4.2 Selective Revision
4.2.1 Selective Multiple Base Revision
We extend the approach of selective revision [96], which is formulated for belief sets and single formulae as input, to selective multiple (base) revision. That is, we consider the new information being represented by sets of formulae ✓ LpropAt instead of a single formula 2 LpropAt .
Even though we target belief base operations our formalization of the postulates and construction of the operation, as well as the results are equally applicable for the belief set case, unless stated differently explicitly. This brings us to the following definition, in accordance with [96], of a change operator ?fB : P(L
prop
At )⇥ P(LpropAt ) ! P(LpropAt )
that is a selective multiple base revision via
B?fB = B⇤ fB( ) (4.2.2)
with a selection function fB : P(LpropAt ) ! P(LpropAt ) and some priori-
tized multiple base revision ⇤ : P(LpropAt )⇥ P(LpropAt )! P(LpropAt ). In the
following we define the general notions of a prioritized multiple base revision (inner revision operator), a non-prioritized multiple base re- vision (outer revision operator), and a selection function by means of postulates based on the considerations in [96]. Then we adapt a result of [96] showing that we obtain an outer revision operator, satisfying all desired properties by a selective revision construction with the de- fined inner revision and selection functions, which satisfy certain sets of properties. For this, we consider postulates from non-prioritized revision and the selective revision framework for the case of multiple base revision. At the same time we slightly adapt them to the case of multiple base revision.
The first property we consider is Extensionality, which we intro- duced for belief sets as the (K*6) postulate in Section 2.5.1 (Page 28). It expresses that equal inputs for the same belief base should lead to equal results, similarly as the Uniformity postulate in belief base revi- sion theory, as introduced in Section 2.5.3 (Page 30). As discussed in Section 2.5, Extensionality is usually not considered for the problem of base revision as base revision is motivated by observing syntax and not (only) semantic contents. We include the postulate here to be compatible with the original formulation of selective revision for belief sets. Further we discuss and modify it to the case of multiple base revision. We phrase Extensionality for multiple base revision as follows:
extensionality If ⌘p , then B ⇤ ⌘p B⇤ .
We denote postulates for multiple revision operations by a subscript . We defined two notions of equivalence for sets of propositional formalae, ⌘p and ⇠=p, in Section 2.2. It holds that ⌘p 0, if and
only if it holds that ` 0 and 0 ` . And it holds that =⇠p 0
2 it holds that ⌘p ( ), i. e., if and 0 are element-wise
equivalent.
For the case of multiple base revision, the satisfaction of Extension- ality imposes that B ⇤ {a, b} ⌘p B⇤ {a ^ b} as {a, b} ⌘p {a ^ b}. It has
been argued, that even for the belief set case {a, b} expresses that a and b are two independent pieces of information while {a ^ b} ex- presses that they are not. Hence, both inputs might, and should, lead to different results, see e. g. [69] for a discussion. For our means and the inner revision operator we define the following weakened form of Extensionality .
weak extensionality If =⇠p 0 then B ⇤ ⌘p B⇤ 0.
The property Weak extensionality only demands that the outcomes of the revisions B ⇤ and B ⇤ 0 are equivalent if and 0 are
element-wise equivalent. This formulation does not have the same problem since {a, b} 6 ⇠=p {a ^ b}. All other postulates are direct gen-
eralizations of the base revision postulates introduced in Section 2.5.3 (Page 30) for a set of sentences as input:
success : ✓ B ⇤
inclusion : B ⇤ ✓ B [
vacuity : If B [ is consistent, then B [ ✓ B ⇤
consistency : If is consistent, then B ⇤ is consistent
relevance : If 0 ✓ (B [ ) \ (B⇤ ), then there is a set H
such that B ⇤ ✓ H ✓ B [ and H is consistent but H [ 0
is inconsistent
Given these postulates we can define the prioritized multiple base revision operator.
Definition 4.2.3. A revision operator ⇤ is called a prioritized multiple base revision operator if ⇤ satisfies Success , Inclusion , Vacuity , Consistency , Relevance , and Weak extensionality .
Having defined the inner revision operator we turn to the outer revision operator, the non-prioritized multiple base revision operator. For non-prioritized multiple base revision the properties Inclusion , Consistency , Relevance , and Weak extensionality are desirable [126]. The Vacuity postulate is also not desirable for selective revision since even if the new information is consistent with the belief base there might be non-logical reasons to reject parts of it. This is not the case for the Success postulate since it demands to give absolute priority to the new information. Success and Vacuity can be replaced by weaker postulates, cf. [126]. Here, we consider the following two. weak success If B [ 6`? then B ?fB ` .
consistent expansion If B 6✓ B ?fB then B [ (B ?fB ) `?.
Note that Weak success follows from Vacuity , and that Consistent expansion follows from Vacuity and Success , as shown in [96]. Definition 4.2.4. A revision operator ?fB is called non-prioritized multi-
ple base revision operator if ?fB satisfies Inclusion , Consistency , Weak
extensionality , Weak success , and Consistent expansion .
The last operator to be defined is the selection function. In [96] several properties for selection functions in the context of belief set revision are discussed. We rephrase some of them here slightly to fit the framework of multiple base revision. Let B ✓ LpropAt be consistent
and let , 0✓ Lprop At .
inclusionf fB( ) ✓
weak inclusionf If B [ is consistent then fB( ) ✓
extensionalityf If ⌘p 0 then fB( ) ⌘p fB( 0)
consistency preservationf If is consistent then fB( ) is
consistent
consistencyf fB( )is consistent
maximalityf fB( ) =
weak maximalityf If B [ is consistent then fB( ) =
In addition to the adapted postulates from [96] we also consider a weakened version of Extensionalityf.
weak extensionalityf If =⇠p 0 then fB( ) ⇠=p fB( 0)
The above properties are considered as possible properties for a selec- tive revision operation and are not assumed to be satisfied by every such operator. For example, the property maximalityf states that fB
should not modify the set . Satisfaction of this property leads to the equivalence of the selective revision operator, as defined in (4.2.2), and the used prioritized revision operator, such that
B?fB = B⇤ fB( ) = B⇤ .
As ⇤ is meant to be a prioritized revision function we lose the possi- bility for non-prioritized revision.
Here, we consider the postulates Inclusionf, Weak extensionalityf,
consistency preservationf, and Weak maximalityf as desirable for a se-
lection function. As for the multiple base revision operator, the satis- faction of Extensionalityf is not desirable for a selection function. The
satisfaction of it would force the results for the inputs = {a, b} and
{a, b} expresses that a and b are two independent pieces of infor- mation while {a ^ b} expresses that they are not. Moreover, for the case of selection functions another problem with the satisfaction of Extensionalityf arises. Consider again = {a, b} and 0 = {a ^ b}.
It follows that ⌘p 0 and if fB satisfies Extensionality
f this re-
sults in fB({a, b}) ⌘p fB({a ^ b}). If fB also satisfies Inclusionf it
follows that fB({a ^ b}) 2 {;, {a ^ b}}, i. e., that the entire input
is accepted or none of it. This might be adequate in this case. How- ever, for {a, b} it follows that fB({a, b}) 2 {;, {a, b}} and surely
the options to keep at least one of the input elements should be also considerable here. Hence, for Weak extensionalityf we demand fB( )
and fB( 0) to be element-wise equivalent (in contrast to the prop-
erty Weak extensionality for revision).
In general, if fB satisfies both Inclusionf and Extensionalityf it fol-
lows that either fB( ) = ; or fB( ) = for every ✓ LpropAt (as
is equivalent to a 0 that consists of a single formula that is the
conjunction of the formulae in such that, by Inclusionf, only the
options fB( 0) = ; or fB( 0) = 0 are possible). As we are inter-
ested in a more graded approach to belief revision we want to be able to accept or reject specific parts of and not just completely. Consequently, we consider Weak extensionalityf as a desirable prop-
erty instead of Extensionalityf. Note that Extensionalityf implies Weak
extensionalityf as =⇠p 0 implies ⌘p 0.
In [96] several representation theorems are given that character- ize non-prioritized belief revision by selective revision via (4.2.2) and specific properties of ⇤ and fB. In particular, it is shown that a rea-
sonable non-prioritized belief revision operator ?fB can be character-
ized by an AGM revision ⇤ and a selection function fB that satisfies
Extensionalityf, Consistency preservationf, and Weak maximality. We can
carry over the results of [96] to the problem of multiple base revision and obtain the following result.
Proposition 4.2.5. Let ⇤ be a prioritized multiple base revision op- erator and let fB satisfy Inclusionf, Weak extensionalityf, Consistency
preservationf, and Weak maximalityf. Then ?fB defined via (4.2.2) is a
non-prioritized multiple base revision operator. Proof. See Appendix A.1.2 on Page 226.
The Relevance postulate does not hold for B ?fB defined via
(4.2.2) in general. It is arguable if relevance should hold, it would constrain the selection function by demanding, roughly, that it can only refuse pieces of information that would lead to inconsistency. However, for a selection function it makes perfect sense to reject the entire input if only a part of it is inconsistent with the belief base. Consider for example a set of convictions BR✓ B of a belief base and
the selection function f0
tent and f0
B( ) = ; otherwise. The selection function f0B satisfies all
properties for selection functions except Maximalityf. But it is easy to
see that B ?fB defined via (4.2.2) using f
0
B and a prioritized multi-
ple base revision operator ⇤ fails to satisfy Relevance . This selection function might be desirable in settings in which the convictions are strongly believed and any information source that contradicts these convictions is regarded as not credible. While such an operation is arguably desirable in all cases, it is clearly desirable that the selective revision framework should be able to capture such an operator.
Up to here we extended the selective revision framework and char- acterized the operators we consider desirable. In the following, we aim at implementing a selective multiple base revision using deduc- tive argumentation.