Dynamic Logics of Belief Change

This chapter gives an overview of current dynamic logics that describe belief update and revision, both for single agents and in multi-agent settings. We employ a mixture of ideas from AGM belief revision theory and dynamic-epistemic logics of information-driven agency. After describing the basic background, we review logics of various kinds of beliefs based on plausibility models, and then go on to various sorts of belief change engendered by changes in current models through hard and soft information. We present matching complete logics with dynamic-epistemic recursion axioms, and develop a very general perspective on belief change by the use of event models and priority Chapter 7 of the Handbook of Logics for Knowledge and Belief, H. van Ditmarsch, J.Y. Halpern, W. van der Hoek and B. Kooi (eds), College Publications, 2015, pp. 299–368. 300 CHAPTER 7. DYNAMIC LOGICS OF BELIEF CHANGE update. The chapter continues with three topics that naturally complement the setting of single steps of belief change: connections with probabilistic approaches to belief change, long-term temporal process structure including links with formal learning theory, and multi-agent scenarios of information flow and belief revision in games and social networks. We end with a discussion of alternative approaches, further directions, and windows to the broader literature, while links with relevant philosophical traditions are discussed throughout. Human cognition and action involve a delicate art of living dangerously. Beliefs are crucial to the way we plan and choose our actions, even though our beliefs can be very wrong and refuted by new information. What keeps the benefits and dangers in harmony is our ability to revise beliefs as the need arises. In this chapter, we will look at the logical structure of belief revision, and belief change generally. But before we can do this, we need background of two kinds: (a) the pioneering AGM approach in terms of postulates governing belief revision which showed that this process has clear formal structures regulating its behavior, and (b) the basics of dynamic-epistemic logics of information flow which showed that change of attitudes for agent and the events triggering such changes are themselves susceptible to exact logical analysis. This is what we will provide in the first two sections of this chapter. With this material in place, Section 7.3 will then start our main topic, the logical treatment of belief revision. 7.1 Basics of belief revision 7.1.1 The AGM account of belief revision What happens when an agent is confronted with a new fact ' that goes against her prior beliefs? If she is to accept the new fact ' and maintain a consistent set of beliefs, she will have to give up some of her prior beliefs. But which of her old beliefs should she give up? More generally, what policy should she follow to revise her beliefs? As we will see in this chapter, several answers to this question are possible. The standard answer in the literature says that our agent should accept the new fact and at the same time maintain as many as possible of her old beliefs without arriving at a contradiction. Making this more precise has been the driving force behind Belief Revision Theory. Standard Belief Revision Theory, also called AGM theory (after the pioneering authors Alchourrón, Gärdenfors and Makinson) has provided us with a series of “rationality conditions”, that are meant to precisely govern the way in which a rational agent should revise her beliefs. AGM theory The AGM theory of belief revision is built up from three basic ingredients: 1) the notion of a theory (or “belief set”) T , which is a logically closed set of sentences { , ...} belonging to a given language L; 2) the input of new information, i.e., a syntactic formula '; and 3) a revision operator ⇤ which is a map associating a theory T ⇤ ' to each pair (T,') consisting of a theory T and an input sentence '. The construct T ⇤ ' is taken to represent the agent’s new 7.1. BASICS OF BELIEF REVISION 301 theory after learning '. Hence T ⇤ ' is the agent’s new set of beliefs, given that the initial set of beliefs is T and that the agent has learnt that '. Expansion The AGM authors impose a number of postulates or rationality conditions on the revision operation ⇤. To state these postulates, we first need an auxiliary belief expansion operator +, that is often considered an unproblematic form of basic update. Belief expansion is intended to model the simpler case in which the new incoming information ' does not contradict the agent’s prior beliefs. The expansion T +' of T with ' is defined as the closure under logical consequence of the set T [ {'}. AGM provides a list of 6 postulates that exactly regulate the expansion operator, but instead of listing them here we will concentrate on belief revision. However, later on, we will see that even expansion can be delicate when complex epistemic assertions are added. Revision Now, belief revision goes beyond belief expansion in its intricacies. It is regulated by the following famous AGM Belief Revision Postulates: (1) Closure T ⇤ ' is a belief set (2) Success ' 2 T ⇤ ' (3) Inclusion T ⇤ ' ✓ T + ' (4) Preservation If ¬' 62 T , then T + ' ✓ T ⇤ ' (5) Vacuity T ⇤ ' is inconsistent iff ` ¬' (6) Extensionality If ` '$ , then T ⇤ ' = T ⇤ (7) Subexpansion T ⇤ (' ^ ) ✓ (T ⇤ ') + (8) Superexpansion If ¬ 62 T ⇤ ', then T ⇤ (' ^ ) ◆ (T ⇤ ') + . These postulates look attractive, though there is more to them than meets the eye. For instance, while the success postulate looks obvious, in our later dynamicepistemic logics, it is the most controversial one in this list. In a logical system allowing complex epistemic formulas, the truth value of the target formula can change in a revision step, and the Success Postulate would recommend incorporating a falsehood ' into the agent’s theory T . One important case in which this can occur is when an introspective agent revises her beliefs on the basis of new information that refers to beliefs or higher-order beliefs (i.e., beliefs about beliefs). Because the AGM setting does not incorporate “theories about theories”, i.e., it ignores an agent’s higher-order beliefs, this problem is side-stepped. All the beliefs covered by AGM are so-called factual beliefs about ontic facts that do not refer to the epistemic state of the agent. However any logic for belief change that does allow explicit belief-operators in the language, will have to pay attention to success conditions for complex updates. A final striking aspect of the Success Postulate is the heavy emphasis placed on the last incoming proposition ', which can abruptly override long accumulated earlier experience against '. This theme, too, will return later when we discuss connections with formal theories of inductive learning. 302 CHAPTER 7. DYNAMIC LOGICS OF BELIEF CHANGE Contraction A third basic operation considered in AGM is that of belief contraction T ', where one removes a given assertion ' from a belief set T , while removing enough other beliefs to make ' underivable. This is harder than expansion, since one has to make sure that there is no other way within the new theory to derive the target formula after all. And while there is no unique way to construct a contracted theory, AGM prescribes the following formal postulates: (1) Closure T ' is a belief set (2) Contraction (T ') ✓ T (3) Minimal Action If ' 62 T , then T ' = T (4) Success If 6` ' then ' 62 (T ') (5) Recovery If ' 2 T , then T ✓ (T ') + ' (6) Extensionality If ` '$ , then T ' = T (7) Min-conjunction T ' \ T ✓ T (' ^ ) (8) Max-conjunction If ' 62 T (' ^ ), then T (' ^ ) ✓ T ' Again, these postulates have invited discussion, with Postulate 5 being the most controversial one. The Recovery Postulate is motivated by the intuitive principle of minimal change, which prescribes that a contraction should remove as little as possible from a given theory T . The three basic operations on theories introduced here are connected in various ways. A famous intuition is the Levi-identity T ⇤ ' = (T ¬') + ' saying that a revision can be obtained as a contraction followed by an expansion. An important result in this area is a theorem by Gärdenfors which shows that if the contraction operation satisfies postulates (1-4) and (6), while the expansion operator satisfies its usual postulates, then the revision operation defined by the Levi-identity will satisfy the revision postulates (1-6). Moreover, if the contraction operation satisfies the seventh postulate, then so does revision, and likewise for the eight postulate. 7.1.2 Conditionals and the Ramsey Test Another important connection runs between belief revision theory and the logic of conditionals. The Ramsey Test is a key ingredient in any study of this link. In 1929, F.P. Ramsey wrote: “If two people are arguing ‘If A, will B?’ and are both in doubt as to A, they are adding A hypothetically to their stock of knowledge and arguing on that basis about B; so that in a sense ‘If A, B’ and ‘If A, B’ are contradictories.” Clearly, this evaluation procedure for conditional sentences A > B uses the notion of belief revision. Gärdenfors formalised the connection with the Ramsey Test as the following statement: A > B 2 T iff B 2 T ⇤A 7.2. MODAL LOGICS OF BELIEF REVISION 303 which should hold for all theories T and sentences A, B. In a famous impossibility result, he then showed that the existence of such Ramsey conditionals is essentially incompatible with the AGM postulates for the belief revision operator ⇤. The standard way out of Gärdenfors’ impossibility result is to weaken the axioms of ⇤, or else to drop the Ramsey test. Most discussions in this line are cast in purely syntactic terms, and in a setting of propositional logic. However, in section 7.3 we will discuss a semantic perspective which saves much of the intuitions underlying the Ramsey Test. This is in fact a convenient point for turning to the modal logic paradigm in studying belief revision. Like we saw with belief expansion, it may help to first introduce a simpler scenario. The second part of this introductory section shows how modal logics can describe information change and its updates1 in what agents know. The techniques found in this realm will then be refined and extended in our later treatment of belief revision. 7.2 Modal logics of belief revision Starting in the 1980s, several authors have been struck by analogies between AGM revision theory and modal logic over suitable universes. Belief and related notions like knowledge could obviously be treated as standard modalities, while the dynamic aspect of belief change suggested the use of ideas from Propositional Dynamic Logic of programs or actions to deal with update, contraction, and revision. There is some interest in seeing how long things took to crystallise into the format used in this chapter, and hence we briefly mention a few of these proposals before introducing our final approach. Propositional dynamic logic over information models Propositional Dynamic Logic (PDL) is a modal logic that has both static propositions ' and programs or actions ⇡. It provides dynamic operators [⇡]' that one can use to reason about what will be true after an action takes place. One special operator of PDL is the “test of a proposition '” (denoted as '?): it takes a proposition ' into a program that tests if the current state satisfies '. Using this machinery over tree-like models of successive information states ordered by inclusion, in 1989, van Benthem introduced dynamic operators that mirror the operations of AGM in a modal framework. One is the addition of ' (also called “update”, denoted as +'), interpreted as moving from any state to a minimal extension satisfying '. Other operators included “downdates” ' moving back to the first preceding state in the ordering where ' is not true. Revision was defined via the Levi-identity. In a modification of this approach by de Rijke in the 1990s, these dynamic operators were taken to work on universes of theories. 1Our use of the term “update” in this chapter differs from a common terminology of “belief update” in AI, due to Katsuno and Mendelzon. The latter notion of update refers to belief change in a factually changing world, while we will mainly (though not exclusively) consider epistemic and doxastic changes but no changes of the basic ontic facts. This is a matter of convenience though, not of principle. 304 CHAPTER 7. DYNAMIC LOGICS OF BELIEF CHANGE Dynamic doxastic logic over abstract belief worlds These developments inspired Segerberg to develop the logical system of Dynamic Doxastic Logic (DDL), which operates at a higher abstraction level for its models. DDL combines a PDL dynamics for belief change with a static logic with modalities for knowledge K and belief B. The main syntactic construct in DDL is the use of the dynamic modal operator [⇤'] which reads “ holds after revision with '”, where ⇤' denotes a relation (often a function) that moves from the current world of the model to a new one. Here ' and were originally taken to be factual formulas only, but in later versions of DDL they can also contain epistemic or doxastic operators. This powerful language can express constructs such as [⇤']B stating that after revision with ' the agent believes . In what follows, we will take a more concrete modal approach to DDL’s abstract world, or state, changes involved in revision – but a comparison will be given in Section 7.9.1. Degrees of belief and quantitative update rules In this chapter, we will mainly focus on qualitative logics for belief change. But historically, the next step were quantitative systems for belief revision in the style of Dynamic Epistemic Logic, where the operations change current models instead of theories or single worlds. Such systems were proposed a decade ago, using labelled operators to express degrees of belief for an agent. In 2003, van Ditmarsch and Labuschagne gave a semantics in which each agent has associated accessibility relations corresponding to labeled preferences, and a syntax that can express degrees of belief. Revision of beliefs with new incoming information was modeled using a binary relation between information states for knowledge and degrees of belief. A more powerful system by Aucher in 2003 had degrees of belief interpreted in Spohn ranking models, and a sophisticated numerical “product update rule” in the style of Baltag, Moss and Solecki (BMS, see Section 7.5.2 below) showing how ranks of worlds change under a wide variety of incoming new information. Belief expansion via public announcement logic An early qualitative approach, due to van Ditmarsch, van der Hoek and Kooi in 2005, relates AGM belief expansion to the basic operation of public announcement in Dynamic Epistemic Logic. The idea is to work with standard relational modal models M for belief (in particular, these need not have a reflexive accessibility relation, since beliefs can be wrong), and then view the action of getting new information ' as a public announcement that takes M to its submodel consisting only of its '-worlds. Thus, an act of belief revision is modeled by a transformation of some current epistemic or doxastic model. The system had some built-in limitations, and important changes were made later by van Benthem and Baltag & Smets to the models and update mechanism to achieve a general theory – but it was on the methodological track that we will follow now for the rest of this chapter. Public announcement logic To demonstrate the methodology of Dynamic Epistemic Logic to be used in this chapter, we explain the basics of Public Announcement Logic (PAL). The language of PAL is built up as follows: ' ::= p | ¬' | ' ^ ' | Ki' | [!']' 7.2. MODAL LOGICS OF BELIEF REVISION 305 Here we read the Ki-modality as the knowledge of agent i and we read the dynamic construct [!'] as “ holds after the public announcement of '”. We think of announcements !' as public events where indubitable hard information that ' is the case becomes available to all agents simultaneously, whether by communication, observation, or yet other means. In what follows, we define and study the corresponding transformations of models, providing a constructive account of how information changes under this kind of update. Semantics for PAL We start with standard modal models M = (W, Ri, V ) where W is a non-empty set of possible worlds. For each agent i, we have an epistemic accessibility relation Ri, while V is a valuation which assigns sets of possible worlds to each atomic sentence p. The satisfaction relation can be introduced as usual in modal logic, making the clauses of the non-dynamic fragment exactly the standard ones for the multi-agent epistemic logic S5. For the case of knowledge (only an auxiliary initial interest in this chapter), we take Ri to be an equivalence relation, so that the underlying base logic is a multi-agent version of the modal logic S5. We now concentrate on the dynamics. The clause for the dynamic modality goes as follows: (M, w) |= [!'] iff (M, w) |= ' implies (M |', w) |= where M |' = (W 0, R0 i, V 0) is obtained by relativising the model M with ' as follows (here, as usual, [[']]M denotes the set of worlds in M where ' is true): W 0 = [[']]M R0 i = Ri \ ([[']]M ⇥ [[']]M ) V 0(p) = V (p) \ [[']]M Example 7.1 (Public announcement by world elimination) Figure 7.1 illustrates the update effect of a public announcement in the state w of model M such that in model M |p only the p-worlds survive: