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Choice revision

  • Li ZhangEmail author
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Abstract

Choice revision is a sort of non-prioritized multiple revision, in which the agent partially accepts the new information represented by a set of sentences. We investigate the construction of choice revision based on a new approach to belief change called descriptor revision. We prove that each of two variants of choice revision based on such construction is axiomatically characterized with a set of plausible postulates, assuming that the object language is finite. Furthermore, we introduce an alternative modelling for choice revision, which is based on a type of relation on sets of sentences, named multiple believability relation. We show without assuming a finite language that choice revision constructed from such relations is axiomatically characterized with the same sets of postulates that we proposed for the choice revision based on descriptor revision, whenever the relations satisfy certain rationality conditions.

Keywords

Choice revision Non-prioritized multiple revision Belief change Descriptor revision Multiple believability relation 

1 Introduction

Belief change1 theory studies how a rational agent changes her belief state when she is exposed to new information. Studies in this field have traditionally had a strong focus on two types of change: contraction in which a specified sentence has to be removed from the original belief state, and revision in which a specified sentence has instead to be consistently added. This paper is mainly concerned with the latter.

Alchourrón, Gärdenfors and Makinson (AGM) performed the pioneering formal study on these two types of change in their seminal paper (Alchourrn et al. 1985). In the AGM theory of belief change, the agent’s belief state is represented by a set of sentences from some formal language \({\mathcal {L}}\), usually denoted by K. The new information is represented by a single sentence in \({\mathcal {L}}\). Belief revision and contraction on K are formally represented by two operations \(*\) and \(\div \), mapping from a sentence \(\varphi \) to a new set of sentences \(K *\varphi \) and \(K \div \varphi \) respectively. Alchourrn et al. (1985) postulated some conditions that a rational revision or contraction operation should satisfy, which are called AGM postulates on revision and contraction.2

Furthermore, Alchourrn et al. (1985) showed that contraction and revision satisfying AGM postulates could be precisely constructed from a model based on partial meet functions on remainder sets. After that, many alternative models (Alchourrón and Makinson 1985; Grove 1988; Gärdenfors and Makinson 1988; Hansson 1994, etc.) have been proposed to construct the operations characterized by these postulates. Although these models look entirely different on the surface, most of them employ the same select-and-intersect strategy (Hansson 2017, p. 19). For example, in partial meet construction for contraction (Alchourrn et al. 1985), a selection is made among remainders and in sphere modelling for revision (Grove 1988), a selection is made among possible worlds. The intersection of the selected objects is taken as the outcome of the operation in both cases.

Although the AGM theory has virtually become a standard model of theory change, many researchers are unsatisfied with its settings in several aspects and have proposed several modifications and generalizations to that framework (see Fermé and Hansson 2011 for a survey). Here we only point out two inadequatenesses of the AGM theory.

On the one hand, in the original AGM model, the input is represented by a single sentence. This is unrealistic since agents often receive more than one piece of information at the same time. In order to cover these cases, we can generalize sentential revision to multiple revision, where the input is a finite or infinite set of sentences. On the other hand, in AGM revision, new information has priority over original beliefs. This is represented by the success postulate: \(\varphi \in K *\varphi \) for all \(\varphi \). The priority means that the new information will always be entirely incorporated, whereas previous beliefs will be discarded whenever the agent need do so in order to incorporate the new information consistently. This is a limitation of AGM theory since in real life it is a common phenomenon that agents do not accept the new information that they receive or only accept it partially. As a modification, we can drop the success postulate and generalize prioritized revision to non-prioritized belief revision.

In this contribution, we will put these two generalizations together and consider the so called non-prioritized multiple belief revision. In Falappa et al. (2012), two different kinds of such generalized revision are distinguished:
  1. 1.

    Merge: K and A are symmetrically treated, i.e., sentences of K and A could be accepted or rejected.

     
  2. 2.

    Choice revision3: some sentences of A could be accepted, some others could be rejected.

     
We use \(*_c\) to denote a choice revision operation. Falappa et al. (2012) investigated the formal properties of merge but left the study on choice revision as future work. As far as we know, little work has been done on this kind of revision in the literature. This fact can be partly explained by that the operation \(*_c\) has the unusual characteristic that the standard select-and-intersect approach is not in general applicable. To see why, let the set K of original beliefs not contain any element of \(A =\{ \varphi , \lnot \varphi \}\). We are going to construct a set \(K *_cA\) which incorporates \(\varphi \) or its negation. Suppose that we do that by first selecting a collection \(\mathbb {X} =\{X_1, X_2, X_3, \ldots \}\) of sets of beliefs, each of which satisfies the success condition for choice revision with A, i.e. \(X_i \cap A \ne \emptyset \) for each \(X_i\). Then it may be the case that \(\varphi \in X_1\) and \(\lnot \varphi \in X_2\). Given that \(X_1\) and \(X_2\) are consistent, it follows that the intersection \(\cap \mathbb {X}\) cannot satisfy the success condition, i.e. it contains neither \(\varphi \) or \(\lnot \varphi \).

Therefore, to develop a modelling for choice revision, we need to choose another strategy than the select-and-intersect method. Hansson (2013) introduced a new approach of belief change named “descriptor revision”, which employs a “select-direct” methodology: it assumes that there is a set of belief sets as potential outcomes of belief change, and the belief change is performed by a direct choice among these potential outcomes. Furthermore, this is a very powerful framework for constructing belief change operations since success conditions for various types of belief changes are described in a general way with the help of a metalinguistic belief operator \({\mathfrak {B}}\). For instance, the success condition of contraction by \(\varphi \) is \(\lnot {\mathfrak {B}} \varphi \), that of revision by \(\varphi \) is \({\mathfrak {B}} \varphi \). Descriptor revision on a belief set K is performed with a unified operator \(\circ \) which applies to any success condition that is expressible with \({\mathfrak {B}}\). Hence, choice revision \(*_c\) with a finite input set can be constructed from descriptor revision in the way of \(K *_c\{\varphi _1, \varphi _2, \ldots , \varphi _n\} = K \circ \{{\mathfrak {B}} \varphi _1 \vee {\mathfrak {B}} \varphi _2 \vee \cdots \vee {\mathfrak {B}} \varphi _n \}\) (Hansson 2017, p. 130).

Although the construction of choice revision in the framework of descriptor revision has been introduced in Hansson (2017), the formal properties of this type of belief change are still in need of investigation. The main purpose of this contribution is to conduct such an investigation. After providing some formal preliminaries in Sect. 2, we will review how to construct choice revision in the framework of descriptor revision in Sect. 3. More importantly, in this section, we will investigate the postulates on choice revision which could axiomatically characterize these constructions. In Sect. 4 we will propose an alternative modelling for choice revision, which is based on multiple believability relations, a generalized version of believability relation introduced in Hansson (2014) and further studied in Zhang (2017). We will investigate the formal properties of this modelling and prove the associated representation theorems. Section 5 concludes and indicates some directions for future work. All proofs of the formal results are placed in the “Appendix”.

2 Preliminaries

The object language \({\mathcal {L}}\) is defined inductively by a set v of propositional variables \(\{p_0 ,\, p_1, \, \ldots , \, p_n, \, \ldots \}\) and the truth-functional operations \(\lnot , \wedge , \vee \) and \(\rightarrow \) in the usual way. \({\scriptstyle \top }\) is a tautology and \({\scriptstyle \perp }\) a contradiction. \({\mathcal {L}}\) is called finite if v is finite. Sentences in \({\mathcal {L}}\) will be denoted by lower-case Greek letters and sets of such sentences by upper-case Roman letters.

\(\mathrm {Cn}\) is a consequence operation for \({\mathcal {L}}\) satisfying supraclassicality: if \(\varphi \) can be derived from A by classical truth-functional logic, then \(\varphi \in \mathrm {Cn}(A)\), compactness: if \(\varphi \in \mathrm {Cn}(A)\), then there exists some finite \(B \subseteq A\) such that \(\varphi \in \mathrm {Cn}(B)\), and the deduction property: \(\varphi \in \mathrm {Cn}(A \cup \{\psi \}\)) if and only if (henceforth iff for short) \(\psi \rightarrow \varphi \in \mathrm {Cn}(A)\). \(X\Vdash \varphi \) and \(X \nVdash \varphi \) are alternative notations for \(\varphi \in \mathrm {Cn}(X)\) and \(\varphi \notin \mathrm {Cn}(X)\) respectively. \(\{ \varphi \} \Vdash \psi \) is rewritten as \(\varphi \Vdash \psi \) for simplicity. And \(\varphi \dashv \Vdash \psi \) means \(\varphi \Vdash \psi \) and \(\psi \Vdash \varphi \). \(A \equiv B\) holds iff for every \(\varphi \in A\), there exists some \(\psi \in B\) such that \(\varphi \dashv \Vdash \psi \) and vice versa.

For all finite A, let \(\&A\) denote the conjunction of all elements in A. For any A and B, Open image in new window . We write Open image in new window and Open image in new window and Open image in new window for simplicity.

The set of beliefs an agent holds is represented by a belief set, which is a set \(X \subseteq {\mathcal {L}}\) such that \(X = \mathrm {Cn}(X)\). K is fixed to denote the original beliefs of the agent. We assume that K is consistent, i.e. \(K \nVdash {\scriptstyle \perp }\), unless stated otherwise.

3 Choice revision based on descriptor revision

Before investigating the properties of choice revision constructed in the framework of descriptor revision, we first present some formal basics of this framework, which is mainly based on Hansson (2013).

3.1 Basics of descriptor revision

An atomic belief descriptor is a sentence \({\mathfrak {B}}\varphi \) with \(\varphi \in {\mathcal {L}}\), which can be interpreted as “believing proposition \(\varphi \)”. A descriptor is a set of truth-functional combinations of atomic descriptors. \({\mathfrak {B}}\varphi \) is satisfied by a belief set X iff \(\varphi \in X\). Conditions of satisfaction for truth-functional combination of atomic descriptors are defined inductively. X satisfies a descriptor \(\varPhi \) (denoted by \(X \vDash \varPhi \) ) iff it satisfies all its elements. Descriptor revision on a belief set K is performed with a unified operator \(\circ \) such that \(K \circ \varPhi \) is an operation with descriptor \(\varPhi \) representing its success condition. Hansson (2013) introduces several constructions for descriptor revision operations, of which the relational model defined as follows has a canonical status.

Definition 1

(Hansson 2013) \((\mathbb {X}, \leqq )\) is a relational select-direct model (in short: relational model) with respect to K iff it satisfies4:
  • \((\mathbb {X} 1)\)\(\mathbb {X}\) is a set of belief sets.

  • \((\mathbb {X} 2)\)\(K \in \mathbb {X}\).

  • \((\leqq 1)\)\(K \leqq X\) for every \(X \in \mathbb {X}\).

  • \((\leqq 2)\) For any descriptor \(\varPhi \), if \(\{X \in \mathbb {X} \mid X \vDash \varPhi \}\) (we denote it as \( {\mathbb {X}}^{\varPhi } \)) is not empty, then it has a unique \(\leqq \)-minimal element denoted by \({\mathbb {X}}^{\varPhi }_{<}\).

A descriptor revision \(\circ \) on K is based on (or determined by) some relational model \(({\mathbb {X}},\leqq )\) with respect to K iff for any descriptor \(\varPhi \),5

\(\mathbb {X}\) is the outcome set which includes all the potential outcomes under various belief change patterns. The ordering \(\leqq \) (with the strict part <) brings out a direct-selection mechanism, which selects the final outcome among candidates satisfying a specific success condition. We call descriptor revision constructed in this way relational descriptor revision. In so far as the selection mechanism is concerned, descriptor revision is at a more abstract level in comparison to the AGM revision. In the construction of descriptor revision \(\circ \), “it assumes that there exists an outcome set which contains all the potential outcomes of the operation \(\circ \), but it says little about what these outcomes should be like” (Zhang 2017, p. 41). In contrast, in the AGM framework, the belief change is supposed to satisfy the principle of consistency preservation and the principle of the informational economy (Gärdenfors 1988). Therefore, the intersection step in the construction of belief change in the AGM framework becomes dispensable in the descriptor revision approach.

3.2 Choice revision constructed from descriptor revision

The success condition for choice revision \(*_c\) with a finite input could be easily expressed by descriptor \(\{ {\mathfrak {B}} \varphi _0 \vee \cdots \vee {\mathfrak {B}} \varphi _n \}\). So, it is straightforward to construct choice revision through descriptor revision as follows.

Definition 2

(Hansson 2017) Let \(\circ \) be some descriptor revision. A choice revision \(*_c\) on K is based on (or determined by) \(\circ \) iff for any finite set A,
$$\begin{aligned} {\langle \circ ~\mathrm {to}~*_c\rangle }\,\,\,\,\,\,\, K *_cA= {\left\{ \begin{array}{ll} K \circ \{ {\mathfrak {B}} \varphi _0 \vee \cdots \vee {\mathfrak {B}} \varphi _n \} &{} \text{ if } A = \{\varphi _0 , \ldots , \varphi _n \} \ne \emptyset ,\\ K &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$

Henceforth, we say \(*_c\) is based on (or determined by) some relational model if it is based on the descriptor revision determined by the same model. The main purpose of this section is to investigate the formal properties of choice revision based on such models.

3.3 Postulates and representation theorem

It is observed that the choice revision determined by relational models should satisfy a set of arguably plausible postulates on choice revision.

Observation 1

Let \(*_c\) be a choice revision determined by any relational descriptor revision \(({\mathbb {X}},\leqq )\). Then it satisfies the following postulates:
  • \(\mathrm {(*_c1)}\)\(\mathrm {Cn}(K *_cA) = K *_cA\). \((*_c\)-closure)

  • \(\mathrm {(*_c2)}\)\(K *_cA = K\) or \(A \cap (K *_cA) \ne \emptyset \). \((*_c\)-relative success)

  • \(\mathrm {(*_c3)}\) If \(A \cap (K *_cB) \ne \emptyset \), then \(A \cap (K *_cA) \ne \emptyset \). \((*_c\)-regularity)

  • \(\mathrm {(*_c4)}\) If \(A \cap K \ne \emptyset \), then \(K *_cA = K\). \((*_c\)-confirmation)

  • \(\mathrm {(*_c5)}\) If \((K *_cA) \cap B \ne \emptyset \) and \((K *_cB) \cap A \ne \emptyset \), then \(K *_cA = K *_cB\). \((*_c\)-reciprocity)

Moreover, another plausible condition on choice revision follows from this set of postulates.

Observation 2

If \(*_c\) satisfies \(*_c\)-closure, relative success, regularity and reciprocity, then \(*_c\) satisfies:
  • If \(A \equiv B\), then \(K *_cA = K *_cB\). \((*_c\)-syntax irrelevance)

It is easy to see that the postulates in above are natural generalizations of the following postulates on sentential revision:
  • \(\mathrm {(*1)}\)\(\mathrm {Cn}(K *\varphi )=K *\varphi \) (\(*\)-closure)

  • \(\mathrm {(*2)}\) If \(K *\varphi \ne K \), then \(\varphi \in K *\varphi \) (\(*\)-relative success)

  • \(\mathrm {(*3)}\) If \(\varphi \in K \), then \(K *\varphi = K\) (\(*\)-confirmation)

  • \(\mathrm {(*4)}\) If \(\psi \in K *\varphi \), then \(\psi \in K *\psi \) (\(*\)-regularity)

  • \(\mathrm {(*5)}\) If \(\psi \in K *\varphi \) and \(\varphi \in K *\psi \), then \(K*\varphi = K *\psi \) (\(*\)-reciprocity6)

and
  • If \(\varphi \dashv \Vdash \psi \), then \(K*\varphi = K *\psi \).7 (\(*\)-extensionality)

The above postulates on choice revision are as intuitively plausible as their correspondents on sentential revision, except that the meaning of \(*_c\)-reciprocity seems not so transparent as that of \(*\)-reciprocity. However, given some weak conditions, we can show that the \(*_c\)-reciprocity postulate is equivalent to a more understandable condition as follows.

Observation 3

Let choice operation \(*_c\) satisfy \(*_c\)-relative success and \(*_c\)-regularity. Then it satisfies \(*_c\)-reciprocity iff it satisfies:
  • If \(A \subseteq B\) and \((K *_cB) \cap A \ne \emptyset \), then \(K *_cA =K *_cB\). \((*_c\)-cautiousness)

The postulate \(*_c\)-cautiousness reflects a distinctive characteristic of choice revision modelled by relational models: the agent who performs this sort of belief change is cautious in the sense of not doing more than what is necessary to incorporate one of the elements of the input. It follows immediately from \(*_c\)-relative success and \(*_c\)-cautiousness that if \(A \cap (K *_cA) \ne \emptyset \), then \(K *_cA = K *_c\{\varphi \}\) for some \(\varphi \in A\). Thus, it is not surprising that the following postulate follows.

Observation 4

If \(*_c\) satisfies \(*_c\)-relative success, regularity and reciprocity, then \(*_c\) satisfies:
  • \(K *_c(A \cup B) = K *_cA\) or \(K *_c(A \cup B) = K *_cB\). \((*_c\)-dichotomy)

In contrast to \((*_c1)\) through \((*_c5)\), postulates \(*_c\)-cautiousness and \(*_c\)-dichotomy do not have directly corresponding postulates in the context of sentential revision. This suggests that though \((*_c1)\) through \((*_c5)\) naturally generalize \((*1)\) through \((*5)\), this sort of generalization is not so trivial as we may think of. As another evidence for this, the following observation shows that the properties of \((*1)\) through \((*5)\) and those of their generalizations are not always paralleled.

Observation 5

Let \(*_c\) satisfy \(*_c\)-regularity. Then it satisfies \(*_c\)-reciprocity iff it satisfies
  • For any \(n \ge 1 \), if \((K *_cA_1 ) \cap A_0 \ne \emptyset \), \(\ldots \), \((K *_cA_{n} ) \cap A_{n-1} \ne \emptyset \), \((K *_cA_{0} ) \cap A_{n} \ne \emptyset \), then \(K *_cA_0 = K *_cA_1 = \cdots = K *_cA_n\). \((*_c\)-strong reciprocity)

\(*\)-strong reciprocity is a generalization of the following postulate on sentential revision:
  • For any \(n \ge 1 \), if \(\varphi _0 \in K \star \varphi _{1}\), \(\ldots \), \(\varphi _{n-1} \in K *\varphi _n\) and \(\varphi _n \in K \star \varphi _0 \), then \(K \star \varphi _0 = K \star \varphi _2= \cdots = K \star \varphi _n\). (\(*\)-strong reciprocity8)

However, in contrast to the result in Observation 5, \(*\)-strong reciprocity is not derivable from \((*1)\) through \((*5)\).9

After an investigation on the postulates \((*_c1)\) through \((*_c5)\) satisfied by choice revision based on rational models, the question raises naturally whether the choice revision could be axiomatically characterized by this set of postulates. We get a partial answer to this question: a representation theorem is obtainable when \({\mathcal {L}}\) is finite.

Theorem 1

Let \({\mathcal {L}}\) be a finite language. Then, \(*_c\) satisfies \((*_c1)\) through \((*_c5)\) iff it is a choice revision based on some relational model.

3.4 More properties of choice revision

In this subsection, we will study additional properties of choice revision from the point of view of postulates. The postulates introduced in the previous subsection do not necessarily cover all the reasonable properties of this operation. In what follows we are going to investigate some additional ones, in particular, the following:
  • If \(A \ne \emptyset \), then \(A \cap (K *_cA) \ne \emptyset \). (\(*_c\)-success)

  • If \(A \not \equiv \{{\scriptstyle \perp }\}\), then \(K *_cA \nVdash {\scriptstyle \perp }\). (\(*_c\)-consistency)

To some extent, \(*_c\)-success is a strengthening of \(*_c\)-relative success and \(*_c\)-regularity, but it does not say anything about the limiting case in which the input is empty. To cover this limiting case, we need the following postulate:
  • If \(A = \emptyset \), then \(K *_cA = K\). (\(*_c\)-vacuity)

The interrelations among \(*_c\)-success, \(*_c\)-relative success and \(*_c\)-regularity are summarized as follows.

Observation 6

Let \(*_c\) be some choice revision on K.
  1. 1.

    If \(*_c\) satisfies relative success, then it satisfies vacuity.

     
  2. 2.

    If \(*_c\) satisfies success and vacuity, then it satisfies relative success.

     
  3. 3.

    If \(*_c\) satisfies success, then it satisfies regularity.

     

\(*_c\)-consistency is a plausible constraint on a rational agent. While accepting \(*_c\)-success and \(*_c\)-consistency as “supplementary” postulates for choice revision \(*_c\), the corresponding relational model on which \(*_c\) is based will also need to satisfy some additional properties. We use the following representation theorem to conclude this subsection.

Theorem 2

Let \({\mathcal {L}}\) be a finite language and \(*_c\) some revision operation on \(K \subseteq {\mathcal {L}}\). Then, \(*_c\) satisfies \(*_c\)-closure, \(*_c\)-success, \(*_c\)-vacuity, \(*_c\)-confirmation, \(*_c\)-reciprocity and \(*_c\)-consistency iff it is a choice revision determined by some relational model which satisfies the following two condition:
  • \((\mathbb {X} 3)\)\(\mathrm {Cn}(\{ {{\scriptstyle \perp }}\}) \in \mathbb {X}\);

  • \((\leqq 3)\)\({\mathbb {X}}^{{\mathfrak {B}} \varphi } \ne \emptyset \) and \({\mathbb {X}}^{{\mathfrak {B}} \varphi }_{<} < \mathrm {Cn}(\{ {{\scriptstyle \perp }}\})\) for every \(\varphi \) such that \(\varphi \nVdash {\scriptstyle \perp }\).

4 An alternative modelling for choice revision

In this section, we propose an alternative modelling for choice revision, which is based on so-called multiple believability relations. A believability relation \(\preceq \) is a binary relation on sentences of \({\mathcal {L}}\). Intuitively, \(\varphi \preceq \psi \) means that the subject is at least as prone to believing \(\varphi \) as to believing \(\psi \).10 We can generalize \(\preceq \) to a multiple believability relation \(\preceq _{*}\) which is a binary relation on the set of all finite subsets of \({\mathcal {L}}\) satisfying:
  • \(\langle \preceq _{*}\)\(\mathrm {to}\)\(\preceq \rangle \)      \(\varphi \preceq \psi \) iff \(\{\varphi \} \preceq _{*} \{\psi \}\).

This kind of generalization can be done in different ways, and at least two distinct relations can be obtained, namely package multiple believability relations, denoted by \(\preceq _{p}\), and choice multiple believability relations, denoted by \(\preceq _{c}\) (with symmetric part \(\simeq _{c}\) and strict part \(\prec _{c}\)). Intuitively, \(A \preceq _{p}B\) means that it is easier for the subject to believe all propositions in A than to believe all propositions in B and \(A \preceq _{c}B\) means that it is easier for the subject to incorporate the plausible information in A than that in B.

\(\preceq _{p}\) is of little interest since \(A \preceq _{p}B\) can be immediately reduced to \(\&A \preceq \&B\), given that A and B are finite. In what follows, multiple believability relations (or multi-believability relations for short) only refer to choice multiple believability relations \(\preceq _{c}\). (\(\{\varphi \} \preceq _{c}A\) and \(A \preceq _{c}\{\varphi \}\) will be written as \(\varphi \preceq _{c}A\) and \(A \preceq _{c}\varphi \) respectively for simplicity.)

4.1 Postulates on multi-believability relations

Recall the following postulates on believability relations \(\preceq \) introduced in Zhang (2017):
  • \(\preceq \)-transitivity If \(\varphi \preceq \psi \) and \(\psi \preceq \lambda \), then \(\varphi \preceq \lambda \).

  • \(\preceq \)-weak coupling If \(\varphi \simeq \varphi \wedge \psi \) and \(\varphi \simeq \varphi \wedge \lambda \), then \(\varphi \simeq \varphi \wedge (\psi \wedge \lambda )\).

  • \(\preceq \)-coupling If \(\varphi \simeq \psi \), then \(\varphi \simeq \varphi \wedge \psi \).

  • \(\preceq \)-counter dominance If \(\varphi \Vdash \psi \), then \(\psi \preceq \varphi \).

  • \(\preceq \)-minimality\(\varphi \in K\) iff \(\varphi \preceq \psi \) for all \(\psi \).

  • \(\preceq \)-maximality If \(\psi \preceq \varphi \) for all \(\psi \), then \(\varphi \dashv \Vdash {\scriptstyle \perp }\).

  • \(\preceq \)-completeness\(\varphi \preceq \psi \) or \(\psi \preceq \varphi \)

Transitivity is assumed for almost all orderings. In virtue of the intuitive meaning of believability relation, \(\varphi \simeq \varphi \wedge \psi \) represents that the agent will accept \(\psi \) in the condition of accepting \(\varphi \). Thus, the rationale for \(\preceq \)-weak coupling is that if the agent will consequently add \(\psi \) and \(\lambda \) to her beliefs when accepting \(\varphi \), then she also adds the conjunction of them to her beliefs in this case. This is reasonable if we assume that the beliefs of the agent are closed under the consequence operation. The justification of \(\preceq \)-counter dominance is that if \(\varphi \) logically entails \(\psi \), then it will be a smaller change and hence easier for the agent to accept \(\psi \) rather than to accept \(\varphi \), because then \(\psi \) must be added too, if we assume that the beliefs of the agent are represented by a belief set. \(\preceq \)-coupling is a strengthening of \(\preceq \)-weak coupling.11 It says that if \(\varphi \) is equivalent to \(\psi \) in believability, then the agent will consequently add \(\psi \) to her beliefs in case of accepting \(\varphi \) and vice versa. \(\preceq _{c}\)-minimality is justifiable since nothing needs to be done to add \(\varphi \) to K if it is already in K. \(\preceq \)-maximality is justifiable since it is reasonable to assume that it is strictly more difficult for a rational agent to accept \({\scriptstyle \perp }\) than to accept any non-falsum. \(\preceq \)-completeness seems a little bit strong. It says that all pairs of sentences are comparable in believability. In accordance with Zhang (2017), we call relations satisfying all these postulates quasi-linear believability relations.
We can generalize these postulates on believability relations in a natural way to postulates multi-believability relations as follows12:
  • \(\preceq _{c}\)-transitivity If \(A \preceq _{c}B\) and \(B \preceq _{c}C\), then \(A \preceq _{c}C\).

  • \(\preceq _{c}\)-weak coupling If Open image in new window and Open image in new window , then Open image in new window .

  • \(\preceq _{c}\)-coupling If \(A \simeq _{c}B\), then Open image in new window .

  • \(\preceq _{c}\)-counter dominance If for every \(\varphi \in B\) there exists \(\psi \in A\) such that \(\varphi \Vdash \psi \), then \(A \preceq _{c}B\).

  • \(\preceq _{c}\)-minimality\(A \preceq _{c}B\) for all B iff \(A \cap K \ne \emptyset \).

  • \(\preceq _{c}\)-maximality If B is not empty and \(A \preceq _{c}B\) for all non-empty A, then \(B \equiv \{{\scriptstyle \perp }\}\).

  • \(\preceq _{c}\)-completeness\(A \preceq _{c}B\) or \(B \preceq _{c}A\).

These postulates on multi-believability relations can be understood in a similar way that their correspondents on believability relations are understood.
Furthermore, we propose the following two additional postulates on multi-believability relations:
  • \(\preceq _{c}\)-determination\(A \prec _{c}\emptyset \) for every non-empty A.

  • \(\preceq _{c}\)-union\(A \preceq A \cup B\) or \(B \preceq A \cup B\).

At least on the surface, these two could not be generalizations of any postulate on believability relation. In some sense the meaning of \(\preceq _{c}\)-determination is correspondent to that of \(*_c\)-success, since if it is a strictly smaller change for the agent to accept some sentences from a non-empty A rather than to take some sentences from the empty set, which is obviously impossible, then it seems to follow that the agent will successfully add some sentences in A to her original beliefs when exposed to the new information represented by A, and vice versa. Similarly, there is an obvious correspondence between the forms and meanings of \(\preceq _{c}\)-union and \(*_c\)-dichotomy. They both suggest that to partially accept a non-empty A is equivalent to accept some single sentence in A. This is plausible if we assume that the agent is extremely cautious to the new information.

Observation 7

Let \(\preceq _{c}\) be some multi-believability relation satisfying \(\preceq _{c}\)-transitivity and \(\preceq _{c}\)-counter dominance. If it satisfies \(\preceq _{c}\)-union in addition, then
  1. 1.

    It satisfies \(\preceq _{c}\)-completeness.

     
  2. 2.

    It satisfies \(\preceq _{c}\)-weak coupling iff it \(\preceq _{c}\)-satisfies coupling.

     

Observation 7 indicates that \(\preceq _{c}\)-union is strong. It should be noted that for a believability relation, neither \(\preceq \)-completeness nor \(\preceq \)-coupling can be derived from \(\preceq \)-transitivity, \(\preceq \)-counter dominance and \(\preceq \)-weak coupling.13

In what follows, we name multi-believability relations satisfying all the above postulates standard multi-believability relations.

4.2 Translations between believability relations and multiple believability relations

In this subsection, we will show that although it is impossible to find a postulate on believability relations that corresponds to \(\preceq _{c}\)-determination or \(\preceq _{c}\)-union, there exists a translation between quasi-linear believability relations and standard multi-believability relations.

Observation 8

Let \(\preceq _{c}\) satisfy \(\preceq _{c}\)-determination, \(\preceq _{c}\)-transitivity and \(\preceq _{c}\)-counter dominance. Then, for any non-empty finite sets A and B,
  1. 1.

    \(A \preceq _{c}B\) iff there exists \(\varphi \in A\) such that \(\varphi \preceq _{c}B\).

     
  2. 2.

    \(A \preceq _{c}B\) iff \(A \preceq _{c}\varphi \) for all \(\varphi \in B\).

     
This observation suggests that \(\preceq \) and \(\preceq _{c}\) can be linked through the following two transitions:
  • \(\langle \preceq _{c}\)\(\mathrm {to}\)\(\preceq \rangle \)      \(\varphi \preceq \psi \) iff \(\{\varphi \} \preceq _{c}\{\psi \}\).

  • \(\langle \preceq \)\(\mathrm {to}\)\(\preceq _{c}\rangle \)      \(A \preceq _{c}B\) iff \(B = \emptyset \) or there exists \(\varphi \in A\) such that \(\varphi \preceq \psi \) for every \(\psi \in B\).

This is confirmed by the following theorem.

Theorem 3

  1. 1.

    If \(\preceq \) is a quasi-linear believability relation and \(\preceq _{c}\) is constructed from \(\preceq \) through the way of \(\langle \preceq \)\(\mathrm {to}\)\(\preceq _{c}\rangle \), then \(\preceq _{c}\) is a standard multi-believability relation and \(\preceq \) can be retrieved from \(\preceq _{c}\) in the way of \(\langle \preceq _{c}\)\(\mathrm {to}\)\(\preceq \rangle \).

     
  2. 2.

    If \(\preceq _{c}\) is a standard multi-believability relation and \(\preceq \) is constructed from \(\preceq _{c}\) through \(\langle \preceq _{c}\)\(\mathrm {to}\)\(\preceq \rangle \), then \(\preceq \) is a quasi-linear believability relation and \(\preceq _{c}\) can be retrieved from \(\preceq \) through \(\langle \preceq \)\(\mathrm {to}\)\(\preceq _{c}\rangle \).

     

4.3 Choice revision constructed from multi-believability relations

Now we turn to the construction of choice revision through multi-believability relations. Recall that a sentential revision \(*\) can be constructed from a believability relation \(\preceq \) in this way (Zhang 2017):
$$\begin{aligned} {\langle \preceq ~\mathrm {to}~*\rangle }\,\,\,\,\,\,\,K *\varphi = \{ \psi \mid \varphi \simeq \varphi \wedge \psi \} \end{aligned}$$
As we have explained, \(\varphi \simeq \varphi \wedge \psi \) could be understood as that the agent will consequently accept \(\psi \) in case of accepting \(\varphi \). So, the set \(\{ \psi \mid \varphi \simeq \varphi \wedge \psi \} \) is just the agent’s new set of beliefs after she performed belief revision with input \(\varphi \). Thus, we can similarly construct choice revision from multi-believability relations in the following way.

Definition 3

Let \(\preceq _{c}\) be some multi-believability relation. A choice revision \(*_c\) on K is based on (or determined by) \(\preceq _{c}\) iff: for any finite A,

The primary results of this section are the following two representation theorems. In comparison with Theorems 1 and 2, these two theorems are applicable to more general cases since they do not assume that the language \({\mathcal {L}}\) is finite. These two theorems demonstrate that multi-believability relations provide a fair modelling for choice revision characterized by the set of postulates mentioned in Sect. 3.

Theorem 4

Let \(*_c\) be some choice revision on K. Then, \(*_c\) satisfies \((*_c1)\) through \((*_c5)\) iff it is determined by some multi-believability relation \(\preceq _{c}\) satisfying \(\preceq _{c}\)-transitivity, \(\preceq _{c}\)-weak coupling, \(\preceq _{c}\)-counter-dominance, \(\preceq _{c}\)-minimality and \(\preceq _{c}\)-union.

Theorem 5

Let \(*_c\) be some choice revision on K. Then, \(*_c\) satisfies \(*_c\)-closure, \(*_c\)- success, \(*_c\)-vacuity, \(*_c\)-confirmation, \(*_c\)-reciprocity and \(*_c\)-consistency iff it is determined by some standard multi-believability relation.

Considering the translation between multi-believability relations and believability relations (Theorem 3), it seems that these results can be easily transferred to the context of believability relations. However, if we drop some postulates on multi-believability relation such as \(\preceq _{c}\)-determination, the translation between multi-believability relation and believability relation will not be so transparent, at least it will not be so straightforward as \(\langle \preceq \)\(\mathrm {to}\)\(\preceq _{c}\rangle \) and \(\langle \preceq _{c}\)\(\mathrm {to}\)\(\preceq \rangle \). As a consequence, the result in Theorem 4 may not be possible to transfer to believability relations in a straightforward way. Moreover, compared with postulates on believability relations, postulates on multi-believability relations such as \(\preceq _{c}\)-determination and \(\preceq _{c}\)-union can present our intuitions on choice revision in a more direct way. Thus, the multi-believability relation is still worth to be studied in its own right.

5 Conclusion and future work

As a generalization of traditional belief revision, choice revision has more realistic characteristics. The new information is represented by a set of sentences and the agent could partially accept these sentences as well as reject the others. From the point of technical view, choice revision is interesting since the standard “select-and-intersect” methodology in modellings for belief change is not suitable for it. But instead, it can be modelled by a newly developed framework of descriptor revision, which employs a “select-direct” approach. After reviewing the construction of choice revision in the framework of descriptor revision, under the assumption that the language is finite, we provided two sets of postulates as the axiomatic characterizations for two variants of choice revision based on such constructions (in Theorems 1 and 2). These postulates, in particular, \(*_c\)-cautiousness and \(*_c\)-dichotomy, point out that choice revision modelled by descriptor revision has the special characteristic that the agent who performs this sort of belief change is cautious in the sense that she only accepts the new information to the smallest possible extent.

For AGM revision and contraction, there are various independently motivated modellings which are equivalent in terms of expressive power. In this contribution, we also propose an alternative modelling for choice revision. We showed that multi-believability relations can also construct the choice revision axiomatically characterized by the sets of postulates proposed for choice revision based on descriptor revision (Theorems 4 and 5). Moreover, these results are obtainable without assuming that the language is finite. This may indicate that multi-believability relations are an even more suitable modelling for choice revision.

The study in this contribution can be developed in at least three directions. First, the cautiousness constraint on choice revision, reflected by \(*_c\)-cautiousness, certainly could be loosened. We think it is an interesting topic for future work to investigate the modeling and axiomatic characterization of more “reckless” variants of choice revision. Secondly, as it was showed in Zhang (2017) that AGM revision could be reconstructed from believability relations satisfying certain conditions, it is interesting to ask which conditions a multi-believability relation should satisfy so that its generated choice revision coincides with an AGM revision when the inputs are limited to singletons. Finally, it is technically interesting to investigate choice revisions with an infinite input set, though they are epistemologically unrealistic.

Footnotes

  1. 1.

    In some literature, the term “belief revision” is used as a synonym for belief change. In what follows, we use belief revision to refer to a particular kind of belief change.

  2. 2.

    In the AGM framework, revision and contraction can be mutually defined through the Levi identity (Levi 1977) and Harper identity (Harpen 1976). In this contribution, we only investigate revision solely without considering its relationship with contraction.

  3. 3.

    Here we use the term “choice revision”, introduced by Fuhrmann (1988), to replace the term “selective change” used in Falappa et al. (2012), for it is easier for us to distinguish it from the “selective revision” introduced in Fermé and Hansson (1999), which is a sort of non-prioritized revision with a single sentence as input. It should be noted that generally choice revision by a finite set A cannot be reduced to selective revision by the conjunction & A of all elements in A. To see this, let \(*_{s}\) be some selective revision. It is assumed that \(*_{s}\) satisfies extensionality, i.e. if \(\varphi \) is logically equivalent to \(\psi \), then \(K *_s \varphi = K *_s \psi \). So, \(K *_s \&\{\varphi , \lnot \varphi \} = K *_s \&\{\psi , \lnot \psi \}\) for all \(\varphi \) and \(\psi \). However, it is implausible that the result of choice revision by \(\{\varphi , \lnot \varphi \} \) should always coincide with that by \( \{\psi , \lnot \psi \}\) for all \(\varphi \) and \(\psi \). Analogously, choice revision cannot be represented by AGM revision by updating with the disjunction of all the formulas of the input, since extensionality also holds for the AGM revision.

  4. 4.

    We will drop the phrase “with respect to K” if this does not affect the understanding, and write \({\mathbb {X}}^{\varphi } \) and \({\mathbb {X}}^{\varphi }_{<}\) instead of \({\mathbb {X}}^{\{{\mathfrak {B}} \varphi \}} \) and \({\mathbb {X}}^{\{{\mathfrak {B}} \varphi \}}_{<}\) for simplicity.

  5. 5.

    Provided that \(({\mathbb {X}},\leqq )\) is a relational model, \(\mathbb {X} \) is equivalent to the domain of \(\leqq \) since \(K \in \mathbb {X}\) and \(K \leqq X\) for all \(X \in \mathbb {X}\). So \(\leqq \) in itself can represent the \(({\mathbb {X}},\leqq )\) faithfully.

  6. 6.

    This postulate is first discussed in Alchourrón and Makinson (1982) in the context of maxichoice revision.

  7. 7.

    It is easy to check that \(*\)-extensionality is derivable from \((*1)\), \((*2)\), \((*3)\) and (\(*5\)).

  8. 8.

    \(*\)-strong reciprocity is closely related to a non-monotonic reasoning rule named “loop” which is first introduced in Kraus et al. (1990). For more discussion on this, see Makinson and Gärdenfors (1991).

  9. 9.

    To see this, let \(p_1\), \(p_2\) and \(p_3\) be pairwise distinct atomic propositions and let \(K = \mathrm {Cn}(\{ {{\scriptstyle \top }}\})\) and revision operation \(*\) on K defined as: (i) if \(p_0 \wedge p_1 \Vdash \varphi \) and \(\varphi \Vdash p_0\), then \(K *\varphi = \mathrm {Cn}(\{ {p_0 \wedge p_1}\})\); (ii) if \(p_1 \wedge p_2 \Vdash \varphi \) and \(\varphi \Vdash p_1\), then \(K *\varphi = \mathrm {Cn}(\{ {p_1 \wedge p_2}\})\); (iii) if \(p_0 \wedge p_2 \Vdash \varphi \) and \(\varphi \Vdash p_2\), then \(K *\varphi = \mathrm {Cn}(\{ {p_0 \wedge p_2}\})\); (iv) otherwise, \(K *\varphi = \mathrm {Cn}(\{ {\varphi }\})\). It is easy to check that \(*\) satisfies \((*1)\) through \((*5)\) but not \(*\)-strong reciprocity. Note that \(*\) defined in such way cannot be derived from any relational descriptor revision \(\circ \) in the way of \(K *\varphi = K \circ {\mathfrak {B}} \varphi \), since \(\leqq \) is linear in every relational model \(({\mathbb {X}},\leqq )\) (see Theorem 2 in Hansson 2013), which cannot model the circle consisting of \(K *p_1\), \(K *p_2\) and \(K *p_3\).

  10. 10.

    For more detailed investigation on believability relations, including its relationship with the epistemic entrenchment relation introduced in Gärdenfors and Makinson (1988), see Hansson (2014) and Zhang (2017).

  11. 11.

    It is easy to see that \(\preceq \)-coupling implies \(\preceq \)-weak coupling, provided that \(\preceq \)-transitivity and \(\preceq \)-counter dominance hold.

  12. 12.

    In what follows, it is always assumed that all sets A and B and C mentioned in postulates on multi-believability relations are finite sets.

  13. 13.

    See Zhang (2017).

  14. 14.

    Note that if we try to reproduce this proof in the sentential revision case, it is impossible to find a counterpart of the set \(A = \bigcup _{0 \le i \le k+1} A_{i}\) there. In particular, neither \(\varphi _0 \wedge \cdots \wedge \varphi _n\) nor \(\varphi _0 \vee \cdots \vee \varphi _n\) is workable.

  15. 15.

    See Jech (2008).

Notes

Acknowledgements

The author would like to thank the two anonymous reviewers for valuable comments. Funding was provided by China Scholarship Council.

References

  1. Alchourrón, C. E., & Makinson, D. (1982). On the logic of theory change: Contraction functions and their associated revision functions. Theoria, 48(1), 14–37.CrossRefGoogle Scholar
  2. Alchourrón, C. E., & Makinson, D. (1985). On the logic of theory change: Safe contraction. Studia Logica, 44(4), 405–422.CrossRefGoogle Scholar
  3. Alchourrn, C. E., Gärdenfors, P., & Makinson, D. (1985). On the logic of theory change: Partial meet contraction and revision functions. The Journal of Symbolic Logic, 50(2), 510–530.CrossRefGoogle Scholar
  4. Falappa, M. A., Kern-Isberner, G., Reis, M. D. L., & Simari, G. R. (2012). Prioritized and non-prioritized multiple change on belief bases. Journal of Philosophical Logic, 41(1), 77–113.CrossRefGoogle Scholar
  5. Fermé, E. L., & Hansson, S. O. (1999). Selective revision. Studia Logica, 63(3), 331–342.CrossRefGoogle Scholar
  6. Fermé, E. L., & Hansson, S. O. (2011). AGM 25 years. Journal of Philosophical Logic, 40(2), 295–331.CrossRefGoogle Scholar
  7. Fuhrmann, A. (1988). Relevant logics, modal logics and theory change. Ph.D. thesis, Australian National University.Google Scholar
  8. Gärdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. Cambridge, MA: The MIT Press.Google Scholar
  9. Gärdenfors, P., & Makinson, D. (1988). Revisions of knowledge systems using epistemic entrenchment. In Proceedings of the 2nd conference on theoretical aspects of reasoning about knowledge, TARK ’88 (pp. 83–95). San Francisco, CA: Morgan Kaufmann Publishers Inc.Google Scholar
  10. Grove, A. (1988). Two modellings for theory change. Journal of Philosophical Logic, 17(2), 157–170.CrossRefGoogle Scholar
  11. Hansson, S. O. (1994). Kernel contraction. Journal of Symbolic Logic, 59(3), 845–859.CrossRefGoogle Scholar
  12. Hansson, S. O. (2013). Descriptor revision. Studia Logica, 102(5), 955–980.CrossRefGoogle Scholar
  13. Hansson, S. O. (2014). Relations of epistemic proximity for belief change. Artificial Intelligence, 217, 76–91.CrossRefGoogle Scholar
  14. Hansson, S. O. (2017). Descriptor revision, Vol. 46 of Trends in Logic. Cham: Springer International Publishing.Google Scholar
  15. Harpen, W. L. (1976). Rational conceptual change. In PSA: Proceedings of the biennial meeting of philosophy of science association, volume two: Symposia and invited papers (pp. 462–494). The Univsersity of Chicago Press.Google Scholar
  16. Jech, T. J. (2008). The axiom of choice. NY: Dover Publications.Google Scholar
  17. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(12), 167–207.CrossRefGoogle Scholar
  18. Levi, I. (1977). Subjunctives, dispositions and chances. Synthese, 34(4), 423–455.CrossRefGoogle Scholar
  19. Makinson, D., & Gärdenfors, P. (1991). Relations between the logic of theory change and nonmonotonic logic. In A. Fuhrmann & M. Morreau (Eds.), The logic of theory change, no. 465 in Lecture Notes in Computer Science (pp. 183–205). Berlin: Springer.Google Scholar
  20. Zhang, L. (2017). Believability relations for select-direct sentential revision. Studia Logica, 105(1), 37–63.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.KTH Royal Institute of TechnologyStockholmSweden

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