We look at multi-agent versions of three different belief contraction operations (severe withdrawal, conservative contraction and moderate contraction), considering them as dynamic operations on plausibility models, meant to represent joint actions of “suspension of belief” by groups of agents (or by individual agents). We provide sound and complete axiomatizations, in the presence of standard static operators such as conditional belief, knowledge and safe belief.

In this paper we present a topological epistemic logic, with modalities for knowledge (modelled as the universal modality), knowability (represented by the topological interior operator), and unknowability of the actual world. The last notion has a non-self-referential reading (modelled by Cantor derivative: the set of limit points of a given set) and a self-referential one (modelled by Cantor's perfect core of a given set: its largest subset without isolated points, where x is isolated iff { x } is open). We completely axiomatize this logic, showing that it is decidable and pspace -complete, and we apply it to the analysis of a famous epistemic puzzle: the Surprise Exam Paradox.

We study the topological μ-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability, and finite model property over general topological spaces, as well as overT0andTDspaces. We also investigate the relational μ-calculus, providing general completeness results for all natural fragments of the μ-calculus over many different classes of relational frames. Unlike most other such proofs for μ-calculi, ours is model theoretic, making an innovative use of a known method from modal logic (the ‘final’ submodel of the canonical model), which has the twin advantages of great generality and essential simplicity.

We study the topological μ-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and FMP over general topological spaces, as well as over T0 and TD spaces. We also investigate relational μ-calculus, providing general completeness results for all natural fragments of μ-calculus over many different classes of relational frames. Unlike most other such proofs for μ-calculus, ours is modeltheoretic, making an innovative use of a known Modal Logic method (-the 'final' submodel of the canonical model), that has the twin advantages of great generality and essential simplicity.

We propose a number of powerful dynamic-epistemic logics for multi-agent information sharing and acts of publicly or privately accessing other agents’ information databases. The static base of our logics is obtained by adding to standard epistemic logic comparative epistemic assertions for groups or individuals, as well as a common distributed knowledge operator (that combines features of both common knowledge and distributed knowledge). On the dynamic side, we introduce actions by which epistemic superiority can be acquired: “sharing all one knows” (by e.g. giving access to one’s information database to all or some of the other agents), as well as more complex informational events, such as hacking. We completely axiomatize several such logics and prove their decidability.

Abstract This article introduces a Probabilistic Logic of Communication and Change , which captures in a unified framework subjective probability, arbitrary levels of mutual knowledge and a mechanism for multi-agent Bayesian updates that can model complex social-epistemic scenarios, such as informational cascades. We show soundness, completeness and decidability of our logic, and apply it to a concrete example of cascade.

In this paper, we propose a unified logical framework for representing and analyzing various forms of correlated information change. Our main thesis is that "logical dynamics," in the sense of van Benthem (Exploring logical dynamics. CSLI Publications, Stanford, 1996; Logical dynamics of information and interaction. Cambridge University Press, Cambridge, 2011), and in particular dynamic epistemic notions of conditional, as developed in Baltag and Smets (Electron Notes Theor Comput Sci 165:5-21, 2006a; Stud Log 89:185-209, 2008a; Texts in logic and games. Amsterdam University Press, Amsterdam, pp 9-58, 2008b), play a central role in understanding and modeling a wide range of apparently very different information-gathering phenomena which do have one specific feature in common, namely the very act of learning new information may directly change the reality that is being learned. On the one hand, we focus on the way in which an introspective agent changes her beliefs when learning new higher-order information, i.e., information that may refer to her own beliefs. On the other hand, we analyze situations in which an observer learns about a phenomenon by performing observations that may perturb the very phenomenon under study, as in the case of quantum measurements, or observations in social sciences, psychology and medicine. Our formal techniques are based on ideas from dynamic logic and on the modeling of "dynamic conditionals." We offer a semantics based on "test frames," i.e., Kripke frames labeled by propositional formulae which yields a unified setting for the two types of correlated information change under study. We show how this framework can be used to analyze the ontic and epistemic-informational aspects of quantum measurements and to compare them with other types of observation, testing, belief revision, counterfactual conditionals, etc.

We analyze the learning power of iterated belief revision methods, and in particular their universality: whether or not they can learn everything that can be learnt. We look in particular at three popular methods: conditioning, lexicographic revision and minimal revision. Our main result is that conditioning and lexicographic revision are universal on arbitrary epistemic states, provided that the observational setting is sound and complete (only true data are observed, and all true data are eventually observed) and provided that a non-standard (non-well-founded) prior plausibility relation is allowed. We show that a standard (well-founded) belief-revision setting is in general too narrow for this. We also show that minimal revision is not universal. Finally, we consider situations in which observational errors (false observations) may occur. Given a fairness condition (saying that only finitely many errors occur, and that every error is eventually corrected), we show that lexicographic revision is still universal in this setting, while the other two methods are not.

What happens if in the Muddy Children story [22] we drop the assumption that the public announcements (made by the father and by the children) are commonly known to be always true, and instead we simply assume that they are true and commonly believed to be true? More generally, 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?

Public announcement logic is an extension of multi-agent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: □φ expresses that φ is true after an arbitrary announcement ψ. As this includes the trivial announcement ⊤, one might as well say that □φ expresses what remains true after any announcement: it therefore corresponds to truth persistence after (definable) relativisation. The dual operation ⋄φ expresses that there is an announcement after which φ. This gives a perspective on Fitch's knowability issues: for which formulas φ does it hold that φ → ⋄Kφ? We give various semantic results, and we show completeness for a Hilbert-style axiomatisation of this logic.

We investigate the discrete (finite) case of the Popper-Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (nonnegatively introspective) type of "knowledge". We develop a probabilistic version of this concept ("degree of safety") and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing "action models" and a notion of probabilistic update product, that comes together with appropriate reduction laws.

We provide an algebraic semantics together with a sound and complete sequent calculus for information update due to epistemic actions. This semantics is flexible enough to accommodate incomplete as well as wrong information e.g. due to secrecy and deceit, as well as nested knowledge. We give a purely algebraic treatment of the muddy children puzzle, which moreover extends to situations where the children are allowed to lie and cheat. Epistemic actions, that is, information exchanges between agents A,B,…∈A⁠, are modeled as elements of a quantale. The quantale (Q,⋁,•) acts on an underlying Q-right module(M,⋁) of epistemic propositions and facts. The epistemic content is encoded by appearance maps, one pair fMA:M→M and fQA:Q→Q of (lax) morphisms for each agent A∈A⁠, which preserve the module and quantale structure respectively. By adjunction, they give rise to epistemic modalities, capturing the agents' knowledge on propositions and actions. The module action is epistemic update and gives rise to dynamic modalities—cf. weakest precondition. This model subsumes the crucial fragment of Baltag, Moss and Solecki's dynamic epistemic logic, abstracting it in a constructive fashion while introducing resource-sensitive structure on the epistemic actions.

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. Here we extend that work 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 characterisations to various forms of entanglement (for example, 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 whole range of other quantum circuits and programs, including other well-known protocols (for example, superdense coding, entanglement swapping, logic-gate teleportation etc.), can be similarly verified using our logic.

In Barwise and van Benthem [6], the authors give a general method for obtaining interpolation and preservation theorems for fragments of L∞ω, those for which there is a co-inductive pebble game Γ characterizing equivalence in the logic. The method is exemplified by an analysis of the following fragments: L∞ω itself, its existential fragment , its positive fragment , the k-variable fragment (and its existential and positive subfragments) and the modal fragment (and its existential and positive subfragments). While most of their method is general, there is one part (showing that Γ has the Scott property relative to the fragment) that required a case-by-case analysis. The purpose of our paper is to replace this case-by-case analysis by a general theorem, and to illustrate this method by obtaining their kinds of results for some additional fragments of L∞ω. Our general problem can be stated in the following way: Given a “nice” fragment F of L∞ω (one satisfying some natural closure conditions), find a pebble game characterization Γ of “preservation of F-formulas” and prove that Γ has the Scott property with respect to F. Applying the Abstract Interpolation Theorem from [6], we can conclude that F has Γ-interpolation, and the corresponding preservation result. In this paper, we shall give an answer to this question. (Our answer is “sufficient” but we don't know if our conditions are necessary.)