Abstract
This is the first of three chapters in which the major traditional belief change operations, namely (sentential) revision and contraction, are constructed as special cases of descriptor revision. In both its local and global forms, sentential revision \((*)\) can be constructed with the simple formula \(K*p= K\circ \mathfrak {B}p\). Two axiomatic characterizations show how the properties of the derived operation \(*\) depend on those of the underlying descriptor revision \(\circ \). The most important results in this chapter are two theorems showing that the major revision operations of AGM, namely partial meet revision and its transitively relational variant, are both reconstructible as subcases of sentential revision in the descriptor framework. In other words, both these classes of sentential revision \((*)\) coincide with operations obtained with the defining formula \(K*p= K\circ \mathfrak {B}p\) from certain classes of descriptor revision \((\circ )\) that are identified here. These results provide important connections between AGM revision and the more general category of descriptor revision. Furthermore, they provide useful insights into the properties of the AGM operations. The chapter also investigates relations of believability, i.e. the relations on sentences that are obtainable from relations of epistemic proximity on descriptors (see Chapter 5) according to the simple principle that p is more believable than q if and only if \(\mathfrak {B}p\) is epistemically more proximate than \(\mathfrak {B}q\). Finally, some results are presented on multiple sentential revision, i.e. simultaneous revision by several sentences, and on the operation of making up one’s mind.
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Notes
- 1.
In other words, \(\widehat{C}\) is K-favouring for atomic descriptors.
- 2.
It follows that there is a one-to-one correspondence between \(\mathbb {X}\) and the set \( \{{ \& }X\mid X\in \mathbb {X}\}\). Since the latter set is countable, so is \(\mathbb {X}\).
- 3.
For an explanation, see footnote 27, p. 110.
- 4.
Zhang Li has proposed a highly illustrative example that shows the need for this postulate: Let the language be based on the three atoms \(\{a_0,a_1,a_2\}\) and let \(K=\text {Cn}(\varnothing )\). Let \(*\) be defined as follows: (1) If \( a_0 \& a_1\vdash p\) and \(p\vdash a_0\), then \( K*p = \text {Cn}(a_0 \& a_1)\), (2) If \( a_1 \& a_2\vdash p\) and \(p\vdash a_1\), then \( K* p= \text {Cn}(a_1 \& a_2)\), (3) If \( a_0 \& a_2\vdash p\) and \(p\vdash a_2\), then \( K*p= \text {Cn}(a_0 \& a_2)\), and (4) otherwise \(K*p= \text {Cn}(\{p\})\).
With the exception of cumulativity, it is easy to check that the first eight postulates of the theorem are satisfied by this construction. For cumulativity, let \(q\in K*p\). There are two cases: (a) \(K*p\) was decided according to one of the first three clauses: We consider clause (1). In this case, \( a_0 \& a_1\vdash p\) and \( a_0 \& a_1\vdash q\), thus \( a_0 \& a_1\vdash p \& q\). Furthermore, \(p\vdash a_0\), thus \( p \& q\vdash a_0\). Consequently, \( K*(p \& q) = \text {Cn}(a_0 \& a_1) = K*p\). (b) \(K*p\) was decided according to the last clause: Then \(q \in K*p= \text {Cn}(\{p\})\), thus \( p \& q\leftrightarrow p\) and \( K*(p \& q) = \text {Cn}(\{p\}) = K*p\). Non-circularity is refuted by the following cycle: \( \text {Cn}(\{a_0 \& a_2\})\) is a specification of \( \text {Cn}(\{a_0 \& a_1\})\) since \( K*a_0=\text {Cn}(\{a_0 \& a_1\})\) and \( K*(a_0 \& a_2)=\text {Cn}(\{a_0 \& a_2\})\). \( \text {Cn}(\{a_1 \& a_2\})\) is a specification of \( \text {Cn}(\{a_0 \& a_2\})\) since \( K*a_2=\text {Cn}(\{a_0 \& a_2\})\) and \( K*(a_1 \& a_2)=\text {Cn}(\{a_1 \& a_2\})\). \( \text {Cn}(\{a_0 \& a_1\})\) is a specification of \( \text {Cn}(\{a_1 \& a_2\})\) since \( K*a_1=\text {Cn}(\{a_1 \& a_2\})\) and \( K*(a_0 \& a_1)=\text {Cn}(\{a_0 \& a_1\})\).
- 5.
This is a slight generalization of a result reported in [73, p. 54].
- 6.
On these postulates, see also [73, p. 54], [99, pp. 270–274], and [217, pp. 107–111].
- 7.
More precisely: The sphere in which there are worlds containing \(X_1\) coincides with the sphere in which there are worlds containing \(X_2\).
- 8.
Lemma (p. 200) is used here.
- 9.
\(\leqq _1\) and \(\leqq _2\) give rise to different descriptor revisions. Let \(\circ _1\) and \(\circ _2\) be the centrolinear descriptor revisions based on \(\leqq _1\) respectively \(\leqq _2\). We then have \(K\circ _1\lnot \mathfrak {B}p=\text {Cn}(\{r\})\) and \(K\circ _2\lnot \mathfrak {B}p=\text {Cn}(\{p\vee r\})\).
- 10.
Some additional results on believability relations can be found in [263].
- 11.
Choice revision with an infinite input set would require an extension of the formal language to include infinite disjunctions. Due to their limited epistemological relevance such constructions will not be discussed here.
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Hansson, S.O. (2017). Sentential Revision. In: Descriptor Revision. Trends in Logic, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-53061-1_8
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