Dynamic Doxastic Logic Research Articles

A Semantics for Hyperintensional Belief Revision Based on Information Bases

I propose a novel hyperintensional semantics for belief revision and a corresponding system of dynamic doxastic logic. The main goal of the framework is to reduce some of the idealisations that are common in the belief revision literature and in dynamic epistemic logic. The models of the new framework are primarily based on potentially incomplete or inconsistent collections of information, represented by situations in a situation space. I propose that by shifting the representational focus of doxastic models from belief sets to collections of information, and by defining changes of beliefs as artifacts of changes of information, we can achieve a more realistic account of belief representation and belief change. The proposed dynamic operation suggests a non-classical way of changing beliefs: belief revision occurs in non-explosive environments which allow for a non-monotonic and hyperintensional belief dynamics. A logic that is sound with respect to the semantics is also provided.

Tracking probabilistic truths: a logic for statistical learning

We propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds (or more generally, truth in all the worlds that are plausible enough). We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: (1) learning observable evidence obtained by repeated sampling from the unknown distribution; and (2) learning higher-order information about the distribution. The first changes only the plausibility map (via a ‘plausibilistic’ version of Bayes’ Rule), but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions (safe belief and statistical knowledge), as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense (statistical) knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning.

Dynamic Preference Logic meets iterated belief change: Representation results and postulates characterization

AGM's belief revision is one of the main paradigms in the study of belief change operations. Recently, several logics for belief and information change have been proposed in the literature and used to encode belief change operations in rich and expressive semantic frameworks. While the connections of AGM-like operations and their encoding in dynamic doxastic logics have been studied before by the work of Segerberg, most works on the area of Dynamic Epistemic Logics (DEL) have not, to our knowledge, attempted to use those logics as tools to investigate mathematical properties of belief change operators. This work investigates how Dynamic Preference Logic, a logic in the DEL family, can be used to study properties of dynamic belief change operators, focusing on well-known postulates of iterated belief change.

Learning Probabilities: Towards a Logic of Statistical Learning

We propose a new model for forming beliefs and learning about unknown probabilities (such as the probability of picking a red marble from a bag with an unknown distribution of coloured marbles). The most widespread model for such situations of 'radical uncertainty' is in terms of imprecise probabilities, i.e. representing the agent's knowledge as a set of probability measures. We add to this model a plausibility map, associating to each measure a plausibility number, as a way to go beyond what is known with certainty and represent the agent's beliefs about probability. There are a number of standard examples: Shannon Entropy, Centre of Mass etc. We then consider learning of two types of information: (1) learning by repeated sampling from the unknown distribution (e.g. picking marbles from the bag); and (2) learning higher-order information about the distribution (in the shape of linear inequalities, e.g. we are told there are more red marbles than green marbles). The first changes only the plausibility map (via a 'plausibilistic' version of Bayes' Rule), but leaves the given set of measures unchanged; the second shrinks the set of measures, without changing their plausibility. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. But our belief change does not comply with standard AGM axioms, since the revision induced by (1) is of a non-AGM type. This is essential, as it allows our agents to learn the true probability: we prove that the beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability). We end by sketching the contours of a dynamic doxastic logic for statistical learning.

A DDL Approach to Pluralistic Ignorance and Collective Belief

A group is in a state of pluralistic ignorance (PI) if, roughly speaking, every member of the group thinks that his or her belief or desire is different from the beliefs or desires of the other members of the group. PI has been invoked to explain many otherwise puzzling phenomena in social psychology. The main purpose of this article is to shed light on the nature of PI states – their structure, internal consistency and opacity – using the formal apparatus of Dynamic Doxastic Logic, and also to study the sense in which such states are “fragile”, i.e. to identify plausible conditions under which a PI state cascades into a state of shared belief as the result of announcement.

A formal model of emotion triggers: an approach for BDI agents

This paper formalizes part of a well-known psychological model of emo- tions.Inparticular,thelogicalstructureunderlyingtheconditionsthattriggeremotions are studied and then hierarchically organized. The insights gained therefrom are used to guide a formalization of emotion triggers, which proceeds in three stages. The first stage captures the conditions that trigger emotions in a semiformal way, i.e., without committing to an underlying formalism and semantics. The second stage cap- tures the main psychological notions used in the emotion model in dynamic doxastic logic. The third stage introduces a BDI-based framework (belief-desire-intention) with achievement goals, which is used to firmly ground the preceding stages. The result is a formalization of emotion triggers for BDI agents with achievement goals. The idea of proceeding in these stages is to provide different levels of commitment to formalisms, so that it remains relatively easy to extend or replace the used formalisms without having to start from scratch. Finally, we show that the formalization renders properties of emotions that are in line with the psychological model on which it is based.

DΔL: a dynamic deontic logic

This paper suggests that it should be possible to develop dynamic deontic logic as a counterpart to the very successful development of dynamic doxastic logic (or dynamic epistemic logic, as it is m ...

SOME COMPLETENESS THEOREMS IN THE DYNAMIC DOXASTIC LOGIC OF ITERATED BELIEF REVISION

The success of the AGM paradigm—the theory of belief change initiated by Alchourrón, Gärdenfors, and Makinson—is remarkable, as even a quick look at the literature it has generated will testify. But it is also remarkable, at least in hindsight, how limited was the original effort. For example, the theory concerns the beliefs of just one agent; all incoming information is accepted; belief change is uniquely determined by the new information; there is no provision for nested beliefs. And perhaps most surprising: there is no analysis of iterated change.In this paper it is that last restriction that is at issue. Our medium of study is dynamic doxastic logic (DDL). The success of the AGM paradigm The particular contribution of the paper is detailed completeness proofs for three dynamic doxastic logics of iterated belief revision.The problem of extending the AGM paradigm to include iterated change has been discussed for years, but systematic discussions have appeared only recently (see Segerberg, 2007 and forthcoming, but also van Benthem, 2007; Rott, 2006; Zvesper, 2007).

The incorporation of Moorean type information by introspective agents

The main task is to discuss the issue in belief dynamics in which philosophical beliefs and rational introspective agents incorporate Moorean type new information. First, a brief survey is conducted on Moore’s Paradox, and one of its solutions is introduced with the help of Update Semantics. Then, we present a Dynamic Doxastic Logic (DDL) which revises the belief of introspective agents put forward by Lindstrom & Rabinowicz. Next, we attempt to incorporate Moorean type new information within the DEL (DDL) framework, as advised by van Benthem, Segerberg et al. Though we maintain the principle of “the primacy of new information” from the literature on traditional belief revision theory, several unsuccessful ways are also presented. We then conclude that some special kind of success (weak success) can still be found in those revision processes although absolute success does not hold. At last, the relevant problem of “learnability” is re-considered through weak success.

Bridging learning theory and dynamic epistemic logic

This paper discusses the possibility of modelling inductive inference (Gold 1967) in dynamic epistemic logic (see e.g. van Ditmarsch et al. 2007). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing (Lange et al. 1996) and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.

Dynamic doxastic logic: why, how, and where to?

We investigate the research programme of dynamic doxastic logic (DDL) and analyze its underlying methodology. The Ramsey test for conditionals is used to characterize the logical and philosophical differences between two paradigmatic systems, AGM and KGM, which we develop and compare axiomatically and semantically. The importance of Gardenfors's impossibility result on the Ramsey test is highlighted by a comparison with Arrow's impossibility result on social choice. We end with an outlook on the prospects and the future of DDL.

Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision

In this paper, we present a semantical approach to multi-agent belief revision and belief update. For this, we introduce relational structures called conditional doxastic models (CDM's, for short). We show this setting to be equivalent to an epistemic version of the classical AGM Belief Revision theory. We present a logic of conditional beliefs that is complete w.r.t. CDM's. Moving then to belief updates (sometimes called “dynamic” belief revision) induced by epistemic actions, we consider two particular cases: public announcements and private announcements to subgroups of agents. We show how the standard semantics for these types of updates can be appropriately modified in order to apply it to CDM's, thus incorporating belief revision into our notion of update. We provide a complete axiomatization of the corresponding dynamic doxastic logics. As an application, we solve a “cheating version” of the Muddy Children Puzzle.

Ddl Unlimited: Dynamic Doxastic Logic for Introspective Agents

The theories of belief change developed within the AGM-tradition are not logics in the proper sense, but rather informal axiomatic theories of belief change. Instead of characterizing the models of belief and belief change in a formalized object language, the AGM-approach uses a natural language — ordinary mathematical English — to characterize the mathematical structures that are under study. Recently, however, various authors such as Johan van Benthem and Maarten de Rijke have suggested representing doxastic change within a formal logical language: a dynamic modal logic.1 Inspired by these suggestions Krister Segerberg has developed a very general logical framework for reasoning about doxastic change: dynamic doxastic logic (DDL).2 This framework may be seen as an extension of standard Hintikka-style doxastic logic (Hintikka 1962) with dynamic operators representing various kinds of transformations of the agent's doxastic state. Basic DDL describes an agent that has opinions about the external world and an ability to change these opinions in the light of new information. Such an agent is non-introspective in the sense that he lacks opinions about his own belief states. Here we are going to discuss various possibilities for developing a dynamic doxastic logic for introspective agents: full DDL or DDL unlimited. The project of constructing such a logic is faced with difficulties due to the fact that the agent’s own doxastic state now becomes a part of the reality that he is trying to explore: when an introspective agent learns more about the world, then the reality he holds beliefs about

Irrevocable Belief Revision in Dynamic Doxastic Logic

In this paper we present a new modeling for belief revision that is what we term irrevocable. This modeling is of philosophical interest since it captures some features of suppositional reasoning, and of formal interest since it is closely connected with AGM, yet provides for iterated belief revision. The analysis is couched in terms of dynamic doxastic logic.