Qualitative Logics for Belief Dynamics by Sonja Smets
We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistem... more We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistemic logic. As the " static " basis for belief revision, we use epistemic plausibility models, together with a modal language based on two epistemic operators: a " knowledge " modality K (the standard S5, fully introspective, notion), and a " safe belief " modality 2 (" weak " , non-negatively-introspective, notion, capturing a version of Lehrer's " indefeasible knowledge "). To deal with " dynamic " belief revision, we introduce action plausibility models, representing various types of " doxastic events ". Action models " act " on state models via a modified update product operation: the " Action-Priority " Update. This is the natural dynamic generalization of AGM revision, giving priority to the incoming information (i.e., to " actions ") over prior beliefs. We completely axiomatize this logic, and show how our update mechanism can " simulate " , in a uniform manner , many different belief-revision policies.
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This chapter gives an overview of current dynamic logics that describe belief update and revision... more 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 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.
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The pre-eminence of logical dynamics, over a static and purely propo-sitional view of Logic, lies... more The pre-eminence of logical dynamics, over a static and purely propo-sitional view of Logic, lies at the core of a new understanding of both formal epistemology and the logical foundations of quantum mechanics. Both areas appear at first sight to be based on purely static propositional formalisms, but in our view their fundamental operators are essentially dynamic in nature. Quantum logic can be best understood as the logic of physically-constrained informational interactions (in the form of measurements and entanglement) between subsystems of a global physical system. Similarly, (multi-agent) epistemic logic is the logic of socially-constrained informational interactions (in the form of direct observations, learning, various forms of communication and testimony) between " subsystems " of a social system. Dynamic Epistemic Logic (DEL) provides us with a unifying setting in which these informational interactions, coming from seemingly very different areas of research, can be fully compared and analyzed. The DEL formalism comes with a powerful set of tools that allows us to make the underlying dynamic/interactive mechanisms fully transparent.
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Table of Contents, Preface written by J. van Benthem and Introduction written by A. Baltag and S.... more Table of Contents, Preface written by J. van Benthem and Introduction written by A. Baltag and S. Smets. BOOK: `Johan van Benthem on Logic and Information Dynamics' , Outstanding Contributions to Logic Series, Volume 4, 55 pages, pp. xv-lxix, Springer, 2014.
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We present a complete, decidable logic for reasoning about a notion of completely trustworthy (" ... more We present a complete, decidable logic for reasoning about a notion of completely trustworthy (" conclusive ") evidence and its relations to justifiable (implicit) belief and knowledge, as well as to their explicit justifications. This logic makes use of a number of evidence-related notions such as availability, admissibility, and " goodness " of a piece of evidence, and is based on an innovative modification of the Fitting semantics for Artemov's Justification Logic designed to preempt Gettier-type counterexamples. We combine this with ideas from belief revision and awareness logics to provide an account for explicitly justified (defeasible) knowledge based on conclusive evidence that addresses the problem of (logical) omniscience.
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We present a logic for reasoning about the evidence-based knowledge and beliefs and the evidentia... more We present a logic for reasoning about the evidence-based knowledge and beliefs and the evidential dynamics of non-logically-omniscient agents. We do this by adapting key tools and techniques from Dynamic Epistemic Logic, Justification Logic, and Belief Revision so as to provide a lightweight, yet fine-grained approach that characterizes well-known epistemic and doxastic attitudes in terms of the evidential reasoning that justifies these attitudes. We then add the dynamic operations of evidence introduction, evidence-based inference, strong acceptance of new evidence (evidential " upgrade "), and irrevocable acceptance of additional evidence (evidential " update "). We exemplify our theory by providing a formal dynamic account of Lehrer's well-known Gettier-type scenario involving the famous Ferrari and the infamous Messrs. Nogot and Havit.
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In this paper we re-evaluate Segerberg's " full DDL " (Dynamic Doxas-tic Logic) from the perspect... more In this paper we re-evaluate Segerberg's " full DDL " (Dynamic Doxas-tic Logic) from the perspective of Dynamic Epistemic Logic (DEL), in its belief-revision-friendly incarnation. We argue that a correct version of full DDL must give up the Success Postulate for dynamic revision. Next, we present (an appropriately generalized and simplified version of) full DDL, showing that it is a generalization of the so-called Topo-logic of Moss and Parikh. We construct AGM-friendly versions of full DDL, corresponding to various revising/contracting operations considered in the Belief Revision literature. We show that DDL can internalize inside one model the " external " doxastic dynamics of DEL, as well as the evidential dynamics investigated by van Benthem and Pacuit. In our Conclusions section, we compare three styles of modeling doxastic dynamics: DDL, DEL and PDL/ETL (the Propo-sitional Dynamic Logic approach, with its Epistemic Temporal Logic variant).
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We investigate the issue of reaching doxastic agreement among the agents of a group by " sharing ... more We investigate the issue of reaching doxastic agreement among the agents of a group by " sharing " information via successive acts of sincere, persuasive and public communication within the group. The topic relates to " preference aggregation " in Social Choice theory, where the problem is to find a natural and fair merge operation for aggregating the agents' preferences into a single group preference. In this paper we study this topic within the setting of Dynamic Epistemic Logic (DEL) and its recent extensions to belief revision theory. First we interpret the agents' preference relations as " doxastic preferences " or " doxastic plausibility orders ". Using these plausibility structures, we can express several types of relevant doxastic attitudes: ranging from an agent's simple beliefs and conditional beliefs to her defeasible knowledge and irrevocable knowledge. Next, we model communication via a series of doxastic/epistemic actions that affect the agent's plausibility structures. Because these structures can change in different ways, a group of agents has to adopt a specific communication protocol in order to aggregate their doxastic attitudes. We study the type of merges that are realizable via specific communication protocols and we highlight important factors which may influence the result, such as " the order of the speakers " , or the listeners' doxastic attitudes towards the speakers (i.e their opinions concerning the reliability of the incoming information).
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While propositional doxastic attitudes , like knowledge and belief, capture an agent's opinion ab... more While propositional doxastic attitudes , like knowledge and belief, capture an agent's opinion about certain propositions, her attitudes towards sources of information express her opinion about the reliability (or trustwor-thiness) of those sources. If an agent trusts a witness, then she will, within certain limits, tend to accept his testimony as veridical. But if she considers the witness to be a notorious liar, she may come to believe the opposite of what he tells her. In this paper, we put such attitudes towards sources (or dynamic (doxastic) attitudes) center stage, and formalize them as belief-revision strategies: policies governing how an agent changess her beliefs whenever new information from a certain (type of) source is received. We present a semantic, qualitative modelling of this notion and investigate its properties .
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We investigate the process of truth-seeking by iterated belief revision with higher-level doxasti... more We investigate the process of truth-seeking by iterated belief revision with higher-level doxastic information. In this paper we elaborate further on the main results and formal settings provided in [8, 7], linking this previous work to the issue of truth approximation. On the one hand our previous results show that on an initial finite Kripke model, a truthful belief upgrade (with the same true sentence) may be repeated 'ad infinitum', without ever reaching a fixed point of the belief-revision process. On the other hand, we proved some positive convergence results: the agent's simple beliefs (and knowledge) will eventually stabilize when iterating updates or truthful radical upgrades. In this paper we apply these results to the problem of convergence to the truth. We study the conditions under which the fixed points of iterated upgrades (if they are reached) would coincide with the truth. We highlight the case in which from some moment onwards, at least if the agent completely or highly trusts the source of new information, she can predict all the future correct answers to a series of questions (but without ever knowing for sure that this is the case). We give two different conditions ensuring that beliefs eventually converge on " full (complete) truth " , as well as conditions ensuring only that they converge to true (but not necessarily complete) beliefs.
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What happens in the long term with a group’s beliefs, knowledge and “epistemic states” (fully des... more What happens in the long term with a group’s beliefs, knowledge and “epistemic states” (fully describable in fact by conditional beliefs), when receiving (or exchanging) a sequence of public announcements of truthful but uncertain information? Do the agents’ beliefs (or knowledge, or conditional beliefs, or other doxastic attitudes such as “strong beliefs”) reach a fixed point? Or do they exhibit instead a cyclic behavior, oscillating forever? These questions are of obvious importance for Belief Revision theory, Learning theory and Social Choice theory,
and may have some relevance to Game Theory as well. In fact, some of these questions came to our attention due to a recent talk by J. van Benthem (Chennai, January 2009), in which he was refining his previous “dynamic” analysis [14] of backward induction solution in perfect information games. This extended abstract provides some partial answers to some of the questions above, as well as a convenient setting for investigating further the ones that are still open.
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We present a semantic analysis of the Ramsey test, pointing out its deep underlying flaw: the ten... more We present a semantic analysis of the Ramsey test, pointing out its deep underlying flaw: the tension between the " static " nature of AGM revision (which was originally tailored for revision of only purely ontic beliefs, and can be applied to higher-order beliefs only if given a " backwards-looking " interpretation) and the fact that, semantically speaking, any Ramsey conditional must be a modal operator (more precisely, a dynamic-epistemic one). Thus, a belief about a Ramsey conditional is in fact a higher-order belief, hence the AGM revision postulates are not applicable to it, except in their " backwards-looking " interpretation. But that interpretation is consistent only with a restricted (weak) version of Ramsey's test (in-applicable to already revised theories). The solution out of the conundrum is twofold: either accept only the weak Ramsey test; or replace the AGM revision operator * by a truly " dynamic " revision operator ⊗, which will not satisfy the AGM axioms, but will do something better: it will " keep up with reality " , correctly describing revision with higher-order beliefs.
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We investigate the long-term behavior of iterated belief revision with higher-level doxastic info... more We investigate the long-term behavior of iterated belief revision with higher-level doxastic information. While the classical literature on iterated belief revision [13, 11] deals only with propositional information, we are interested in learning (by an introspective agent, of some partial information about the) answers to various questions Q 1, Q 2, ..., Q n , ... that may refer to the agent’s own beliefs (or even to her belief-revision plans). Here, “learning” can be taken either in the “hard” sense (of becoming absolutely certain of the answer) or in the “soft” sense (accepting some answers as more plausible than others). If the questions are binary (“is φ true or not?”), the agent “learns” a sequence of true doxastic sentences φ 1, ..., φ n , .... “Investigating the long-term behavior” of this process means that we are interested in whether or not the agent’s beliefs, her “knowledge” and her conditional beliefs stabilize eventually or keep changing forever.
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We present a logic of conditional doxastic actions, obtained by incorporating ideas from belief r... more We present a logic of conditional doxastic actions, obtained by incorporating ideas from belief revision theory into the usual dynamic logic of epistemic actions. We do this by extending to actions the setting of epistemic plausibility models, developed in Baltag and Smets (2006) for representing (static) conditional beliefs. We introduce
a natural extension of the notion of update product from Baltag and Moss (2004) to plausibility models.
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Electronic Notes in Theoretical Computer Science, 2006
In this paper, we present a semantical approach to multi-agent belief revision and belief update.... more 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.
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In this paper, we develop a notion of doxastic actions, general enough to cover all examples of c... more In this paper, we develop a notion of doxastic actions, general enough to cover all examples of communication actions and most other belief-changing actions encountered in the literature, but also flexible enough to deal with the issue of (static and dynamic) revision of beliefs. This can be seen as a natural extension of the work in [3, 4] on “epistemic actions”, incorporating ideas from the semantics of belief revision and of conditional belief, along the lines pioneered in [2] and [11], but using the conditional belief approach adopted in [22, 10, 9] and adapted in [25] to the context of dynamic belief revision.
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Quantum Logic and Quantum Foundations by Sonja Smets
Preprint IQSA 2023, 2023
We take a fresh look at Wigner's Friend thought-experiment and some of its more recent variants a... more We take a fresh look at Wigner's Friend thought-experiment and some of its more recent variants and extensions, such as the Frauchiger-Renner (FR) Paradox. We discuss various solutions proposed in the literature, focusing on a few questions: what is the correct epistemic interpretation of the multiplicity of state assignments in these scenarios; under which conditions can one include classical observers into the quantum state descriptions, in a way that is still compatible with traditional Quantum Mechanics?; under which conditions can one system be admitted as an additional 'observer' from the perspective of another background observer?; when can the standard axioms of multi-agent Epistemic Logic (that allow "knowledge transfer" between agents) be applied to quantum-physical observers? In the last part of the paper, we propose a new answer to these questions, sketch a particular formal implementation of this answer, and apply it to obtain a principled solution to Wigner Friend-type paradoxes.
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We introduce a probabilistic modal (dynamic-epistemic) quantum logic PLQP for reasoning about qua... more We introduce a probabilistic modal (dynamic-epistemic) quantum logic PLQP for reasoning about quantum algorithms. We illustrate its expressivity by using it to encode the correctness of the well-known quantum search algorithm, as well as of a quantum protocol known to solve one of the paradigmatic tasks from classical distributed computing (the leader election problem). We also provide a general method (extending an idea employed in the decidability proof in [12]) for proving the decidability of a range of quantum logics, interpreted on finite-dimensional Hilbert spaces. We give general conditions for the applicability of this method, and in particular we apply it to prove the decidability of PLQP.
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We address the old question whether a logical understanding of Quantum Mechanics requires abandon... more We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others 1 , our answer is a clear " no ". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with an empirical-experimental approach to Logic, as advocated by both Beth and Put-nam, but understood by us in the view of the operational-realistic tradition of Jauch and Piron, i.e. as an investigation of " the logic of yes-no experiments " (or " questions "). Technically, we use the recently-developed setting of Quantum Dynamic Logic [4, 6] to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results [4], we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the " non-classicality " of quantum behavior than any perspective based on static Propositional Logic.
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Mathematical Structures in Computer Science, 2006
The main contribution of this paper is the introduction of a dynamic logic formalism for reasonin... more The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. We extend that work here to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the Quantum Logic tradition with concepts from (dynamic) Modal Logic and from Quantum Computation. This logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterizations to various forms of entanglement (e.g. the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a while range of other quantum circuits and programs, including other known protocols (e.g. Superdense Coding, Entanglement Swapping, Logic-Gate Teleportation etc.), can be similarly verified using our logic.
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Qualitative Logics for Belief Dynamics by Sonja Smets
and may have some relevance to Game Theory as well. In fact, some of these questions came to our attention due to a recent talk by J. van Benthem (Chennai, January 2009), in which he was refining his previous “dynamic” analysis [14] of backward induction solution in perfect information games. This extended abstract provides some partial answers to some of the questions above, as well as a convenient setting for investigating further the ones that are still open.
a natural extension of the notion of update product from Baltag and Moss (2004) to plausibility models.
Quantum Logic and Quantum Foundations by Sonja Smets
and may have some relevance to Game Theory as well. In fact, some of these questions came to our attention due to a recent talk by J. van Benthem (Chennai, January 2009), in which he was refining his previous “dynamic” analysis [14] of backward induction solution in perfect information games. This extended abstract provides some partial answers to some of the questions above, as well as a convenient setting for investigating further the ones that are still open.
a natural extension of the notion of update product from Baltag and Moss (2004) to plausibility models.
quantum measurements, unitary evolutions and entanglements in compound quantum systems. We give a syntax and a relational semantics in which we abstract away from phases and probabilities. We present a sound proof system for this logic, and we show how to characterize by logical means various forms of entanglement (e.g. the Bell states) and various linear operators. As an example we sketch an analysis of the teleportation protocol.
still universal in this setting, while the other two methods are not.