This book develops a view of logic as a theory of information-driven agency and intelligent interaction between many agents - with conversation, argumentation and games as guiding examples. It provides one uniform account of dynamic logics for acts of inference, observation, questions and communication, that can handle both update of knowledge and revision of beliefs. It then extends the dynamic style of analysis to include changing preferences and goals, temporal processes, group action and strategic interaction in games. Throughout, the book (...) develops a mathematical theory unifying all these systems, and positioning them at the interface of logic, philosophy, computer science and game theory. A series of further chapters explores repercussions of the 'dynamic stance' for these areas, as well as cognitive science. (shrink)

In _Modal Logic for Open Minds,_ Johan van Benthem provides an up-to-date introduction to the field of modal logic, outlining its major ideas and exploring the numerous ways in which various academic fields have adopted it. Van Benthem begins with the basic theories of modal logic, semantics, bisimulation, and axiomatics, and also covers more advanced topics, such as expressive power and computational complexity. The book then moves to a wide range of applications, including new developments in information flow, (...) intelligent agency, and games. Taken together, the chapters show modal logic at the crossroads of philosophy, mathematics, linguistics, computer science, and economics. Most of the chapters are followed by exercises, making this volume ideal for undergraduate and graduate students in philosophy, computer science, symbolic systems, cognitive science, and linguistics. (shrink)

In this article, we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modelling for classical propositional and modal logics. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms (...) under comparison both semantically and syntactically at the level of derivations. Moreover, our analysis clarifies the interplay between the complexity of translations and axiomatizations of the corresponding logics: adding axioms to the target bimodal logic simplifies translations, or vice versa, complex translations can simplify frame conditions. We also investigate a transfer of first-order correspondence theory between possibility semantics and its bimodal counterpart. Finally, we discuss the conceptual trade-off between giving translations and giving new semantics for logical systems, and we identify a number of further research directions to which our analysis gives rise. (shrink)

Language in Action demonstrates the viability of mathematical research into the foundations of categorial grammar, a topic at the border between logic and linguistics. Since its initial publication it has become the classic work in the foundations of categorial grammar. A new introduction to this paperback edition updates the open research problems and records relevant results through pointers to the literature. Van Benthem presents the categorial processing of syntax and semantics as a central component in a more general dynamic (...) logic of information flow, in tune with computational developments in artificial intelligence and cognitive science. Using the paradigm of categorial grammar, he describes the substructural logics driving the dynamics of natural language syntax and semantics. This is a general type-theoretic approach that lends itself easily to proof-theoretic and semantic studies in tandem with standard logic. The emphasis is on a broad landscape of substructural categorial logics and their proof-theoretical and semantic peculiarities. This provides a systematic theory for natural language understanding, admitting of significant mathematical results. Moreover, the theory makes possible dynamic interpretations that view natural languages as programming formalisms for various cognitive activities. (shrink)

REFERENCES Barwise, J. & R. Cooper (1981) — 'Generalized Quantifiers and Natural Language', Linguistics and Philosophy 4:2159-219. Van Benthem, J. (1983a) — ' Five Easy Pieces', in Ter Meulen (ed.), 1-17. Van Benthem, J. (1983b) ...

In _Modal Logic for Open Minds,_ Johan van Benthem provides an up-to-date introduction to the field of modal logic, outlining its major ideas and exploring the numerous ways in which various academic fields have adopted it. Van Benthem begins with the basic theories of modal logic, semantics, bisimulation, and axiomatics, and also covers more advanced topics, such as expressive power and computational complexity. The book then moves to a wide range of applications, including new developments in information flow, (...) intelligent agency, and games. Taken together, the chapters show modal logic at the crossroads of philosophy, mathematics, linguistics, computer science, and economics. Most of the chapters are followed by exercises, making this volume ideal for undergraduate and graduate students in philosophy, computer science, symbolic systems, cognitive science, and linguistics. (shrink)

A comprehensive examination of the interfaces of logic, computer science, and game theory, drawing on twenty years of research on logic and games.

We investigate the modal logic of stepwise removal of objects, both for its intrinsic interest as a logic of quantification without replacement, and as a pilot study to better understand the complexity jumps between dynamic epistemic logics of model transformations and logics of freely chosen graph changes that get registered in a growing memory. After introducing this logic (MLSR) and its corresponding removal modality, we analyze its expressive power and prove a bisimulation characterization theorem. We then provide a complete Hillbert-style (...) axiomatization for the logic of stepwise removal in a hybrid language enriched with nominals and public announcement operators. Next, we show that model-checking for MLSR is PSPACE-complete, while its satisfiability problem is undecidable. Lastly, we consider an issue of fine-structure: the expressive power gained by adding the stepwise removal modality to fragments of first-order logic. (shrink)

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)

Current dynamic epistemic logics for analyzing effects of informational events often become cumbersome and opaque when common knowledge is added for groups of agents. Still, postconditions involving common knowledge are essential to successful multi-agent communication. We propose new systems that extend the epistemic base language with a new notion of ‘relativized common knowledge’, in such a way that the resulting full dynamic logic of information flow allows for a compositional analysis of all epistemic postconditions via perspicuous ‘reduction axioms’. We also (...) show how such systems can deal with factual alteration, rather than just information change, making them cover a much wider range of realistic events. After a warm-up stage of analyzing logics for public announcements, our main technical results are expressivity and completeness theorems for a much richer logic that we call LCC. This is a dynamic epistemic logic whose static base is propositional dynamic logic (PDL), interpreted epistemically. This system is capable of expressing all model-shifting operations with finite action models, while providing a compositional analysis for a wide range of informational events. This makes LCC a serious candidate for a standard in dynamic epistemic logic, as we illustrate by analyzing some complex communication scenarios, including sending successive emails with both ‘cc’ and ‘bcc’ lines, and other private announcements to subgroups. Our proofs involve standard modal techniques, combined with a new application of Kleene’s theorem on finite automata, as well as new Ehrenfeucht games of model comparison. (shrink)

Minimal predicates P satisfying a given first-order description φ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ φ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal (...) in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond. (shrink)

Statements not only update our current knowledge, but also have other dynamic effects. In particular, suggestions or commands ?upgrade' our preferences by changing the current order among worlds. We present a complete logic of knowledge update plus preference upgrade that works with dynamic-epistemic-style reduction axioms. This system can model changing obligations, conflicting commands, or ?regret'. We then show how to derive reduction axioms from arbitrary definable relation changes. This style of analysis also has a product update version with preferences between (...) actions, as well as worlds. Some illustrations are presented involving defaults and obligations. We conclude that our dynamic framework is viable, while admitting a further extension to more numerical ?utility update'. (shrink)

This paper adds evidence structure to standard models of belief, in the form of families of sets of worlds. We show how these more fine-grained models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. We show how this perspective leads to new richer languages for existing neighborhood semantics for modal logic. Our main results are relative completeness theorems for the resulting dynamic logic of evidence.

We discuss games of both perfect and imperfect information at two levels of structural detail: players’ local actions, and their global powers for determining outcomes of the game. We propose matching logical languages for both. In particular, at the ‘action level’, imperfect information games naturally model a combined ‘dynamic-epistemic language’ – and we find correspondences between special axioms and particular modes of playing games with their information dynamics. At the ‘outcome level’, we present suitable notions of game equivalence, plus some (...) simple representation results. (shrink)

A variety of logical frameworks have been developed to study rational agents interacting over time. This paper takes a closer look at one particular interface, between two systems that both address the dynamics of knowledge and information flow. The first is Epistemic Temporal Logic (ETL) which uses linear or branching time models with added epistemic structure induced by agents’ different capabilities for observing events. The second framework is Dynamic Epistemic Logic (DEL) that describes interactive processes in terms of epistemic event (...) models which may occur inside modalities of the language. This paper systematically and rigorously relates the DEL framework with the ETL framework. The precise relationship between DEL and ETL is explored via a new representation theorem characterizing the largest class of ETL models corresponding to DEL protocols in terms of notions of Perfect Recall , No Miracles , and Bisimulation Invariance . We then focus on new issues of completeness . One contribution is an axiomatization for the dynamic logic of public announcements constrained by protocols, which has been an open problem for some years, as it does not fit the usual ‘reduction axiom’ format of DEL. Finally, we provide a number of examples that show how DEL suggests an interesting fine-structure inside ETL. (shrink)

Questions are triggers for explicit events of ‘issue management’. We give a complete logic in dynamic-epistemic style for events of raising, refining, and resolving an issue, all in the presence of information flow through observation or communication. We explore extensions of the framework to multi-agent scenarios and long-term temporal protocols. We sketch a comparison with some alternative accounts.

This Handbook documents the main trends in current research between logic and language, including its broader influence in computer science, linguistic theory and cognitive science. The history of the combined study of Logic and Linguistics goes back a long way, at least to the work of the scholastic philosophers in the Middle Ages. At the beginning of this century, the subject was revitalized through the pioneering efforts of Gottlob Frege, Bertrand Russell, and Polish philosophical logicians such as Kazimierz Ajdukiewicz. Around (...) 1970, the landmark achievements of Richard Montague established a junction between state-of-the-art mathematical logic and generative linguistic theory. Over the subsequent decades, this enterprise of Montague Grammar has flourished and diversified into a number of research programs with empirical and theoretical substance. This appears to be the first Handbook to bring logic-language interface to the fore. Both aspects of the interaction between logic and language are demonstrated in the book i.e. firstly, how logical systems are designed and modified in response to linguistic needs and secondly, how mathematical theory arises in this process and how it affects subsequent linguistic theory. The Handbook presents concise, impartial accounts of the topics covered. Where possible, an author and a commentator have cooperated to ensure the proper breadth and technical content of the papers. The Handbook is self-contained, and individual articles are of the highest quality. (shrink)

We give in this paper a sufficient condition, cast in semantic terms, for Hallden-completeness in normal modal logics, a modal logic being said to be Hallden-complete (or Ήallden-reasonable') just in case for any disjunctive formula provable in the logic, where the disjuncts have no propositional variables in common, one or other of those disjuncts is provable in the logic.

Modern logic is undergoing a cognitive turn, side-stepping Frege’s ‘antipsychologism’. Collaborations between logicians and colleagues in more empirical fields are growing, especially in research on reasoning and information update by intelligent agents. We place this border-crossing research in the context of long-standing contacts between logic and empirical facts, since pure normativity has never been a plausible stance. We also discuss what the fall of Frege’s Wall means for a new agenda of logic as a theory of rational agency, and what (...) might then be a viable understanding of ‘psychologism’ as a friend rather than an enemy of logical theory. (shrink)

Game-theoretic solution concepts describe sets of strategy profiles that are optimal for all players in some plausible sense. Such sets are often found by recursive algorithms like iterated removal of strictly dominated strategies in strategic games, or backward induction in extensive games. Standard logical analyses of solution sets use assumptions about players in fixed epistemic models for a given game, such as mutual knowledge of rationality. In this paper, we propose a different perspective, analyzing solution algorithms as processes of learning (...) which change game models. Thus, strategic equilibrium gets linked to fixed-points of operations of repeated announcement of suitable epistemic statements. This dynamic stance provides a new look at the current interface of games, logic, and computation. (shrink)

Where is logic heading today? There is a general feeling that the discipline is broadening its scope and agenda beyond classical foundational issues, and maybe even a concern that, like Stephen Leacock’s famous horseman, it is ‘riding off madly in all directions’. So, what is the resultant vector? There seem to be two broad answers in circulation today. One is logical pluralism, locating the new scope of logic in charting a wide variety of reasoning styles, often marked by non-classical structural (...) rules of inference. This is the new program that I subscribed to in my work on sub-structural logics around 1990, and it is a powerful movement today. But gradually, I have changed my mind about the crux of what logic should become. I would now say that the main issue is not variety of reasoning styles and notions of consequence, but the variety of informational tasks performed by intelligent interacting agents, of which inference is only one among many, involving observation, memory, questions and answers, dialogue, or general communication. And logical systems should deal with a wide variety of these, making information-carrying events first-class citizens in their set-up. The purpose of this brief paper is to contrast and compare the two approaches, drawing freely on some insights from earlier published papers. In particular, I will argue that logical dynamics sets itself the more ambitious diagnostic goal of explaining why substructural phenomena occur, by ‘deconstructing’ them into classical logic plus an explicit account of the relevant informational events. (shrink)

Preference is a basic notion in human behaviour, underlying such varied phenomena as individual rationality in the philosophy of action and game theory, obligations in deontic logic (we should aim for the best of all possible worlds), or collective decisions in social choice theory. Also, in a more abstract sense, preference orderings are used in conditional logic or non-monotonic reasoning as a way of arranging worlds into more or less plausible ones. The field of preference logic (cf. Hansson [10]) studies (...) formal systems that can express and analyze notions of preference between various sorts of entities: worlds, actions, or propositions. The art is of course to design a language that combines perspicuity and low complexity with reasonable expressive power. In this paper, we take a particularly simple approach. As preferences are binary relations between worlds, they naturally support standard unary modalities. In particular, our key modality ♦ϕ will just say that is ϕ true in some world which is at least as good as the current one. Of course, this notion can also be indexed to separate agents. The essence of this language is already in [4], but our semantics is more general, and so are our applications and later language extensions. Our modal language can express a variety of preference notions between propositions. Moreover, as already suggested in [9], it can “deconstruct” standard conditionals, providing an embedding of conditional logic into more standard modal logics. Next, we take the language to the analysis of games, where some sort of preference logic is evidently needed ([23] has a binary modal analysis different from ours). We show how a qualitative unary preference modality suffices for defining Nash Equilibrium in strategic games, and also the Backward Induction solution for finite extensive games. Finally, from a technical perspective, our treatment adds a new twist. Each application considered in this paper suggests the need for some additional access to worlds before the preference modality can unfold its true power.. (shrink)

Logic and philosophy of science share a long history, though contacts have gone through ups and downs. This paper is a brief survey of some major themes in logical studies of empirical theories, including links to computer science and current studies of rational agency. The survey has no new results: we just try to make some things into common knowledge.

Contemporary historians of logic tend to credit Bernard Bolzano with the invention of the semantic notion of consequence, a full century before Tarski. Nevertheless, Bolzano's work played no significant rôle in the genesis of modern logical semantics. The purpose of this paper is to point out three highly original, and still quite relevant themes in Bolzano's work, being a systematic study of possible types of inference, of consistency, as well as their meta-theory. There are certain analogies with Tarski's concerns here, (...) although the main thrust seems to be different, both philosophically and technically. Thus, if only obliquely, we also provide some additional historical perspective on Tarski's achievement. (shrink)

Games, Norms, and Reasons: Logic at the Crossroads provides an overview of modern logic focusing on its relationships with other disciplines, including new interfaces with rational choice theory, epistemology, game theory and informatics. This book continues a series called "Logic at the Crossroads" whose title reflects a view that the deep insights from the classical phase of mathematical logic can form a harmonious mixture with a new, more ambitious research agenda of understanding and enhancing human reasoning and intelligent interaction. The (...) editors have gathered together articles from active authors in this new area that explore dynamic logical aspects of norms, reasons, preferences and beliefs in human agency, human interaction and groups. The book pays a special tribute to Professor Rohit Parikh, a pioneer in this movement. (shrink)

Proof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book (...) takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other. (shrink)

Information is a notion of wide use and great intuitive appeal, and hence, not surprisingly, different formal paradigms claim part of it, from Shannon channel theory to Kolmogorov complexity. Information is also a widely used term in logic, but a similar diversity repeats itself: there are several competing logical accounts of this notion, ranging from semantic to syntactic. In this chapter, we will discuss three major logical accounts of information.

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)

The complexity of any logical modeling reflects both the intrinsic structure of a topic described and the weight of the formal tools. Some of this weight seems inherent in even the most basic logical systems. Notably, standard predicate logic is undecidable. In this paper, we investigate ‘lighter’ versions of this general purpose tool, by modally ‘deconstructing’ the usual semantics, and locating implicit choice points in its set up. The first part sets out the interest of this program and the modal (...) techniques employed, while the second part provides technical elaborations demonstrating its viability. (shrink)

Current dynamic-epistemic logics model different types of information change in multi-agent scenarios. We generalize these logics to a probabilistic setting, obtaining a calculus for multi-agent update with three natural slots: prior probability on states, occurrence probabilities in the relevant process taking place, and observation probabilities of events. To match this update mechanism, we present a complete dynamic logic of information change with a probabilistic character. The completeness proof follows a compositional methodology that applies to a much larger class of dynamic-probabilistic (...) logics as well. Finally, we discuss how our basic update rule can be parameterized for different update policies, or learning methods. (shrink)

The general verificationist thesis says that What is true can be known or formally: φ → ◊Kφ VT Fitch's argument trivializes this principle. It uses a weak modal epistemic logic to show that VT collapses truth and knowledge, by taking a clever substitution instance for φ: P ∧ ¬KP → ◊ K(P ∧ ¬KP) Then we have the following chain of three conditionals (a) ◊ K(P ∧ ¬KP) → ◊ (KP ∧ K¬KP) in the minimal modal logic for the knowledge (...) operator K, (b) ◊ (KP ∧ K¬KP) → ◊ (KP ∧¬KP) in the modal logic T, and finally (c) ◊ (KP ∧¬KP) → ⊥ in the minimal modal logic for. (shrink)

Rational agents base their actions on information from observation, inference, introspection, or other sources. But this information comes in different kinds, and it is usually handled by different logical mechanisms. We discuss how to integrate external ‘updating information’ and internal ‘elucidating information’ into one system of dynamic epistemic logic, by distinguishing two basic informational actions: ‘bare seeing’ versus ‘conscious realization’.

The most widely used attractive logical account of knowledge uses standard epistemic models, i.e., graphs whose edges are indistinguishability relations for agents. In this paper, we discuss more general topological models for a multi-agent epistemic language, whose main uses so far have been in reasoning about space. We show that this more geometrical perspective affords greater powers of distinction in the study of common knowledge, defining new collective agents, and merging information for groups of agents.

We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 ⊕ S4. We axiomatize the modal logic of products of spaces with horizontal, vertical, and standard product topologies.We prove that both of these logics are complete for the product of rational numbers ℚ × ℚ with the appropriate topologies.

Dynamic update of information states is a new paradigm in logicalsemantics. But such updates are also a traditional hallmark ofprobabilistic reasoning. This note brings the two perspectives togetherin an update mechanism for probabilities which modifies state spaces.

The combination of logic and game theory provides a fine-grained perspective on information and interaction dynamics, a Theory of Play. In this paper we lay down the main components of such a theory, drawing on recent advances in the logical dynamics of actions, preferences, and information. We then show how this fine-grained perspective has already shed new light on the long-term dynamics of information exchange, as well as on the much-discussed question of extensive game rationality.