reasoning 136 For the case Ò ½, assume that ½ ¼ and ½ ½. By RM on each sequent we have that ½ ¼ ½ and ½ ¼ ½. A CC gives the desired result. Assume the claim for Ò; We prove it for Ò ½: Suppose that for ½ Ò ½ and ½ Ò ½ ¼. By induction hypothesis applied to the last sequent and, for ½ Ò, we get ¼ ½ Ò Ò ½. From this and Ò ½ Ò ½ we get that ¼ ½ Ò ½ like in the case of Ò ½. Note By Proposition 2.20, the single conclusion counterpart of CC Ò is valid for any sccr (not only the cumulative or preferential ones). Proposition 2.52 Let be an scr with a combining disjunction. A weakly -cumulative sccr satisfies Or iff it is closed under CC Ò for every finite Ò. PROOF. Suppose first that satisfies Or. Then from ½ Òµ it easily follows that ½ Ò. On the other hand, ½ Ò follows from ½ Ò by IDR and Proposition Thus, by CC ½. For the converse, suppose that is a weakly -cumulative sccr that satisfies CC Ò for every finite Ò, and suppose that and. Now, since then by Cum, and CM with yields [1]:. Similarly, since then by Cum and CM with we have [2]:. Also, since then by Cum, [3]:. A CC ¾ of [1], [2], and [3] yields. Corollary 2.53 et be an scr with a combining disjunction. An sccr is weakly preferential iff it is weakly -preferential. PROOF. By Propositions 2.51 and Proposition 2.54 Let be an scr. Then is weakly -preferential iff it satisfies Cum, CM, CC, and RM. PROOF. One direction is obvious. For the other direction, we have to show that if satisfies the above conditions then it also satisfies RW ½ and CC ½. For RW ½, assume that and. By Cum and RM on the first assumption,. By RM on the second assumption,. A CC on the last two sequents yields. We leave the proof of CC ½ to the reader. Corollary 2.55 Let be an scr. A relation is a weakly -preferential iff it satisfies Cum, CM, and the following rule: s-ac strong additive cut: if ½ and ¾ then ½ ¾ PROOF. Suppose first that satisfies Cum, CM, and s-ac. By Proposition 2.54 we have to show that satisfies CC and RM. CC is obtained by taking ½ ¾ in s-ac. For RM, Suppose that and let Æ ¾. Then Æ. On the other hand, since Æ Æ, then by Cum, Æ Æ. s-ac with Æ yields. For the converse, suppose that is a weakly -preferential sccr for which ½ and ¾. By RM, ½ ¾ and ½ ¾. Thus, ½ ¾, by CC. We are now ready to introduce our strongest notions of nonmonotonic Scott consequence relation: Definition 2.56 Let be an scr. An sccr is called -preferential iff it satisfies Cum, CM, CC, RM, and RW Ò for every Ò.

19 General Patterns for Nonmonotonic Reasoning 137 Proposition 2.57 Let be an scr. The following conditions are equivalent: a) is -preferential, b) is a -cumulative sccr that satisfies CC, c) is a weakly -preferential sccr that satisfies RW Ò for every Ò. The proof is left to the reader. Proposition 2.58 Let be an scr and let be a -preferential sccr. a) A combining conjunction w.r.t. is also an internal conjunction and a combining conjunction w.r.t.. b) A combining disjunction w.r.t. is also an internal disjunction and half combining disjunction w.r.t.. 16 PROOF. Part (a) follows from Corollary Part (b) follows from Corollary 2.41 and Corollary CC Ò Ò ½µ is a natural generalization of cautious cut. A dual generalization, which seems equally natural, is given in the following rule from [25]: LCC Ò ½ Ò ½ Ò Definition 2.59 [25] A binary relation is a plausibility logic if it satisfies Inclusion ( ), CM, RM, and LCC Ò (Ò½). Definition 2.60 Let be an scr. A relation is called -plausible if it is a -preferential sccr and a plausibility logic. A more concise characterization of a -plausible relation is given in the following proposition: Proposition 2.61 Let be an scr. A relation is -plausible iff it satisfies Cum, CM, RM, and LCC Ò for every Ò. PROOF. Since CC is just LCC ½, we only need to show the derivability for all Ò of RW Ò. So assume that ½ Òµ and ½ Ò. By Cum and RM this implies that ½ Òµ and ½ Ò. Hence follows by LCC Ò. Proposition 2.62 Let be an scr with a combining conjunction. A relation is - preferential iff it is -plausible. PROOF. One direction is obvious. By the last proposition, for showing the converse we have to prove that if is -preferential and has a combining conjunction, then satisfies LCC Ò for every finite Ò. This follows from Corollary 2.47 and the following lemma: Lemma 2.62-A: Let be a -preferential sccr, where is an scr with a combining conjunction. Then Á is equivalent to LCC Ò. Proof: µµ If ½ Ò then by Á, ½ Ò. Also, if 16 I.e., satisfies left -introduction (but not necessarily left -elimination).

20 General Patterns for Nonmonotonic Reasoning 138 ½ Ò then by ICR (see Proposition 2.42(a)), ½ Ò. By CC, then,. µ Suppose that and. By RM, and. Also, by Cum on we have that. By LCC ¾ on these three sequents,. Table 1 and Figure 1 summarize the various types of Scott relations considered in this section and their relative strengths. is assumed there to be an scr, and, are combining disjunction and conjunction (respectively) w.r.t., whenever they are mentioned. consequence relation sccr TABLE 1. Scott relations general conditions valid conditions with and s-r, CM, CC ½ weakly -cumulative Cum, CM, CC ½, RW ½, RM sccr Á,,, Á, -cumulative sccr Cum, CM, CC ½, RW Ò, RM Á,, Á,, Á, weakly -preferential Cum, CM, CC, RM sccr Á,,, Á, Á,, -preferential sccr Cum, CM, CC, RW Ò, RM Á,, Á,, Á, Á, -plausible sccr scr extending Cum, CM, LCC Ò, RM Á,, Á,, Á, Á, Cum, M, C Á,, Á,, Á,, Á, 3 A semantical point of view In this section we present a general method of constructing nonmonotonic consequence relations of the strongest type considered in the previous section, i.e.: preferential and plausible sccrs. Our approach is based on a multiple-valued semantics. This will allow us to define in a natural way consequence relations that are not only nonmonotonic, but also paraconsistent (i.e.: capable of reasoning with inconsistency in a nontrivial way). A basic idea behind our method is that of using a set of preferential models for making inferences. Preferential models were introduced by McCarthy [30] and later by Shoham [40, 41] as a generalization of the notion of circumscription. The essential idea is that only a subset of models should be relevant for making inferences from a given theory. These models are the most preferred ones according to some conditions that can be specified syntactically by a set of (usually second-order) propositions, the satisfaction of which yields the exact kind of preference one wants to work with. Here we choose the preferred models according to preference criteria, specified by preorders on the set of models of a given theory. The resulting consequence relations are shown to be plausible Scott relations.

21 General Patterns for Nonmonotonic Reasoning 139 weakly -preferential sccr Ù À ÀÀ weakly -cumulative sccr Ù + CC À ÀÀ + RW Ò À ÀÀ (if a combining conjunction is available) À ÀÀ À ÀÀ Ù -preferential sccr Ù -plausible sccr Ù An scr that extends À ÀÀ + LCC Ò + M + RW Ò À ÀÀ + CC À À À Ù -cumulative sccr FIG. 1. Relative strength of the Scott relations 3.1 Multiple-valued models and Scott consequence relations Definition 3.1 Let be an arbitrary propositional language. A multiple-valued structure for is a triple Ä Ëµ, where Ä is set of elements ( truth values ), is a nonempty proper subset of Ä, and Ë is a set of operations on Ä that correspond to the connectives in. The set consists of the designated values of Ä, i.e.: those that represent true assertions. In what follows we shall assume that Ä contains at least the classical values Ø, and that ؾ, ¾. Definition 3.2 Let Ä Ëµ be a multiple-valued structure, and let a language. be a set of formulae in a) A (multiple-valued) valuation is a function that assigns an element of Ä to each atomic formula. A valuation is extended to complex formulae in the standard way. The set of all the valuations into Ä is denoted by Î. b) A valuation satisfies a formula (notation: Ä ) if µ ¾. The relation Ä ¾Î is called a satisfaction relation. c) A valuation is a model of (notation: Ä ) if it satisfies every formula in. The set of the models of is denoted by ÑÓ µ.

22 General Patterns for Nonmonotonic Reasoning 140 Definition 3.3 Let Ä Ëµ be a multiple-valued structure. Denote Ä if every model of is a model of some formula in. Example 3.4 Many well-known formalisms correspond to Definition 3.3, especially when a lattice structure is defined on the elements of Ä, and the elements of form a filter in this lattice. Classical logic, for instance, is obtained by taking the two-valued lattice Ø Ä Øµ with Ø. For Kleene three-valued logic [22] take Ä Ø with Ø. The connectives in Ë correspond to the lattice operations of a lattice in which Ä Ä Ø together with a negation operation defined by: Ø Ø. Belnap four-valued logics [9, 10] is obtained from Ä Ø, Ø, and Ë that contains the lattice operations of the the four-valued lattice in which Ä µ Ä Ø, and a negation operation defined by: Ø Ø. Proposition 3.5 Ä is an scr. PROOF. Reflexivity and Monotonicity immediately follow from the definition of Ä. For cut, assume that Å ¾ ÑÓ ½ ¾µ. In particular, Å ¾ ÑÓ ½µ, and since ½ Ä ½, either Å Ä Æ for some Æ ¾ ½, or Å Ä. In the former case we are done. In the latter case Å ¾ ÑÓ ¾ µ and since ¾ Ä ¾, we have that Å Ä Æ for some Æ ¾ ¾. Definition 3.6 Let Ä Ëµ be a multiple-valued structure. a) A binary operation ¾Ë is conjunctive if for all Ü Ý ¾Ä, ÜÝ ¾ iff ܾ and Ý ¾. b) A binary operation Ö¾Ë is disjunctive if for all Ü Ý ¾Ä, ÜÖÝ ¾ iff ܾ or Ý ¾. The following result is immediate from the definitions: Proposition 3.7 Let Ä Ëµ be a multiple-valued structure for a language. a) If is a connective of s.t. the corresponding operation of Ë is conjunctive, then is a combining conjunction and an internal conjunction w.r.t. Ä. b) If is a connective of s.t. the corresponding operation of Ë is disjunctive, then is a combining disjunction and an internal disjunction w.r.t. Ä. 3.2 Preferential models and Scott cautious consequence relations The relation Ä Definition 3.8 A preferential system in a structure Ä Ëµ is a triple È Î Ä µ, where Î is the set of all the valuations on Ä, Ä ¾Î is the satisfaction relation defined in 3.2, and is a preorder on Î. Definition 3.9 Let È Î Ä µ be a preferential system in Ä Ëµ. A valuation Å ¾ ÑÓ µ is a È-preferential model of if there is no other valuation Å ¼ ¾ ÑÓ µ s.t. Å ¼ Å. The set of all the preferential models of in È is denoted by ȵ. Definition 3.10 [29] A preferential system È is called stoppered 17 if for every set of formulae and every Å ¾ÑÓ µ there is an Å ¼ ¾ ȵ s.t. Å ¼ Å. 17 In [24] the same property is called smoothness.

23 General Patterns for Nonmonotonic Reasoning 141 Note that if Î is well-founded under (i.e., Î does not have an infinitely descending chain under ), then È is stoppered. Definition 3.11 Let È Î Ä µ be a preferential system in Ä Ëµ. A set of formulae È-preferentially entails a set of formulae (notation: Ä ) if for every Å ¾ ȵ there is a Æ ¾ s.t. Å Ä Æ. 18 We say that Ä is the consequence relation 19 induced by È. Proposition 3.12 If È Î Ä µ is a stoppered preferential system in Ä Ëµ, then is a Ä -plausible sccr. Ä For proving Proposition 3.12 we first show the following lemma: Lemma 3.13 Let È be a preferential system and let ½ ¾ be two sets of formulae s.t. ÑÓ ½µÑÓ ¾µ. Then ¾ ȵ ÑÓ ½µ ½ ȵ. PROOF. Suppose that Å ¾ ¾ ȵ ÑÓ ½µ, but Å ¾ ½ ȵ. Then there is an Æ ¾ ÑÓ ½µ s.t. Æ Å. But ÑÓ ½µÑÓ ¾µ so Æ ¾ÑÓ ¾µ, therefore Å ¾ ¾ ȵ. PROOF. [of Proposition 3.12] The validity of Cum immediately follows from the definition of Ä. This is also the case with RM. By Proposition 2.61 it remains to show CM, and LCC Ò : Ä Suppose that satisfies cautious monotonicity:, and Ä. Let Å ¾ ȵ. In particular, Å is a model of. Moreover, Å ¾ ȵ, since otherwise by the fact that È is stoppered, there would Ä have been a model Æ ¾ ȵ that is strictly -smaller than Å. Since, this Æ would have been a model of, which is -smaller than Å a contradiction. Thus Å ¾ ȵ. Now, since Ä, Å is a model of some Æ ¾. Hence Ä. Ä Ä satisfies LCCÒ for every Ò: Let Å ¾ ȵ. If Å is a model of some Æ ¾ we are done. Otherwise, since Ä for ½ Ò, Å is a model of ½ Ò. By Lemma 3.13, Å ¾ ½ Ò Èµ. Since ½ Ò Ä, there exists Æ ¾ s.t. Å ¾ÑÓ Æµ in this case as well. Corollary 3.14 Let È Î Ä µ be a stoppered preferential system in Ä Ëµ. a) If is a connective s.t. the corresponding operation of Ë is conjunctive, then is an internal conjunction and a combining conjunction w.r.t. Ä. b) If is a connective s.t. the corresponding operation of Ë is disjunctive, then is an internal disjunction w.r.t. Ä, which satisfies left -introduction. PROOF. By Propositions 3.12 Ä is Ä -plausible, and so it is obviously a Ä -preferential sccr. The claim now follows from Proposition Note that we do not require that Å ¾ Æ Èµ, or that Å ¾ Æ Èµ. 19 Here and in what follows we use the notion consequence relation in a wider sense than that of Tarski and Scott. In particular, we don t assume monotonicity.