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Summary Epistemic logics are logics that allow one to reason about knowledge in some way. Doxastic logics are similar, but allow one to reason about belief rather than knowledge. The languages of these logics contain knowledge or belief operators or predicates, governed by appropriate axioms or rules. (Just what axioms and rules are appropriate is a controversial matter, however.) Many epistemic and doxastic logics are modal logics, whose languages contains one or more knowledge or belief operators, and whose semantics is given in terms of relational Kripke models. In such models, the points or states can be thought of as epistemically (or doxastically) possible worlds, related to one another by epistemic (or doxastic) accessibility relations. This modal approach to epistemic and doxastic logic has been widely adopted in formal logic, philosophy, computer science, artificial intelligence, economics and game theory. The category doxastic logic also includes work on belief revision and belief update, which addresses the question: how should an agent revise or update its beliefs, on receiving conflicting information?
Key works Modern epistemic logic began with Hintikka 1962, who developed Kripke-style semantics for epistemic notions and discussed appropriate axioms for knowledge and belief. Hintikka proposes a solution to the logical omniscience problem, whereby agents are treated as automatically knowing all consequences of what they know, in Hintikka 1975. Hintikka's approach is developed and applied to problems in computer science in Fagin et al 1995. The leading theory of belief revision, the ‘AGM’ theory, was first presented in Alchourrón et al 1985
Introductions Hintikka 1962 is a great introduction to epistemic and doxastic logics; Hendricks 2008 briefly surveys the area. Huber 2013 introduces and discusses AGM theories of belief revision.
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  1. Carlos E. Alchourrón (1972). The Intuitive Background of Normative Legal Discourse and its Formalization. Journal of Philosophical Logic 1 (3/4):447 - 463.
  2. Natasha Alechina, Brian Logan, Hoang Nga Nguyen & Abdur Rakib (2009). Verifying Time, Memory and Communication Bounds in Systems of Reasoning Agents. Synthese 169 (2):385 - 403.
    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 (...)
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  3. Patrick Allo (2013). The Many Faces of Closure and Introspection. Journal of Philosophical Logic 42 (1):91-124.
    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.
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  4. Patrick Allo (2011). The Logic of 'Being Informed' Revisited and Revised. Philosophical Studies 153 (3):417-434.
    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 (...)
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  5. Patrick Allo (2009). Reasoning About Data and Information. Synthese 167 (2):231-249.
    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 (...)
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  6. Patrick Allo (2008). Formalising the 'No Information Without Data-Representation' Principle. In P. Brey, A. Briggle & K. Waelbers (eds.), Current Issues in Computing and Philosophy. IOS Press.
    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 (...)
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  7. Patrick Allo (2008). Vincent Hendricks, Mainstream and Formal Epistemology. [REVIEW] Erkenntnis 69 (3):427-432.
    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 (...)
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  8. Giambattista Amati, Luigia Carlucci-Aiello & Fiora Pirri (1997). Intuitionistic Autoepistemic Logic. Studia Logica 59 (1):103-120.
    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.
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  9. C. Anthony Anderson (1993). Toward a Logic of A Priori Knowledge. Philosophical Topics 21 (2):1-20.
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  10. Lennart Aqvist (1971). Revised Foundations for Imperative-Epistemic and Interrogative Logic. Theoria 37 (1):33-73.
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  11. Dmitri A. Archangelsky & Mikhail A. Taitslin (1997). A Logic for Information Systems. Studia Logica 58 (1):3-16.
    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 (...)
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  12. Horacio Arló-Costa & Cristina Bicchieri (2007). Knowing and Supposing in Games of Perfect Information. Studia Logica 86 (3):353 - 373.
    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 (...)
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  13. Sergei Artemov (forthcoming). Justification Logic. Stanford Encyclopedia of Philosophy.
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  14. Sergei Artemov (2008). The Logic of Justification. Review of Symbolic Logic 1 (4):477-513.
    We describe a general logical framework, Justification Logic, for reasoning about epistemic justification. Justification Logic is based on classical propositional logic augmented by justification assertions t: F that read t is a justification for F. Justification Logic absorbs basic principles originating from both mainstream epistemology and the mathematical theory of proofs. It contributes to the studies of the well-known Justified True Belief vs. Knowledge problem. We state a general Correspondence Theorem showing that behind each epistemic modal logic, there is a (...)
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  15. Sergei N. Artemov (2001). Explicit Provability and Constructive Semantics. Bulletin of Symbolic Logic 7 (1):1-36.
    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 (...)
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  16. Sergei N. Artemov & Lev D. Beklemishev (1993). On Propositional Quantifiers in Provability Logic. Notre Dame Journal of Formal Logic 34 (3):401-419.
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  17. Sergei Artemov & Giorgie Dzhaparidze (1990). Finite Kripke Models and Predicate Logics of Provability. Journal of Symbolic Logic 55 (3):1090-1098.
    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 (...)
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  18. Sergei Artemov & Rosalie Iemhoff (2007). The Basic Intuitionistic Logic of Proofs. Journal of Symbolic Logic 72 (2):439 - 451.
    The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.
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  19. Sergei Artëmov & Franco Montagna (1994). On First-Order Theories with Provability Operator. Journal of Symbolic Logic 59 (4):1139-1153.
    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.
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  20. Sergei Artemov & Tudor Protopopescu (2013). Discovering Knowability: A Semantic Analysis. Synthese 190 (16):3349-3376.
    In this paper, we provide a semantic analysis of the well-known knowability paradox stemming from the Church–Fitch observation that the meaningful knowability principle /all truths are knowable/, when expressed as a bi-modal principle F --> K♢F, yields an unacceptable omniscience property /all truths are known/. We offer an alternative semantic proof of this fact independent of the Church–Fitch argument. This shows that the knowability paradox is not intrinsically related to the Church–Fitch proof, nor to the Moore sentence upon which it (...)
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  21. Guillaume Aucher (2010). An Internal Version of Epistemic Logic. Studia Logica 94 (1):1 - 22.
    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 (...)
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  22. Philippe Balbiani, Alexandru Baltag, Hans van Ditmarsch, Andreas Herzig, Tomohiro Hoshi & Tiago de Lima (2008). Knowable' as 'Known After an Announcement. Review of Symbolic Logic 1 (3):305-334.
    Public announcement logic is an extension of multiagent epistemic logic with dynamic operators to model the informational consequences of announcements to the entire group of agents. We propose an extension of public announcement logic with a dynamic modal operator that expresses what is true after any announcement: after which , does it hold that Kφ? We give various semantic results and show completeness for a Hilbert-style axiomatization of this logic. There is a natural generalization to a logic for arbitrary events.
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  23. Diderik Batens (2004). The Need for Adaptative Logics in Epistemology. In Shadid Rahman, John Symons, Dov Gabbay & Jean Bendegem (eds.), Logic, Epistemology, and the Unity of Science. Kluwer. 459-485.
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  24. H. Battaly (2008). Review: Nicholas Rescher: Cognitive Harmony: The Role of Systematic Harmony in the Constitution of Knowledge; Epistemic Logic: A Survey of the Logic of Knowledge; and Realism and Pragmatic Epistemology. [REVIEW] Mind 117 (465):205-210.
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  25. Johan Van Benthem (2011). Logic in a Social Setting. Episteme 8 (3):227-247.
    Taking Backward Induction as its running example, this paper explores avenues for a logic of information-driven social action. We use recent results on limit phenomena in knowledge updating and belief revision, procedural rationality, and a ‘Theory of Play’ analyzing how games are played by different agents.
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  26. Francesco Berto (2014). On Conceiving the Inconsistent. Proceedings of the Aristotelian Society 114 (1pt1):103-121.
    I present an approach to our conceiving absolute impossibilities—things which obtain at no possible world—in terms of ceteris paribus intentional operators: variably restricted quantifiers on possible and impossible worlds based on world similarity. The explicit content of a representation plays a role similar in some respects to the one of a ceteris paribus conditional antecedent. I discuss how such operators invalidate logical closure for conceivability, and how similarity works when impossible worlds are around. Unlike what happens with ceteris paribus counterfactual (...)
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  27. Francesco Berto (2012). Non-Normal Worlds and Representation. In Michal Peliš & Vít Punčochář (eds.), The Logica Yearbook. College Publications.
    World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic (...)
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  28. Floris Bex, Henry Prakken, Chris Reed & Douglas Walton (2003). Towards a Formal Account of Reasoning About Evidence: Argumentation Schemes and Generalisations. [REVIEW] Artificial Intelligence and Law 11 (2-3):125-165.
    This paper studies the modelling of legal reasoning about evidence within general theories of defeasible reasoning and argumentation. In particular, Wigmore's method for charting evidence and its use by modern legal evidence scholars is studied in order to give a formal underpinning in terms of logics for defeasible argumentation. Two notions turn out to be crucial, viz. argumentation schemes and empirical generalisations.
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  29. Jean-Yves Béziau & Alexandre Costa-Leite (eds.) (2007). Perspectives on Universal Logic.
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  30. Jens Christian Bjerring (2012). Problems in Epistemic Space. Journal of Philosophical Logic (1):1-18.
    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 (...)
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  31. Ivan Boh (2000). Four Phases of Medieval Epistemic Logic. Theoria 66 (2):129-144.
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  32. Ivan Boh (1993). Epistemic Logic in the Later Middle Ages. Routledge.
    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 (...)
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  33. Slavko Brkic (2004). Presuppositions, Logic, and Dynamics of Belief. Prolegomena 3 (2):151-177.
    In researching presuppositions dealing with logic and dynamic of belief we distinguish two related parts. The first part refers to presuppositions and logic, which is not necessarily involved with intentional operators. We are primarily concerned with classical, free and presuppositonal logic. Here, we practice a well known Strawson’s approach to the problem of presupposition in relation to classical logic. Further on in this work, free logic is used, especially Van Fraassen’s research of the role of presupposition in supervaluations logical systems. (...)
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  34. Berit Brogaard & Joe Salerno, Fitch's Paradox of Knowability. The Stanford Encyclopedia of Philosophy.
    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 (...)
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  35. Berit Brogaard & Joe Salerno (2006). Knowability and a Modal Closure Principle. American Philosophical Quarterly 43 (3):261-270.
    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 (...)
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  36. Michael E. Byrd (1973). Knowledge and True Belief in Hintikka's Epistemic Logic. Journal of Philosophical Logic 2 (2):181 - 192.
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  37. Lauri Carlson (1988). Quantified Hintikka-Style Epistemic Logic. Synthese 74 (2):223 - 262.
    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 (...)
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  38. L. S. Carrier (1977). The Irreducibility of Knowledge. Logique Et Analyse 77 (Sommaire):167-176.
    In this article it is argued that it is impossible to give a reductive analysis of knowledge, given that knowledge is an "epistemic" concept with these marks: (1) like necessity, it is only partially truth-functional; and, (2) unlike necessity, it includes an "intentional" component (belief) which is completely non-truth-functional. a reductive analysis would have to contain at least one extensional component, one intentional component, and none that is itself epistemic. but any plausible analysis then turns out either to be non-reductive, (...)
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  39. John M. Collins (2002). Ryan on Epistemic Closure Principles. Philosophia 29 (1-4):371-376.
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  40. C. B. Cross (2012). The Paradox of the Knower Without Epistemic Closure -- Corrected. Mind 121 (482):457-466.
    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 (...)
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  41. Charles B. Cross (2004). More on the Paradox of the Knower Without Epistemic Closure. Mind 113 (449):109-114.
    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 (...)
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  42. Charles B. Cross (2001). The Paradox of the Knower Without Epistemic Closure. Mind 110 (438):319-333.
    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.
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  43. Boudewijn de Bruin (2009). The Logic of Valuing. In Thomas Boylan & Ruvin Gekker (eds.), Economics, Rational Choice and Normative Philosophy. Routledge.
    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 (...)
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  44. Boudewijn de Bruin (2008). Epistemic Logic and Epistemology. In Vincent F. Hendricks & Duncan Pritchard (eds.), New Waves in Epistemology. Palgrave Macmillan.
    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 (...)
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  45. Boudewijn de Bruin (2008). Common Knowledge of Payoff Uncertainty in Games. Synthese 163 (1):79-97.
    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.’.
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  46. Boudewijn de Bruin (2008). Common Knowledge of Rationality in Extensive Games. Notre Dame Journal of Formal Logic 49 (3):261-280.
    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 (...)
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  47. Cédric Dégremont & Nina Gierasimczuk (2011). Finite Identification From the Viewpoint of Epistemic Update. Information And Computation 209 (3):383-396.
    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 (...)
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  48. Cédric Dégremont, Lena Kurzen & Jakub Szymanik (2011). On theTractability of Comparing Informational Structures. In J. van Eijck & R. Verbrugge (eds.), Proceedings of the Workshop 'Reasoning about other minds: Logical and cognitive perspectives.
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  49. Stéphane Demri (1999). A Logic with Relative Knowledge Operators. Journal of Logic, Language and Information 8 (2):167-185.
    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.
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  50. Max Deutscher (1969). Hintikka's Conception of Epistemic Logic. Australasian Journal of Philosophy 47 (2):205 – 208.
    "hintikka's conception of epistemic logic" is a critical comment on hintikka's defence of his philosophical method ("epistemic logic and the methods of philosophical analysis", "a.J.P." no.1, 1968). There is a discussion of the symbolization and analysis of "a knows that p", "a knows that he knows that p", And the notions of virtual equivalence and virtual implication. The conclusion drawn is that whereas hintikka thought his critics misunderstood his method, In fact they were attacking his employment of it.
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