We suggest that the issue of constructing andverifying social procedures, which we suggestively call socialsoftware, be pursued as systematically as computer software is pursued by computer scientists. Certain complications do arise withsocial software which do not arise with computer software, but thesimilarities are nonetheless strong, and tools already exist which wouldenable us to start work on this important project. We give a variety ofsuggestive examples and indicate some theoretical work which alreadyexists.

Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science, and as a means of reasoning about the knowledge and belief of agents. This book provides a broad introduction to the subject, along with many exercises and their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the context of distributed systems and (...) artificial intelligence. These include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The authors also discuss extensively the problem of logical omniscience. They cover Halpern & Moses' theory of honest formulas, and they make a digression into the realm of nonmonotonic reasoning and preferential entailment. They discuss Moore's autoepistemic logic, together with Levesque's related logic of "all I know". Furthermore, they show how one can base default and counterfactual reasoning on epistemic logic. Graduate students in philosophy or in computer science, especially those with an interest in AI, will find this book useful. (shrink)

In ”Formal Problems about Knowledge,” Roy Sorensen examines epistemological issues that have logical aspects. He uses Fitch's proof for unknowables and the surprise test paradox to illustrate the hopes of the modal logicians who developed epistemic logic, and he considers the epistemology of proof with the help of the knower paradox. One solution to this paradox is that knowledge is not closed under deduction. Sorensen reviews the broader history of this maneuver along with the relevant alternatives model of knowledge which (...) assumes that ”know” is an absolute term like ”flat.” Sorensen argues that the difference between epistemic absolute terms and extensional absolute terms gives rise to an asymmetry that undermines recent claims that there is a structural parallel between the supervaluational and epistemicist theories of vagueness, and he suggests that we have overestimated the ability of logical demonstration to produce knowledge. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

What role, if any, does formal logic play in characterizing epistemically rational belief? Traditionally, belief is seen in a binary way - either one believes a proposition, or one doesn't. Given this picture, it is attractive to impose certain deductive constraints on rational belief: that one's beliefs be logically consistent, and that one believe the logical consequences of one's beliefs. A less popular picture sees belief as a graded phenomenon.

Counterfactuals is David Lewis' forceful presentation of and sustained argument for a particular view about propositions which express contrary to fact conditionals, including his famous defense of realism about possible worlds and his theory of laws of nature.

In ‘belief revision’ a theory is revised with a formula φ resulting in a revised theory . Typically, is in , one has to give up belief in by a process of retraction, and φ is in . We propose to model belief revision in a dynamic epistemic logic. In this setting, we typically have an information state (pointed Kripke model) for the theory wherein the agent believes the negation of the revision formula, i.e., wherein is true. The revision with (...) φ is a program *φ that transforms this information state into a new information state. The transformation is described by a dynamic modal operator [*φ], that is interpreted as a binary relation [ [*φ] ] between information states. The next information state is computed from the current information state and the belief revision formula. If the revision is successful, the agent believes φ in the resulting state, i.e., Bφ is then true. To make this work, as information states we propose ‘doxastic epistemic models’ that represent both knowledge and degrees of belief. These are multi-modal and multi-agent Kripke models. They are constructed from preference relations for agents, and they satisfy various characterizable multi-agent frame properties. Iterated, revocable, and higher-order belief revision are all quite natural in this setting. We present, for an example, five different ways of such dynamic belief revision. One can also see that as a non-deterministic epistemic action with two alternatives, where one is preferred over the other, and there is a natural generalization to general epistemic actions with preferences. (shrink)

David Lewis (1941-2001) was Class of 1943 University Professor of Philosophy at Princeton University. His contributions spanned philosophical logic, philosophy of language, philosophy of mind, philosophy of science, metaphysics, and epistemology. In On the Plurality of Worlds, he defended his challenging metaphysical position, "modal realism." He was also the author of the books Convention, Counterfactuals, Parts of Classes, and several volumes of collected papers.

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of (...) its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate "partial meet contraction functions", which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are "relational" and "transitively relational", are studied in detail, and their connections with certain "supplementary postulates" of Gardenfors investigated, with a further representation theorem established. (shrink)

This paper provides a complete characterization of epistemic models in which distributed knowledge complies with the principle of full communication (van der Hoek et al., 1999; Gerbrandy, 1999). It also introduces an extended notion of bisimulation and corresponding model comparison games that match the expressive power of distributed knowledge operators.

We present a way of classifying the logically possible ways out of Gärdenfors' inconsistency or triviality result on belief revision with conditionals. For one of these ways—conditionals which are not descriptive but which only have an inferential role as being given by the Ramsey test—we determine which of the assumptions in three different versions of Gärdenfors' theorem turn out to be false. This is done by constructing ranked models in which such Ramsey-test conditionals are evaluated and which are subject to (...) natural postulates on belief revision and acceptability sets for conditionals. Along the way we show that in contrast with what Gärdenfors himself proposed, there is no dichotomy of the form: either the Ramsey test has to be given up or the Preservation condition. Instead, both of them follow from our postulates. (shrink)

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

Using formal systems to represent and reason about uncertainty.

Reasoning About Knowledge is the first book to provide a general discussion of approaches to reasoning about knowledge and its applications to distributed ...

David Lewis is widely credited with the first formulation of common knowledge and the first rigorous analysis of convention. However, common knowledge and convention entered mainstream game theory only when they were formulated, later and independently, by other theorists. As a result, some of the most distinctive and valuable features of Lewis' game theory have been overlooked. We re-examine this theory by reconstructing key parts in a more formal way, extending it, and showing how it differs from more recent game (...) theory. In contrast to current theories of common knowledge, Lewis' theory is based on an explicit analysis of the modes of reasoning that are accessible to rational individuals and so can be used to analyse the genesis of common knowledge. Lewis' analysis of convention emphasises the role of inductive reasoning and of salience in the maintenance of conventions over time. Footnotes Earlier versions of this paper were presented at the 13th Amsterdam Colloquium at the University of Amsterdam, at a workshop on social norms at Wissenschaftskolleg zu Berlin, and at seminars at Tilburg University and the University of Bristol. We are grateful for comments from participants at those meetings, from two anonymous referees, and from Michael Bacharach, Nick Bardsley, Cristina Bicchieri, Luc Bovens, Simon Grant, David McCarthy, Shepley Orr, Brian Skyrms, Peter Vanderschraaf, Peter Wakker and Jörgen Weibull. Robert Sugden's work was supported by the Leverhulme Trust. (shrink)

Epistemology and epistemic logic At first sight, the modern agenda of epistemology has little to do with logic. Topics include different definitions of knowledge, its basic formal properties, debates between externalist and internalist positions, and above all: perennial encounters with sceptics lurking behind every street corner, especially in the US. The entry 'Epistemology' in the Routledge Encyclopedia of Philosophy (Klein 1993) and the anthology (Kim and Sosa 2000) give an up-to-date impression of the field. Now, epistemic logic started as a (...) contribution to epistemology, or at least a tool in its modus operandi, with the seminal book Knowledge and Belief (Hintikka's 1962, 2005). Formulas like Ki for "the agent i knows that " Bi for "the agent i believes that " provided logical forms for stating and analyzing philosophical propositions and arguments. And more than that, their model-theoretic semantics in terms of ranges of alternatives provided an appealing extensional way of thinking about what agents know or believe in a given situation. In particular, on Hintikka's view, an agent knows those propositions which are true in all situations compatible with what she knows about the actual world; i.e., her current range of uncertainty. (shrink)

This paper supersedes an ealier version, entitled "A Non-Standard Semantics for Inexact Knowledge with Introspection", which appeared in the Proceedings of "Rationality and Knowledge". The definition of token semantics, in particular, has been modified, both for the single- and the multi-agent case.

We investigate an axiomatization of the notion of common belief that makes use of no rules of inference and highlight the property of the set of accessibility relations that characterizes each axiom.

We show how belief revision can be treated systematically in the format of dynamicepistemic logic, when operators of conditional belief are added. The core engine consists of definable update rules for changing plausibility relations between worlds, which have been proposed independently in the dynamic-epistemic literature on preference change. Our analysis yields two new types of modal result. First, we obtain complete logics for concrete mechanisms of belief revision, based on compositional reduction axioms. Next, we show how various abstract postulates for (...) belief revision can be analyzed by standard modal frame correspondences for model-changing operations. (shrink)

On the basis of impossibility results on probability, belief revision, and conditionals, it is argued that conditional beliefs differ from beliefs in conditionals qua mental states. Once this is established, it will be pointed out in what sense conditional beliefs are still conditional, even though they may lack conditional contents, and why it is permissible to still regard them as beliefs, although they are not beliefs in conditionals. Along the way, the main logical, dispositional, representational, and normative properties of conditional (...) beliefs are studied, and it is explained how the failure of not distinguishing conditional beliefs from beliefs in conditionals can lead philosophical and empirical theories astray. (shrink)

The appropriateness of S5 as a logic of knowledge has been attacked at some length in the philosophical literature. Here one particular attack based on the interplay between knowledge and belief is considered: Suppose that knowledge satisfies S5, belief satisfies KD45, and both the entailment property (knowledge implies belief) and positive certainty (if the agent believes something, she believes she knows it) hold. Then it can be shown that belief reduces to knowledge: it is impossible to have false beliefs. While (...) the entailment property has typically been viewed as perhaps the least controversial of these assumptions, an argument is presented that it can plausibly be viewed as the culprit. More precisely, it is shown that this attack fails if we weaken the entailment property so that it applies only to objective (nonmodal) formulas, rather than to arbitrary formulas. Since the standard arguments in favor of the entailment property are typically given only for objective formulas, this observation suggests that care must be taken in applying intuitions that seem reasonable in the case of objective formulas to arbitrary formulas. (shrink)

In ‘belief revision’ a theory\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal K}$$\end{document} is revised with a formula φ resulting in a revised theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal K}\ast\varphi$$\end{document}. Typically, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\neg\varphi$$\end{document} is in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal K}$$\end{document}, one has to give up belief in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\neg\varphi$$\end{document} by a process (...) of retraction, and φ is in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal K}\ast\varphi$$\end{document}. We propose to model belief revision in a dynamic epistemic logic. In this setting, we typically have an information state (pointed Kripke model) for the theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal K}$$\end{document} wherein the agent believes the negation of the revision formula, i.e., wherein \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\neg\varphi$$\end{document} is true. The revision with φ is a program *φ that transforms this information state into a new information state. The transformation is described by a dynamic modal operator [*φ], that is interpreted as a binary relation [ [*φ] ] between information states. The next information state is computed from the current information state and the belief revision formula. If the revision is successful, the agent believes φ in the resulting state, i.e., Bφ is then true. To make this work, as information states we propose ‘doxastic epistemic models’ that represent both knowledge and degrees of belief. These are multi-modal and multi-agent Kripke models. They are constructed from preference relations for agents, and they satisfy various characterizable multi-agent frame properties. Iterated, revocable, and higher-order belief revision are all quite natural in this setting. We present, for an example, five different ways of such dynamic belief revision. One can also see that as a non-deterministic epistemic action with two alternatives, where one is preferred over the other, and there is a natural generalization to general epistemic actions with preferences. (shrink)

John Locke proposed a straightforward relationship between qualitative and quantitative doxastic notions: belief corresponds to a sufficiently high degree of confidence. Richard Foley has further developed this Lockean thesis and applied it to an analysis of the preface and lottery paradoxes. Following Foley's lead, we exploit various versions of these paradoxes to chart a precise relationship between belief and probabilistic degrees of confidence. The resolutions of these paradoxes emphasize distinct but complementary features of coherent belief. These features suggest principles that (...) tie together qualitative and quantitative doxastic notions. We show how these principles may be employed to construct a quantitative model - in terms of degrees of confidence - of an agent's qualitative doxastic state. This analysis fleshes out the Lockean thesis and provides the foundation for a logic of belief that is responsive to the logic of degrees of confidence. (shrink)

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal Logic (ATL). The key insight (...) in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as "Alternating-time Temporal Logic". The key insight in (...) ATL is that the path quantifiers of CTL could be replaced by "cooperation modalities", of the form $\langle \langle \Gamma \rangle \rangle $, where Γ is a set of agents. The intended interpretation of an ATL formula $\langle \langle \Gamma \rangle \rangle \varphi $ is that the agents Γ can cooperate to ensure that φ holds. In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as "group Γ can cooperate to bring about φ iff it is common knowledge in Γ that ψ". The resulting logic -- Alternating-time Temporal Epistemic Logic -- shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems. (shrink)

We show that several logics of common belief and common knowledge are not only complete, but also strongly complete, hence compact. These logics involve a weakened monotonicity axiom, and no other restriction on individual belief. The semantics is of the ordinary fixed-point type.

In this paper we study substantive assumptions in social interaction. By substantive assumptions we mean contingent assumptions about what the players know and believe about each other’s choices and information. We first explain why substantive assumptions are fundamental for the analysis of games and, more generally, social interaction. Then we show that they can be compared formally, and that there exist contexts where no substantive assumptions are being made. Finally we show that the questions raised in this paper are related (...) to a number of issues concerning “large” structures in epistemic game theory. (shrink)

Taking Löb's Axiom in modal provability logic as a running thread, we discuss some general methods for extending modal frame correspondences, mainly by adding fixed-point operators to modal languages as well as their correspondence languages. Our suggestions are backed up by some new results – while we also refer to relevant work by earlier authors. But our main aim is advertizing the perspective, showing how modal languages with fixed-point operators are a natural medium to work with.