Possible Worlds Semantics for Partial Meet Multiple Contraction

In the logic of theory change, the standard model is AGM, proposed by Alchourrón et al. (J Symb Log 50:510–530, 1985). This paper focuses on the extension of AGM that accounts for contractions of a theory by a set of sentences instead of only by a single sentence. Hansson (Theoria 55:114–132, 1989), Fuhrmann and Hansson (J Logic Lang Inf 3:39–74, 1994) generalized Partial Meet Contraction to the case of contractions by (possibly non-singleton) sets of sentences. In this paper we present the possible worlds semantics for partial meet multiple contractions.

receives a sentence or a set of sentences that is intended to be removed from it and returns a new set of beliefs that no longer contains the given sentence(s).
All earlier investigations of this processes [1,[9][10][11][12] dealt essentially with the removal of a specific sentence from a belief set -a set of sentences closed under logical consequence. The first and most widely known constructive model of such an operation is the partial meet contraction [1]. The possible worlds semantics for this operation was introduced in [12].
However, in a subsequent stage of the development of the theory of belief contraction, several researchers (e.g. [6-8, 13, 16]) pointed out the need for defining operations that could account for changes consisting of the removal of sets with more than one element. In this respect, Fuhrmann and Hansson [8] remarked that contracting a belief set K by a set of sentences B can mean either to remove all elements of B from K (package contractions), or to remove at least one of the elements of B from K (choice contractions). In the present paper we will only consider the first of those two kinds of operations and we will use the expression multiple contraction to refer to such functions. Furthermore, we will restrict our attention to the case when the sets to be contracted from K are finite. Hence, throughout this paper whenever we refer to the multiple contraction of a belief set K by a set of sentences B, such set B is assumed to be finite. We will use the expression singleton contraction to designate contractions by a single sentence.
The first constructive model of multiple contractions to be presented was the partial meet multiple contraction in [13,14]. The operations of multiple contraction defined within such model are essentially the generalization to the case of contraction by sets of sentences (rather than by a single sentence) of the partial meet (singleton) contraction functions. Nevertheless, a possible worlds semantics has not yet been provided for such generalization.
In this work we aim to provide such possible worlds semantics for the partial meet multiple contractions, inspired in the analogous model for the singleton case proposed by Grove [12] and generalized by Hansson [15].
The paper is organized in the following way: In Section 2 we provide the notation and background needed for the rest of the paper. In Section 3 we provide the possible worlds semantics for the partial meet multiple contractions. The Appendix contains the proofs of all the original results.

Formal Preliminaries
We will assume a language L that is closed under truth-functional operations. We shall make use of a consequence operation Cn that takes sets of sentences to sets of sentences and which satisfies the following properties: We will sometimes use Cn(α) for Cn({α}), A α for α ∈ Cn(A), α for α ∈ Cn(∅), A α for α ∈ Cn(A), α for α ∈ Cn(∅). The letters α, α i , β, . . . (except for γ ) will be used to denote sentences. stands for an arbitrary tautology and ⊥ for an arbitrary contradiction. A, A i , B, . . . shall denote subsets of sentences of L. K is reserved to represent a set of sentences that is closed under logical consequence (i.e. K = Cn(K)) -such a set is called a belief set or theory. The conjunction of all elements of a finite and non-empty set A shall be denoted by &A. We shall denote the set of all maximal consistent subsets of L by M L . We will use the expression possible world (or just world) to designate an element of M L . M, N i , W, . . ., (except for L and P), shall be used to denote subsets of M L . Such sets are called propositions. Given a set of sentences R, the set consisting of all the possible worlds that contain R is denoted by R . The elements of R are the R-worlds. ϕ is an abbreviation of {ϕ} and the elements of ϕ are the ϕ-worlds. To any set of possible worlds V we associate a belief set Th(V) given by Th(V) = V -under the assumption that ∅ = L. M, N i , W, . . . shall be used to denote subsets of P(M L ).

Partial Meet Contractions
A Partial Meet Contraction function [1,3] is built upon a selection from the maximal subsets of K that do not imply the sentence to be contracted. In what follows we present the formal definition of such functions.
Given a belief set K and a set of sentences B, the remainder set of K by B is the set of maximal subsets of K that do not imply any element of B and is denoted by K⊥B. Its elements are the remainders (of K, by B). K⊥α is an abbreviation of K⊥{α} and is called the remainder set of K by α. It is also convenient to notice here that, according to [ An operation−is a partial meet contraction on K if and only if there is a selection function γ for K such that for all sentences α: K − α = γ (K⊥α).
Next we present the possible worlds semantics for partial meet contractions. To do that we first notice that propositions (sets of possible worlds) can be used to represent belief sets and sentences. More precisely, a belief set K can be represented by the set K and a sentence α can be identified with the set α .
The following result clarifies more formally the above mentioned interrelation between belief sets and sets of possible worlds: ([4]) Let H, K be belief sets and U, V be sets of possible worlds. Then: When considering a propositional approach to represent sentences and belief sets, the contraction of K by α takes the form of the addition of some ¬αworlds to the set K . In order to clarify this statement we start by exposing, in the following observation, the very tight interconnection between the sets K⊥ϕ and ¬ϕ , for some belief set K and some sentence ϕ ∈ K: Observation 2.3 (Grove's bijection [12,Section 4]) Let K be a belief set and ϕ be a sentence such that ϕ ∈ K. Then the function g from ¬ϕ to K⊥ϕ def ined by Now we introduce the concept of propositional selection function, which we will use afterwords to present the possible worlds semantics for the partial meet contractions.
We are now in position to present the following observation, which illustrates how the partial meet contractions can be described in terms of possible worlds.
Observation 2.5 ([12], [15,Observation 2.69]) Let K be a belief set. An operation−on K is a partial meet contraction if and only if there is a propositional selection function f for K such that for all sentences α: K − α = Th( K ∪ f ( ¬α )).

Partial Meet Multiple Contractions
Now we recall the definition of the partial meet multiple contractions, which are the generalization of the partial meet contraction functions to the case of contractions by (possibly non-singleton) sets. Definition 2.6 (Partial meet multiple contraction [8,13]) Let K be a belief set. A package selection function for K is a function γ such that for all sets of sentences B: ∅ = γ (K⊥B) ⊆ K⊥B, if K⊥B = ∅, and γ (K⊥B) = {K}, otherwise.
An operation ÷ is a partial meet multiple contraction on K if and only if there is some package selection function γ for K, such that for all sets of sentences B: K÷B = γ (K⊥B).
An operation ÷ on K is a maxichoice multiple contraction if and only if it is a partial meet multiple contraction generated by a package selection function γ such that for all sets B, the set γ (K⊥B) has exactly one element. The full meet multiple contraction on K is the partial meet multiple contraction · ∼ · that is generated by the package selection function γ such that for all sets B, if K⊥B = ∅, then γ (K⊥B) = K⊥B, i.e., K · ∼ · B = K⊥B, if B ∩ Cn(∅) = ∅, and K · ∼ · B = K, otherwise.

Possible Worlds Semantics for Partial Meet Multiple Contractions
In this section our main goal is to obtain the possible worlds semantics for the partial meet multiple contractions. In the case of singleton contraction the possible worlds semantics proposed by Grove [12] and generalized by Hansson [15] is essentially based on the relation between remainder sets (by a single sentence) and sets of possible worlds which is expressed by Grove's bijection (Observation 2.3). Having this in mind, to achieve our goal, we will start by finding out the relation between remainder sets by sets of sentences and sets of possible worlds, which is the generalization of Grove's bijection to the multiple case. Afterwards, by making use of such relation we present the characterization of the partial meet multiple contractions in terms of possible worlds.

Possible Worlds Semantics for Remainders
A partial meet contraction K − α is defined as the intersection of some of the elements of the remainder set K⊥α, where each element of the remainder set is a maximal subset of K that does not entail α. In the possible worlds model this corresponds to adding to K the minimal number of worlds such that K − α ⊆ α . According Grove's bijection this corresponds simply to adding one and only one ¬α-world to K . By this simple relation, partial meet singleton contraction in terms of possible worlds is defined as K plus some (selected) ¬α-worlds.
The definition of remainder in multiple contraction is similar to the singleton case, i.e., maximal subsets of K that do not entail any element of B. This corresponds in terms of possible worlds to adding to K the minimal number of worlds such that they guarantee that no element of B can be inferred from the remainder set, i.e., K÷B ⊆ α i for all α i ∈ B. More precisely, in what follows we will show that if X ∈ K⊥B, then X = Th( K ∪ W), for some W ⊆ M L such that, on the one hand W contains at least one ¬α i -world for all α i ∈ B ∩ K and, on the other hand, if W ∈ W then W is the only ¬α j -world included in W, for some α j ∈ B ∩ K.
To see that this indeed holds, we start by formally introducing, in the following definition, the set W K⊥B , composed by all the sets of possible worlds that satisfy the above mentioned conditions. Definition 3.1 Let K be a belief set and B be a set of sentences. We denote by W K⊥B the subset of P { ¬α i : α i ∈ B ∩ K} such that W ∈ W K⊥B if and only if: Although this definition is clear, it is not easy to understand how a remainder set can be "constructed" in terms of possible worlds. The most intuitive approach is to assume that each remainder set corresponds to K plus one and only one ¬α i -world for each α i ∈ B, which basically correspond to take individual remainders from each member of B. However, this is not the case. On Fig. 1 we can see a remainder set which includes more than one ¬α 1 -world. On the other hand, Figs. 2 and 3 show that we cannot define a remainder  Now we investigate the composition of the set W K⊥B : Observation 3.2 Let K be a belief set and B be a set of sentences. Then the following statements hold: Notice that according to the above observation (and excluding the limiting cases 1 and 2) the set W K⊥B contains at least one non-empty set of worlds, from which it follows that we can use W K⊥B in the construction of partial meet multiple contraction by means of a propositional package selection function (as in the singleton case): In what follows we present the relation between W K⊥B and K⊥B. Such relationship will play an analogous role to the one played by Grove's bijection in the singleton case.

Theorem 3.4 Let K be a belief set and B be a f inite set of sentences. Then:
The first statement of the theorem means that the sets in W K⊥B are minimal sets of worlds in the sense that the addition of each of those sets to K corresponds to a maximal subset of K that does not imply any element of B. The second statement yields that each of the remainders can be constructed by means of a set of W K⊥B .
As an immediate consequence of the above observation we can define K⊥B in terms of possible worlds:

Possible Worlds Semantics for Partial Meet Multiple Contraction
Having characterized the remainders in terms of possible worlds, our next goal is to obtain the possible worlds semantics for partial meet multiple contractions. As in the singleton case, the lower bound of partial meet multiple contraction would be full meet multiple contraction, hence we will start by obtaining the possible worlds semantics for that limiting case of partial meet contractions. Since full meet multiple contraction is given by the intersection of all the element of the remainder set, the corresponding construction in the semantic level is by adding to K all the elements of W K⊥B as it is stated in the following observation: Observation 3.6 Let K be a belief set, B a set of sentences and · ∼ · be the full meet multiple contraction on K. Then Having seen this, we will now investigate which worlds are included in the set W K⊥B .
Observation 3.7 Let K be a belief set and B be a set of sentences. Then: The above observation reveals a big difference between the singleton and the multiple case: In singleton contraction, the full meet contraction K ∼ α corresponds to the addition of the whole set ¬α to K . However Point 3 of the previous observation clarifies that the full meet multiple contraction does not correspond, in general, to the addition of all the ¬α i -worlds to K .
In order to find out which worlds of the set { ¬α i : α i ∈ B ∩ K} are indeed included in W K⊥B , we shall start by finding a way of defining a subset of { ¬α i : α i ∈ B ∩ K} that does not contain any world M satisfying (simultaneously) conditions (i)−(iii) mentioned in Point 3 of the previous observation.
Having this in mind, we now notice that, if we consider a subset B of B consisting of those and only those sentences α i in B such that for any sentence α j ∈ B, it holds that ¬α j ⊂ ¬α i , then the elements of the set { ¬α i : α i ∈ B ∩ K} do not simultaneously satisfy the conditions (i)−(iii) mentioned above.
We are now in a position to explicitly indicate which worlds compose the set W K⊥B :

Observation 3.9 Let K be a belief set and B be a set of sentences. Then
Combining the identity just presented in the above observation with the result of Observation 3.6 we can obtain a characterization of the full meet multiple contraction: 1 Observation 3.10 Let K be a belief set, B be a set of sentences and · ∼ · be the operator of full meet multiple contraction on K. Then, under the assumption that ∅ = L, it holds that Next we introduce one last observation concerning the full meet multiple contraction, that explicitly indicates which worlds compose the set K · ∼ · B : Observation 3.11 Let K be a belief set, B be a set of sentences and · ∼ · be the operator of full meet multiple contraction on K. Then Now that we have thoroughly exposed the characterization in terms of possible worlds of the full meet multiple contraction, we turn our attention to obtain the possible worlds semantics of the partial meet multiple contractions in general. We start by using Observation 3.11 to obtain a superset of the set K÷B , where ÷ is a partial meet multiple contraction on K: Observation 3.12 Let K be a belief set and B be a set of sentences. If an operation ÷ is a partial meet multiple contraction on K then Our next goal is to obtain the possible worlds semantics for the partial meet multiple contractions. In order to do that we will present first the following observation:

Observation 3.13 Let K be a belief set and B and C be sets of sentences. Then
The above observation essentially states (through the representation of remainders in terms of possible worlds) that the remainder set of K by B coincides with the remainder set of K by B N ∩ K. Therefore K÷B is identical to K÷(B N ∩ K) whenever ÷ is a partial meet multiple contraction. We can now present the main result of the paper, which provides the characterization of partial meet multiple contractions in terms of possible worlds. Theorem 3.14 Let K be a belief set. An operation ÷ is a partial meet multiple contraction on K if and only if there exists a propositional package selection function f for K such that, for any set of sentences B: Using the above theorem and Theorem 3.4 it can be easily shown that if f is such that, for any set of sentences B, f (W K⊥B ) contains at most one element, then the operation ÷ defined by K÷B = Th K ∪ f (W K⊥B ) is a maxichoice multiple contraction on K. On the other hand, it is also worth noticing that it follows immediately from Observation 3.6 that the operation · ∼ · of full meet multiple contraction on K is such that

Conclusions
We have presented the possible worlds semantics for the partial meet multiple contraction which generalizes the possible worlds semantics for partial meet (singleton) contraction presented in [12,15].
To be more precise we have shown that every remainder X ∈ K⊥B is such that X = Th( K ∪ W) for some set of possible worlds W such that (i) W contains at least one ¬α i -world for every α i ∈ B ∩ K and (ii) for every world in W there is a sentence α ∈ B ∩ K for which that one is the only ¬α-world in W. Subsequently we have proven that partial meet multiple contractions can be defined by means of intersections of sets of the form K ∪ M , where M consists of a selection of sets of possible worlds that satisfy the above conditions (i) and (ii).
In particular, we have also characterized in terms of possible worlds the upper and lower bounds of that class of functions, i.e., full meet and maxichoice multiple contractions.
Based on these results, in [5] we introduced a new class of multiple contraction functions -the system of spheres-based multiple contractions -which are a generalization to the case of contraction by (possibly non-singleton) sets of sentences of Grove's system of spheres-based contraction functions.
Acknowledgements We wish to thank the three referees of the JPL and also Jim Delgrande for their valuable comments on a previous version of this paper which have led to improvements in exposition.

Appendix: Proofs
Previous Lemmas

Lemma 4.1 Let K be a belief set and B be a set of sentences such that B
Proof If B ∩ K = ∅ then K⊥B = {K} and the rest of the proof follows trivially. Now assume B ∩ K = ∅ and let X ∈ K⊥B.
1. Since X ⊆ K, it follows from Proposition 2.2-(iii) that K ⊆ X . Now we prove that X ⊆ K ∪ { ¬α i : α i ∈ B} . Let M be an arbitrary element of X . Assume by redutio that M ∈ ( K ∪ { ¬α i : α i ∈ B} , then there is some δ ∈ K such that ¬δ ∧ &B ∈ M. Therefore, because M is consistent, it follows that δ ∨ ¬&B ∈ M. Then, on the one hand, from M ∈ X it follows that δ ∨ ¬&B ∈ X and, on the other hand, since δ ∈ K, then δ ∨ ¬&B ∈ K. So it holds that δ ∨ ¬&B ∈ K \ X. Now, since X ∈ K⊥B, it follows that there is some α j ∈ B such that α j ∈ Cn(X ∪ {δ ∨ ¬&B}). Then, by deduction it follows that (¬δ ∧ &B) ∨ α j ∈ Cn(X). But, from α j ∈ B it follows that (¬δ ∧ &B) ∨ α j ↔ α j . Therefore we obtain that α j ∈ Cn(X), which contradicts the fact that X ∈ K⊥B. Hence we can conclude that 2. Let α i ∈ B and assume by redutio that X ∩ ¬α i = ∅. Then it follows that α i ∈ Cn(X) which contradicts the fact that X ∈ K⊥B. Therefore we have that X ∩ ¬α i = ∅, for all α i ∈ B, hence N ∩ ¬α i = ∅, for all α i ∈ B ∩ K.

Lemma 4.2 Let K be a belief set, B be a f inite set of sentences and N be a subset of M
Proof If B ∩ K = ∅ then it is enough to define N = ∅ and the required conditions are trivially satisfied. So, we assume B ∩ K = ∅. For each α i ∈ B ∩ K let M α i be one (arbitrarily chosen) world in N ∩ ¬α i (recall that N ∩ ¬α i = ∅, for all α i ∈ B ∩ K) and consider the set Notice that, since B is finite, B ∩ K also is and, therefore, we may assume without loss of generality that N 0 = {M 1 , . . . , M n }, with n ≥ 1. Next, for each m ∈ {1, . . . n}, let N m be the set defined in the following way: It follows immediately from the above construction that: So according to Definition 3.1, N n ∈ W K⊥B . So, we put N = N n and this finishes the proof.

Lemma 4.3 Let B be a set of sentences. If α l ∈ B and α l ∈ B N then there is some
Proof Let α l ∈ B \ B N . It follows from the definition of B N that there is some α s ∈ B such that ¬α s ⊂ ¬α l . If α s ∈ B N this ends the proof. Assume α s ∈ B N , then using the same arguments, there is some α s 1 ∈ B such that ¬α s 1 ⊂ ¬α s . Now, on the one hand, if α s 1 ∈ B N this ends the proof because under this conditions we have that ¬α s 1 ⊂ ¬α l . On the other hand, if α s 1 ∈ B N , reasoning in an analogous way we can find a α s 2 ∈ B which is related to α s 1 in precisely in the same way that α s 1 was related to α s and so on. Now consider the sequence of sentences (δ n ) defined in the following way δ 1 = α s , δ 2 = α s 1 , δ 3 = α s 2 , . . ., where the next terms of this sequence are the consecutive elements of B obtained using the process just described.
Note that ¬δ j ⊆ ¬δ i whenever j < i. Since B is finite there must be some δ m ∈ B such that for all α j ∈ B, if ¬α j ⊆ ¬δ m , then ¬δ m ⊆ ¬α j , which means that δ m ∈ B N . And it follows from the construction of the sequence (δ n ) that ¬δ m ⊂ ¬α l , and this ends the proof.

Lemma 4.4 Let K be a belief set, B be a set of sentences such that B ∩ Cn(∅)
= ∅ and f : { α : α ∈ L and ¬α} → P(M L ) be a function such that if α ∈ L and ¬α then ∅ = f ( α ) ⊆ α . If B f ⊆ B is such that: Proof Let B f ⊆ B be such that conditions (i) and (ii) are satisfied.
1. If B ∩ K = ∅ then B f ∩ K = ∅ and 1. is vacuously true. Let B ∩ K = ∅. Let α j be an arbitrary element of B f ∩ K and M be an arbitrary element of Furthermore, in these conditions, it holds that f ( ¬α m ) ⊆ ¬α j . To prove this we will show that (a) f ( ¬α m ) = ¬α j and (b) f ( ¬α m ) ⊂ ¬α j .
(a) Since α l ∈ M it follows that M ∈ ¬α l and because f ( ¬α m ) ⊆ ¬α l , we obtain that M ∈ f ( ¬α m ). On the other hand, from M ∈ f ( ¬α j ) and f ( ¬α j ) ⊆ ¬α j it follows that M ∈ ¬α j and we are done.

Lemma 4.5 Let K be a belief set and B be a set of sentences such that B ∩
Proof We will prove this lemma using Case 1 α l ∈ B N . Then α l ∈ B N ∩ K and ¬α l ⊆ ¬α l , and we are done. Case 2 α l ∈ B N . Then, it follows from Lemma 4.3 that there is some α m ∈ B N such that ¬α m ⊂ ¬α l . On the other hand, from α l ∈ K and ¬α m ⊂ ¬α l we can conclude that α m ∈ K. Hence, we have that α m ∈ B N ∩ K and ¬α m ⊆ ¬α l , as required and this finishes the proof.

Lemma 4.6 Let K be a belief set and G and H be sets of sentences. If every subset X of K implies some element of G if and only if X implies some element of H, then for all
Proof Assume that every subset X of K implies some element of G if and only if X implies some element of H and let β i ∈ G N ∩ K. We will start by showing that there is an element of H N ∩ K which is logically equivalent to β i . Since {β i } ⊆ K and {β i } β i ∈ G, it follows from the hypothesis that there is some ς i ∈ H such that {β i } ς i . Then, by deduction we have that β i → ς i which is equivalent to ¬ς i ⊆ ¬β i . Since K is a belief set, we can conclude that ς i ∈ K, hence, proceeding analogously, we can conclude that there is some β j ∈ G such that ¬β j ⊆ ¬ς i . Then we have that ¬β j ⊆ ¬β i , and from the fact that β i ∈ G N we can conclude that ¬β j = ¬β i . So, it holds that ¬β i = ¬ς i . It remains to prove that ς i ∈ H N ∩ K. We already have that ς i ∈ H ∩ K. To show that ς i ∈ H N we must prove that for all ς j ∈ H it holds that if ¬ς j ⊆ ¬ς i then ¬ς i ⊆ ¬ς j . So, let ς j ∈ H be such that ¬ς j ⊆ ¬ς i . Since ¬β i = ¬ς i , it follows that ¬ς j ⊆ ¬β i . Reasoning as above we can conclude that there is some β l ∈ G such that ¬β l ⊆ ¬ς j . Then we have that ¬β l ⊆ ¬β i , and again from the fact that β i ∈ G N we obtain that ¬β l = ¬β i , which allows us to conclude that ¬ς i ⊆ ¬ς j as required.

Proofs
Proof of Observation 3.2 In order to prove that the statements 1.
Then α j ∈ B ∩ K and ¬α j = ∅. Therefore, there is no W such that W ∩ ¬α i = ∅, for all α i ∈ B ∩ K. Hence W K⊥B = ∅.
Condition (i) is trivially satisfied. Regarding (ii), let α i be an arbitrary element of B, we will show that α i ∈ Cn K ∩ W . If α i ∈ B \ K, it is obvious that α i ∈ Cn K ∩ W . If α i ∈ B ∩ K, then, according to Definition 3.1, there is some world M ∈ W such that M ∈ ¬α i . Therefore Hence there is some M ∈ W such that β ∈ M. Therefore, because M ∈ M L , it holds that ¬β ∈ M. Since M ∈ W, according to Definition 3.1, there is some α j ∈ B ∩ K such that W ∩ ¬α j = {M}. Now, from ¬β ∈ M, α j ∈ K and α j ∈ W i , for every W i ∈ W \ M, it follows that ¬β ∨ α j ∈ K ∩ W . Therefore α j ∈ Cn K ∩ W ∪ {β} , and then Cn K ∩ W ∪ {β} ∩ B = ∅ as required. 2. We will divide the proof in cases: On the other hand, in the above conditions, according to Lemma 4.2, there is some set N such that N ⊆ N and N ∈ W K⊥B . Let X = Th( K ∪ N ). To finish the proof it is enough to show that X = X . Now, to see that this is indeed the case, we start by observing that, Furthermore, according to point 1. proven above, because N ∈ W K⊥B , we also have that X ∈ K⊥B. Hence, in particular it holds that X ⊆ K and Cn(X ) ∩ B = ∅. From this and the fact that X ∈ K⊥B it follows that X ⊂ X . Therefore, since X ⊆ X , we can conclude that X = X , and this finishes the proof.
Proof of Observation 3.6 If B ∩ Cn(∅) = ∅, according to Corollary 3.5 it Then, on the one hand, K · ∼ · B = K and, on the other hand, W K⊥B = ∅ (according to Observation 3.2-1.) and, consequently, W K⊥B = ∅. Hence we only need to show that K = ( K ∪ ∅) and K = K ∩ ∅ . But, the former identity follows immediately from Proposition 2.2-(i), and the latter equality is an obvious consequence of our assumption that Th(∅) = ∅ = L.
Proof of Observation 3.9 If B ∩ Cn(∅) = ∅ or B ∩ K = ∅ then using Observation 3.2-1. and 2. it can be trivially shown that W K⊥B = { ¬α i : α i ∈ B N ∩ K} = ∅. Now we assume that B ∩ Cn(∅) = ∅ and B ∩ K = ∅, and we prove the equality by double inclusion: Then, according to Definition 3.1 it holds that: (i) W ∩ ¬α i = ∅, for all If α j ∈ B N . Then, α j ∈ B N ∩ K and M ∈ ¬α j . Therefore we can conclude that M ∈ { ¬α i : α i ∈ B N ∩ K}, and we are done. If α j ∈ B N . Then, it follows from Lemma 4.3 that there is some α k ∈ B N such that ¬α k ⊂ ¬α j . So, from (i) and (ii) we can conclude that M ∈ ¬α k . Furthermore, from α j ∈ K and ¬α k ⊂ ¬α j we can conclude that α k ∈ K (since from ¬α j ∩ K = ∅ and ¬α k ⊂ ¬α j it follows that ¬α k ∩ K = ∅). Hence, M ∈ { ¬α i : α i ∈ B N ∩ K}, as required. (⊇) Follows trivially from Lemma 4.5.
Proof of Observation 3.11 The equality of sets K ∪ W K⊥B = K ∪ { ¬α i : α i ∈ B N ∩ K} follows immediately from Observation 3.9.
On the other hand, according to Observation 3.10, it holds that K · ∼ · B = K ∪ { ¬α i : α i ∈ B N ∩ K} . Therefore it follows that K · ∼ · B ⊇ K ∪ { ¬α i : α i ∈ B N ∩ K} . Now we prove the converse inclusion. If B ∩ K = ∅, then K · ∼ · B = K ⊆ K ∪ { ¬α i : α i ∈ B N ∩ K} and we are done. So, let B ∩ K = ∅. Then, making use of Lemma 4.3, it follows trivially that B N ∩ K = ∅. Now let M ∈ K · ∼ · B and assume by redutio that M ∈ K ∪ { ¬α i : α i ∈ B N ∩ K}). Then there is some δ ∈ K such that ¬δ ∧ &(B N ∩ K) ∈ M and therefore, because M is consistent, δ ∨ ¬&(B N ∩ K) ∈ M.
Proof of Observation 3.12 Let · ∼ · the operator of full meet multiple contraction on K. In order to show that ÷ is a partial meet multiple contraction we must show that there is some package selection function γ for K, such that K÷B = γ (K⊥B). Now let γ be the operator defined by γ (K⊥B) = {K} if K⊥B = ∅, and γ (K⊥B) = {Th( K ∪ W) : W ∈ f (W K⊥B )} if K⊥B = ∅. We will show that (a) γ is a package selection function for K and (b) K÷B = γ (K⊥B).
In order to prove (a) we must prove first that γ is a function; i.e., if K⊥C = K⊥B then γ (K⊥C) = γ (K⊥B) which follows immediately from Observation 3.13-2 and the definition of γ . Now we must prove that γ is a package selection function for K. If We will show that f is a propositional package selection function for K . Since when W K⊥B = W K⊥C , it follows from Observation 3.13-2 and the definition of f that f (W K⊥B ) = f (W K⊥C ), we can conclude that f is a function. Now we must prove that f satisfies conditions (1) and (2) , and this finishes the proof.