The impossibility theorem of Dekel, Lipman and Rustichini has been thought to demonstrate that standard state-space models cannot be used to represent unawareness. We first show that Dekel, Lipman and Rustichini do not establish this claim. We then distinguish three notions of awareness, and argue that although one of them may not be adequately modeled using standard state spaces, there is no reason to think that standard state spaces cannot provide models of the other two notions. In fact, standard space (...) models of these forms of awareness are attractively simple. They allow us to prove completeness and decidability results with ease, to carry over standard techniques from decision theory, and to add propositional quantifiers straightforwardly. (shrink)

Scientists and philosophers frequently speak about levels of description, levels of explanation, and ontological levels. In this paper, I propose a unified framework for modelling levels. I give a general definition of a system of levels and show that it can accommodate descriptive, explanatory, and ontological notions of levels. I further illustrate the usefulness of this framework by applying it to some salient philosophical questions: (1) Is there a linear hierarchy of levels, with a fundamental level at the bottom? And (...) what does the answer to this question imply for physicalism, the thesis that everything supervenes on the physical? (2) Are there emergent properties? (3) Are higher-level descriptions reducible to lower-level ones? (4) Can the relationship between normative and non-normative domains be viewed as one involving levels? Although I use the terminology of “levels”, the proposed framework can also represent “scales”, “domains”, or “subject matters”, where these are not linearly but only partially ordered by relations of supervenience or inclusion. (shrink)

Scientists and philosophers frequently speak about levels of description, levels of explanation, and ontological levels. This paper presents a framework for studying levels. I give a general definition of a system of levels and discuss several applications, some of which refer to descriptive or explanatory levels while others refer to ontological levels. I illustrate the usefulness of this framework by bringing it to bear on some familiar philosophical questions. Is there a hierarchy of levels, with a fundamental level at the (...) bottom? And what does the answer to this question imply for physicalism, the thesis that everything supervenes on the physical? Are there emergent higher-level properties? Are higher-level descriptions reducible to lower-level ones? Can the relationship between normative and non-normative domains be viewed as one involving levels? And might a levelled framework shed light on the relationship between third-personal and first-personal phenomena? (shrink)

This article takes off from Johan van Benthem’s ruminations on the interface between logic and cognitive science in his position paper “Logic and reasoning: Do the facts matter?”. When trying to answer Van Benthem’s question whether logic can be fruitfully combined with psychological experiments, this article focuses on a specific domain of reasoning, namely higher-order social cognition, including attributions such as “Bob knows that Alice knows that he wrote a novel under pseudonym”. For intelligent interaction, it is important that the (...) participants recursively model the mental states of other agents. Otherwise, an international negotiation may fail, even when it has potential for a win-win solution, and in a time-critical rescue mission, a software agent may depend on a teammate’s action that never materializes. First a survey is presented of past and current research on higher-order social cognition, from the various viewpoints of logic, artificial intelligence, and psychology. Do people actually reason about each other’s knowledge in the way proscribed by epistemic logic? And if not, how can logic and cognitive science productively work together to construct more realistic models of human reasoning about other minds? The paper ends with a delineation of possible avenues for future research, aiming to provide a better understanding of higher-order social reasoning. The methodology is based on a combination of experimental research, logic, computational cognitive models, and agent-based evolutionary models. (shrink)

Recent literature has examined the problem facing decision makers with bounded awareness, who may be unaware of some states of nature. A question that naturally arises here is whether a value of awareness, analogous to value of information, can be attributed to changes in awareness. In this paper, such a value is defined. It is shown that the sum VOA +\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+$$\end{document} VOI is constant and, except for scale effects, independent of the (...) choice set. It follows that the larger is VOA, the smaller is VOI. This point is illustrated for a simple two-state case, then proved for general classes of compact convex choice sets, and for alternative interpretations of the concept of unawareness. (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)

Decision-making typically requires judgments about causal relations: we need to know the causal effects of our actions and the causal relevance of various environmental factors. We investigate how several individuals' causal judgments can be aggregated into collective causal judgments. First, we consider the aggregation of causal judgments via the aggregation of probabilistic judgments, and identify the limitations of this approach. We then explore the possibility of aggregating causal judgments independently of probabilistic ones. Formally, we introduce the problem of causal-network aggregation. (...) Finally, we revisit the aggregation of probabilistic judgments when this is constrained by prior aggregation of qualitative causal judgments. (shrink)

Possible worlds models of belief have difficulties accounting for unawareness, the inability to entertain (and hence believe) certain propositions. Accommodating unawareness is important for adequately modelling epistemic states, and representing the informational content to which agents have in principle access given their explicit beliefs. In this paper, I develop a model of explicit belief, awareness, and informational content, along with an sound and complete axiomatisation. I furthermore defend the model against the seminal impossibility result of Dekel, Lipman and Rustichini, according (...) to which three intuitive conditions preclude non-trivial unawareness on any possible worlds model of belief. (shrink)

Most models of rational action assume that all possible states and actions are pre-defined and that preferences change only when beliefs do. But several decision and game problems lack these features, calling for a dynamic model of preferences: preferences can change when unforeseen possibilities come to light or when there is no specifiable or measurable change in belief. We propose a formally precise dynamic model of preferences that extends an existing static model. Our axioms for updating preferences preserve consistency while (...) minimising change, like Hansson’s :1–28, 1995). But unlike prior models of preference change, ours supports default reasoning with partial preference information, which is essential to handle decision problems where the decision tree isn’t surveyable. We also show that our model avoids problems for other models of preference change discussed in Spohn. (shrink)

Over the past 20 years or so, a small but growing literature has emerged with the aim of modeling agents who are unaware of certain things. In this paper we compare two different approaches to modeling unawareness: the object-based approach of Board and Chung (Object-based unawareness: theory and applications. University of Minnesota, Mimeo, 2008) and the subjective-state-space approach of Heifetz et al. (J Econ Theory 130: 78-94,2006). In particular, we show that subjectivestate-space models (henceforth HMS structures) can be embedded within (...) object-based models (henceforth OBU structures), demonstrating that the latter are at least as expressive. As long as certain restrictions are imposed on the form of the OBU structure, the embedding can also go the other way. A generalization of HMS structures (relaxing the partitional properties of knowledge) gives us a full converse. (shrink)

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. (...) In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics. (shrink)

In the real world, there can be constraints on rational decision-making: there can be limitations on what I can know and on what you can know. There can also be constraints on my ability to deliberate or on your ability to deliberate. It is useful to know what the norms of rational deliberation should be in ideal contexts, for fully informed agents, in an ideal world. But it is also useful to know what the norms of rational deliberation should be (...) in the actual world, in non-ideal contexts, for imperfectly informed agents, especially for big, life-changing decisions. That is, we want to know how to deliberate as best we can, given the real-world limitations on what we can know, and given real-world limitations on how we are able to deliberate. In this paper, our concern is with the norms of rational deliberation in certain, important, non-ideal contexts, where the reasoning occurs from the agent’s first person, subjective point of view. The norms governing the process of deliberation for real people in the sorts of non-ideal contexts we’ll consider need to reflect the way that real agents, with an incomplete grasp on the facts and an imperfect ability to deliberate, can be expected to proceed. Our central contention is that framing and exploring the deliberative process from the first person perspective allows us to uncover and explore important, real-world constraints on boundedly rational agents deliberating from the subjective perspective. (shrink)

We examine various logics that combine knowledge, awareness, and change of awareness. An agent can become aware of propositional propositions but also of other agents or of herself. The dual operation to becoming aware, forgetting, can also be modelled. Our proposals are based on a novel notion of structural similarity that we call awareness bisimulation, the obvious notion of modal similarity for structures encoding knowledge and awareness.

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)

We provide a syntactic model of unawareness. By introducing multiple knowledge modalities, one for each sub-language, we specifically model agents whose only mistake in reasoning (other than their unawareness) is to underestimate the knowledge of more aware agents. We show that the model is a complete and sound axiomatization of the set-theoretic model of Galanis (University of Southampton Discussion paper 709, 2007) and compare it with other unawareness models in the literature.

Formal learning theory is the mathematical embodiment of a normative epistemology. It deals with the question of how an agent should use observations about her environment to arrive at correct and informative conclusions. Philosophers such as Putnam, Glymour and Kelly have developed learning theory as a normative framework for scientific reasoning and inductive inference.

The paper develops an interface between syntax-based logical models of awareness and dynamic epistemic logic. The framework is shown to be able to accommodate a variety of notions of awareness and knowledge, as well as their dynamics. This, it is argued, offers a natural formal environment for the analysis of epistemic phenomena typical of multi-agent information exchange, such as how agents become aware of relevant details, how they perform inferences and how they share their information within a group. Technically, the (...) logics presented are all simple refinements of the logic of public announcements. (shrink)

In the most popular logics combining knowledge and awareness, it is not possible to express statements about knowledge of unawareness such as “Ann knows that Bill is aware of something Ann is not aware of”—without using a stronger statement such as “Ann knows that Bill is aware of \(p\) and Ann is not aware of \(p\) ”, for some particular \(p\) . In Halpern and Rêgo (Proceedings of KR 2006; Games Econ Behav 67(2):503–525, 2009b) Halpern and Rêgo introduced a logic (...) in which such statements about knowledge of unawareness can be expressed. The logic extends the traditional framework with quantification over formulae, and is thus very expressive. As a consequence, it is not decidable. In this paper we introduce a decidable logic which can be used to reason about certain types of unawareness. Our logic extends the traditional framework with an operator expressing full awareness, i.e., the fact that an agent is aware of everything, and another operator expressing relative awareness, the fact that one agent is aware of everything another agent is aware of. The logic is less expressive than Halpern’s and Rêgo’s logic. It is, however, expressive enough to express all of the motivating examples in Halpern and Rêgo (Proceedings of KR 2006; Games Econ Behav 67(2):503–525, 2009b). In addition to proving that the logic is decidable and that its satisfiability problem is PSPACE-complete, we present an axiomatisation which we show is sound and complete. (shrink)

Abstract:One view of practical reasoning is that it involves the weighing of reasons. It is not clear, however, how the weights of reasons combine, especially given the logical and substantive relations among different reasons. Nor is it clear how the weighing of reasons relates to decision theoretic maximization of expected value. This paper constructs a formal model of reasons and their weight in order to shed light on these issues. The model informs philosophical debates about reasons, such as the question (...) of whether reasons or values are more fundamental and the opposing doctrines of atomism and holism about reasons. (shrink)

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)

In recent years, much work has been dedicated by logicians, computer scientists and economists to understanding awareness, as its importance for human behaviour becomes evident. Although several logics of awareness have been proposed, little attention has been explicitly dedicated to change in awareness. However, one of the most crucial aspects of awareness is the changes it undergoes, which have countless important consequences for knowledge and action. The aim of this paper is to propose a formal model of awareness change, and (...) to derive from it logics of awareness change. In the first part of the paper, the model of epistemic states of bounded agents proposed in Hill (Stud Log 89(1):81–109, 2008a ) is extended with operations modelling awareness change. In the second part of the paper, it is shown how this model naturally extends the “standard” logic of awareness to yield a logic of awareness change. (shrink)

The integration of learning cognitive agents in the research agenda is an important step in the evolution of economics. However, relying on a retrospective analysis of the treatment of decision making in economics, this article argues that the cognitive programme aims to justify rational behaviour in an equilibrium framework rather than to integration an interpretative conception of agents' behaviour. As a consequence, the level of generality of analytical results remains limited and economists miss the opportunity to establish a discussion with (...) other social sciences. In particular, it fails to analyse a class of situations related to the problem of knowledge creation and the subsequent learning paradox. Building on recent advances in heterodox cognitive science, we suggest a behavioural approach that discriminates different learning situations. If applied, this approach would permit a better understanding of human behaviour and a richer analysis of coordination mechanisms in unfamiliar environment with no loss of rigour. (shrink)

This paper focuses on radical pooling, or the question of how to aggregate credences when there is a fundamental disagreement about which is the relevant logical space for inquiry. The solution advanced is based on the notion of consensus as common ground, where agents can find it by suspending judgment on logical possibilities. This is exemplified with cases of scientific revolution. On a formal level, the proposal uses algebraic joins and imprecise probabilities; which is shown to be compatible with the (...) principles of marginalization, rigidity, reverse bayesianism, and minimum divergence commonly endorsed in these contexts. Furthermore, I extend results from previous work by to show that pooling sets of imprecise probabilities can satisfy important pooling axioms. (shrink)

This volume concerns Rational Agents - humans, players in a game, software or institutions - which must decide the proper next action in an atmosphere of partial information and uncertainty. The book collects formal accounts of Uncertainty, Rationality and Agency, and also of their interaction. It will benefit researchers in artificial systems which must gather information, reason about it and then make a rational decision on which action to take.

Selfish, dishonest or malicious agents may find an interest in manipulating others. While many works deal with designing robust systems or manipulative strategies, few works are interested in defining in a broad sense what is a manipulation and how we can reason with such a notion. In this article, based on a social science literature, we give a general definition of manipulation for multi-agent systems. A manipulation is a deliberate effect of an agent – called manipulator – to instrumentalize another (...) agent – called victim – while making sure to conceal that effect. We present then a logical framework, called KBE, to express and reason about manipulations. Since manipulation relies on deliberate effects, KBE introduces a deliberate BIAT operator which abstracts deliberate consequences of actions. We prove that this logic is sound and complete. Furthermore,we express related notions such as coercion, persuasion, or deception. (shrink)

If we express our knowledge in sentences, we will find that these sentences are linked in complex patterns governed by our observations and our inferences from these observations. These inferences are to a large extent driven by logical rules. We ask whether the structure logic imposes on our knowledge restricts what we forget and what we remember. The model is a two period S5 logic. In this logic, we propose a memory loss operator: the agent forgets a sentence pif and (...) only if he knows pat time 1 and he does not know pat time 2. Equipped with the operator, we prove theorems on the relation between knowledge and memory loss. The main results point to classes of formulas that an agent cannot forget, and classes of formulas he must forget. A desirable feature is that most results hold in the S4 logic. The results illustrate bounds to memory loss, and thus to bounded rationality. We apply the model to single-agent conventions: conventions made between an agent and himself. (shrink)

The standard model of knowledge, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}, consists of state space, Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}, and possibility correspondence, P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P$$\end{document}. Usually, it is assumed that P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P$$\end{document} satisfies all knowledge axioms. Violating at least one of these axioms is defined as epistemic bounded rationality. If this happens, a researcher may (...) try to look for another model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}, which generates the initial model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}, while satisfying all knowledge axioms. Rationalizing EBR means that the researcher finds such a model. I determine when rationalization of EBR is possible. I also investigate when a model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}, which satisfies all knowledge axioms, generates a model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document}, which satisfies these axioms as well. (shrink)