Belief Update Through Semiorders

Abstract

Belief change is a core component of intelligent reasoning, enabling agents to adapt their beliefs in response to new information. A prominent form of belief change is belief revision, which involves altering an agent’s beliefs about a static (unchanging) world in light of new evidence. A foundational framework for modeling rational belief revision was introduced by Alchourrón, Gärdenfors, and Makinson (AGM), who formalized revision functions based on total preorders over possible worlds—that is, orderings that encode the relative plausibility of alternative states of affairs. Building on this, Peppas and Williams later characterized AGM-style revision functions using weaker preference structures known as semiorders, which, unlike total preorders, permit intransitive indifference between alternatives. In this article, we extend the framework of Peppas and Williams to the context of belief update. In contrast to belief revision, belief update concerns maintaining coherent beliefs in response to actual changes in a dynamic, evolving environment. We provide both axiomatic and semantic characterizations of update functions derived from semiorders, establishing corresponding representation theorems. These results essentially generalize the classical belief-update framework of Katsuno and Mendelzon, which relies on total preorders, thereby offering a broader and more flexible perspective. The intransitivity of indifference inherent in semiorders plays a central role in our framework, enabling the representation of nuanced plausibility distinctions between possible states of affairs—an essential feature for realistically modeling belief dynamics.

1. Introduction

The AGM paradigm, introduced by Alchourrón, Gärdenfors, and Makinson [1], is a widely recognized approach for rational belief change (revision and contraction)—the process by which an agent modifies her beliefs about a static (unchanged) world, in the light of new evidence [2,3,4]. Central to this framework is a set of formal postulates that govern the rational behavior of belief-change functions. The semantic interpretation of these postulates is provided by constructive models, which typically rely on plausibility rankings over possible states of affairs, such as possible worlds or sentences [5,6]. These rankings reflect the comparative plausibility of alternative scenarios, allowing the agent to identify the most credible ones. The modified belief corpus is then determined by the information shared across these most plausible alternatives.

A cornerstone assumption of the AGM paradigm is that indifference of comparative plausibility is transitive. To illustrate this transitivity, let us adopt the constructive model proposed by Katsuno and Mendelzon, which is based on a special type of total preorder over possible worlds, called faithful preorders [5]. These total preorders represent the agent’s plausibility ordering over possible worlds, relative to a given state of belief. Now, consider three distinct possible worlds $w_1$, $w_2$, and $w_3$. If $w_1$ and $w_2$ are equally plausible, and $w_2$ and $w_3$ are also equally plausible relative to the agent’s state of belief, then it follows by the transitivity of indifference inherent in total preorders that $w_1$ and $w_3$ must likewise be equally plausible.

In the field of preference modeling, however, it has long been recognized that transitivity is not always a natural characteristic of indifference in preferences. The following passage from Luce [7] illustrates the issue.

“Find a subject who prefers a cup of coffee with one cube of sugar to one with five cubes (this should not be difficult). Now, prepare 401 cups of coffee with $\left(1+{\displaystyle \frac{i}{100}}\right)·x$ grams of sugar, $i=0,1,\dots ,400$, where x is the weight of one cube of sugar. It is evident that he will be indifferent between cup i and cup $i+1$, for any i, but by choice he is not indifferent between $i=0$ and $i=400$.”

For similar reasons, Peppas and Williams argued that the assumption of transitive indifference in the plausibility of possible states of affairs, as applied in the context of belief change, is not always valid [8,9]. To illustrate this point, consider the following scenario.

Example 1

(Transitive Plausibility Indifference). Imagine an agent who believes his grandmother, whom he never met, had a huge collection of stamps from many different countries. Based on this belief, the agent would consider the possible world $w_{1000}$, where she owns 1000 stamps, more plausible than the possible world $w_{100}$, where she only owns 100 stamps. However, the agent might be indifferent between possible worlds like $w_{999}$ and $w_{998}$, where the only difference is one stamp less in the latter possible world. If indifference were transitive (as it is the case in the context of the AGM paradigm), then after a series of applications of transitivity, the agent would become indifferent between $w_{1000}$ and $w_{100}$, which contradicts his initial consideration that his grandmother’s collection was much larger and more diverse than just a hundred stamps.

In light of such scenarios, Peppas and Williams [8,9] proposed a reconstruction of the classical AGM paradigm by replacing the total preorders of Katsuno and Mendelzon [5] with weaker preference structures, known as semiorders [10,11]. The new approach captures the intransitivity of indifference inherent in semiorders, and gives rise to a broader class of revision (and contraction) functions, grounded in a set of weaker postulates and supported by a constructive model based on semiorders.

In this article, we address an unexplored gap by extending the semiorder-based framework of Peppas and Williams, originally developed for belief revision, to the domain of belief update, for which no analogous formal treatment has previously been offered. Belief update is a process distinct from belief revision, where a rational agent keeps her beliefs up to date with a dynamic (evolving) world. In this sense, belief update focuses on changes in the state of the world, while belief revision deals with changes in the agent’s perception or description of the world. Although the difference between belief revision and belief update was first recognized by Keller and Winslett [12] in the context of relational databases, belief update was formally defined by Katsuno and Mendelzon [13].

Herein, we generalize (i.e., weaken) Katsuno and Mendelzon’s model for belief update to accommodate intransitive plausibility indifference, which is captured by semiorders. Accordingly, we provide both a novel axiomatic characterization by introducing new update-specific postulates, and a semantic characterization, via a corresponding construction based on semiorders over possible worlds. Together, these yield formal representation results that extend classical belief update. By relaxing the assumption of transitive indifference inherent in total preorders, our approach enhances the expressive power of belief update, allowing for the representation of states of belief in which certain alternatives are judged equally plausible, even though no transitive plausibility chain connects them. This captures more realistic belief dynamics in cases where agents make fine-grained distinctions among possible worlds.

(Importantly, indifference between two choices should not be confused with incomparability. Incomparability is captured by—not necessarily total—preorders, from which belief-change operators can be induced, as demonstrated by Katsuno and Mendelzon—see [5] for belief revision and [13] for belief update.)

The remainder of this study is structured as follows: The next section outlines the formal prerequisites for our discussion. Following that, Section 3 provides a concise overview of the axiomatic and semantic characterizations of the AGM paradigm. Section 4 and Section 5 present the semiorders-based construction for belief revision, as proposed by Peppas and Williams. Section 6 briefly outlines the process of belief update, and Section 7 contains our main contribution, where we extend the Peppas-Williams construction to the context of belief update. The final section offers concluding remarks and outlines directions for future work.

2. Formal Background

This section is dedicated to establishing the formal foundations essential to our discussion.

• Logic Language: Throughout this study, we work with a propositional language $\mathcal{L}$, built over finitely many propositional variables (atoms), using the standard Boolean connectives ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (equivalence), ¬ (negation), and governed by classical propositional logic. The classical inference relation is denoted by ⊧.
• Belief Sets: For a set of sentences $\Gamma$ of $\mathcal{L}$, $Cn(\Gamma )$ denotes the set of all logical consequences of $\Gamma$; i.e., $Cn(\Gamma )=\left\{\phi \in \mathcal{L}:\Gamma \vDash \phi \right\}$. The set of beliefs of an agent will be modelled by a belief set, which is a theory of the language. A theory K is a deductively closed set of sentences of $\mathcal{L}$; that is, $K=Cn\left(K\right)$. A theory K is complete iff, for any sentence $\phi \in \mathcal{L}$, either $\phi \in K$ or $\neg \phi \in K$. For two sets of sentences $\Gamma$ and $\Delta$ of $\mathcal{L}$, we denote by $\Gamma +\Delta$ the deductive closure of the union $\Gamma \cup \Delta$; that is, $\Gamma +\Delta =Cn(\Gamma \cup \Delta )$. For a theory K and a sentence $\phi$ of $\mathcal{L}$, we write $K+\phi$ as an abbreviation of $K+\left\{\phi \right\}$.
• Possible Worlds: A literal is a propositional variable or its complement (negation). A possible world (or simply world) r is an inclusion-maximal consistent set of literals; hence, for any propositional variable p of $\mathcal{L}$, either $p\in r$ or $\neg p\in r$. The set of all possible worlds is denoted by $\mathbb{M}$. For a sentence or set of sentences $\phi$ of $\mathcal{L}$, $\left[\phi \right]$ is the set of worlds implying $\phi$; that is, $\left[\phi \right]=\left\{r\in \mathbb{M}:r\vDash \phi \right\}$. By the definition of complete theories, it follows that, for any consistent complete theory K, there is a single world $w\in \mathbb{M}$ such that $\left[K\right]=\left\{w\right\}$. In other words, K is satisfied uniquely by w, and it is true that $K=Cn\left(w\right)$. In this case, theory K is maximally informative, as it fully specifies—within the expressive limits of the language $\mathcal{L}$—what holds true or false in a given state of the world.
• Binary Relations. Let V be a non-empty set, and let R be a binary relation on V. For any subset S of V, the minimal elements in S, with respect to R, are contained in a set $min(S,R)$, such that
  $$min(S,R)=\left\{w\in S:\mathrm{for}\mathrm{all}{w}^{\prime }\in S,w^{\prime }Rw\mathrm{entails}wR{w}^{\prime }\right\}.$$

It is easy to verify that, if R is irreflexive and anti-symmetric (i.e., asymmetric), then

$$min(S,R)=\left\{w\in S:\mathrm{there}\mathrm{is}\mathrm{no}{w}^{\prime }\in S,\mathrm{such}\mathrm{that}{w}^{\prime }Rw\right\}.$$

A binary relation R on V is called:

• a preorder iff it is reflexive and transitive.
• a strict order iff it is irreflexive and transitive.
• total iff, for all $w,w^{\prime }\in V$, $wR{w}^{\prime }$ or $w^{\prime }Rw$. The strict part of a total preorder ⪯ over V is denoted by ≺; that is, $w\prec w^{\prime }$ iff $w⪯w^{\prime }$ and $w^{\prime }\cancel{⪯}w$.

Lastly, we denote the set of real numbers by $\mathbb{R}$, as usual.

3. The AGM Paradigm: Postulates for Belief Revision

In their seminal article [1], Alchourrón, Gärdenfors, and Makinson introduced a formal framework for modeling rational belief revision, which has since become known as the AGM paradigm. This framework remains a widely accepted foundation for the formal study and implementation of belief revision. Within the AGM paradigm, the process of belief revision is formalized through a binary function, called revision function.

Definition 1

(Revision Function, ∗). A revision function ∗ is a binary function that maps a theory K and a sentence φ of $\mathcal{L}$ to a new theory $K\ast \phi$, representing the result of revising K by φ.

Alchourrón, Gärdenfors, and Makinson emphasized that not every such function qualifies as rational; only those that satisfy the following set of rationality postulates are considered acceptable within the AGM paradigm.

$(\mathbf{K}\ast \mathbf{1})$	$K\ast \phi$ is a theory.
$(\mathbf{K}\ast \mathbf{2})$	$\phi \in K\ast \phi$.
$(\mathbf{K}\ast \mathbf{3})$	$K\ast \phi \subseteq K+\phi$.
$(\mathbf{K}\ast \mathbf{4})$	If $\neg \phi \notin K$, then $K+\phi \subseteq K\ast \phi$.
$(\mathbf{K}\ast \mathbf{5})$	If $\phi$ is consistent, then $K\ast \phi$ is consistent.
$(\mathbf{K}\ast \mathbf{6})$	If $Cn\left(\left\{\phi \right\}\right)=Cn\left(\left\{\psi \right\}\right)$, then $K\ast \phi =K\ast \psi$.
$(\mathbf{K}\ast \mathbf{7})$	$K\ast (\phi \wedge \psi )\subseteq (K\ast \phi )+\psi$.
$(\mathbf{K}\ast \mathbf{8})$	If $\neg \psi \notin K\ast \phi$, then $(K\ast \phi )+\psi \subseteq K\ast (\phi \wedge \psi )$.

The guiding intuition behind postulates $(K\ast 1)$–$(K\ast 8)$ is the principle of minimal change: the theory K should be modified as conservatively as possible to incorporate the new information $\phi$ in a consistent manner. A thorough examination of $(K\ast 1)$–$(K\ast 8)$ and their implications is provided in the work of Peppas [3] (Section 8.3.1) and Gärdenfors [2] (Section 3.3).

In a subsequent work [5], Katsuno and Mendelzon showcased that the revision functions satisfying postulates $(K\ast 1)$–$(K\ast 8)$ are precisely those that are induced by total preorders over possible worlds.

Definition 2

(Total Preorders Faithful to Theories, [5]). A total preorder ${⪯}_K$ over $\mathbb{M}$ is faithful to a theory K iff, for any $r,r^{\prime }\in \mathbb{M}$, the following conditions hold:

(i) 

If $r\in \left[K\right]$, then $r{⪯}_K{r}^{\prime }$.

(ii) 

If $r\in \left[K\right]$ and $r^{\prime }\notin \left[K\right]$, then $r{\prec }_K{r}^{\prime }$.

Intuitively, $r{⪯}_K{r}^{\prime }$ expresses that the world r is at least as plausible as the world $r^{\prime }$, relative to theory K. Hence, it follows from Definition 2 that all K-worlds (worlds in $\left[K\right]$) are equally plausible with respect to ${⪯}_K$, and strictly more plausible than all non-K-worlds (worlds not in $\left[K\right]$). The worlds not in $\left[K\right]$ may themselves be totally preordered under ${⪯}_K$, reflecting distinctions in their relative plausibility.

Katsuno and Mendelzon established the following representation theorem characterizing the class of revision functions induced from total preorders over worlds [5].

Theorem 1

([5]). Let K be a theory. A revision function ∗ satisfies postulates $(K\ast 1)$–$(K\ast 8)$ at K iff there exists a total preorder over $\mathbb{M}$, faithful to K, such that, for any $\phi \in \mathcal{L}$:

$$\mathbf{(}\mathbf{T}\mathbf{\ast }\mathbf{)} \left[K\ast \phi \right]=min(\left[\phi \right],{⪯}_K).$$

The above theorem shows that the revised belief set $K\ast \phi$ can be specified in terms of the $\phi$-worlds that are most plausible relative to the agent’s current belief set K.

4. Semiorders

Semiorders were introduced by Scott and Suppes [10] as a more natural alternative to total preorders for modeling preference. In this section, we briefly review the main definitions and results related to semiorders, some of which are included—following Peppas and Williams [8]—for completeness.

Definition 3

(Semiorder). Let V be a finite set of alternatives between which an agent makes choices. A semiorder ⊏ on V is a binary relation over V that satisfies the following axioms, for any $r_1,r_2,r_3,r_4\in V$:

(SO1) 

$r_1\cancel{⊏}{r}_1$.

(SO2) 

If $r_1⊏r_2$ and $r_2⊏r_3$, then $r_1⊏r_4$ or $r_4⊏r_3$.

(SO3) 

If $r_1⊏r_2$ and $r_3⊏r_4$, then $r_1⊏r_4$ or $r_3⊏r_2$.

Axiom (SO1) excludes self-preference, ensuring that the relation ⊏ is irreflexive. Axiom (SO2) ensures that, if we observe a chain of preferences, other options cannot be entirely disconnected—they must enter the ordering ⊏ in a coherent way. Lastly, axiom (SO3) avoids having completely unrelated comparisons by requiring that preferences across different pairs still interact.

For any two choices $r_1,r_2\in V$, we shall say that an agent is indifferent between $r_1$ and $r_2$, denoted $r_1\sim r_2$, iff $r_1\cancel{⊏}{r}_2$ and $r_2\cancel{⊏}{r}_1$. It turns out that indifference is not, in general, transitive in the context of semiorders. To see this, consider, for example, the semiorder ⊏ over the set $V=\{r_1,r_2,r_3,r_4,r_5\}$ depicted in Figure 1. It is easy to verify that ⊏ satisfies all axioms (SO1)–(SO3). Moreover, observe that $r_3\sim r_4$, $r_4\sim r_5$, and yet $r_3⊏r_5$, violating transitive indifference.

A key result concerning semiorders, which offers insight into their underlying structure, pertains to their numerical representation. It has been demonstrated that every semiorder ⊏ on V can be mapped onto a utility function $u:V\mapsto \mathbb{R}$, such that, for all $r_1,r_2\in V$, $r_1⊏r_2$ iff $u\left(r_2\right)-u\left(r_1\right)⩾1$, and $r_1\sim r_2$ iff $|u\left(r_2\right)-u\left(r_1\right)|<1$ [7,11]. Intuitively, this means that the agent differentiates between two alternatives $r_1$ and $r_2$ whenever the difference in their corresponding utilities exceeds a certain threshold (set to 1 in this case). For example, the semiorder ⊏ of Figure 1 can be represented by a utility function u, such that $u\left(r_1\right)=0$, $u\left(r_2\right)=0.5$, $u\left(r_3\right)=1.5$, $u\left(r_4\right)=2$, and $u\left(r_5\right)=2.5$.

Other useful facts about semiorders are summarised in the following remark, reported as a lemma in [8] (p. 164).

Remark 1

([8]). Let V be a non-empty set, let ⊏ be a semiorder over V, and let S be a non-empty subset of V. Then, the following statements are true:

(i) 

⊏ is transitive.

(ii) 

$min(S,⊏)\ne ⌀$.

(iii) 

If $r\in S$ and $r\notin min(S,⊏)$, then there exists an $r^{\prime }\in min(S,⊏)$ such that $r^{\prime }⊏r$.

The next observations follow directly from the above analysis.

Remark 2.

Let V be a non-empty set, and let ⊏ be a semiorder over V. Since ⊏ is irreflexive and transitive, ⊏ is a strict order over V. This in turn entails that ⊏ is asymmetric. Since a relation is asymmetric iff it is both anti-symmetric and irreflexive, it follows that ⊏ is anti-symmetric as well.

Remark 3.

Let V be a non-empty set. The binary relation over V defined as the union $(⊏)\cup (\sim )$ is reflexive and total, but not necessarily transitive.

We note at this point that, while Figure 1 presents a semiorder with a nearly linear structure for illustrative simplicity, semiorders can exhibit a much broader range of topologies beyond linear chains. In particular, they can support tree-like branching or layered hierarchies, as long as the defining axioms (SO1)–(SO3) are satisfied. This structural flexibility stems from the fact that indifference in semiorders is not transitive, allowing for richer configurations. In contrast, total preorders induce a more constrained structure, composed of indifference classes totally ordered among themselves. Each class forms a flat cluster of indistinguishable elements, while the overall ordering remains linear. Thus, whereas total preorders enforce a rigid topological form, semiorders offer greater expressive power, particularly for modeling nuanced or threshold-based preferences.

To eliminate the requirement of transitivity in plausibility indifference, Peppas and Williams [8] replace total preorders with semiorders in Katsuno and Mendelzon’s possible-worlds construction of a revision function [5]. To introduce the construction of semiorders, we first present the following definition of their faithfulness, as formulated in [8] (cf. Definition 2 of Section 3).

Definition 4

(Semiorders Faithful to Theories, [8]). A semiorder ${⊏}_K$ over $\mathbb{M}$ is faithful to a theory K iff, for any $r,r^{\prime }\in \mathbb{M}$, the following conditions hold:

(i) 

If $r\in \left[K\right]$, then there is no $r^{\prime }\in \mathbb{M}$ such that $r^{\prime }{⊏}_{K}r$.

(ii) 

If $r\in \left[K\right]$ and $r^{\prime }\notin \left[K\right]$, then $r{⊏}_K{r}^{\prime }$.

Intuitively, a semiorder ${⊏}_K$ over $\mathbb{M}$, faithful to K, captures the comparative plausibility of possible worlds relative to theory K. Specifically, the more plausible a world is, the lower it appears in the ordering. Therefore, in revising K by a sentence $\phi$, it is reasonable to assume that the resulting belief set $K\ast \phi$ is defined in terms of the most K-plausible $\phi$-worlds. As shown in Theorem 1, this is precisely the construction proposed by Katsuno and Mendelzon [5], with the key difference that their framework employed total preorders rather than semiorders. Within the context of semiorders, this idea is formally expressed by the following semantic condition (S∗).

$$\mathbf{(}\mathbf{S}\mathbf{\ast }\mathbf{)} [K\ast \phi ]=min(\left[\phi \right],{⊏}_K).$$

5. Weaker Postulates for Belief Revision

To characterize the class of semiorder-based revision functions, Peppas and Williams proposed the following set of postulates [8].

$(\mathbf{K}\ast \mathbf{8}\mathbf{a})$	$K\ast (\phi \vee \psi )\subseteq (K\ast \phi )+(K\ast \psi )$.
$(\mathbf{K}\ast \mathbf{8}\mathbf{b})$	If $K\ast \psi ⊈(K\ast \phi )+\psi$, then $K\ast \phi \subseteq (K\ast \psi )+\phi$.
$(\mathbf{K}\ast \mathbf{8}\mathbf{c})$	If $\neg \psi \in K\ast \phi$ and $\neg \chi \notin K\ast \phi$, then $K\ast (\phi \wedge \psi )\subseteq (K\ast \chi )+(\phi \wedge \psi )$.

The following observation from [8] is important.

Remark 4

([8]). The conjunction of postulates $(K\ast 8a)$–$(K\ast 8c)$ is a strict weakening of postulate $(K\ast 8)$.

On that basis, the representation Theorems 2 and 3 of [8] showcase that postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ constitute a precise axiomatic characterization of revision functions induced from semiorders over possible worlds.

Theorem 2

([8]). Let K be a theory, and let ∗ be a revision function satisfying postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ at K. There exists a semiorder ${⊏}_K$ over $\mathbb{M}$, faithful to K, that satisfies condition (S*).

Theorem 3

([8]). Let K be a theory, and let ${⊏}_K$ be a semiorder over $\mathbb{M}$, faithful to K. The revision function ∗ induced from ${⊏}_K$, via condition (S∗), satisfies postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ at K.

The following observation follows immediately from Theorems 2 and 3, and Remark 4.

Remark 5.

The revision functions induced from total preorders form a proper subclass of the revision functions induced from semiorders.

Finally, the classes of revision functions induced from total preorders and semiorders over possible worlds are depicted in Figure 2.

6. Postulates for Belief Update

Belief revision concerns how an agent changes her beliefs in response to new information, assuming the world itself remains static (unchanged). In contrast, belief update addresses how an agent maintains an accurate representation of a world that is itself dynamic and constantly evolving. The following real-world scenario, inspired by Winslett [14], illustrates the conceptual difference between these two processes of belief change.

Example 2

(Smart Home Sensor). Initially, the smart home agent knows that either the window is open (represented by the atom a) or the air conditioner is on (represented by the atom b), but not both.

• Case 1 (Revision): The agent receives a message saying “The window is open”. Based on her prior knowledge, according to which $(a\vee b)\wedge \neg (a\wedge b)$, she concludes that the air conditioner is off. This is a case of revision, where the world has not changed—just the agent’s knowledge of it.
• Case 2 (Update): The agent is told that “The window has just been opened”. Since this implies a change in the world, the agent does not necessarily infer that the air conditioner is off—it might still be running. This is an update, reflecting a new situation.

The distinction between belief revision and belief update was first identified by Keller and Winslett [12] in the context of relational databases. However, it was Katsuno and Mendelzon who provided a formal foundation for belief update, introducing a comprehensive axiomatic and semantic framework—analogous in spirit to the AGM paradigm—that rigorously characterizes this process [13]. This section is devoted to the presentation of this framework.

Like in belief revision, the process of belief update is formalized through a binary function, called update function.

Definition 5

(Update Function, ⋄). An update function ⋄ is a binary function that maps a theory K and a sentence φ of $\mathcal{L}$ to a new theory $K\diamond \phi$, representing the result of updating K by φ.

Katsuno and Mendelzon proposed the following set of rationality postulates for update functions [13].

$(\mathbf{K}\diamond \mathbf{1})$	$K\diamond \phi$ is a theory.
$(\mathbf{K}\diamond \mathbf{2})$	$\phi \in K\diamond \phi$.
$(\mathbf{K}\diamond \mathbf{3})$	If $\phi \in K$, then $K\diamond \phi =K$.
$(\mathbf{K}\diamond \mathbf{4})$	$K\diamond \phi =\mathcal{L}$ iff K or $\phi$ is inconsistent.
$(\mathbf{K}\diamond \mathbf{5})$	If $Cn\left(\left\{\phi \right\}\right)=Cn\left(\left\{\psi \right\}\right)$, then $K\diamond \phi =K\diamond \psi$.
$(\mathbf{K}\diamond \mathbf{6})$	$K\diamond (\phi \wedge \psi )\subseteq (K\diamond \phi )+\psi$.
$(\mathbf{K}\diamond \mathbf{7})$	If K is complete and $\neg \psi \notin K\diamond \phi$, then $(K\diamond \phi )+\psi \subseteq K\diamond (\phi \wedge \psi )$.
$(\mathbf{K}\diamond \mathbf{8})$	If K is consistent, then $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\diamond \phi$.

For ease of comparison, the Katsuno and Mendelzon postulates have been reformulated using AGM-style notation. Hence, in this presentation, postulates $(K\diamond 1)$–$(K\diamond 8)$ correspond to postulates (U1)–(U9) as originally introduced in [13], under the assumption that states of belief are represented as theories rather than individual sentences.

Katsuno and Mendelzon demonstrated that the update functions satisfying postulates $(K\diamond 1)$–$(K\diamond 8)$ are precisely those that are induced by total preorders over possible worlds [13].

Definition 6

(Total Preorders Faithful to Possible Worlds, [13]). A total preorder ${⪯}_w$ over $\mathbb{M}$ is faithful to a possible world w iff, for any $r\in \mathbb{M}$, $w\ne r$ implies $w{\prec }_{w}r$.

Definition 6 introduces the notation ${⪯}_w$ to denote a total preorder that is faithful to a particular world w, in direct analogy with Definition 2 of Section 3, where ${⪯}_K$ denotes a total preorder faithful to a theory K. As with faithful preorders associated with theories (which govern belief revision), $r{⪯}_w{r}^{\prime }$ states that the world r is at least as plausible as the world $r^{\prime }$, relative to the world w.

Katsuno and Mendelzon established the following representation theorem characterizing the class of update functions induced from total preorders over worlds [13].

Theorem 4

([13]). Let K be a theory. An update function ⋄ satisfies postulates $(K\diamond 1)$–$(K\diamond 8)$ at K iff, for each world $w\in \left[K\right]$, there exists a total preorder ${⪯}_w$ over $\mathbb{M}$, faithful to w, such that, for any $\phi \in \mathcal{L}$:

$$\mathbf{(}\mathbf{T}\mathbf{\diamond }\mathbf{)} \left[K\diamond \phi \right]=\bigcup _{w\in \left[K\right]}min(\left[\phi \right],{⪯}_w).$$

The semantic characterizations of belief revision and belief update, while structurally similar, reveal a key technical distinction between these two modes of belief change. In belief revision, a single total preorder over possible worlds is associated with a theory K, capturing the agent’s preferences among worlds. In contrast, belief update assigns a family of total preorders to a theory K—specifically, one preorder for each world satisfying K. This pointwise nature of belief update arises from postulate $(K\diamond 8)$, which mandates that the deductive closure of each K-world—essentially a consistent complete theory—is revised independently of the others.

7. Weaker Postulates for Belief Update

Having established the foundations of belief update, we now extend the analysis of Section 5 to this new setting. In particular, our aim is to provide a novel axiomatic and semantic characterization of update functions induced from semiorders over possible worlds. This constitutes a genuine extension of the belief-update framework beyond its classical formulation. To that end, consider the following postulates first.

$(\mathbf{K}\diamond \mathbf{7}\mathbf{a})$	If K is complete, then $K\diamond (\phi \vee \psi )\subseteq (K\diamond \phi )+(K\diamond \psi )$.
$(\mathbf{K}\diamond \mathbf{7}\mathbf{b})$	If K is complete and $K\diamond \psi ⊈(K\diamond \phi )+\psi$, then $K\diamond \phi \subseteq (K\diamond \psi )+\phi$.
$(\mathbf{K}\diamond \mathbf{7}\mathbf{c})$	If K is complete, $\neg \psi \in K\diamond \phi$ and $\neg \chi \notin K\diamond \phi$, then
	$K\diamond (\phi \wedge \psi )\subseteq (K\diamond \chi )+(\phi \wedge \psi )$.

Owing to the structural parallelism between postulates $(K\diamond 7a)$–$(K\diamond 7c)$ and $(K\ast 8a)$–$(K\ast 8c)$ of Section 5, Remark 6 follows immediately from Remark 4.

Remark 6.

The conjunction of postulates $(K\diamond 7a)$–$(K\diamond 7c)$ is a strict weakening of postulate $(K\diamond 7)$.

Definition 7 presents a straightforward adaptation of the notion of faithfulness for total preorders (Definition 6) to the context of semiorders.

Definition 7

(Semiorders Faithful to Possible Worlds). A semiorder ${⊏}_w$ over $\mathbb{M}$ is faithful to a possible world w iff, for any $r\in \mathbb{M}$, $w\ne r$ implies $w{⊏}_{w}r$.

In what follows, we shall confine our analysis to the interesting case of consistent theories. On that basis, consider the following semantic condition, which defines the updated belief set $K\diamond \phi$ as the theory corresponding to the union of all ${⊏}_w$-minimal $\phi$-worlds, for each world $w\in \left[K\right]$.

$$\mathbf{(}\mathbf{S}\mathbf{\diamond }\mathbf{)} \left[K\diamond \phi \right]=\bigcup _{w\in \left[K\right]}min(\left[\phi \right],{⊏}_w).$$

Obviously, condition (S⋄) serves as the semiorder-based counterpart to condition (T⋄) in Theorem 4 of Section 6.

Given this foundation, the subsequent representation Theorems 5 and 6—which serve as counterparts to Theorems 2 and 3 of Section 5—prove that postulates $(K\diamond 1)$–$(K\diamond 6)$, $(K\diamond 7a)$–$(K\diamond 7c)$ and $(K\diamond 8)$ precisely characterize the update functions induced (via (S⋄)) from semiorders over possible worlds.

Theorem 5.

Let K be a theory, and let ⋄ be an update function satisfying postulates $(K\diamond 1)$–$(K\diamond 6)$, $(K\diamond 7a)$–$(K\diamond 7c)$ and $(K\diamond 8)$ at K. There exists a family of semiorders ${\left\{{⊏}_w\right\}}_{w\in \left[K\right]}$ over $\mathbb{M}$ that satisfy condition (S⋄).

Proof. 

This proof partially draws upon arguments used in the proof of Theorem 2.1 in [15] (p. 123).

Let w be a world of $\left[K\right]$, let $H=Cn\left(w\right)$ be the consistent complete theory corresponding to w (i.e., $\left[H\right]=\left\{w\right\}$), and let $\phi$ be an arbitrary sentence of $\mathcal{L}$. Define a revision function ∗ such that $H\ast \phi =H\diamond \phi$. First, we show that ∗ satisfies postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ at theory H.

Postulates $(K\ast 1)$, $(K\ast 2)$, $(K\ast 5)$, $(K\ast 6)$, $(K\ast 7)$, and $(K\ast 8a)$–$(K\ast 8c)$ follow directly from $(K\diamond 1)$, $(K\diamond 2)$, $(K\diamond 4)$, $(K\diamond 5)$, $(K\diamond 6)$, and $(K\diamond 7a)$–$(K\diamond 7c)$, respectively. For postulate $(K\ast 3)$, first assume that $\phi \notin H$. Since H is complete, it follows that $\neg \phi \in H$, and therefore, $H+\phi =\mathcal{L}\supseteq H\ast \phi$. Next, assume that $\phi \in H$. Then, from postulate $(K\diamond 3)$ we derive that $H\diamond \phi =H=H+\phi$. Given that $H\ast \phi =H\diamond \phi$ (by definition), we have that $H\ast \phi =H+\phi$. Therefore, postulate $(K\ast 3)$ is satisfied at H in both cases. For postulate $(K\ast 4)$, assume that $\neg \phi \notin H$. Since H is complete, it follows that $\phi \in H$, and therefore, we derive from postulate $(K\diamond 3)$ that $H\diamond \phi =H=H+\phi$. Given that $H\ast \phi =H\diamond \phi$ (by definition), we have that $H+\phi =H\ast \phi$, from which we derive that $(K\ast 4)$ is satisfied at H.

Now, since ∗ satisfies postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ at H, it follows from Theorem 2 of Section 5 that there exists a semiorder ${⊏}_H$ over $\mathbb{M}$, faithful to H, such that $[H\ast \phi ]=min(\left[\phi \right],{⊏}_H)$. Since $\left[H\right]=\left\{w\right\}$ and $[H\ast \phi ]=[H\diamond \phi ]$, we derive that there exists a semiorder ${⊏}_w$ over $\mathbb{M}$, faithful to w, such that $[H\diamond \phi ]=min(\left[\phi \right],{⊏}_w)$. As the K-world w has been selected arbitrarily, we conclude, in view of postulate $(K\diamond 8)$, that $[K\diamond \phi ]={\displaystyle \bigcup _{w\in \left[K\right]}}min(\left[\phi \right],{⊏}_w)$. Consequently, there exists a family of semiorders ${\left\{{⊏}_w\right\}}_{w\in \left[K\right]}$ over $\mathbb{M}$ that satisfy condition (S⋄), as desired. □

Theorem 6.

Let K be a theory, and let ${\left\{{⊏}_w\right\}}_{w\in \left[K\right]}$ be a family of semiorders over $\mathbb{M}$. The update function ⋄ induced from ${\left\{{⊏}_w\right\}}_{w\in \left[K\right]}$, via condition (S⋄), satisfies postulates $(K\diamond 1)$–$(K\diamond 6)$, $(K\diamond 7a)$–$(K\diamond 7c)$ and $(K\diamond 8)$ at K.

Proof. 

Certain parts of this proof draw on reasoning from the proof of Theorem 2.2 in [15] (p. 124).

Let ⋄ be an update function such that, for any $\phi \in \mathcal{L}$, $[K\diamond \phi ]={\displaystyle \bigcup _{w\in \left[K\right]}}min(\left[\phi \right],{⊏}_w)$. Let w be a world of $\left[K\right]$, let $H=Cn\left(w\right)$ be the consistent complete theory corresponding to w (i.e., $\left[H\right]=\left\{w\right\}$), and let $\phi ,\chi ,\psi$ be arbitrary sentences of $\mathcal{L}$. Clearly then, $[H\diamond \phi ]=min(\left[\phi \right],{⊏}_w)$. Define a revision function ∗ such that $H\ast \phi =H\diamond \phi$. Since $\left[H\right]=\left\{w\right\}$ and $[H\ast \phi ]=[H\diamond \phi ]$, it follows that there exists a semiorder ${⊏}_H$ over $\mathbb{M}$, faithful to H, such that $[H\ast \phi ]=min(\left[\phi \right],{⊏}_H)$. Then, it follows from Theorem 3 of Section 5 that ∗ satisfies postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$ at H. Furthermore, combining the above, and since the K-world w has been selected arbitrarily, we derive that $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$. Hence, postulate $(K\diamond 8)$ is satisfied at K. (Interestingly, the identity $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$ also appears in a slightly different form in [15] (p. 123). In that work, Peppas and Williams referred to it as the Winslett Identity, owing to its close relationship with an identity introduced by Winslett in [14].)

Next, we show that ⋄ satisfies postulates $(K\diamond 1)$–$(K\diamond 6)$ and $(K\diamond 7a)$–$(K\diamond 7c)$ at K as well. Postulates $(K\diamond 1)$–$(K\diamond 5)$ follow directly from postulates $(K\ast 1)$–$(K\ast 7)$ and $(K\ast 8a)$–$(K\ast 8c)$.

For postulate $(K\diamond 6)$, given that $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$, for any $\phi \in \mathcal{L}$, we have that $K\diamond (\phi \wedge \psi )=$${\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast (\phi \wedge \psi )$. Moreover, by postulate $(K\ast 7)$, for every $w\in \left[K\right]$, $Cn\left(w\right)\ast (\phi \wedge \psi )\subseteq \left(Cn\left(w\right)\ast \phi \right)+\psi$, and therefore, ${\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast (\phi \wedge \psi )\subseteq$${\displaystyle \bigcap _{w\in \left[K\right]}}\left(Cn\left(w\right)\ast \phi +\psi \right)\subseteq \left({\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi \right)+\psi$. Combining the above, we derive that $K\diamond (\phi \wedge \psi )\subseteq \left(\bigcap _{w\in \left[K\right]}Cn\left(w\right)\ast \phi \right)+\psi$, and since ${\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi =K\diamond \phi$, we have that $K\diamond (\phi \wedge \psi )\subseteq (K\diamond \phi )+\psi$, as desired.

For postulate $(K\diamond 7a)$, assume that K is complete. Then, from $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$, we derive that $K\diamond \phi =K\ast \phi$. Since, from postulate $(K\ast 8a)$, $K\ast (\phi \vee \psi )\subseteq (K\ast \phi )+(K\ast \psi )$, it follows directly that $K\diamond (\phi \vee \psi )\subseteq (K\diamond \phi )+(K\diamond \psi )$, as desired.

For postulate $(K\diamond 7b)$, assume that K is complete and $K\diamond \psi ⊈(K\diamond \phi )+\psi$. Then, from $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$, we derive that $K\diamond \phi =K\ast \phi$. Hence, $K\ast \psi \neg \subseteq (K\ast \phi )+\psi$, which, in view of postulate $(K\ast 8b)$, entails that $K\ast \phi \subseteq (K\ast \psi )+\phi$. Therefore, we conclude that $K\diamond \phi \subseteq (K\diamond \psi )+\phi$, as desired.

Lastly, for postulate $(K\diamond 7c)$, assume that K is complete, $\neg \psi \in K\diamond \phi$ and $\neg \chi \notin K\diamond \phi$. Then, from $K\diamond \phi ={\displaystyle \bigcap _{w\in \left[K\right]}}Cn\left(w\right)\ast \phi$, we derive again that $K\diamond \phi =K\ast \phi$. Hence, $\neg \psi \in K\ast \phi$ and $\neg \chi \notin K\ast \phi$. Therefore, from postulate $(K\ast 8c)$, we have that $K\ast (\phi \wedge \psi )\subseteq (K\ast \chi )+(\phi \wedge \psi )$. Consequently, $K\diamond (\phi \wedge \psi )\subseteq (K\diamond \chi )+(\phi \wedge \psi )$, as desired. □

The following observation follows directly from Theorems 5 and 6, along with Remark 6, and highlights that the provided representation theorems essentially generalize the belief-update framework of Katsuno and Mendelzon [13].

Remark 7.

The update functions induced from total preorders form a proper subclass of the update functions induced from semiorders.

Finally, the classes of update functions induced from total preorders and semiorders over possible worlds are depicted in Figure 3 (cf. Figure 2 of Section 5).

Example 3 concludes this section by concretely demonstrating the process of belief update by means of semiorders over possible worlds.

Example 3

(Belief Update through Semiorders). Let $V=\{r_1,r_2,r_3,r_4,r_5\}\subseteq \mathbb{M}$ be a set of possible worlds, let K be a theory such that $\left[K\right]=\{r_1,r_2\}$, and let φ be a sentence of $\mathcal{L}$ such that $\left[\phi \right]=\{r_3,r_4,r_5\}$. Moreover, let ⊏ denote the semiorder over V depicted in Figure 1 of Section 4, and reproduced in Figure 4 for convenience. Recall that $r_3\sim r_4$, $r_4\sim r_5$, and $r_3⊏r_5$.

Let ⋄ be an update function satisfying postulates $(K\diamond 1)$–$(K\diamond 6)$, $(K\diamond 7a)$–$(K\diamond 7c)$ and $(K\diamond 8)$, which assigns at $r_1$ and $r_2$ two semiorders ${⊏}_{r_1}$, ${⊏}_{r_2}$, respectively, both over $\mathbb{M}$, and each faithful to its corresponding world. Assume further that ${⊏}_{r_1}$ and ⊏ coincide on their subdomains $\left(V\setminus \left\{r_2\right\}\right)\times \left(V\setminus \left\{r_2\right\}\right)$, and likewise ${⊏}_{r_2}$ and ⊏ coincide on their subdomains $\left(V\setminus \left\{r_1\right\}\right)\times \left(V\setminus \left\{r_1\right\}\right)$. Under these assumptions, it is straightforward to verify that $min(\left[\phi \right],{⊏}_{r_1})=min(\left[\phi \right],{⊏}_{r_2})=\{r_3,r_4\}$. Observe that the φ-world $r_5$ is not contained in $min(\left[\phi \right],{⊏}_{r_1})$ and $min(\left[\phi \right],{⊏}_{r_2})$, since $r_3⊏r_5$, and thus $r_3{⊏}_{r_1}{r}_5$ and $r_3{⊏}_{r_2}{r}_5$. Consequently, we derive from condition (S⋄) that

$$[K\diamond \phi ]=\bigcup _{w\in \left[K\right]}min(\left[\phi \right],{⊏}_w)=min(\left[\phi \right],{⊏}_{r_1})\cup min(\left[\phi \right],{⊏}_{r_2})=\{r_3,r_4\}.$$

It is important to note that the update scenario described in Example 3 cannot be adequately captured using total preorders over possible worlds, as defined in the belief-update framework of Katsuno and Mendelzon [13]. Under a total preorder, the $\phi$-worlds $r_2$, $r_3$, and $r_5$ would all be considered equally plausible, both from the perspective of the K-world $r_1$ and from that of $r_2$. This is because total preorders enforce transitive indifference: the indifference between $r_3$ and $r_4$, combined with the indifference between $r_4$ and $r_5$, would necessitate indifference between $r_3$ and $r_5$. As a result, in view of condition (T⋄) of Section 6, the set $[K\diamond \phi ]$ corresponding to the updated belief set would be $\{r_3,r_4,r_5\}$—a set that differs from the outcome $\{r_3,r_4\}$ derived in Example 3 under our generalized framework. Crucially, the equal plausibility of the $\phi$-worlds $r_2$, $r_3$, and $r_5$, relative to each K-world, can indeed be captured using an appropriate semiorder, since semiorders permit indifference to be intransitive, though they do not require it. These observations confirm that, while every belief-update scenario representable by total preorders can also be captured by semiorders, the converse does not hold (cf. Remark 7).

8. Conclusions

In this article, we extended the work of Peppas and Williams [8] on characterizing AGM-style revision functions induced from semiorders, applying it to the domain of belief update. Specifically, we provided both axiomatic and semantic characterizations of update functions derived from semiorders over possible worlds by establishing corresponding representation results (Theorems 5 and 6). Our approach constitutes a natural generalization of the classical belief-update model proposed by Katsuno and Mendelzon [13], which is based on total preorders over possible worlds. By replacing total preorders with semiorders, we weakened the underlying structure to allow for intransitive indifference—a more realistic feature in preference modeling that better captures the nuances of plausibility and belief dynamics. This generalization broadens the expressive power of belief-update frameworks (possibly at the cost of increased complexity in reasoning or algorithmic implementation), enabling the representation of states of belief where agents are indifferent between some alternatives, without requiring that this indifference be transitive.

As future work, we intend to explore applications of semiorder-based update functions in practical reasoning systems. In dynamic decision-making agents, such as autonomous robots or adaptive control systems, the ability to represent intransitive indifference allows agents to make context-sensitive updates without oversimplifying belief structures—for example, treating sensor readings that fall within indistinguishable thresholds as distinct only when differences become meaningful. In epistemic planning, where agents reason about sequences of actions and their informational effects, semiorder-based updates can support more realistic models of evolving knowledge, particularly when agents operate under uncertain or imprecise observations that resist total plausibility ordering. In multiagent environments, semiorders offer a principled way to model individual agents with distinct, threshold-based plausibility assessments, enabling richer belief-monitoring mechanisms that capture partial agreement, indifference gaps, and evolving consensus. Undoubtedly, these directions hold promise for expanding the expressiveness and robustness of belief-change systems under real-world constraints.