Restricting attention to the class of extensive games defined by von Neumann and Morgenstern with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information (...) partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of “maximum information” we are able to provide an alternative characterization of the meet of the information partitions. (shrink)

An obituary of Philippe Mongin (1950-2020).

Since Lewis’s (1969) and Aumann’s (1976) pioneering contributions, the concepts of common knowledge and common belief have been discussed extensively in the literature, both syntactically and semantically1. At the individual level the difference between knowledge and belief is usually identified with the presence or absence of the Truth Axiom ( iA → A), which is interpreted as ”if individual i believes that A, then A is true”. In such a case the individual is often said to know that A (thus (...) it is possible for an individual to believe a false proposition but she cannot know a false proposition). Going to the interpersonal level, the literature then distinguishes between common knowledge and common belief on the basis of whether or not the Truth Axiom is postulated at the individual level. However, while at the individual level the Truth Axiom captures merely a relationship between the individuals’ beliefs and the external world, at the interpersonal level it has very strong implications. For example, the following is a consequence of the Truth Axiom: i jA → iA, that is, if individual i believes that individual j believes that A, then individual i herself believes that A. Thus, in contrast to other axioms, the Truth Axiom does not merely reflect individual agents’ “logic of belief”. (The reason why the Truth Axiom is much stronger in an interpersonal context than appears at first glance is that it amounts to assuming that agreement of any individual’s belief with the truth is common knowledge). Given its logical force, it is not surprising to find that it has strong implications for the logic of common knowledge. In particular, if each individual’s beliefs satisfy the strongest logic of knowledge (namely S5 or KT5), the associated common knowledge operator satisfies this logic too. Such is not the case for belief: bereft of the Truth Axiom, even the strongest logic for individual belief (KD45) is insufficient to ensure the satisfaction of the “Negative Introspection” axiom for common belief: ¬ ∗A → ∗¬ ∗A (where ∗ denotes the common belief operator).. (shrink)

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.

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, (...) we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II. (shrink)

This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models and other model-theoretic (...) constructions; comparisons with other semantics for modal logic ; neighborhood semantics for first-order modal logic, applications in game theory ; applications in epistemic logic ; and non-normal modal logics with dynamic modalities. The book can be used as the primary text for seminars on philosophical logic focused on non-normal modal logics; as a supplemental text for courses on modal logic, logic in AI, or philosophical logic ; or as the primary source for researchers interested in learning about the uses of neighborhood semantics in philosophical logic and game theory. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the (...) iterative notion of common belief. (shrink)

We show that if an agent reasons according to standard inference rules, the truth and introspection axioms extend from the set of non-epistemic propositions to the whole set of propositions. This implies that the usual axiomatization of partitional possibility correspondences is redundant, and provides a justification for truth and introspection that is partly based on reasoning.

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.

Dynamic epistemic logic, broadly conceived, is the study of logics of information change. This is the first paper in a two-part series introducing this research area. In this paper, I introduce the basic logical systems for reasoning about the knowledge and beliefs of a group of agents.

This paper investigates a limitation of the model of belief and knowledge prevailing in mainstream economics, namely the state-space model. Because of its set-theoretic nature, this model has difficulties in capturing the difference between expressions that designate the same object but have different meanings, i.e., expressions with the same extension but different intensions. This limitation generates puzzling results concerning what individuals believe or know about the world as well as what individuals believe or know about what other individuals believe or (...) know about the world. The paper connects these puzzling results to two issues that are relevant for economic theory beyond the state-space model, namely, framing effects and the distinction between the model-maker and agents that appear in the model. Finally, the paper discusses three possible solutions to the limitations of the state-space model, and concludes that the two alternatives that appear practicable also have significant drawbacks. (shrink)

We show the faithful embedding of common knowledge logic CKL into game logic GL, that is, CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pure strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and (...) its cut-elimination. The faithful embedding theorem is proved for the KD4-type propositional CKL and GL, but it holds for some variants of them. (shrink)