Reflective Luck and Belief Ownership

A belief is reflectively lucky if it is a matter of luck that the belief is true, given what a subject is aware of on reflection alone. Various epistemologists have argued that any adequate theory of knowledge should eliminate reflective luck, but doing so has proven difficult. This article distinguishes between two kinds of reflective luck arguments in the literature: local arguments and global arguments. It argues that local arguments are best interpreted as demanding, not that one be reflectively aware of the reliability of the sources of one’s beliefs, but that one’s beliefs be attributable to one as one’s own. The article then argues that global arguments make illegitimate demands, because they require that we be ultimately answerable for our beliefs. In the end, the article argues that epistemologists should shift their focus away from reflective luck and toward the conditions under which beliefs are attributable to cognitive agents.     

Reliabilism and Brains in Vats

According to epistemic internalism, the only facts that determine the justificational status of a belief are facts about the subject’s own mental states, like beliefs and experiences. Externalists instead hold that certain external facts, such as facts about the world or the reliability of a belief-producing mechanism, affect a belief’s justificational status. Some internalists argue that considerations about evil demon victims and brains in vats provide excellent reason to reject externalism: because these subjects are placed in epistemically unfavorable settings, externalism seems unable to account for the strong intuition that these subjects’ beliefs are nonetheless justified. I think these considerations do not at all help the internalist cause. I argue that by appealing to the anti-individualistic nature of perception, it can be shown that skeptical scenarios provide no reason to prefer internalism to externalism.     

The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries

Beliefs constitute a central part of a person’s professional competencies and are crucial to the perception of situations as they influence our choice of actions. This paper focuses on epistemological beliefs about the nature of mathematics of future primary teachers from an international perspective. The data reported are part of a larger sample originating from the TEDS-M study which compares primary mathematics teacher education in 15 countries. In this paper we examine the pattern of beliefs of future teachers aiming to teach mathematics at primary level. We explore whether and to what extent beliefs concerning the nature of mathematics are influenced by cultural factors, in our case the extent to which a country’s culture can be characterized by an individualistic versus collectivistic orientation according to Hofstede’s terminology. In the first part of the paper, the literature on epistemological beliefs is reviewed and the role of culture and individualism/collectivism on the formation of beliefs concerning the nature of mathematics will be discussed. In the empirical part, means and distributions of belief ratings will be reported. Finally, multilevel analyses explore how much of the variation of belief preferences between countries can be explained by the individualistic orientation of a country.     

Convex Sets in the Plane with Three of Every Four Meeting

Dedicated to the memory of Paul Erdős Suppose we have a finite collection of closed convex sets in the plane, (which without loss of generality we can take to be polygons). Suppose further that among any four of them, some three have non-empty intersection. We show that 13 points are sufficient to meet every one of the convex sets. Received October 27, 1999/Revised April 11, 2000 RID="*" ID="*" Supported by grant OTKA-T-029074. RID="†" ID="†" Supported by NSF grant DMS-99-70071, OTKA-T-020914 and OTKA-F-22234.     

In Defense of Epistemic Circularity

In this paper I defend epistemic circularity by arguing that the “No Self-Support” principle (NSS) is false. This principle, ultimately due to Fumerton (1995), states that one cannot acquire a justified belief in the reliability of a source of belief by trusting that very source. I argue that NSS has the skeptical consequence that the trustworthiness of all of our sources ultimately depends upon the trustworthiness of certain fundamental sources – sources that we cannot justifiably believe to be reliable. This is a problem, I claim, because if the trustworthiness of all of our sources depends upon sources that we should not believe to be reliable, then a reflective individual should not trust any of his sources at all. The hidden cost of rejecting epistemic circularity is thus the unacceptable skeptical thesis that reflective individuals like you and I have no justified beliefs whatsoever.     

Interactive epistemology II: Probability

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or “player.” One is interested not only in each person's knowledge and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space Ω of states of the world, and for each player i, partitions ℐ_i of Ω and probability distributions π_i(·; ω) on Ω for each state ω of the world. The atom of ℐ_i containing a given state ω represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω; the probability distributions π_i(·; ω) represents i's beliefs at the state ω. The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent.  In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space Ω, the partitions ℐ_i, and the probability distributions π_i(·; ω) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.     

Market selection: hungry misers and bloated bankrupts

The Market Selection Hypothesis is a principle which (informally) proposes that ‘less knowledgeable’ agents are eventually eliminated from the market. This elimination may take the form of starvation (the proportion of output consumed drops to zero), or may take the form of going broke (the proportion of asset held drops to zero), and these are not the same thing. Starvation may result from several causes, diverse beliefs being only one. We firstly identify and exclude these other possible causes, and then prove that starvation is equivalent to inferior belief, under suitable technical conditions. On the other hand, going broke cannot be characterized solely in terms of beliefs, as we show. We next present a remarkable example with two agents with different beliefs, in which one agent starves yet amasses all the capital, and the other goes broke yet consumes all the output—the hungry miser and the bloated bankrupt. This example also serves to show that although an agent may starve, he may have long-term impact on the prices. This relates to the notion of price impact introduced by Kogan et al. (Market selection, working paper, 2009), which we correct in the final section, and then use to characterize situations where asymptotically equivalent pricing holds.     

The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting”.     

Generalized Goal Programming: polynomial methods and applications

 In this paper we address a general Goal Programming problem with linear objectives, convex constraints, and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets. In particular, classical Linear Goal Programming problems, as well as several models in Location and Regression Analysis are modeled within this framework. In spite of its generality, this problem can be analyzed from a geometrical and a computational viewpoint, and a unified solution methodology can be given. Indeed, a dual is derived, enabling us to describe the set of optimal solutions geometrically. Moreover, Interior-Point methods are described which yield an $\varepsilon$-optimal solution in polynomial time. Received: February 1999 / Accepted: March 2002 Published online: September 5, 2002 Key words. Goal programming – closest points – interior point methods – location – regression     

Perception and Virtue Reliabilism

In some recent work, Ernest Sosa rejects the “perceptual model” of rational intuition, according to which intuitive beliefs (e.g., that 2 + 2 = 4 2 + 2 = 4 ) are justified by standing in the appropriate relation to a nondoxastic intellectual experience (a seeming-true, or the like), in much the way that perceptual beliefs are often held to be justified by an appropriate relation to nondoxastic sense experiential states. By extending some of Sosa’s arguments and adding a few of my own, I argue that Sosa is right to reject the perceptual model of intuition, and that we should reject the “perceptual model” of perception as well. Rational intuition and perception should both receive a virtue theoretic (e.g., reliabilist) account, rather than an evidentialist one. To this end, I explicitly argue against the Grounds Principle, which holds that all justified beliefs must be based on some adequate reason, or ground.     

Approximating zero-variance importance sampling in a reliability setting

We consider a class of Markov chain models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multicomponent systems in reliability settings. We are interested in the design of efficient importance sampling (IS) schemes to estimate the reliability of such systems by simulation. For these models, there is in fact a zero-variance IS scheme that can be written exactly in terms of a value function that gives the expected cost-to-go (the exact reliability, in our case) from any state of the chain. This IS scheme is impractical to implement exactly, but it can be approximated by approximating this value function. We examine how this can be effectively used to estimate the reliability of a highly-reliable multicomponent system with Markovian behavior. In our implementation, we start with a simple crude approximation of the value function, we use it in a first-order IS scheme to obtain a better approximation at a few selected states, then we interpolate in between and use this interpolation in our final (second-order) IS scheme. In numerical illustrations, our approach outperforms the popular IS heuristics previously proposed for this class of problems. We also perform an asymptotic analysis in which the HRMS model is parameterized in a standard way by a rarity parameter ε, so that the relative error (or relative variance) of the crude Monte Carlo estimator is unbounded when ε→0. We show that with our approximation, the IS estimator has bounded relative error (BRE) under very mild conditions, and vanishing relative error (VRE), which means that the relative error converges to 0 when ε→0, under slightly stronger conditions.     

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.     

Development and Initial Validation of the Beliefs About Reformed Science Teaching and Learning (BARSTL) Questionnaire

This article describes the development and the initial validation of an instrument that can be used to assess teachers' beliefs about science teaching and learning. The instrument, which is called the Beliefs About Reformed Science Teaching and Learning (BARSTL) questionnaire, draws on the current national science education reform efforts in order to define a traditional‐reformed teaching and learning belief continuum that can be used to map teachers' beliefs. The reliability and validity of the instrument were examined using a multiple perspective approach. The psychometric properties of the BARSTL suggest that it is a valid and reliable instrument for measuring prospective elementary teachers' beliefs about science education.     

Two Characterizations of Optimality in Dynamic Programming

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations.     

Basic reductions in the description of normal subgroups

Classification of subgroups in a Chevalley group G(Φ, R) over a commutative ring R, normalized by the elementary subgroup E(Φ, R), is well known. However, for exceptional groups, in the available literature neither the parabolic reduction nor the level reduction can be found. This is due to the fact that the Abe-Suzuki-Vaserstein proof relied on localization and reduction modulo the Jacobson radical. Recently, for the groups of types E ₆, E ₇, and F ₄, the first-named author, M. Gavrilovich, and S. Nikolenko have proposed an even more straightforward geometric approach to the proof of structure theorems, similar to that used for exceptional cases. In the present paper, we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank ≥ 2. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralizers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems. Bibliography: 64 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 30–52.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

Exponential attractors for a class of reaction-diffusion problems with time delays

We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain. The reaction term depends on the population densities as well as on their past histories in a very general way. This class of systems is widely used in population dynamics modelling. Due to its generality, the longtime behavior of the solutions can display a certain complexity. Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models. More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast. This result will be achieved by means of a suitable adaptation of the l-trajectory method coming back to the seminal paper of Málek and Nečas. The first author was partially supported by the Italian PRIN 2006 research project Problemi a frontiera libera, transizioni di fase e modelli di isteresi. The second author was supported by the research project MŠM 0021620839 and by the project LC06052 (Jindřich Nečas Center for Mathematical Modeling).     

General affine adjunctions,Nullstellensätze,and dualities

We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert's Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.     

A new perspective to stress–strength models

The stress–strength models have been widely used for reliability design of systems. In these models the reliability is defined as the probability that the strength is larger than the stress. The analysis is then based on the binary reliability theory since there are two possible states for the system. In this paper, we study the stress–strength reliability in a different framework assigning more than two states to the system depending on the difference between strength and stress values. In other words, the stress–strength reliability is studied under multi-state system modeling. System state probabilities are evaluated and estimated under various assumptions on the system. The multicomponent form is also studied and some results are provided for large systems.     

The island confinement method for reducing search space in local search methods

Typically local search methods for solving constraint satisfaction problems such as GSAT, WalkSAT, DLM, and ESG treat the problem as an optimisation problem. Each constraint contributes part of a penalty function in assessing trial valuations. Local search examines the neighbours of the current valuation, using the penalty function to determine a “better” neighbour valuation to move to, until finally a solution which satisfies all the constraints is found. In this paper we investigate using some of the constraints as “hard” constraints, that are always satisfied by every trial valuation visited, rather than as part of a penalty function. In this way these constraints reduce the possible neighbours in each move and also the overall search space. The treating of some constraints as hard requires that the space of valuations that are satisfied is “connected” in order to guarantee that a solution can be found from any starting position within the region; thus the concept of islands and the name “island confinement method” arises. Treating some constraints as hard provides new difficulties for the search mechanism since the search space becomes more jagged, and there are more deep local minima. A new escape strategy is needed. To demonstrate the feasibility and generality of our approach, we show how the island confinement method can be incorporated in, and significantly improve, the search performance of two successful local search procedures, DLM and ESG, on SAT problems arising from binary CSPs. A preliminary version of this paper appeared in AAAI’2002.