In this paper, I develop an algorithmic impossible-worlds model of belief and knowledge that provides a middle ground between models that entail that everyone is logically omniscient and those that are compatible with even the most egregious kinds of logical incompetence. In outline, the model entails that an agent believes (knows) φ just in case she can easily (and correctly) compute that φ is true and thus has the capacity to make her actions depend on whether φ. The model thereby (...) captures the standard view that belief and knowledge ground are constitutively connected to dispositions to act. As I explain, the model improves upon standard algorithmic models developed by Parikh, Halpern, Moses, Vardi, and Duc, among other ways, by integrating them into an impossible-worlds framework. The model also avoids some important disadvantages of recent candidate middle-ground models based on dynamic epistemic logic or step logic, and it can subsume their most important advantages. (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)

When we say “I know why he was late”, we know not only the fact that he was late, but also an explanation of this fact. We propose a logical framework of “knowing why” inspired by the existing formal studies on why-questions, scientific explanation, and justification logic. We introduce the Kyi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {K}}{}\textit{y}}_i$$\end{document} operator into the language of epistemic logic to express “agent i knows why φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}” and propose a Kripke-style semantics of such expressions in terms of knowing an explanation of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}. We obtain two sound and complete axiomatizations w.r.t. two different model classes depending on different assumptions about introspection. Finally we connect our logic with justification logic technically by providing an alternative semantics and an in-depth comparison on various design choices. (shrink)

We present a framework for epistemic logic, modeling the logical aspects of System 1 and System 2 cognitive processes, as per dual process theories of reasoning. The framework combines non-normal worlds semantics with the techniques of Dynamic Epistemic Logic. It models non-logically-omniscient, but moderately rational agents: their System 1 makes fast sense of incoming information by integrating it on the basis of their background knowledge and beliefs. Their System 2 allows them to slowly, step-wise unpack some of the logical consequences (...) of such knowledge and beliefs, by paying a cognitive cost. The framework is applied to three instances of limited rationality, widely discussed in cognitive psychology: Stereotypical Thinking, the Framing Effect, and the Anchoring Effect. (shrink)

Logical omniscience states that the knowledge set of ordinary rational agents is closed for its logical consequences. Although epistemic logicians in general judge this principle unrealistic, there is no consensus on how it should be restrained. The challenge is conceptual: we must find adequate criteria for separating obvious logical consequences from non-obvious ones. Non-classical game-theoretic semantics has been employed in this discussion with relative success. On the one hand, with urn semantics [15], an expressive fragment of classical game semantics that (...) weakens the dependence relations between quantifiers occurring in a formula, we can formalize, for a broad array of examples, epistemic scenarios in which an individual ignores the validity of some first-order sentence. On the other hand, urn semantics offers a disproportionate restriction of logical omniscience. Therefore, an improvement of this system is needed to obtain a better solution of the problem. In this paper, I argue that our linguistic competence in using quantifiers requires a sort of basic hypothetical logical knowledge that can be formulated as follows: when inquiring after the truth-value of ∀xφ, an individual might be unaware of all substitutional instances this sentence accepts, but at least she must know that, if an element a is given, then ∀xφ holds only if φ is true. This thesis accepts game-theoretic formalization in terms of a refinement of urn semantics. I maintain that the system so obtained affords an improved solution of the logical omniscience problem. To do this, I characterize first-order theoremhood in US+. As a consequence of this result, we will see that the ideal reasoner depicted by US+ only knows the validity of first-order formulas whose Herbrand witnesses can be trivially found, a fact that provides strong evidence that our refinement of urn semantics captures a relevant sense of logical obviousness. (shrink)

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.

Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of them updates (...) on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational. (shrink)

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do (...) not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic. This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study. (shrink)

The Interrogative Model of Inquiry and Dynamic Epistemic Logics are two central paradigms in formal epistemology. This paper is motivated by the observation of a significant complementarity between them: on the one hand, the IMI provides a framework for investigating inquiry represented as an idealized game between an Inquirer and Nature, along with an account of the interaction between questions and inferences in information-seeking processes, but is lacking a formulation in the multi-agent case; on the other hand, DELs model various (...) operations of information change in multi-agent systems, but the field is lacking a proper integration of question and inference dynamics, along with an application to the investigation of inquiry processes. The goal of this paper is to integrate the two paradigms in such a way as to combine their respective insights. To this end, we develop a formal system called DEL $$_\mathrm{IMI }$$ IMI which aims to represent the interaction between question and inference dynamics in inquiry—as described by the IMI—in a multi-agent setting, and this in such a way as to enable an investigation of inquiry games with multi-agent dimensions. The DEL $$_\mathrm{IMI }$$ IMI system is designed to represent the possible moves of such inquiry games through three types of epistemic actions: agents addressing questions to Nature, agents addressing questions to other agents, agents drawing logical inferences. We then show how the resulting framework can be used to formally define multi-agent inquiry games. We conclude by evaluating the interest of the DEL $$_\mathrm{IMI }$$ IMI system for the IMI and DELs paradigms. (shrink)

The aim of this paper is to explore the advantages deriving from the application of relating semantics in epistemic logic. As a first step, I will discuss two versions of relating semantics and how they can be differently exploited for studying modal and epistemic operators. Next, I consider several standard frameworks which are suitable for modelling knowledge and related notions, in both their implicit and their explicit form and present a simple strategy by virtue of which they can be associated (...) with intuitive systems of relating logic. As a final step, I will focus on the logic of knowledge based on justification logic and show how relating semantics helps us to provide an elegant solution to some problems related to the standard interpretation of the explicit epistemic operators. (shrink)

Language use and interpretation is heavily contingent on context. But human interlocutors need not always agree what the actual context is. In game theoretic approaches to language use and interpretation, interlocutors’ beliefs about the context are the players’ beliefs about the game that they are playing. Together this entails that we need to consider cases in which interlocutors have different subjective conceptualizations of the game they are in. This paper therefore extends iterated best response reasoning, as an established model for (...) pragmatic reasoning, to games with unawareness. This extension not only leads to more plausible context models for many communicative situations, but also to improved predictions for otherwise problematic cases and an extension of the scope of pragmatic phenomena that can be captured by game theoretic analysis. (shrink)

Comparativism is the view that comparative confidences (e.g., being more confident that P than that Q) are more fundamental than degrees of belief (e.g., believing that P with some strength x). In this paper, I outline the basis for a new, non-probabilistic version of comparativism inspired by a suggestion made by Frank Ramsey in `Probability and Partial Belief'. I show how, and to what extent, `Ramseyan comparativism' might be used to weaken the (unrealistically strong) probabilistic coherence conditions that comparativism traditionally (...) relies on. (shrink)

One response to the problem of logical omniscience in standard possible worlds models of belief is to extend the space of worlds so as to include impossible worlds. It is natural to think that essentially the same strategy can be applied to probabilistic models of partial belief, for which parallel problems also arise. In this paper, I note a difficulty with the inclusion of impossible worlds into probabilistic models. Under weak assumptions about the space of worlds, most of the propositions (...) which can be constructed from possible and impossible worlds are in an important sense inexpressible; leaving the probabilistic model committed to saying that agents in general have at least as many attitudes towards inexpressible propositions as they do towards expressible propositions. If it is reasonable to think that our attitudes are generally expressible, then a model with such commitments looks problematic. (shrink)

According to comparativism, degrees of belief are reducible to a system of purely ordinal comparisons of relative confidence. (For example, being more confident that P than that Q, or being equally confident that P and that Q.) In this paper, I raise several general challenges for comparativism, relating to (i) its capacity to illuminate apparently meaningful claims regarding intervals and ratios of strengths of belief, (ii) its capacity to draw enough intuitively meaningful and theoretically relevant distinctions between doxastic states, and (...) (iii) its capacity to handle common instances of irrationality. (shrink)

It would be good to have a Bayesian decision theory that assesses our decisions and thinking according to everyday standards of rationality — standards that do not require logical omniscience (Garber 1983, Hacking 1967). To that end we develop a “fragmented” decision theory in which a single state of mind is represented by a family of credence functions, each associated with a distinct choice condition (Lewis 1982, Stalnaker 1984). The theory imposes a local coherence assumption guaranteeing that as an agent's (...) attention shifts, successive batches of "obvious" logical information become available to her. A rule of expected utility maximization can then be applied to the decision of what to attend to next during a train of thought. On the resulting theory, rationality requires ordinary agents to be logically competent and to often engage in trains of thought that increase the unification of their states of mind. But rationality does not require ordinary agents to be logically omniscient. (shrink)

Security properties naturally combine temporal aspects of protocols with aspects of knowledge of the agents. Since BAN-logic, there have been several initiatives and attempts to incorpórate epistemics into the analysis of security protocols. In this paper, we give an overview of work in the field and present it in a unified perspective, with comparisons on technical subtleties that have been employed in different approaches. Also, we study to which degree the use of epistemics is essential for the analysis of security (...) protocols. We look for formal conditions under which knowledge modalities can bring extra expressive power to pure temporal languages. On the other hand, we discuss the cost of the epistemic operators in terms of model checking complexity. (shrink)

Logical theories for representing knowledge are often plagued by the so-called Logical Omniscience Problem. The problem stems from the clash between the desire to model rational agents, which should be capable of simple logical inferences, and the fact that any logical inference, however complex, almost inevitably consists of inference steps that are simple enough. This contradiction points to the fruitlessness of trying to solve the Logical Omniscience Problem qualitatively if the rationality of agents is to be maintained. We provide a (...) quantitative solution to the problem compatible with the two important facets of the reasoning agent: rationality and resource boundedness. More precisely, we provide a test for the logical omniscience problem in a given formal theory of knowledge. The quantitative measures we use are inspired by the complexity theory. We illustrate our framework with a number of examples ranging from the traditional implicit representation of knowledge in modal logic to the language of justification logic, which is capable of spelling out the internal inference process. We use these examples to divide representations of knowledge into logically omniscient and not logically omniscient, thus trying to determine how much information about the reasoning process needs to be present in a theory to avoid logical omniscience. (shrink)

Epistemic logic is the logic of knowledge and belief. It provides insight into the properties of individual knowers, has provided a means to model complicated scenarios involving groups of knowers and has improved our understanding of the dynamics of inquiry.

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)