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)

Our preferences determine how we act and think, but exactly what the mechanics are and how they work is a central cause of concern in many disciplines. This book uses techniques from modern logics of information flow and action to develop a unified new theory of what preference is and how it changes. The theory emphasizes reasons for preference, as well as its entanglement with our beliefs. Moreover, the book provides dynamic logical systems which describe the explicit triggers driving preference (...) change, including new information, suggestions, and commands. In sum, the book creates new bridges between many fields, from philosophy and computer science to economics, linguistics, and psychology. For the experienced scholar access to a large body of recent literature is provided and the novice gets a thorough introduction to the action and techniques of dynamic logic. (shrink)

The logical omniscience problem, whereby standard models of epistemic logic treat an agent as believing all consequences of its beliefs and knowing whatever follows from what else it knows, has received plenty of attention in the literature. But many attempted solutions focus on a fairly narrow specification of the problem: avoiding the closure of belief or knowledge, rather than showing how the proposed logic is of philosophical interest or of use in computer science or artificial intelligence. Sentential epistemic logics, as (...) opposed to traditional possible worlds approaches, do not suffer from the problems of logical omniscience but are often thought to lack interesting epistemic properties. In this paper, I focus on the case of rule-based agents, which play a key role in contemporary AI research but have been neglected in the logical literature. I develop a framework for modelling monotonic, nonmonotonic and introspective rule-based reasoners which have limited cognitive resources and prove that the resulting models have a number of interesting properties. An axiomatization of the resulting logic is given, together with completeness, decidability and complexity results. (shrink)

Justification Logic provides an axiomatic description of justifications and delegates the question of their nature to semantics. In this note, we address the conceptual issue of the logical type of justifications: we argue that justifications in the logical setting are naturally interpreted as sets of formulas which leads to a class of epistemic models that we call modular models . We show that Fitting models for Justification Logic naturally encode modular models and can be regarded as convenient pre-models of the (...) former. (shrink)

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement (...) result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements. (shrink)

We study term modal logics, where modalities can be indexed by variables that can be quantified over. We suggest that these logics are appropriate for reasoning about systems of unboundedly many reasoners and define a notion of bisimulation which preserves propositional fragment of term modal logics. Also we show that the propositional fragment is already undecidable but that its monodic fragment is decidable, and expressive enough to include interesting assertions.

Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer for one of its flagship events: the Augustus de Morgan Workshop of 2005. Benedikt Loewe and I proposed the topic of 'interactive logic' for this occasion, with an (...) emphasis on social software – the logical analysis and design of social procedures – and on games, arguably the formal interactive setting par excellence. This choice reflects current research interests in our logic community at ILLC Amsterdam and beyond. In this broad area of interfaces between logic, computer science, and game theory, this paper is my own attempt at playing Dov. I am, perhaps not telling, but at least asking other people to find out for me what I myself cannot. (shrink)

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on plausibility models. Similarly to other epistemic statements, 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 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 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 believe in a player's rationality after updating with his irrationality. (shrink)

We develop a multi-agent deontic action logic to study the logical behaviour of two types of deontic conditionals: (1) conditional obligations, having the form "If group H were to perform action aH, then, in group F's interest, group G ought to perform action aG" and (2) conditional permissions, having the form "If group H were to perform action aH, then, in group F's interest, group G may perform action aG". First, we define a formal language for multi-agent deontic action logic (...) and a class of consequentialist models to interpret the formulas of the language. Second, we define a transformation that converts any strategic game into a consequentialist model. Third, we show that an outcome a* is a Nash equilibrium of a strategic game if and only if a conjunction of certain conditional permissions is true in the consequentialist model that results from the transformation of that strategic game. (shrink)

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.

In this paper I study two ways of transforming decision problems on the basis of previously adopted intentions, ruling out incompatible options and imposing a standard of relevance, with a particular focus on situations of strategic interaction. I show that in such situations problems arise which do not appear in the single-agent case, namely that transformation of decision problems can leave the agents with no option compatible with what they intend. I characterize conditions on the agents’ intentions which avoid such (...) problematic scenarios, in a way that requires each agent to take account of the intentions of others. (shrink)

Understanding human behaviour involves "why"'s as well as "how"'s. Rational people have good reasons for acting, but it can be hard to find out what these were and how they worked. In this Note, we discuss a few ways in which actions, preferences, and expectations are intermingled. This mixture is especially clear with the well-known solution procedure for extensive games called 'Backward Induction'. In particular, we discuss three scenarios for analyzing behaviour in a game. One can rationalize given moves as (...) revealing agents' preferences, one can also rationalize them as revealing agents' beliefs about others, but one can also change a predicted pattern of behaviour by making promises. All three scenarios transform given games to new ones, and we prove some results about their scope. A more general view of relevant game transformations would involve dynamic and epistemic game logics. Finally, our analysis describes and disentangles matters: but it will not tell you what to do! (shrink)

In this paper I present a dynamic-epistemic hybrid logic for reasoning about information and intention changes in situations of strategic interaction. I provide a complete axiomatization for this logic, and then use it to study intentions-based transformations of decision problems.

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)

Using epistemic logic, we provide a non-probabilistic way to formalise payoff uncertainty, that is, statements such as ‘player i has approximate knowledge about the utility functions of player j.’ We show that on the basis of this formalisation common knowledge of payoff uncertainty and rationality (in the sense of excluding weakly dominated strategies, due to Dekel and Fudenberg (1990)) characterises a new solution concept we have called ‘mixed iterated strict weak dominance.’.

This paper proposes a logical framework for representing static and dynamic properties of different kinds of individual and collective attitudes. A complete axiomatization as well as a decidability result for the logic are given. The logic is applied to game theory by providing a formal analysis of the epistemic conditions of iterated deletion of weakly dominated strategies (IDWDS), or iterated weak dominance for short. The main difference between the analysis of the epistemic conditions of iterated weak dominance given in this (...) paper and other analysis is that we use a semi-qualitative approach to uncertainty based on the notion of plausibility first introduced by Spohn, whereas other analysis are based on a quantitative representation of uncertainty in terms of probabilities. (shrink)

In this paper, we investigate processes involving iterative information updating due to van Benthem (Int Game Theory Rev 9: 13—45, 2007), who characterized existent game-theoretic solution concepts by such processes in the framework of Plaza's public announcement logic. We refine this approach and clarify the relationship between stable strategies and information update processes. After extending Plaza's logic, we then derive the conditions under which a stable outcome is determined independently of the order of the iterative information updates. This result gives (...) an epistemic foundation for the order independence of iterated elimination of disadvantageous strategies. (shrink)

Logics of joint strategic ability have recently received attention, with arguably the most influential being those in a family that includes Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). Notably, both CL and ATL bypass the epistemic issues that underpin Schelling-type coordination problems, by apparently relying on the meta-level assumption of (perfectly reliable) communication between cooperating rational agents. Yet such epistemic issues arise naturally in settings relevant to ATL and CL: these logics are standardly interpreted on structures where agents move (...) simultaneously, opening the possibility that an agent cannot foresee the concurrent choices of other agents. In this paper we introduce a variant of CL we call Two-Player Strategic Coordination Logic (SCL2). The key novelty of this framework is an operator for capturing coalitional ability when the cooperating agents cannot share strategic information. We identify significant differences in the expressive power and validities of SCL2 and CL2, and present a sound and complete axiomatization for SCL2. We briefly address conceptual challenges when shifting attention to games with more than two players and stronger notions of rationality. (shrink)

In this paper I study intentions of the form, that is, intentions with a we-content, and their role in interpersonal coordination. I focus on the notion of epistemic support for such intentions. Using tools from epistemic game theory and epistemic logic, I cast doubt on whether such support guarantees the other agents' conditional mediation in the achievement of such intentions, something that appears important if intentions with a we-content are to count as genuine intentions. I then formulate a stronger version (...) of epistemic support, one that does indeed ensure the required mediation, but I then argue that it rests on excessively strong informational conditions. In view of this I provide an alternative set of conditions that are jointly sufficient for coordination in games, and I argue that these conditions constitute a plausible alternative to the proposed notion of epistemic support. (shrink)

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)

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 formulate a formal framework in which we combine the theory of dynamic epistemic logic and the theory of games. In particular, we show how we can use tools of dynamic epistemic logic to reason about information change ? and in particular, the effect of communication acts ? in such a game of imperfect information. We show how this framework allows for the formulation of specific assumptions in pragmatics of communication, as well as the formulation of general results about the (...) value of information. (shrink)

``Pluralistic ignorance'' is a phenomenon mainly studied in social psychology. Viewed as an epistemic phenomenon, one way to define it is as a situation where ``no one believes, but everyone believes that everyone else believes''. In this paper various versions of pluralistic ignorance are formalized using epistemic/doxastic logic. The motive is twofold. Firstly, the formalizations are used to show that the various versions of pluralistic ignorance are all consistent, thus there is nothing in the phenomenon that necessarily goes against logic. (...) Secondly, pluralistic ignorance, is on many occasions, assumed to be fragile. In this paper, however, it is shown that pluralistic ignorance need not be fragile to announcements of the agents' beliefs. Hence, to dissolve pluralistic ignorance in general, something more than announcements of the subjective views of the agents is needed. Finally, suggestions to further research are outlined. (shrink)

I propose a modal logic that enables to reason about self-regarding and otherregarding motivations in strategic games. This logic integrates the concepts of joint action, belief, individual and group payoff. The first part of the article is focused on self-regarding agents. A self-regarding agent decides to perform a certain action only if he believes that this action maximizes his own personal benefit. The second part of the article explores different kinds of other-regarding motivations such as fairness and reciprocity. Differently from (...) self-regarding agents, other-regarding agents also consider the benefits of their choices for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. The analysis also considers team-directed reasoning, i.e. the mode of reasoning that people use when they take themselves to be acting as members of a group or a team. (shrink)

IntroductionA quick glance at the opening paragraphs in many of the classic logic textbooks reveals a common view: Logical methods highlight the reasoning patterns of a single agent engaged in some form of mathematical thinking.A sampling from my bookshelf: Shoenfield’s Mathematical Logic: “Logic is the study of reasoning; and mathematical logic is the study of the type of reasoning done by mathematicians”; Enderton’s A Mathematical Introduction of Logic: “Symbolic logic is a mathematical model of deductive thought”; and Chiswell and Hodges (...) Mathematical Logic: “In this course we shall study some ways of proving statements.” However, this traditional view of the “subject matter” of logic is expanding. There is a growing literature using phrases such as “rational interaction” or “information flow” to describe its subject matter while still employing traditional logical methods. The clearest example of this can be found in the work of Johan van Benthem and others on l .. (shrink)

This volume seeks to further the use of formal methods in clarifying one of the central problems of philosophy: that of our free human agency and its place in our indeterministic world. It celebrates the important contributions made in this area by Nuel Belnap, American logician and philosopher. Philosophically, indeterminism and free action can seem far apart, but in Belnap’s work, they are intimately linked. This book explores their philosophical interconnectedness through a selection of original research papers that build forth (...) on Belnap’s logical and philosophical work. Some contributions take the form of critical discussions of Belnap's published work, some develop points made in his publications in new directions, and others provide additional insights on the topics of indeterminism and free action. In Nuel Belnap’s work on indeterminism and free action, three formal frameworks figure prominently: the simple branching histories framework known as "branching time;" its relativistic spatio-temporal extension, branching space-times; and the “seeing to it that” logic of agency. As those frameworks provide the formal background for the contributed papers, the volume introduction gives an overview of the current state of their development. It also introduces case-intensional first order logic, a general intensional logic offering resources for a first-order extension of the mentioned frameworks and a recent research focus of Belnap’s. The volume also contains an extended biographical interview with Nuel Belnap. (shrink)