A Topology-Inspired Approach to AGM Belief Change

Abstract

AGM’s belief revision is one of the main paradigms in the study of belief change operations. Despite its popularity, it is well recognised that AGM’s work relies on a strong idealisation of the agent’s capabilities and on the supporting logic. Recent results in the literature have shed light on the definability of AGM-like belief change operations in general logics, showing how the properties of a logic can be leveraged to construct these operations. Exploring well-known connections between Logic, Algebra and Topology, our work explores a general notion of belief change contraction, which can be connected to well-studied operations in the literature, such as AGM rational contractions, partial meet contractions, and multiple contractions, employing the framework of Abstract Model Theory and Topological Semantics. We show that AGM rational contractions can be characterised in our framework, and we can establish connections to different results in the literature.

1 Introduction

Belief Change is the area that studies how doxastic agents change their minds after acquiring new information. One of the most influential approaches in the literature, namely the AGM framework [1], studies rational constraints, or postulates, that characterise rational ways of changing beliefs, given a representation of one’s doxastic state.

While the seminal work of AGM focuses on belief change for classical logic, recently, several works investigated its applications to a wide range of non-classical logics of interest in Artificial Intelligence [4, 16]. Investigating the definability of AGM belief change operations on abstract logics, Flouris [5] shows necessary and sufficient conditions for a logic to allow the definition of these operations and that several logics did not satisfy these requirements.

Flouris employs machinery from abstract logics and topology to prove their results. Similarly, investigating model-theoretic approaches to define forms of belief change, Souza and Wasserman [18, p.13] point out that “different notions of minimality arise as the reflection of topological properties of the model space”, which indicates a deeper connection between AGM belief change and the characteristics of topological space defined by the logic.

The literature on Abstract Logic and Abstract Model Theory has long established deep connections between properties of a logic and of the topology of the space of models that defines it [11, 12, 21]. Surprisingly, although topological approaches to the study of Formal Epistemology and Epistemic Logic exist in the literature [2, 3, 10], to our knowledge, very little work has been dedicated to the study of topological aspects of belief change. In fact, to our knowledge, only the work of Özgün [13] explicitly explores topological models to study belief change. Topological notions, however, seem to infiltrate model-theoretic approaches to belief change, such as that of Peppas [14] and of Ribeiro et al. [15] on the generalisation of AGM belief change to other logics than those considered by the original authors.

In this work, we investigate a general notion of belief change contraction and the definability of AGM belief contraction operators based on results from abstract model theory and its connections to topology (Theorems 1 and 2, and Lemma 5), generalising the results of Ribeiro et al. [15] on the definability of AGM contraction for non-monotonic logics (Proposition 6 and Corollary 2). We employ our characterisation to show, in Example 1, that Propositional Intuitionistic Logic is not AGM-compatible.

This work is structured as follows: in Sect. 2 we present preliminary notions related to Abstract Logic and Belief Change, which will be the base of our work; in Sect. 3, we develop the semantic framework employed in this work, investigating the connections between Logic, Abstract Model Theory and Topology; in Sect. 4, we employ the investigated framework to define belief change operations, and study their properties; in Sect. 5, we discuss the related work. Finally, we present some final considerations on the significance of our results and pathways for future research.

2 Preliminaries

In this work, we employ the tools from Abstract Logic and Abstract Model Theory to study classes of belief change operations and their definability in non-classical logics.

We will call a logic any pair $\mathcal {L} = \langle L, Cn \rangle$, where L is a non-empty set, called the logical language, and $Cn:2^L\rightarrow 2^L$ is a function called a consequence operator satisfying the following properties¹:

• inclusion: $A \subseteq Cn(A)$.

• idempotence: $Cn(A) = Cn(Cn(A))$.

• monotonicity: If $A\subseteq A'$ then $Cn(A) \subseteq Cn(A')$.

Given a logic $\mathcal {L} = \langle L, Cn\rangle$, we will employ the notation $Th_\mathcal {L}$ to describe the set of all theories of $\mathcal {L}$, i.e. $Th_\mathcal {L}= \{A\subseteq L~|~ Cn(A) = A\}$. Furthermore, we say a logic is compact if, for any $A \subseteq L$ and $\varphi \in L$, it holds that if $\varphi \in Cn(A)$, then there is some finite $A'\subseteq A$ s.t. $\varphi \in Cn(A')$.

Given a logic $\mathcal {L}$ as above, we call a belief change operation any function $\star : 2^L \times L \rightarrow 2^L$, which maps pairs of sets of sentences and a sentence, called a set of beliefs and a piece of input information, to a set of sentences, the resulting beliefs.

AGM [1] investigate three basic belief change operations: expansions, contractions and revisions. Belief expansion blindly integrates a new piece of information into the agent’s beliefs. Belief contraction removes a currently held belief from the agent’s set of beliefs with minimal alterations. Finally, belief revision is the operation of integrating new information into an agent’s beliefs while maintaining consistency.

Among these basic operations, only expansion can be univocally defined. The other two are characterised by a set of postulates, which define a class of suitable change operators representing different rational ways in which an agent can change their beliefs. Given a closed set of beliefs², i.e. $K\subseteq L$ s.t. $K = Cn(K)$, we say a belief change operation $\dot{-}$ is an AGM rational contraction on K if for any $\varphi ,\psi \in L$, it satisfies:

• (closure) $K\dot{-}\varphi = Cn(K\dot{-}\varphi )$

• (success) If $\varphi \not \in Cn(\emptyset )$ then $\varphi \notin K \dot{-}\varphi$

• (inclusion) $K\dot{-}\varphi \subseteq K$

• (vacuity) If $\varphi \not \in K$ then $K\dot{-}\varphi = K$

• (recovery) $K \subseteq Cn(K\dot{-}\varphi \cup \{\varphi \})$

• (extensionality) If $Cn(\varphi ) = Cn(\psi )$ then $K\dot{-}\varphi = K\dot{-}\psi$

To characterise their rational contractions, AGM propose the notion of partial meet belief contraction, an operation that preserves a maximal amount of “safe” information from the agent’s beliefs, i.e., information that cannot be used to derive what the agent has ceased to believe.

Definition 1

Let $B\subseteq L$ be a set of formulas and $\varphi \in L$ be a formula of L, the remainder set $B\bot _\mathcal {L} \varphi$ is the set of sets $B'$ satisfying:

• $B'\subseteq B$

• $\varphi \not \in Cn(B')$

• $B'\subset B''\subseteq B$ implies $\varphi \in Cn(B'')$.

A partial meet contraction $\dot{-}$ is an operation for which there is a selection function $\gamma$, that characterises this operation. By selection function, we mean that the function $\gamma$ satisfies (i) $\emptyset \ne \gamma (B\bot _\mathcal {L} \varphi ) \subseteq B\bot _\mathcal {L} \varphi$ if $B\bot _\mathcal {L} \varphi \ne \emptyset$ and (ii) $\gamma (B\bot _\mathcal {L} \varphi ) = \{B\}$ otherwise.

Definition 2

We say a belief base change operator $\dot{-}$ is a partial meet contraction on a set $B \subseteq L$ if there is a selection function $\gamma$, s.t. for any $\varphi$

$$B\dot{-}\varphi = \bigcap \gamma (B\bot _\mathcal {L} \varphi ).$$

The authors show that for any compact logic closed by classical disjunction and negation, a rational contraction on a closed set K is a partial meet contraction on K and vice versa. This equivalence, however, does not hold in general for any abstract logic [17].

On the other hand, Flouris [5] studied the definability of rational contraction operations, i.e. operations satisfying the original AGM postulates, in tarskian logics, obtaining sufficient and necessary conditions for such definability. Similarly, Hansson and Wassermann [9] showed that partial meet contractions are definable in any monotonic and compact (not necessarily taskian) logic. In this work, we will focus on AGM rational contractions; as such, in the following, we will discuss only Flouris’ characterisation.

Definition 3

Let $\mathcal {L}=\langle L, Cn\rangle$ be a logic, a set $B\subseteq L$ is said to be decomposable in $\mathcal {L}$, if for any $\varphi \in L$, with $Cn(\emptyset ) \subset Cn(\varphi ) \subseteq Cn(B)$, the set

$$\varphi ^-(B) = \{B' \subseteq B~| \varphi \not \in Cn(B') ~\text{ and }~ Cn(B)=Cn(B'\cup \{\varphi \})\}$$

is not empty. A logic is said to be decomposable if every $B\subseteq L$ is decomposable.

Flouris [5] show that decomposability is a necessary and sufficient condition for the definability of AGM contraction operations.

Proposition 1

(Adapted from [5]). Let $\mathcal {L}=\langle L, Cn\rangle$ be a logic, $K\subseteq L$ be a closed set of formulas, which is decomposable in $\mathcal {L}$, and $\dot{-}$ be a belief change operation, then $\dot{-}$ is a rational contraction on K iff for any $\varphi \in L$ it holds that (i) $K\dot{-}\varphi = Cn(K')$ for some $K' \in \varphi ^-(K)$, if $Cn(\emptyset ) \subset Cn(\varphi ) \subseteq Cn(K)$, and (ii) $K\dot{-}\varphi = K$, otherwise.

Our aim in this work is to study a general notion of belief change that subsumes different notions, such as partial meet and rational contraction, in a single semantic framework and characterise them. Here, we will focus on characterising rational contractions and using our framework to study how the topological properties of the interpretation space of the logic allow the construction of rational contractions. In doing so, we generalise Ribeiro et al.’s results on the characterisation of rational contractions for non-compact logics.

3 Logics, Abstract Model Theories, and Topological Spaces

The deep connection between Logic, Algebra, and Topology is well-known since, at least, the works of Stone [20]. This connection has given rise to fruitful research in areas such as Algebraic Logic, Abstract Model Theory, among others. Here, we will briefly explore some connections between logic and topology by means of an abstract model-theoretic approach, which will inform our results in Sect. 4.

The reasoning for focusing on a model-theoretic approach for belief change is not only based on the fact that this approach, in our opinion, provides an interesting framework with which we can connect and contrast the fundamental assumptions of our theory with those of related fields, such as Epistemic Logic, Conditional Logic and Non-monotonic Logics, but also a semantic approach helps us abstract from concrete realisations of these operations for particular logics, based on notions such as remainder sets, sets of complements, etc., to a general understanding of AGM Belief Change as a phenomenon.

As such, given a logical language L, we will employ the notion of an abstract model theory to prescribe how these formulas can be interpreted. This notion will then be used to formally define an abstract logic - a more common terminology to those working on AGM-based Belief Change, and with which we can establish connections between our results and those in the literature.

Definition 4

Let L be a logical language, we call a tuple $\mathcal {M} = \langle M, L, \vDash \rangle$ an abstract model theory (AMT) if

• M is a non-empty set of models or interpretations;

• L is a non-empty set of formulas called a logical language;

• $\vDash : M \mapsto 2^L$ is a satisfaction function.

Given a model $w\in M$ and a formula $\varphi \in L$, we say w satisfies $\varphi$, denoted $w\vDash \varphi$, if $\varphi \in \vDash (w)$. Similarly, for a set $\varGamma \subseteq L$, $w\vDash \varGamma$ if $w\vDash \varphi$ for all $\varphi \in \varGamma$.

Given a set of interpretations $X\subseteq M$ for some abstract model theory, we will often employ the notation Th(X) for the set of all formulas satisfied by the interpretations of X - also referred to as the ‘theory of X.’ If $w\in M$, we will denote $Th(\{w\})$ by Th(w). We also define, as usual, $\llbracket \varGamma \rrbracket$ as the set of all interpretations satisfying it. In the following, given a set of interpretations X, we will usually denote the complement of X, i.e. $M\setminus X$, by $\overline{X}$.

Definition 5

Let $\mathcal {L} = \langle L, Cn\rangle$ be a logic and $\mathcal {M} = \langle M, L, \vDash \rangle$ be an abstract model theory. We say the logic $\mathcal {L}$ is induced by $\mathcal {M}$ if for any $\varGamma \subseteq L$, it holds that $Cn(\varGamma ) = \{\psi \in L~|~ \text{ for } \text{ all }~w\in M~\text{ s.t. }~w\vDash \varGamma , w\vDash \psi \}$. We will often denote by $\mathcal {M} = Mod(\mathcal {L})$ the fact that the model theory $\mathcal {M}$ induces the logic $\mathcal {L}$.

Notice that in Definition 5, since we establish that $\varphi \in Cn(\varGamma )$ iff $\llbracket \varGamma \rrbracket \subseteq \llbracket \varphi \rrbracket$, it is easy to see that any logic induced by some AMT trivially satisfies inclusion, monotonicity and idempotency, i.e., it is tarskian.

Lemma 1

Let $\mathcal {M} = \langle M, L, \vDash \rangle$ be an AMT, it induces a tarskian logic $\mathcal {L}$.

More yet, given that any logic defines a lattice of its theories, we can use that structure as a domain of interpretation of the formulas. As such, all tarskian logics are induced by some AMT.

Proposition 2

Any tarskian logic $\mathcal {L}$ is induced by some abstract model theory $\mathcal {M}$

Proof

Let $\mathcal {L} = \langle L, Cn\rangle$, recall that the set of all theories is defined as $Th_\mathcal {L} = \{\varGamma \subseteq L~|~ Cn(\varGamma ) = \varGamma \}$. Let us define $\mathcal {M} = \langle Th_\mathcal {L}, \vDash \rangle$ s.t. $\varGamma \vDash \varphi$ iff $\varphi \in \varGamma$.

Well, let $\varGamma \subseteq L$ be a set of $\mathcal {L}$-formulas. Take $\psi \in Cn(\varGamma )$, then for any $\varGamma '\in Th_\mathcal {L}$, if $\varGamma '\vDash \varGamma$, then $\varGamma \subseteq \varGamma '$ and, since $\mathcal {L}$ is tarskian, $\varGamma '\vDash \psi$. Now, take $\psi \in L$ s.t. for any $\varGamma '\in Th_\mathcal {L}$ with $\varGamma '\vDash \varGamma$, it holds that $\varGamma '\vDash \psi$. Since $\mathcal {L}$ is tarskian, $\psi \in \varGamma '$. As $Cn(\varGamma )\in Th_\mathcal {L}$, then $\psi \in Cn(\varGamma )$, as well. Thus $Cn(\varGamma ) = \{\psi \in L~|~ \text{ for } \text{ all }~w\in M~\text{ s.t. }~w\vDash \varGamma ~\text{ implies }~ w\vDash \psi \}.$    $\square$

Next, we introduce an important concept to our theory, which will be used to characterise AGM rational contractions in our framework.

Definition 6

Let $Mod(\mathcal {L}) = \mathcal {M} = \langle M, \vDash \rangle$ be an abstract model theory, we say a set of interpretations $O \subseteq M$ is a basic open set (b.o.s.) of $\mathcal {M}$ if $\llbracket Th (O) \rrbracket = O$. We denote by $BO_{\mathcal {M}}$ the set of all b.o.s.’s of $\mathcal {M}$.

A similar notion to our basic open sets appears in the work of Ribeiro [15] as complete interpretation sets and plays a similar role in the semantic machinery we will employ in this work. Notice that the b.o.s.’s are exactly those sets which are the interpretation of a theory.

Lemma 2

Let $\mathcal {M} = \langle M, \vDash \rangle$ be an abstract model theory inducing a logic $\mathcal {L}$, and $O \subseteq M$. $O \in BO_\mathcal {M}$ iff $O = \llbracket K\rrbracket$ for some $K\in Th_\mathcal {L}$.

With the notion of a basic open set, we can introduce a notion of closure in our models, which complements a set of interpretations.

Definition 7

Let $\mathcal {M} = \langle M, \vDash \rangle$ be an abstract model theory inducing a logic $\mathcal {L}$, and $X \subseteq M$ a set of interpretations. We define the closure of X, denoted by Cl(X), the biggest set $O \subseteq \mathcal {L}$, s.t. $Th(O) = Th(X)$.

It is not difficult to see that Cl(X) exists and is unique, and $Cl(X) = \llbracket Th(X)\rrbracket$, as $\mathcal {L}$ is tarskian. More so, Cl is, in fact, a closure operator.

As stated before, since at least the works of Stone [20], it is well-known that logics have a deep connection with topology. This connection is the subject of study in areas such as Abstract Model Theory, Topological Logic, etc. Here, we will present how topological spaces can be thought of as spaces of interpretations for a logic and, in doing so, help shed light on the theoretical inspirations for our constructions.

Definition 8

We call topological space any tuple $\mathcal {W} = \langle W, \tau \rangle$, where W is a non-empty set and $\tau$ is a collection of subsets of W satisfying:

• 1. $\emptyset , W\in \tau$;

• 2. If $O,O'\in \tau$, then $O\cap O' \in \tau$;

• 3. If $\mathcal {O}\subseteq \tau$, then $\bigcup \mathcal {O} \in \tau$.

The elements of $\tau$ are called open sets of $\mathcal {W}$.

Definition 9

Let $\mathcal {L} = \langle L, Cn\rangle$ be a logic and $\mathcal {W} = \langle W, \tau \rangle$ a topological space. We say $\mathcal {W}$ induces the logic $\mathcal {L}$ if there is some function $f: \tau \rightarrow Th_\mathcal {L}$ s.t.

• If $O \subseteq O'$ then $f(O') \subseteq f(O)$;

• $f(\bigcup \mathcal {O}) = \bigcap _{O\in \mathcal {O}} f(O)$

• For any theories $K,K'\in Th_\mathcal {L}$, it holds $f(\bigcup f^{-1}(K) \cap \bigcup f^{-1}(K')) = Cn(K \cup K')$

In this case, we say f is an $\mathcal {L}$-interpretation on $\mathcal {W}$.

With that, it is easy to see that any tarskian logic is induced by some topological space since we can use the lattice of theories to construct it, in a similar fashion to what we have done in Proposition 2.

Proposition 3

Let $\mathcal {L} = \langle L, Cn\rangle$ be a logic, there is a topological space $\mathcal {W} = \langle W, \tau \rangle$ inducing $\mathcal {L}$.

Proof

Take $\mathcal {W} = \langle Th_\mathcal {L}, \tau \rangle$ s.t. that topology $\tau$ is the upset topology from the lattice of theories, i.e. the topology obtained from the subbase $\mathcal {B} = \{\uparrow T~|~T\in Th_\mathcal {L}\}$, where $\uparrow T = \{T'\in Th_\mathcal {L}~|~ T\subseteq T'\}$. It is trivial to verify that the function $f(O) = \bigcap O$ is an $\mathcal {L}$-interpretation on $\mathcal {W}$.

By contrasting Propositions 2 and 3, we are able to see that there is a strong connection between abstract model theories and topological spaces equipped with interpretation functions.

Lemma 3

Let $\mathcal {W} = \langle W, \tau \rangle$ be a topological space inducing a logic $\mathcal {L}$ and $f:\tau \mapsto Th_\mathcal {L}$ an $\mathcal {L}$-interpretation on $\mathcal {W}$. The abstract model theory $\mathcal {M}_\mathcal {W} = \langle W, \vDash \rangle$ s.t. $w\vDash \varphi$ if $\varphi \in f(\bigcup \{O\in \tau : w\in O\})$ induces $\mathcal {L}$. Furthermore, $BO_{\mathcal {M}_\mathcal {W}} = \{\cup f^{-1}(K)~|~K \in Th_\mathcal {L}\}$.

Notice that Lemma 3 above tells us that the language of AMTs and topological spaces are interchangeable to study the structure of the logic. Moreover, it is easy to see that any AMT defines a topological space inducing the same logic.

Lemma 4

Let $\mathcal {M} = \langle M, L, \vDash \rangle$ be an abstract model theory inducing a logic $\mathcal {L}$, the topological model $\mathcal {W}_{\mathcal {M}} = \langle M, 2^M\rangle$ induces $\mathcal {L}$ and $f: 2^M\mapsto Th_\mathcal {L}$ s.t. $f(O) = \{\varphi \in L~|~ \forall w\in O:~w\vDash \varphi \}$ is an $\mathcal {L}$-interpretation on $\mathcal {W}$.

We employed the terminology of open basic sets to the sets $\llbracket \varGamma \rrbracket$ as a foreshadowing of the fact that we employ exactly those sets to provide a basis for the canonical topology inducing the logic. In a sense, basic open sets are the ‘stable’ interpretations, meaning that they are completely definable within the logical language.

4 Topological Belief Change and AGM Rationality

We begin our investigation of belief change by proposing a semantic model for belief change, which is agnostic to the underlying logic. Notice that, differently from Sect. 2, we will consider belief contractions as operations which take two sets of formulas and return a set of formulas, i.e. a function of the form $\star : 2^L \times 2^L \mapsto 2^L$, as done by Ribeiro and Wasserman [16]. The reason for that is some non-classical logic may not possess conjunctions in its language. This means that we are dealing with multiple contractions, rather than single contractions. More precisely with choice contractions [6].

The central notion underlying most belief change operations is that of choice. Different constructions [1, 8, 15] represent the agent’s epistemic commitments by means of which beliefs the agent chooses to preserve (or discard) given new information. In connection with the literature on conditional logic and semantic modelling, we will represent this choice element as a function selecting elements from a set. We will focus our presentation on abstract model theories. However, given Lemmas 3 and 4, the same notions could be equivalently described employing topological spaces and interpretation functions.

Definition 10

Let $Mod(\mathcal {L}) = \langle M, \vDash \rangle$ be an AMT and $X\subseteq M$ a set of points (interpretations), we call a semantic selection function (SSF) on X any mapping $f: 2^M \times 2^M \rightarrow 2^M$, such that for any $Y,Z\subseteq M$, it holds that:

• i) $X \subseteq f(X,Y) \subseteq X\cup Y$

• ii) if $Y\ne \emptyset$, then $f(X,Y)\cap Y\ne \emptyset$

• iii) if $Y\cap X \ne \emptyset$, then $f(X,Y) \subseteq X$

Additionally, we say an SSF is uniform if it satisfies:

• iv) if $f(X,Y)\cap Z \ne \emptyset$ and $f(X,Z)\cap Y \ne \emptyset$ then, $f(X,Y) = f(X,Z)$.

With semantic selection functions, we can finally define a form of belief contraction, which we will study in this work. We aim to show that our semantic selection contractions generalise different notions of belief contractions in the literature and characterise AGM contractions by means of properties on the selection function and the space of models of the logic.

Definition 11

Let $\mathcal {L} = \langle L, Cn\rangle$ be a logic, we say a belief change operator $\dot{-} : 2^L \times 2^L \rightarrow 2^L$ is a semantic selection contraction operation on a set of formulas $B\subseteq L$, if there is an AMT $Mod(\mathcal {L})$ and an SSF f on $\llbracket B\rrbracket$ s.t. for any set of formulas $A \subseteq L$, it holds:

$$B\dot{-}A = Th(f(\llbracket B\rrbracket , \overline{\llbracket A\rrbracket } )) \cap B.$$

In such a case, we say f is a characteristic function for $\dot{-}$.

First, we show that semantic selection contractions satisfy the basic postulates for belief contraction.

Proposition 4

Let $\mathcal {L}$ be a logic, $B,A,A'\subseteq L$ be sets of formulas and $\dot{-} : 2^L \times 2^L \rightarrow 2^L$ is a semantic selection contraction operation on B, then $\dot{-}$ satisfies:

• (closure) If $Cn(B)=B$, then $Cn(B\dot{-}A) = B\dot{-}A$

• (success) If $A \not \subseteq Cn(\emptyset )$, then $A\not \subseteq Cn(B \dot{-}A)$

• (inclusion) $B\dot{-}A\subseteq B$

‘Additionally, if the characteristic function of $\dot{-}$ is uniform, then $\dot{-}$ satisfies

• (uniformity) If for any $B'\subseteq B$ it holds that $A \subseteq Cn(B')$ iff $A' \subseteq Cn(B')$, then it holds that $B\dot{-}A = B\dot{-}A'$

Proof

If $\dot{-}$ is a semantic selection contraction, then there is some AMT $Mod(\mathcal {L}) = \langle M,\vDash \rangle$ and a semantic selection function f, s.t., f is a characteristic function for $\dot{-}$. The satisfaction of the closure and inclusion postulates is trivial by definition of a semantic selection contraction. Let us show that the other two hold.

(success): If $A \not \subseteq Cn(\emptyset )$, then $\llbracket A\rrbracket \ne M$, i.e. $\overline{\llbracket A\rrbracket } \ne \emptyset$. Then $f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket })\cap \overline{\llbracket A\rrbracket } \ne \emptyset$. As such, there is some $m \in f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket })$, s.t. $m\not \vDash A$ and $A\not \subseteq Th(f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket }))$ In other words, $A \not \subseteq B\dot{-}A$.

(uniformity) Let $A,A' \subseteq L$ s.t. for any $B'\subseteq B$ it holds that $A \subseteq Cn(B')$ iff $A' \subseteq Cn(B')$. If $A\not \subseteq B$, then $\overline{\llbracket A\rrbracket }\cap \llbracket B\rrbracket \ne \emptyset$ and, by condition iii) of Definition 10, $f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket }) = \llbracket B\rrbracket$. As such, we only need to consider the case in which $A,A'\subseteq Cn(B)$. There are two cases to consider: $A \subseteq Cn(B\dot{-} A)$ and $A \not \subseteq Cn(B\dot{-} A)$. In the first case, $A \subseteq Cn(\emptyset )$, by (success), thus, $f(\llbracket B\rrbracket , \overline{\llbracket A\rrbracket }) = \llbracket B\rrbracket = f(\llbracket B\rrbracket , \overline{\llbracket A'\rrbracket })$ and $A'\subseteq Cn(B\dot{-}A')$, by condition i) of Definition 10. In the second case, as $B\dot{-}A \subseteq B$, then $A\not \subseteq Cn(B\dot{-}A)$ implies $A'\not \subseteq Cn(B\dot{-}A)$ and, in particular, $A' \not \subseteq Th(f(\llbracket B\rrbracket ,\overline{\llbracket A'\rrbracket }))$. As such, it must be the case that $f(\llbracket B\rrbracket ,\overline{\llbracket A'\rrbracket }) \cap \overline{\llbracket A\rrbracket } \ne \emptyset$. Conversely, we may also conclude that $f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket }) \cap \overline{\llbracket A'\rrbracket } \ne \emptyset$, thus, $f(\llbracket B\rrbracket ,\overline{\llbracket A\rrbracket }) = f(\llbracket B\rrbracket ,\overline{\llbracket A'\rrbracket })$, since f is uniform, and $B\dot{-} A = B\dot{-}A'$.    $\square$

Our interest now is investigating which properties of semantic selection functions imply adequate behaviours in their respective selection contractions.

Proposition 5

Let $Mod(\mathcal {L}) = \langle M, \vDash \rangle$ be an AMT inducing logic $\mathcal {L} = \langle L, Cn\rangle$, and $B,A \subseteq Th_\mathcal {L}$ theories s.t. $A \not \subseteq B$.

• if Cn is compact, for any $B' \in B\bot A$, there is some $w\in \overline{\llbracket A\rrbracket }$ s.t. $B' = Th(\llbracket B\rrbracket \cup \{m\})$.

• if $\mathcal {L}$ is AGM-compliant, for any $B' \in B^{-(A)}$ there is $X \subseteq \overline{\llbracket A\rrbracket }$ s.t. $B' = Th(\llbracket B\rrbracket \cup X)$.

The above result indicates that both AGM’s rational contractions and Hansson’s belief base contraction could, in principle, be defined by means of a semantic selection function that chooses an appropriate set of models to define the contraction. The question that now remains is: which conditions could be enforced on SSFs to define these operations? From now on, we will focus on studying the conditions of an SSF to ensure the respective contraction is rational. The study of partial meet contractions in the framework of semantic selection contractions will be relegated to future work.

Definition 12

Let $\mathcal {M} = \langle M, L, \vDash \rangle$ be an AMT, $X \in BO_{\mathcal {M}}$ be a b.o.s, and f be an SSF on X. We say f is topological in X if for any Y s.t. $\overline{Y} \in BO_{\mathcal {M}}$, it holds that $f(X,Y) \in BO_{\mathcal {M}}$.

We say a selections contraction operator $\dot{-}$ on a set of formulas B is topological if there is some topological SSF f on $\llbracket B\rrbracket$.

We now show that topological contractions are AGM rational contractions.

Theorem 1

Let $\mathcal {L}$ be a logic, $K\in Th_\mathcal {L}$ be a theory and $\dot{-} : 2^L \times 2^L \rightarrow 2^L$ be a topological semantic selection contraction on K. Then, for any $A,A'\subseteq L$, it holds that:

• (closure) $K\dot{-}A = Cn(K\dot{-}A)$

• (success) If $A \not \subseteq Cn(\emptyset )$, then $A\not \subseteq Cn(B \dot{-}A)$

• (inclusion) $B\dot{-}A\subseteq B$

• (vacuity) If $A\not \subseteq K$ then $K\dot{-}A = K$

• (recovery) $K \subseteq Cn(K\dot{-}A \cup A)$

• (extensionality) If $Cn(A) = Cn(A')$ then $K\dot{-}A = K\dot{-}A'$

Proof

Notice that (extensionality) and (closure) follow directly from the fact that K is a theory and from the definition of a selection contraction. For the other postulates, let us take a model theory $Mod(\mathcal {L})$ and an SSF f which characterise $\dot{-}$ on the set B. As (success) and (inclusion) have been proved in Proposition 4, we will focus on the remaining postulates.

We briefly discussed (vacuity) in the proof of (uniformity) in Proposition 4, but let us show the complete reasoning here. Take $A \not \subseteq K$, then $\llbracket K\rrbracket \cap \overline{\llbracket A\rrbracket } \ne \emptyset$. As such, $f(\llbracket K\rrbracket ,\llbracket K\rrbracket ) \cap \overline{\llbracket A\rrbracket } = \llbracket K\rrbracket \cap \overline{\llbracket A\rrbracket } \ne \emptyset$. On the other hand, $f(\llbracket K\rrbracket , \overline{\llbracket A\rrbracket }) \cap \llbracket K\rrbracket \ne \emptyset$. Thus, $f(\llbracket K\rrbracket ,\overline{\llbracket A\rrbracket }) = \llbracket K\rrbracket$ and $K\dot{-}A = K$.

To prove (recovery), we just need to observe that, since f is topological and $\llbracket K\dot{-}A\rrbracket = f(\llbracket K\rrbracket , \overline{\llbracket A\rrbracket })$, it holds that

$$\begin{array}{ll} \llbracket Cn( K\dot{-}A \cup A)\rrbracket & = \llbracket K\dot{-}A\rrbracket \cap \llbracket A\rrbracket \\ & = f(\llbracket K\rrbracket , \overline{\llbracket A\rrbracket }) \cap \llbracket A\rrbracket \\ & = (\llbracket K\rrbracket \cup X) \cap \llbracket A\rrbracket ,~\text{ for } \text{ some }~X\subseteq \overline{\llbracket A\rrbracket } \\ & = \llbracket K\rrbracket \cap \llbracket A\rrbracket \end{array}$$

Since $\llbracket Cn( K\dot{-}A \cup A)\rrbracket \subseteq \llbracket K\rrbracket$, we conclude $K \subseteq Cn(K\dot{-}A \cup A)$.    $\square$

More yet, any semantic selection contraction which is rational is topological.

Theorem 2

Let $\mathcal {L}$ be a logic, $K\in Th_\mathcal {L}$ be a theory and $\dot{-} : 2^L \times 2^L \mapsto 2^L$ be a semantic selection contraction on K satisfying (closure), (success),(inclusion), (vacuity),(recovery), and (extensionality) for any $A\subseteq L$, then $\dot{-}$ is topological.

Proof

As $\dot{-}$ is a semantic selection contraction, it means that there is some $Mod(\mathcal {L}) = \langle M, \vDash \rangle$ and an SSF f s.t. for any $A \in Th_\mathcal {L}$, $K\dot{-}A = Th(f(\llbracket K\rrbracket , \overline{\llbracket A\rrbracket })$.

Let us construct a function $f'$ as

$$ f'(X,Y) = {\left\{ \begin{array}{ll} f(X,Y) & \text{ if }~X\ne \llbracket K\rrbracket \\ Cl(f(X,Y))\cap (X\cup Y) & \text{ if }~X= \llbracket K\rrbracket \\ \end{array}\right. }$$

We need to show that $f'$ is SSF on $X=\llbracket K\rrbracket$ and that it is topological on X.

• i) Notice $X \subseteq f'(X,Y) \subseteq X\cup Y$ is immediate by construction, as $X \subseteq f(X,Y) \subseteq X\cup Y$, since f is an SSF, and $Z \subseteq Cl(Z)$.

• ii) If $Y\ne \emptyset$, then $f(X,Y)\cap Y\ne \emptyset$, thus $Cl(f(X,Y))\cap Y \ne \emptyset$, and $f'(X,Y)\cap Y \ne \emptyset$.

• iii) Take $Y\subseteq M$, s.t. $X\cap Y \ne \emptyset$, then $f(X,Y) =X$, thus $f'(X,Y) = Cl(X) = X$ since $X= \llbracket K\rrbracket$.

• iv) Take $Y \in BO_{Mod(\mathcal {L})}$, $f'(X,\overline{Y}) = Cl(f(X,\overline{Y})) \cap (X\cup Y)$. First, let us see that $Th(f'(X,Y)) = Th(f(X,Y))$. We known that $Th(f(X,Y)) = Th(Cl(f(X,Y))$, by definition, and $f'(X,Y) \subseteq Cl(f(X,Y))$, thus it holds that $Th(f(X,Y)) \subseteq Th(f'(X,Y))$. Furthermore, as $f(X,Y) \subseteq Cl(f(X,Y))$ and $f(X,Y)\subseteq X\cup Y$, it holds that $f(X,Y) \subseteq f'(X,Y)$. Thus $Th(f'(X,Y)) \subseteq Th(f(X,Y))$. Let $K' = Th(f'(X,Y))$. Since $Th(f'(X,Y)) = Th(f(X,Y))$, $X = \llbracket K\rrbracket$, $\overline{Y}\in BO_{Mod(\mathcal {L})}$, and f is the characteristic SSF of $\dot{-}$, then $K' = K \dot{-} Th(\overline{Y})$. Now, take $w\in \llbracket K'\rrbracket$ and suppose $w\not \in X\cup Y$, then $w\in \overline{X} \cap \overline{Y}$. From $w\vDash K'$, we conclude that $w\vDash K \dot{-} Th(\overline{Y})$ and, from $w\in \overline{Y}$, we conclude that $w\vDash Th(\overline{Y})$. Since $\dot{-}$ is a rational contraction, thus satisfying (recovery), it must hold that $w\vDash K$, i.e. $w\in \llbracket K\rrbracket = X$. But this contradicts the fact that $w\in \overline{X}$. Then it must hold that $\llbracket K'\rrbracket \subseteq X\cup Y$ and $f'(X,Y) = \llbracket K'\rrbracket \in BO_{Mod(\mathcal {L})}$. Thus $f'$ is topological.

   $\square$

Notice in the proof above that the requirement of satisfying the (recovery) postulate is essential to construct a topological SSF. As (recovery) is a central property of rational contractions, i.e. it is the postulate differentiating these operations from others in the literature, such as withdraws, partial meet, etc., it indicates that our notion of topological encodes the central aspects of AGM rationality.

4.1 On the Existence of Topological Contractions

We now turn our attention to investigating sufficient and necessary conditions for the definability of topological contractions in a given logic. This investigation is closely related to those of Flouris [5] and Ribeiro et al.  [15], for example, but employing the technical framework of Abstract Model Theory and Topology. First, an obvious observation is that topological functions can be constructed whenever there are appropriate basic open sets.

Lemma 5

Let $Mod(\mathcal {L})$ be a regular abstract model theory and $X \in BO_{Mod(\mathcal {L})}$ be a b.o.s. There is a topological semantic selection function f on X iff for any b.o.s. $Y\in BO_{Mod(\mathcal {L})}$ s.t. $\overline{Y} \cap X=\emptyset$, there is some b.o.s. $Z\in BO_{Mod(\mathcal {L})}$ s.t. $X \subset Z \subseteq X \cup \overline{Y}$.

Notice that Lemma 5 above is directly related to Proposition 1. In fact, the results can be directly translated into each other. As such, this informs us that any AGM rational contraction is, in fact, a topological semantic selection contraction. Also, in the face of Flouris’ [5, Theorem 4.2] characterisation of AGM-compliance by cuts, a similar characterisation could be encoded within our language. Since this result does not increase our understanding of AGM compliance and the definability of topological functions, we will omit it in this work. Lemma 5, however, can be employed to show that some logics are not AGM-compatible.

Example 1

Let us consider the intuitionistic propositional logic with one propositional symbol p, $\mathcal {L}_I = \langle L_0, Cn\rangle$ and $Mod(\mathcal {L}_I)$ its canonical model theory, i.e. the model theory obtained from its theory space, as in the proof of Proposition 2. The b.o.s.’s of $Mod(\mathcal {L}_I)$ form a complete lattice regarding the inclusion order, corresponding to the Rieger-Nishimura lattice of intuitionistic theories, depicted in Fig. 1. Any b.o.s. in it contains all sets below it in the lattice. Consider now the theories $K = Cn(\lnot p)$ and set $A = Cn(p\vee \lnot p)$, as one can see in Fig. 1, there is no theory $\varGamma '$ s.t. $Cn(\emptyset ) \subset K'$, $A \not \subseteq K'$, and $Cn(K' \cup A) = Cn(\emptyset )$. In another way, given the open set $X = \llbracket \lnot p\rrbracket$, for $Y = \llbracket p \vee \lnot p\rrbracket$, there is no b.o.s. Z s.t. $X\subset Z \subseteq X \cup \overline{Y}$, as any such element would necessarily include $Cn(p\vee \lnot p)\not \in X \cup \overline{Y}$. Thus, by Lemma 5, there is no topological semantic selection function on $\llbracket \lnot p\rrbracket$.

Fig. 1.

The Rieger-Nishimura lattice of intuitionistic basic open sets

Notice that the topological contractions provided above are defined to contract a (possibly infinite) set of formulas from a theory - which coincides with the notion of choice multiple contractions [6]. It is useful, however, to weaken our definition to account for a smaller family of theories considered as inputs to our contraction functions. For example, we can restrict our analysis to contracting a finite amount of information (or equivalently a finitely representable theory), as investigated by authors such as Ribeiro and Wassermann [16] in the context of belief change for description logics. To provide constructions for rational contractions on a limited class of inputs, we need to introduce the notion of a restricted semantic selection function.

Definition 13

Let $Mod(\mathcal {L}) = \langle M, \vDash \rangle$ be an abstract model theory , $\mathcal {O} \subseteq 2^M$ a family of subsets of M, $X\in \mathcal {O}$ a set of interpretations, and $f: 2^M \times 2^M \rightarrow 2^M$ a function. We say f is an SSF restricted to $\mathcal {O}$ if its restriction to $\mathcal {O}$ is an SSF on X, i.e. of the function $f\restriction _\mathcal {O}$ satisfies the conditions in Definition 10.

Notice the above definition is akin to defining selection functions on the sublogic defined by the family $\mathcal {O}$. Furthermore, we say an operator is a semantic contraction operator on a set of sentences B, restricted to a family of theories $\mathcal {T}$, if there is an SSF on $\llbracket B\rrbracket$ restricted to $\overline{\llbracket \mathcal {T}\rrbracket } = \{\overline{\llbracket K\rrbracket }~|~ K\in \mathcal {T}\}$, satisfying the condition in Definition 11 for any $A \subseteq \mathcal {T}$. With that, it is easy to see that the following holds.

Corollary 1

Let $\mathcal {L}$ be a logic, $K\in Th_\mathcal {L}$ be a theory, $\mathcal {T}\subseteq Th_\mathcal {L}$ be a family of theories of $\mathcal {L}$, and $\dot{-} : 2^L \times 2^L \rightarrow 2^L$ be a semantic selection contraction on K restricted to $\mathcal {T}$. Then, $\dot{-}$ satisfies (closure), (success),(inclusion), (vacuity),(recovery), and (extensionality) for all $A\subseteq L$ with $Cn(A) \in \mathcal {T}$, if, and only if, it is topological on $\mathcal {T}$.

We employ the result above to show that our construction subsumes those proposed by authors such as Ribeiro et al. [15] for rational contractions of finitely representable theories on boolean non-compact logics. Let us first define an appropriate property of this family of theories that is appropriate for the construction of a topological function.

Definition 14

Let $\mathcal {L}$ be a logic and $\mathcal {T} \subseteq Th_\mathcal {L}$ be a family of $\mathcal {L}$-theories. We say $\mathcal {T}$ is closed under classical negation if for any $\varGamma \in \mathcal {T}$, there is some $neg(\varGamma ) \in \mathcal {T}$ s.t. $neg(\varGamma )\cap \varGamma = Cn(\emptyset )$ and $Cn(\varGamma \cup neg(\varGamma )) = L$.

It is easy to see that we can always construct topological functions restricted to a family of theories which is closed under classical negation.

Proposition 6

Let $\mathcal {L}$ be a logic and $\mathcal {T} \subseteq Th_\mathcal {L}$ a family of theories closed under classical negation, there is a topological selection function f restricted to the set $\llbracket \mathcal {T}\rrbracket = \{\llbracket \varGamma \rrbracket ~|~\varGamma \in \mathcal {T}\}$.

Proof

It suffices to see that for any family of theories $\mathcal {T}$ closed under classical negation, if $\varGamma \in \mathcal {T}$, there is some theory $neg(\varGamma ) \in \mathcal {T}$ s.t. $neg(\varGamma ) \cap \varGamma = Cn(\emptyset )$ and $Cn(neg(\varGamma ) \cup \varGamma ) = L$. As such, $\overline{\llbracket \varGamma \rrbracket }$ is a b.o.s. and, thus, there is a function $f(X,Y) = X$ if $X\cap Y\ne \emptyset$ and $f(X,Y) = X\cup Y$, which is topological for all $Y = \overline{\llbracket \varGamma \rrbracket } = \llbracket neg(\varGamma )\rrbracket$ with $\varGamma \in \mathcal {T}$.   $\square$

The above result generalises Ribeiro et al. [15] construction for non-compact logics, showing that we can extend their construction to consider other logics and inputs - as long as we restrict the input to cases in which the logic admits classical negation.

Corollary 2

[15] Let $\mathcal {L}$ be a logic closed under disjunction and classical negation, there is a topological semantic selection contraction $\dot{-}$ restricted to finitely representable theories.

5 Related Work

Work on the definability of belief change operators in non-classical logics remounts at least to the work of Hansson and Wassermann [9], in which the authors show that partial meet contractions are definable for any monotonic and compact logic.

While most work on belief change for non-classical logics, such as that of Delgrande [4] or Girard and Tanaka [7], focused on defining appropriate operations in these logics, only recent work in the literature has focused on investigating which classes of logics allow the definition of a given operator.

Famously, Flouris [5] has shown a characterisation of the definability of AGM rational contractions employing the framework of order theory and its connections to Abstract Logic. This study has been explored further by Ribeiro et al. [15] showing constructions of AGM rational contractions for non-compact logics, and by Souza and Wassermann [18] showing that a semantically motivated framework, which they employ to study hyperintensional belief change, provides a rich foundation to encode and study different notions of belief change.

Our work develops on these ideas, connecting the work on Conditional Logics, Abstract Model Theory and Belief Change to investigate a general notion of belief contraction, namely semantic selection contraction, which can be used to unify different results in the area.

6 Final Considerations

In this work, we investigated the use of abstract model theory and topology to describe belief change operators, showing that our notion subsumes different operations in the literature, such as partial meet contractions and AGM rational contractions. Furthermore, we characterise rational contractions within our framework and show that other characterisation results, such as that of Flouris [5] and Ribeiro et al. [15], are captured within it.

This investigation provides greater evidence to the conjecture raised by Souza and Wassermann [19] that “different notions of minimality arise as the reflection of topological properties of the model space”, not necessarily as intrinsic distinction in their constructions.

For future work, we intend to study different belief change operations in this framework, such as partial meet contractions, revisions and semi-revisions and their relations, aiming to provide a unified view of definability and constructibility of these operations based on a single semantic framework.