A Map of Common Knowledge Logics

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole.     

Common knowledge of payoff uncertainty in games

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.’     

Understanding the Brandenburger-Keisler Paradox

Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic. Special Issue Formal Epistemology II. Edited by Branden Fitelson     

First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas

The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).     

Cut-free common knowledge

Starting off from the infinitary system for common knowledge over multi-modal epistemic logic presented in [L. Alberucci, G. Jäger, About cut elimination for logics of common knowledge, Annals of Pure and Applied Logic 133 (2005) 73–99], we apply the finite model property to “finitize” this deductive system. The result is a cut-free, sound and complete sequent calculus for common knowledge.     

Inexact Knowledge with Introspection

Standard Kripke models are inadequate to model situations of inexact knowledge with introspection, since positive and negative introspection force the relation of epistemic indiscernibility to be transitive and euclidean. Correlatively, Williamson’s margin for error semantics for inexact knowledge invalidates axioms 4 and 5. We present a new semantics for modal logic which is shown to be complete for K45, without constraining the accessibility relation to be transitive or euclidean. The semantics corresponds to a system of modular knowledge, in which iterated modalities and simple modalities are not on a par. We show how the semantics helps to solve Williamson’s luminosity paradox, and argue that it corresponds to an integrated model of perceptual and introspective knowledge that is psychologically more plausible than the one defended by Williamson. We formulate a generalized version of the semantics, called token semantics, in which modalities are iteration-sensitive up to degree n and insensitive beyond n. The multi-agent version of the semantics yields a resource-sensitive logic with implications for the representation of common knowledge in situations of bounded rationality.     

Substructural Logics in Natural Deduction

Extensions of Natural Deduction to Substructural Logics of IntuitionisticLogic are shown: Fragments of Intuitionistic Linear, Relevantand BCK Logic. Rules for implication, conjunction, disjunctionand falsum are defined, where conjunction and disjunction respectcontexts of assumptions. So, conjunction and disjunction areadditive in the terminology of linear logic. Explicit contractionand weakening rules are given. It is shown that conversionsand permutations can be adapted to all these rules, and thatweak normalisation and subformula property holds. The resultsgeneralise to quantification.     

Strong Completeness Theorems for Weak Logics of Common Belief

We show that several logics of common belief and common knowledge are not only complete, but also strongly complete, hence compact. These logics involve a weakened monotonicity axiom, and no other restriction on individual belief. The semantics is of the ordinary fixed-point type.     

Valuing Knowledge: A Deontological Approach

The fact that we ought to prefer what is comparatively more likely to be good, I argue, does, contrary to consequentialism, not rest on any evaluative facts. It is, in this sense, a deontological requirement. As such it is the basis of our valuing those things which are in accordance with it. We value acting (and believing) well, i.e. we value acting (and believing) as we ought to act (and to believe). In this way, despite the fact that our interest in justification depends on our interest in truth, we value believing with justification on non-instrumental grounds. A deontological understanding of justification, thus, solves the Value of Knowledge Problem.

Common Knowledge and Reductionism about Shared Agency

Most reductionist accounts of intentional joint action include a condition that it must be common knowledge between participants that they have certain intentions and beliefs that cause and coordinate the joint action. However, this condition has typically simply been taken for granted rather than argued for. The condition is not necessary for ensuring that participants are jointly responsible for the action in which each participates, nor for ensuring that each treats the others as partners rather than as social tools. It is thus something of a mystery why the condition is so widely accepted. By rejecting three arguments that could potentially support it, I argue that reductionists should get rid of the condition. I show that two of the arguments fail. While the third argument is intuitively compelling, it builds on key premises that are unavailable to the reductionist.     

Knowing full well: the normativity of beliefs as performances

Belief is considered a kind of performance, which attains one level of success if it is true (or accurate), a second level if competent (or adroit), and a third if true because competent (or apt). Knowledge on one level (the animal level) is apt belief. The epistemic normativity constitutive of such knowledge is thus a kind of performance normativity. A problem is posed for this account by the fact that suspension of belief seems to fall under the same sort of epistemic normativity as does belief itself, yet to suspend is of course precisely not to perform, certainly not with the aim of truth. The paper takes up this problem, and proposes a solution that distinguishes levels of performance norrmativity, including a first order where execution competence is in play, and a second order where the performer must assess the risks attendant on issuing a first-order performance. This imports a level of reflective knowledge that ascends above the animal level.

Shortcuts and Dynamic Marking in the Tableau Method for Adaptive Logics

Adaptive logics typically pertain to reasoning procedures for which there is no positive test. In [7], we presented a tableau method for two inconsistency-adaptive logics. In the present paper, we describe these methods and present several ways to increase their efficiency. This culminates in a dynamic marking procedure that indicates which branches have to be extended first, and thus guides one towards a decision — the conclusion follows or does not follow — in a very economical way.     

A Dynamic-Logical Perspective on Quantum Behavior

In this paper we show how recent concepts from Dynamic Logic, and in particular from Dynamic Epistemic logic, can be used to model and interpret quantum behavior. Our main thesis is that all the non-classical properties of quantum systems are explainable in terms of the non-classical flow of quantum information. We give a logical analysis of quantum measurements (formalized using modal operators) as triggers for quantum information flow, and we compare them with other logical operators previously used to model various forms of classical information flow: the “test” operator from Dynamic Logic, the “announcement” operator from Dynamic Epistemic Logic and the “revision” operator from Belief Revision theory. The main points stressed in our investigation are the following: (1) The perspective and the techniques of “logical dynamics” are useful for understanding quantum information flow. (2) Quantum mechanics does not require any modification of the classical laws of “static” propositional logic, but only a non-classical dynamics of information. (3) The main such non-classical feature is that, in a quantum world, all information-gathering actions have some ontic side-effects. (4) This ontic impact can affect in its turn the flow of information, leading to non-classical epistemic side-effects (e.g. a type of non-monotonicity) and to states of “objectively imperfect information”. (5) Moreover, the ontic impact is non-local: an information-gathering action on one part of a quantum system can have ontic side-effects on other, far-away parts of the system.     

知识表征的新观点--知觉符号理论

在认知科学中一个经典的争论是知识在人的头脑中是如何表征的。其中占主流的理论一直是命题符号理论,而最近Barsalou又提出了一种新的观点——知觉符号理论。本文对这一新理论做了介绍,阐述了它的主要特征,比较了它与命题符号理论的差异,并提供了现今对这一理论的实验证实。希望借此能启发国内的研究者也来关注和研究这一理论。     

Logical Knowledge

It seems obvious that our beliefs are logically imperfect in two ways: they are neither deductively closed nor logically consistent. But this common-sense truism has been judged erroneous by some philosophers in the light of various arguments. In defence of common sense I consider and rebut interpretative arguments for logical perfection and show that the assumption espoused by common sense is theoretically superior, and capable - unlike its rival - of accounting for the informativeness of mathematics. Finally, I suggest that common sense opens the way to genuine disputes about the correct logic.     

Determining the environment: a modal logic for closed interaction

The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to reason about games and their regulation.     

婴儿关于物体支撑的直觉性物理知识

支撑问题是儿童朴素物理理论知识结构中的重要部分, 是指关于一个物体放在固体平面上是否稳定以及如何才能稳定的问题。支撑问题目前有两种主要实验方法。此领域中的主要实验研究使用这两种方法, 探索了婴儿获得与支撑相关的直觉性物理知识的发展过程及“先建立核心性的初始概念, 再以此为基础增加各类变量”的规律。目前研究中存在“范式单一、缺乏各类研究相互印证”的问题, 有待未来的研究进一步发展。