Epistemic logic in the tradition of Hintikka provides, as one of its many applications, a toolkit for the precise analysis of certain epistemological problems. In recent years, dynamic epistemic logic has expanded this toolkit. Dynamic epistemic logic has been used in analyses of well-known epistemic “paradoxes”, such as the Paradox of the Surprise Examination and Fitch’s Paradox of Knowability, and related epistemic phenomena, such as what Hintikka called the “anti-performatory effect” of Moorean announcements. In this paper, we explore a variation (...) on basic dynamic epistemic logic—what we call sequential epistemic logic—and argue that it allows more faithful and fine-grained analyses of those epistemological topics. (shrink)

This work proposes an understanding of deductive, default and abductive reasoning as different instances of the same phenomenon: epistemic dynamics. It discusses the main intuitions behind each one of these reasoning processes, and suggest how they can be understood as different epistemic actions that modify an agent’s knowledge and/or beliefs in a different way, making formal the discussion with the use of the dynamic epistemic logic framework. The ideas in this paper put the studied processes under the same umbrella, thus (...) highlighting their relationship and allowing a better understanding of how they interact together. (shrink)

In this work we define contingency logic with arbitrary announcement. In contingency logic, the primitive modality contingency formalises that a proposition may be true but also may be false, so that if it is non-contingent then it is necessarily true or necessarily false. To this logic one can add dynamic operators to describe change of contingency. Our logic has operators for public announcement and operators for arbitrary public announcement, as in the dynamic epistemic logic called arbitrary public announcement logic. However, (...) our language primitive is the more suitable notion of public announcement whether, instead of the public announcement that. We compare the expressive power of our logic and its various fragments to related dynamic epistemic logics. We further present an axiomatisation and show its completeness by adapting a method to demonstrate completeness of arbitrary public announcement logic. Various extensions are also shown to be complete with respect to the corresponding frame classes. (shrink)

Arbitrary public announcement logic ) reasons about how the knowledge of a set of agents changes after true public announcements and after arbitrary announcements of true epistemic formulas. We consider a variant of arbitrary public announcement logic called positive arbitrary public announcement logic ), which restricts arbitrary public announcements to announcement of positive formulas. Positive formulas prohibit statements about the ignorance of agents. The positive formulas correspond to the universal fragment in first-order logic. As two successive announcements of positive formulas (...) need not correspond to the announcement of a positive formula, \ is rather different from \. We show that \ is more expressive than public announcement logic \, and that \ is incomparable with \. We also provide a sound and complete infinitary axiomatisation. (shrink)

We propose a multi-agent logic of knowledge, public announcements and arbitrary announcements, interpreted on topological spaces in the style of subset space semantics. The arbitrary announcement modality functions similarly to the effort modality in subset space logics, however, it comes with intuitive and semantic differences. We provide axiomatizations for three logics based on this setting, with S5 knowledge modality, and demonstrate their completeness. We moreover consider the weaker axiomatizations of three logics with S4 type of knowledge and prove soundness and (...) completeness results for these systems. (shrink)

This paper provides a sound and complete proof system for a language that adds to Dynamic Epistemic Logic a discrete previous-time operator as well as single symbol formulas that partially reveal the most recent event that occurred. The completeness theorem is by filtration followed by model unraveling and other model transformations. Decidability follows from the completeness proof. The degree to which it is important to include the additional single symbol formulas is addressed in a discussion about the difficulties of the (...) completeness for a language that only adds the previous-time operator to DEL. Discussion is also given regarding the completeness for a language obtained by removing common knowledge operators from. (shrink)

We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multi-agent Kripke models that contain a designated temporal relation. These operations are natural extensions of the well-known “action models” from Dynamic Epistemic Logic. Our “temporal action models” may be used to define a number of informational actions that can modify the “objective” temporal structure of a model along with the agents’ basic and higher-order knowledge and beliefs about this structure, including their beliefs about the time. In essence, (...) this approach provides one way to extend the domain of action model-style operations from atemporal Kripke models to temporal Kripke models in a manner that allows actions to control the flow of time. We present a number of examples to illustrate the subtleties involved in interpreting the effects of our extended action models on temporal Kripke models. We also study preservation of important epistemic-temporal properties of temporal Kripke models under temporal action model-induced operations, provide complete axiomatizations for two theories of temporal action models, and connect our approach with previous work on time in DEL. (shrink)

This work proposes an understanding of deductive, default and abductive reasoning as different instances of the same phenomenon: epistemic dynamics. It discusses the main intuitions behind each one of these reasoning processes, and suggest how they can be understood as different epistemic actions that modify an agent’s knowledge and/or beliefs in a different way, making formal the discussion with the use of the dynamic epistemic logic framework. The ideas in this paper put the studied processes under the same umbrella, thus (...) highlighting their relationship and allowing a better understanding of how they interact together. (shrink)

This is the second paper in a two-part series introducing logics for reasoning about the dynamics of knowledge and beliefs. Part I introduced different logical systems that can be used to reason about the knowledge and beliefs of a group of agents. In this second paper, I show how to adapt these logical systems to reason about the knowledge and beliefs of a group of agents during the course of a social interaction or rational inquiry. Inference, communication and observation are (...) typical examples of informative events, which have been subjected to a logical analysis. The main goal of this article is to introduce the key conceptual and technical issues that drive much of the research in this area. (shrink)

This paper surveys the interface between the two major logical trends that describe agents’ intelligent interaction over time: dynamic epistemic logic (DEL) and epistemic temporal logic (ETL). The initial attempt to “merge” DEL and ETL was made in van Benthem et al. (Merging frameworks for interaction: DEL and ETL, 2007) and followed up by van Benthem et al. (J Phil Logic 38(5):491–526, 2009) and Hoshi (Epistemic dynamics and protocol information. Ph.D. thesis, Stanford University Stanford, 2009a). The merged framework provides a (...) systematic comparison between these two logical systems and studies new logics of intelligent interaction. This paper presents the main results and the recent developments at the interface between DEL and ETL. (shrink)

The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform substitution. (...) First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution. (shrink)

Marton ( 2019 ) argues that that it follows from the standard antirealist theory of truth, which states that truth and possible knowledge are equivalent, that knowing possibilities is equivalent to the possibility of knowing, whereas these notions should be distinct. Moreover, he argues that the usual strategies of dealing with the Church–Fitch paradox of knowability are either not able to deal with his modal-epistemic collapse result or they only do so at a high price. Against this, I argue that (...) Marton’s paper does not present any seriously novel challenge to anti-realism not already found in the Church–Fitch result. Furthermore, Edgington ( 1985 ) reformulated antirealist theory of truth can deal with his modal-epistemic collapse argument at no cost. (shrink)

Dynamic epistemic logic describes the possible information-changing actions available to individual agents, and their knowledge pre-and post conditions. For example, public announcement logic describes actions in the form of public, truthful announcements. However, little research so far has considered describing and analysing rational choice between such actions, i.e., predicting what rational self-interested agents actually will or should do. Since the outcome of information exchange ultimately depends on the actions chosen by all the agents in the system, and assuming that agents (...) have preferences over such outcomes, this is a game theoretic scenario. This is, in our opinion, an interesting general research direction, combining logic and game theory in the study of rational information exchange. In this article we take some first steps in this direction: we consider the case where available actions are public announcements, and where each agent has a (typically epistemic) goal formula that she would like to become true. What will each agent announce? The truth of the goal formula also depends on the announcements made by other agents. We analyse such public announcement games. (shrink)

This paper demonstrates the undecidability of a number of logics with quantification over public announcements: arbitrary public announcement logic, group announcement logic, and coalition announcement logic. In APAL we consider the informative consequences of any announcement, in GAL we consider the informative consequences of a group of agents all of which are simultaneously making known announcements. So this is more restrictive than APAL. Finally, CAL is as GAL except that we now quantify over anything the agents not in that group (...) may announce simultaneously as well. The logic CAL therefore has some features of game logic and of ATL. We show that when there are multiple agents in the language, the satisfiability problem is undecidable for APAL, GAL, and CAL. In the single agent case, the satisfiability problem is decidable for all three logics. (shrink)

Public announcement logic is an extension of epistemic logic with dynamic operators that model the effects of all agents simultaneously and publicly acquiring the same piece of information. One of the extensions of PAL, group announcement logic, allows quantification over announcements made by agents. In GAL, it is possible to reason about what groups can achieve by making such announcements. It seems intuitive that this notion of coalitional ability should be closely related to the notion of distributed knowledge, the implicit (...) knowledge of a group. Thus, we study the extension of GAL with distributed knowledge, and in particular possible interaction properties between GAL operators and distributed knowledge. The perhaps surprising result is that, in fact, there are no interaction properties, contrary to intuition. We make this claim precise by providing a sound and complete axiomatisation of GAL with distributed knowledge. We also consider several natural variants of GAL with distributed knowledge, as well as some other related logic, and compare their expressive power. (shrink)

There are several ways to quantify over public announcements. The most notable are reflected in arbitrary, group, and coalition announcement logics, with the latter being the least studied so far. In the present work, we consider coalition announcements through the lens of group announcements, and provide a complete axiomatisation of a logic with coalition announcements. To achieve this, we employ a generalisation of group announcements. Moreover, we study some logical properties of both coalition and group announcements that have not been (...) studied before. (shrink)

A proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you know whether the proposition is true. In this paper, we study contingency logic with the noncontingency operator? but without the necessity operator 2. This logic is not a normal modal logic, because?→ is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over (...) classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization over symmetric frames. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap. (shrink)

Logic is breaking out of the confines of the single-agent static paradigm that has been implicit in all formal systems until recent times. We sketch some recent developments that take logic as an account of information-driven interaction. These two features, the dynamic and the social, throw fresh light on many issues within logic and its connections with other areas, such as epistemology and game theory.

We analyse the computational complexity of comparing informational structures. Intuitively, we study the complexity of deciding queries such as the following: Is Alice’s epistemic information strictly coarser than Bob’s? Do Alice and Bob have the same knowledge about each other’s knowledge? Is it possible to manipulate Alice in a way that she will have the same beliefs as Bob? The results show that these problems lie on both sides of the border between tractability (P) and intractability (NP-hard). In particular, we (...) investigate the impact of assuming information structures to be partition-based (rather than arbitrary relational structures) on the complexity of various problems. We focus on the tractability of concrete epistemic tasks and not on epistemic logics describing them. (shrink)

is an extension of public announcement logic. It is based on a modal operator that expresses what is true after any arbitrary announcement. An incorrect Truth Lemma has been stated and ‘demonstrated’ in Balbiani et al. . In this paper, we put right the wording and the proof of the Truth Lemma for.

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation (...) when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality. (shrink)

We introduce Arbitrary Public Announcement Logic with Memory (APALM), obtained by adding to the models a ‘memory’ of the initial states, representing the information before any communication took place (“the prior”), and adding to the syntax operators that can access this memory. We show that APALM is recursively axiomatizable (in contrast to the original Arbitrary Public Announcement Logic, for which the corresponding question is still open). We present a complete recursive axiomatization, that includes a natural finitary rule, and study this (...) logic’s expressivity and the appropriate notion of bisimulation. We then examine Group Announcement Logic with Memory (GALM), the extension of APALM obtained by adding to its syntax group announcement operators, and provide a complete finitary axiomatization (again in contrast to the original Group Announcement Logic, for which the only known axiomatization is infinitary). We also show that, in the memory-enhanced context, there is a natural reduction of the so-called coalition announcement modality to group announcements (in contrast to the memory-free case, where this natural translation was shown to be invalid). (shrink)

In this paper I present a more refined analysis of the principles of deductive closure and positive introspection. This analysis uses the expressive resources of logics for different types of group knowledge, and discriminates between aspects of closure and computation that are often conflated. The resulting model also yields a more fine-grained distinction between implicit and explicit knowledge, and places Hintikka’s original argument for positive introspection in a new perspective.

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the (...) Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational models with infinitely (...) many agents, PAL-K_omega, as well as standard extensions thereof, e.g., PAL-T_omega, PAL-S4_omega, and PAL-S5_omega. We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. (shrink)