We present a framework for verifying systems composed of heterogeneous reasoning agents, in which each agent may have differing knowledge and inferential capabilities, and where the resources each agent is prepared to commit to a goal (time, memory and communication bandwidth) are bounded. The framework allows us to investigate, for example, whether a goal can be achieved if a particular agent, perhaps possessing key information or inferential capabilities, is unable (or unwilling) to contribute more than a given portion of its (...) available computational resources or bandwidth to the problem. We present a novel temporal epistemic logic, BMCL-CTL , which allows us to describe a set of reasoning agents with bounds on time, memory and the number of messages they can exchange. The bounds on memory and communication are expressed as axioms in the logic. As an example, we show how to axiomatise a system of agents which reason using resolution and prove that the resulting logic is sound and complete. We then show how to encode a simple system of reasoning agents specified in BMCL-CTL in the description language of the Mocha model checker (Alur et al., Proceedings of the tenth international conference on computer-aided verification (CAV) , 1998), and verify that the agents can achieve a goal only if they are prepared to commit certain time, memory and communication resources. (shrink)

In this paper I present a more refined analysis of the principles of deductive closure and positive introspection. This analysis uses the expressive resources of logics for different types of group knowledge, and discriminates between aspects of closure and computation that are often conflated. The resulting model also yields a more fine-grained distinction between implicit and explicit knowledge, and places Hintikka’s original argument for positive introspection in a new perspective.

The logic of ‘being informed’ gives a formal analysis of a cognitive state that does not coincide with either belief, or knowledge. To Floridi, who first proposed the formal analysis, the latter is supported by the fact that unlike knowledge or belief, being informed is a factive, but not a reflective state. This paper takes a closer look at the formal analysis itself, provides a pure and an applied semantics for the logic of being informed, and tries to find out (...) to what extent the formal analysis can contribute to an information-based epistemology. (shrink)

Cognitive states as well as cognitive commodities play central though distinct roles in our epistemological theories. By being attentive to how a difference in their roles affects our way of referring to them, we can undoubtedly accrue our understanding of the structure and functioning of our main epistemological theories. In this paper we propose an analysis of the dichotomy between states and commodities in terms of the method of abstraction, and more specifically by means of infomorphisms between different ways to (...) classify states of information, information-bases, and evidential situations. (shrink)

As Vincent Hendricks remarks early on in this book, the formal and mainstream traditions of epistemic theorising have mostly evolved independently of each other. This initial impression is confirmed by a comparison of the main problems and methods practitioners in each tradition are concerned with. Mainstream epistemol- ogy engages in a dialectical game of proposing and challenging definitions of knowledge. Formal epistemologists proceed differently, as they design a wide variety of axiomatic and model-theoretic methods whose consequences they investigate independently of (...) the need of giving counterexample-free definitions of knowledge. Or at least, this is a common way to explain where both disciplines stand in the larger landscape of epistemic theorising, and why interactions between them remain scarce. The main ambition of this book is to show that the distinction between formal and mainstream approaches should not preclude a fruitful interaction, and that it only takes the right outlook on their respective practices to disclose plenty of room for interaction. (shrink)

One of the basic principles of the general definition of information is its rejection of dataless information, which is reflected in its endorsement of an ontological neutrality. In general, this principles states that “there can be no information without physical implementation” (Floridi (2005)). Though this is standardly considered a commonsensical assumption, many questions arise with regard to its generalised application. In this paper a combined logic for data and information is elaborated, and specifically used to investigate the consequences of restricted (...) and unrestricted data-implementation-principles. (shrink)

In this paper we address the problem of combining a logic with nonmonotonic modal logic. In particular we study the intuitionistic case. We start from a formal analysis of the notion of intuitionistic consistency via the sequent calculus. The epistemic operator M is interpreted as the consistency operator of intuitionistic logic by introducing intuitionistic stable sets. On the basis of a bimodal structure we also provide a semantics for intuitionistic stable sets.

A conception of an information system has been introduced by Pawlak. The study has been continued in works of Pawlak and Orlowska and in works of Vakarelov. They had proposed some basic relations and had constructed a formal system of a modal logic that describes the relations and some of their Boolean combinations. Our work is devoted to a generalization of this approach. A class of relation systems and a complete calculus construction method for these systems are proposed. As a (...) corollary of our main result, our paper contains a solution of a Vakarelov's problem: how to construct a formal system that describes all the Boolean combinations of the basic relations. (shrink)

The paper provides a framework for representing belief-contravening hypotheses in games of perfect information. The resulting t-extended information structures are used to encode the notion that a player has the disposition to behave rationally at a node. We show that there are models where the condition of all players possessing this disposition at all nodes (under their control) is both a necessary and a sufficient for them to play the backward induction solution in centipede games. To obtain this result, we (...) do not need to assume that rationality is commonly known (as is done in [Aumann (1995)]) or commonly hypothesized by the players (as done in [Samet (1996)]). The proposed model is compared with the account of hypothetical knowledge presented by Samet in [Samet (1996)] and with other possible strategies for extending information structures with conditional propositions. (shrink)

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...) resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus. (shrink)

The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite Kripke model, then there exists an arithmetical interpretation f such that $PA \nvdash fR$ . This result implies the arithmetical completeness of arithmetically correct modal predicate logics with the finite model property (including the one-variable fragments of QGL and QS). The proof was obtained by adding "the predicate part" as a specific (...) addition to the standard Solovay construction. (shrink)

In this paper the modal operator "x is provable in Peano Arithmetic" is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.

Representing an epistemic situation involving several agents obviously depends on the modeling point of view one takes. We start by identifying the types of modeling points of view which are logically possible. We call the one traditionally followed by epistemic logic the perfect external approach, because there the modeler is assumed to be an omniscient and external observer of the epistemic situation. In the rest of the paper we focus on what we call the internal approach, where the modeler is (...) one of the agents involved in the situation. For this approach we propose and axiomatize a logical formalism based on epistemic logic. This leads us to formalize some intuitions about the internal approach and about its connections with the external ones. Finally, we show that our internal logic is decidable and PSPACE-complete. (shrink)

When a proposition might be the case, for all an agent knows, we can say that the proposition is epistemically possible for the agent. In the standard possible worlds framework, we analyze modal claims using quantification over possible worlds. It is natural to expect that something similar can be done for modal claims involving epistemic possibility. The main aim of this paper is to investigate the prospects of constructing a space of worlds—epistemic space—that allows us to model what is epistemically (...) possible for ordinary, non-ideally rational agents like you and me. I will argue that the prospects look dim for successfully constructing such a space. In turn, this will make a case for the claim that we cannot use the standard possible worlds framework to model what is epistemically possible for ordinary agents. (shrink)

Epistemic logic is one of the most exciting areas in medieval philosophy. Neglected almost entirely after the end of the Middle Ages, it has been rediscovered by philosophers of the twentieth century. Epistemic Logic in the Later Middle Ages provides the first comprehensive study of the subject. Ivan Boh explores the contrast between epistemic and alethic conceptions of consequence, the general epistemic rules of consequence, the search for conditions of knowing contingent propositions, the problems of substitutivity in intentional contexts, the (...) considerations of epistemic/doxastic iterated modalities, and the problems of composite and divided senses in authors ranging from Abelard to Frachantian. Boh concludes with a comparison between medieval endeavors and the epistemic logic of our own times. Written in a clear and readable style with minimal symbolic apparatus, this book employs modern symbolism and conceptual frameworks, and complements the studies of the syntacticand semantic dimensions of medieval logic. (shrink)

The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn't possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known. The proof has been used to argue (...) against versions of anti-realism committed to the thesis that all truths are knowable. For clearly there are unknown truths; individually and collectively we are non-omniscient. So, by the main result, it is false that all truths are knowable. The result has also been used to draw more general lessons about the limits of human knowledge. Still others have taken the proof to be fallacious, since it collapses an apparently moderate brand of anti-realism into an obviously implausible and naive idealism. (shrink)

Does a factive conception of knowability figure in ordinary use? There is some reason to think so. ‘Knowable’ and related terms such as ‘discoverable’, ‘observable’, and ‘verifiable’ all seem to operate factively in ordinary discourse. Consider the following example, a dialog between colleagues A and B: A: We could be discovered. B: Discovered doing what? A: Someone might discover that we're having an affair. B: But we are not having an affair! A: I didn’t say that we were. A’s remarks (...) sound contradictory. In this context the factivity of ‘someone might discover that’ explains this fact. So there is some reason to believe that knowability and related modalities are factive in ordinary use. For factive treatments of knowability in the context of epistemic theories of truth, compare Tennant (2000: 829) and Wright (2001: 59-60, n. 17). (shrink)

This paper contains a formal treatment of the system of quantified epistemic logic sketched in Appendix II of Carlson (1983). Section 1 defines the syntax and recapitulates the model set rules and principles of the Appendix system. Section 2 defines a possible worlds semantics for this system, and shows that the Appendix system is complete with respect to this semantics. Section 3 extends the system by an explicit truth operatorT it is true that and considers quantification over nonexistent individuals. Section (...) 4 formalizes the idea of variable identity criteria typical of Hintikkian epistemic logic. (shrink)

This essay corrects an error in the presentation of the Paradox of the Knowledge-Plus Knower, which is the variant of Kaplan and Montague’s Knower Paradox presented in C. Cross 2001: ‘The Paradox of the Knower without Epistemic Closure,’ MIND, 110, pp. 319–33. The correction adds a universally quantified transitivity principle for derivability as an additional assumption leading to paradox. This correction does not affect the status of the Knowledge-Plus paradox as a rebuttal to an argument against epistemic closure, since the (...) quantified transitivity principle is true in the standard model of arithmetic and therefore innocuous. (shrink)

In “The Paradox of the Knower without Epistemic Closure”, MIND 110:319-33, 2001, I develop a version of the Knower Paradox which does not assume epistemic closure, and I use it to argue that the original Knower Paradox does not support an argument against epistemic closure. In “The Paradox of the Knower without Epistemic Closure?”, MIND 113:95-107, 2004, Gabriel Uzquiano, using his own result, argues that my rebuttal to the anti-closure argument is not successful. I respond here by arguing that in (...) order to use Uzquiano’s result in an argument against closure, one must assume an implausible skepticism about arithmetic. (shrink)

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

This paper analyzes the logical form of valuing. I argue that valuing a concept or property is a universal statement qua logical form, that valuing an object is an existential statement qua logical form, and, furthermore, that a correct analysis of the logical form of valuing contains doxastic operators. I show that these ingredients give rise to an interesting interplay between uniform and ununiform quantification, on the one hand, and de dicto and de re beliefs, on the other. I apply (...) this analysis to the value of political freedom. The received view is that the value of freedom lies in the value of the specific things one is free to do. But Ian Carter has recently shown that freedom has irreducible, "non-specific" value, too. I show that underlying the debate between the proponents of the received view and their critics is a disagreement about logical form: ununiform de dicto beliefs about freedom as a concept, for the received view, and uniform half-de dicto-half-de re beliefs about freedom as an object, for its critics. (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.’.

We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination with a principle (...) of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility. (shrink)

This paper contributes to an increasing literature strengthening the connection between epistemic logic and epistemology (Van Benthem, Hendricks). I give a survey of the most important applications of epistemic logic in epistemology. I show how it is used in the history of philosophy (Steiner's reconstruction of Descartes' sceptical argument), in solutions to Moore's paradox (Hintikka), in discussions about the relation between knowledge and belief (Lenzen) and in an alleged refutation of verificationism (Fitch) and I examine an early argument about the (...) (im)possibility of epistemic logic (Hocutt). Subsequently, I deal with interpretive questions about epistemic logic that, although implicitly, already appeared in the first section. I contend that a conception of epistemic logic as a theory of knowledge assertions is incoherent, and I argue that it does not make sense to adopt a normative interpretation of epistemic logic. Finally, I show ways to extend epistemic logic with other branches of philosophical logic so as to make it useful for some epistemological questions. Conditional logics and logics of public announcement are used to understand causal theories of knowledge and versions of reliabilism. Temporal logic helps understand some dynamic aspects of knowledge as well as the verificationist thesis. (shrink)

Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of (...) learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work. (shrink)

We study a knowledge logic that assumes that to each set of agents, an indiscernibility relation is associated and the agents decide the membership of objects or states up to this indiscernibility relation. Its language contains a family of relative knowledge operators. We prove the decidability of the satisfiability problem, we show its EXPTIME-completeness and as a side-effect, we define a complete Hilbert-style axiomatization.

In this paper we compare different models of vagueness viewed as a specific form of subjective uncertainty in situations of imperfect discrimination. Our focus is on the logic of the operator “clearly” and on the problem of higher-order vagueness. We first examine the consequences of the notion of intransitivity of indiscriminability for higher-order vagueness, and compare several accounts of vagueness as inexact or imprecise knowledge, namely Williamson’s margin for error semantics, Halpern’s two-dimensional semantics, and the system we call Centered semantics. (...) We then propose a semantics of degrees of clarity, inspired from the signal detection theory model, and outline a view of higher-order vagueness in which the notions of subjective clarity and unclarity are handled asymmetrically at higher orders, namely such that the clarity of clarity is compatible with the unclarity of unclarity. (shrink)

Compositional verification aims at managing the complexity of theverification process by exploiting compositionality of the systemarchitecture. In this paper we explore the use of a temporal epistemiclogic to formalize the process of verification of compositionalmulti-agent systems. The specification of a system, its properties andtheir proofs are of a compositional nature, and are formalized within acompositional temporal logic: Temporal Multi-Epistemic Logic. It isshown that compositional proofs are valid under certain conditions.Moreover, the possibility of incorporating default persistence ofinformation in a system, is (...) explored. A completion operation on aspecific type of temporal theories, temporal completion, is introducedto be able to use classical proof techniques in verification withrespect to non-classical semantics covering default persistence. (shrink)

When reasoning about complex domains, where information available is usually only partial, nonmonotonic reasoning can be an important tool. One of the formalisms introduced in this area is Reiter's Default Logic (1980). A characteristic of this formalism is that the applicability of default (inference) rules can only be verified in the future of the reasoning process. We describe an interpretation of default logic in temporal epistemic logic which makes this characteristic explicit. It is shown that this interpretation yields a semantics (...) for default logic based on temporal epistemic models. A comparison between the various semantics for default logic will show the differences and similarities of these approaches and ours. (shrink)

Several justification logics have been created, starting with the logic LP, [1]. These can be thought of as explicit versions of modal logics, or of logics of knowledge or belief, in which the unanalyzed necessity (knowledge, belief) operator has been replaced with a family of explicit justification terms. We begin by sketching the basics of justification logics and their relations with modal logics. Then we move to new material. Modal logics come in various strengths. For their corresponding justification logics, differing (...) strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, various familiar extensions are actually conservative with respect to each other. Our method of proof is very simple, and general enough to handle several justification logics not directly corresponding to distinct modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open. (shrink)

I want to look at recent developments of representing AGM-style belief revision in dynamic epistemic logics and the options for doing something similar for ranking theory. Formally, my aim will be modest: I will define a version of basic dynamic doxastic logic using ranking functions as the semantics. I will show why formalizing ranking theory this way is useful for the ranking theorist first by showing how it enables one to compare ranking theory more easily with other approaches to belief (...) revision. I will then use the logic to state an argument for defining ranking functions on larger sets of ordinals than is customary. Secondly, I will argue that the only way to extend the account of belief revision given by ranking theory to higher-order beliefs and revisions is by continuing the approach taken by me and defining ranking theoretical equivalents of dynamic epistemic logics. For proponents of dynamic epistemic logic, such logics will naturally be of interest provided they are convinced of the revision operator defined by ranking theory. (shrink)

We examine the paradox of the surprise examination using dynamic epistemic logic. This logic contains means of expressing epistemic facts as well as the effects of learning new facts, and is therefore a natural framework for representing the puzzle. We discuss a number of different interpretations of the puzzle in this context, and show how the failure of principle of success, that states that sentences, when learned, remain to be true and come to be believed, plays a central role in (...) understanding the puzzle. (shrink)

This paper discusses the possibility of modelling inductive inference (Gold 1967) in dynamic epistemic logic (see e.g. van Ditmarsch et al. 2007). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing (Lange et al. 1996) and iterated epistemic (...) update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective. (shrink)

We study a generalization of the Muddy Children puzzle by allowing public announcements with arbitrary generalized quantifiers. We propose a new concise logical modeling of the puzzle based on the number triangle representation of quantifi ers. Our general aim is to discuss the possibility of epistemic modeling that is cut for specifi c informational dynamics. Moreover, we show that the puzzle is solvable for any number of agents if and only if the quanti fier in the announcement is positively active (...) (satis es a form of variety). (shrink)

The paper presents two case studies of multi-agent information exchange involving generalized quantifiers. We focus on scenarios in which agents successfully converge to knowledge on the basis of the information about the knowledge of others, so-called Muddy Children puzzle and Top Hat puzzle. We investigate the relationship between certain invariance properties of quantifiers and the successful convergence to knowledge in such situations. We generalize the scenarios to account for public announcements with arbitrary quantifiers. We show that the Muddy Children puzzle (...) is solvable for any number of agents if and only if the quantifier in the announcement is positively active (satisfies a version of the variety condition). In order to get the characterization result, we propose a new concise logical modeling of the puzzle based on the number triangle representation of generalized quantifiers. In a similar vein, we also study the Top Hat puzzle. We observe that in this case an announcement needs to satisfy stronger conditions in order to guarantee solvability. Hence, we introduce a new property, called bounded thickness, and show that the solvability of the Top Hat puzzle for arbitrary number of agents is equivalent to the announcement being 1-thick. (shrink)

In the present paper we propose a system of propositional logic for reasoning about justification, truthmaking, and the connection between justifiers and truthmakers. The logic of justification and truthmaking is developed according to the fundamental ideas introduced by Artemov. Justifiers and truthmakers are treated in a similar way, exploiting the intuition that justifiers provide epistemic grounds for propositions to be considered true, while truthmakers provide ontological grounds for propositions to be true. This system of logic is then applied both for (...) interpreting the notorious definition of knowledge as justified true belief and for advancing a new solution to Gettier counterexamples to this standard definition. (shrink)

In a penetrating investigation of the relationship between belief and quantitative degrees of confidence (or degrees of belief) Richard Foley (1992) suggests the following thesis: ... it is epistemically rational for us to believe a proposition just in case it is epistemically rational for us to have a sufficiently high degree of confidence in it, sufficiently high to make our attitude towards it one of belief. Foley goes on to suggest that rational belief may be just rational degree of confidence (...) above some threshold level that the agent deems sufficient for belief. He finds hints of this view in Locke’s discussion of probability and degrees of assent, so he calls it the Lockean Thesis.1 The Lockean Thesis has important implications for the logic of belief. Most prominently, it implies that even a logically ideal agent whose degrees of confidence satisfy the axioms of probability theory may quite rationally believe each of a large body of propositions that are jointly inconsistent. For example, an agent may legitimately believe that on each given occasion her well-maintained car will start, but nevertheless believe that she will eventually encounter a.. (shrink)

John Locke proposed a straightforward relationship between qualitative and quantitative doxastic notions: belief corresponds to a sufficiently high degree of confidence. Richard Foley has further developed this Lockean thesis and applied it to an analysis of the preface and lottery paradoxes. Following Foley's lead, we exploit various versions of these paradoxes to chart a precise relationship between belief and probabilistic degrees of confidence. The resolutions of these paradoxes emphasize distinct but complementary features of coherent belief. These features suggest principles that (...) tie together qualitative and quantitative doxastic notions. We show how these principles may be employed to construct a quantitative model - in terms of degrees of confidence - of an agent's qualitative doxastic state. This analysis fleshes out the Lockean thesis and provides the foundation for a logic of belief that is responsive to the logic of degrees of confidence. (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.

This book provides a valuable look at the work of up and coming epistemologists. The topics covered range from the central issues of mainstream epistemology to the more formal issues in epistemic logic and confirmation theory. This book should be read by anyone interested in seeing where epistemology is currently focused and where it is heading. - Stewart Cohen , Arizona State University..

Epistemic logic begins with the recognition that our everyday talk about knowing and believing has some systematic features that we can track and re‡ect upon. Epistemic logicians have studied and extended these glints of systematic structure in fascinating and important ways since the early 1960s. However, for one reason or another, mainstream epistemologists have shown little interest. It is striking to contrast the marginal role of epistemic logic in contemporary epistemology with the centrality of modal logic for metaphysicians. This article (...) is intended to help in correcting this oversight by presenting some important developments in epistemic logic and suggesting ways to understand their applicability to traditional epistemological problems. Obviously, by itself, tweaking the formal apparatus of epistemic logic does not solve traditional epistemological problems. Epistemic logic can help us to navigate through problems in a systematic fashion by unpacking the logic of the problematic concepts, it can also lead us to recognize problems that we had not anticipated. This is basically analogous to the role that modal logic has played in contemporary metaphysics. (shrink)

The following analysis shows how developments in epistemic logic can play a nontrivial role in cognitive neuroscience. We argue that the striking correspondence between two modes of identification, as distinguished in the epistemic context, and two cognitive systems distinguished by neuroscientific investigation of the visual system (the "where" and "what" systems) is not coincidental, and that it can play a clarificatory role at the most fundamental levels of neuroscientific theory.

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the (...) Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

Epistemic modal predicate logic raises conceptual problems not faced in the case of alethic modal predicate logic: Frege’s “Hesperus-Phosphorus” problem—how to make sense of ascribing to agents ignorance of necessarily true identity statements—and the related “Hintikka-Kripke” problem—how to set up a logical system combining epistemic and alethic modalities, as well as others problems, such as Quine’s “Double Vision” problem and problems of self-knowledge. In this paper, we lay out a philosophical approach to epistemic predicate logic, implemented formally in Melvin Fitting’s (...) First-Order Intensional Logic, that we argue solves these and other conceptual problems. Topics covered include: Quine on the “collapse” of modal distinctions; the rigidity of names; belief reports and unarticulated constituents; epistemic roles; counterfactual attitudes; representational vs. interpretational semantics; ignorance of co-reference vs. ignorance of identity; two-dimensional epistemic models; quantification into epistemic contexts; and an approach to multi-agent epistemic logic based on centered worlds and hybrid logic. (shrink)

van Bentham et al. (Merging frameworks for interaction: DEL and ETL, 2007) provides a framework for generating the models of Epistemic Temporal Logic ( ETL : Fagin et al., Reasoning about knowledge, 1995; Parikh and Ramanujam, Journal of Logic, Language, and Information, 2003) from the models of Dynamic Epistemic Logic ( DEL : Baltag et al., in: Gilboa (ed.) Tark 1998, 1998; Gerbrandy, Bisimulations on Planet Kripke, 1999). We consider the logic TDEL on the merged semantic framework, and its extension (...) with the labeled past-operator “ P ϵ ” (“The event ϵ has happened before which. . .”). To axiomatize the extension, we introduce a method for transforming a given model into a normal form in a suitable sense. These logics suggest further applications of DEL in the theory of agency, the theory of learning, etc. (shrink)

Various theories try to give accounts of how measures of this confidence do or ought to behave, both as far as the internal mental consistency of the agent as ...

Kaplan says that monsters violate Principle 2 of his theory. Principle 2 is that indexicals, pure and demonstrative alike, are directly referential. In providing this explanation of there being no monsters, Kaplan feels his theory has an advantage over double-indexing theories like Kamp’s or Segerberg’s (or Stalnaker’s), which either embrace monsters or avoid them only by ad hoc stipulation, in the sharp conceptual distinction it draws between circumstances of evaluation and contexts of utterance. We shall argue that Kaplan’s prohibition is (...) also essentially stipulative, and that it is too general. The main difference between ourselves and Kaplan is that the basic carriers of a truth-value is a sentence-in-a-context; our account is utterance-based. (shrink)

Real-world agents do not know all consequences of what they know. But we are reluctant to say that a rational agent can fail to know some trivial consequence of what she knows. Since every consequence of what she knows can be reached via chains of trivial consequences of what she knows, we have a paradox. I argue that the problem cannot be dismissed easily, as some have attempted to do. Rather, a solution must give adequate weight to the normative requirements (...) on rational agents’ epistemic states, without treating those agents as mathematically ideal reasoners. I’ll argue that agents can fail to know trivial consequences of what they know, but never determinately. Such cases are epistemic oversights on behalf of the agent in question, and the facts about epistemic oversights are always indeterminate facts. As a result, we are never in a position to assert that such-and-such constitutes an epistemic oversight for agent i (for we may rationally assert only determinate truths). I then develop formal epistemic models according to which epistemic accessibility relations are vague. Given these models, we can show that epistemic oversights always concern indeterminate cases of knowledge. (shrink)

For deductive reasoning to be justified, it must be guaranteed to preserve truth from premises to conclusion; and for it to be useful to us, it must be capable of informing us of something. How can we capture this notion of information content, whilst respecting the fact that the content of the premises, if true, already secures the truth of the conclusion? This is the problem I address here. I begin by considering and rejecting several accounts of informational content. I (...) then develop an account on which informational contents are indeterminate in their membership. This allows there to be cases in which it is indeterminate whether a given deduction is informative. Nevertheless, on the picture I present, there are determinate cases of informative (and determinate cases of uninformative) inferences. I argue that the model I offer is the best way for an account of content to respect the meaning of the logical constants and the inference rules associated with them without collapsing into a classical picture of content, unable to account for informative deductive inferences. (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)

I discuss three ways of responding to the logical omniscience problems faced by traditional ‘possible worlds’ epistemic logics. Two of these responses were put forward by Hintikka and the third by Cresswell; all three have been influential in the literature on epistemic logic. I show that both of Hintikka's responses fail and present some problems for Cresswell’s. Although Cresswell's approach can be amended to avoid certain unpalatable consequences, the resulting formal framework collapses to a sentential model of knowledge, which defenders (...) of the ‘possible worlds’ approach are frequently critical of. (shrink)

Information is often modelled as a set of relevant possibilities, treated as logically possible worlds. However, this has the unintuitive consequence that the logical consequences of an agent's information cannot be informative for that agent. There are many scenarios in which such consequences are clearly informative for the agent in question. Attempts to weaken the logic underlying each possible world are misguided. Instead, I provide a genuinely psychological notion of epistemic possibility and show how it can be captured in a (...) formal model, which I call a fan. I then show how to use fans to build formal models of being informed, as well as knowledge, belief and information update. (shrink)

Syntactic logics do not suffer from the problems of logical omniscience but are often thought to lack interesting properties relating to epistemic notions. By focusing on the case of rule-based agents, I develop a framework for modelling resource-bounded agents and show that the resulting models have a number of interesting properties.

One of the fundamental properties inclassical equational reasoning isLeibniz's principle of substitution. Unfortunately, this propertydoes not hold instandard epistemic logic. Furthermore,Herbrand's lifting theorem which isessential to thecompleteness ofresolution andParamodulation in theclassical first order logic (FOL), turns out to be invalid in standard epistemic logic. In particular, unlike classical logic, there is no skolemization normal form for standard epistemic logic. To solve these problems, we introduce anintensional epistemic logic, based on avariation of Kripke's possible-worlds semantics that need not have a constant (...) domain. We show how a weaker notion of substitution through indexed terms can retain the Herbrand theorem. We prove how the logic can yield a satisfibility preserving skolemization form. In particular, we present an intensional principle for unifing indexed terms. Finally, we describe asound andcomplete inference system for a Horn subset of the logic withequality, based onepistemic SLD-resolution. (shrink)

This paper deals with the problem of verification of game-like structures by means of symbolic model checking. Alternating-time Temporal Epistemic Logic (ATEL) is used for expressing properties of multi-agent systems represented by alternating epistemic temporal systems as well as concurrent epistemic game structures. Unbounded model checking (a SAT based technique) is applied for the first time to verification of ATEL. An example is given to show an application of the technique.

In this paper I combine the dynamic epistemic logic ofGerbrandy (1999) with the probabilistic logic of Fagin and Halpern (1994). The resultis a new probabilistic dynamic epistemic logic, a logic for reasoning aboutprobability, information, and information change that takes higher orderinformation into account. Probabilistic epistemic models are defined, and away to build them for applications is given. Semantics and a proof systemis presented and a number of examples are discussed, including the MontyHall Dilemma.

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau (...) system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences. (shrink)

We show how some model-theoretical devices (local reasoning, modes of presentation, an additional accessibility relation) can be combined in first-order modal logic to formalize the consequence relation that includes de dicto and de re contradictory beliefs. Instead of special ``sense objects'', appearances of objects in an agent's belief are introduced and presented as ordered pairs consisting of an object and an individual constant. A non-classical identity relation is applied. A relation S on the set of possible worlds is introduced, which (...) models possible distortions in an agent's picture of a (real) world. The application of such models in deontic logic is illustrated by a characteristic example. , , , , , , , ,. (shrink)

We investigate the research programme of dynamic doxastic logic (DDL) and analyze its underlying methodology. The Ramsey test for conditionals is used to characterize the logical and philosophical differences between two paradigmatic systems, AGM and KGM, which we develop and compare axiomatically and semantically. The importance of Gärdenfors’s impossibility result on the Ramsey test is highlighted by a comparison with Arrow’s impossibility result on social choice. We end with an outlook on the prospects and the future of DDL.

In this paper, we first propose a simple formal language to specify types of agents in terms of necessary conditions for their announcements. Based on this language, types of agents are treated as ‘first-class citizens’ and studied extensively in various dynamic epistemic frameworks which are suitable for reasoning about knowledge and agent types via announcements and questions. To demonstrate our approach, we discuss various versions of Smullyan’s Knights and Knaves puzzles, including the Hardest Logic Puzzle Ever (HLPE) proposed by Boolos (...) (in Harv Rev Philos 6:62–65, 1996). In particular, we formalize HLPE and verify a classic solution to it. Moreover, we propose a spectrum of new puzzles based on HLPE by considering subjective (knowledge-based) agent types and relaxing the implicit epistemic assumptions in the original puzzle. The new puzzles are harder than the previously proposed ones in the literature, in the sense that they require deeper epistemic reasoning. Surprisingly, we also show that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. Our formalism paves the way for studying these new puzzles using automatic model checking techniques. (shrink)

Practical reasoners are resource-bounded—in particular they require time to derive consequences of their knowledge. Building on the Timed Reasoning Logics (TRL) framework introduced in [1], we show how to represent the time required by an agent to reach a given conclusion. TRL allows us to model the kinds of rule application and conflict resolution strategies commonly found in rule-based agents, and we show how the choice of strategy can influence the information an agent can take into account when making decisions (...) at a particular point in time. We prove general completeness and decidability results for TRL, and analyse the impact of communication in an example system consisting of two agents which use different conflict resolution strategies. (shrink)

This work explores the conceptual and empirical issues of the concept of knowledge and its relation to the pattern of our belief change, from formal and experimental perspectives. Part I gives an analysis of knowledge (called Sustainability) that is formally represented and naturalistically plausible at the same time, which is claimed to be a synthesized view of knowledge, covering not only empirical knowledge, but also knowledge of future, practical knowledge, mathematical knowledge, knowledge of general facts. Part II tries to formalize (...) the natural pattern of belief change assumed in Sustainability in terms of a specific formal theory of belief change, after carefully examining the notions of belief, belief change, and Information, from which the cognitive function F in Chapter 3 is actually constructed, which is later implemented by a computer program and its behavior against random input is demonstrated. In Part III we proceed to examine the analysis empirically. In particular, we will investigate Sustainability from the developmental point of view. We first justify experimental approach in philosophy as a legitimate method of philosophical investigation, and then developmental approach in particular. The specific proposal of experiment here is what we call the Gettier Task, an analogue of the famous false-belief task in developmental psychology, which has been much discussed in philosophy of mind. Two versions of the Gettier Task were tested on children aged from 6 to 12, and we claim that the data obtained empirically supports our analysis of knowledge, or Sustainability. (shrink)

Contribution to a symposium on Timothy Williamson's "Gettier Cases in Epistemic Logic".