A characterization of ‘min’ as a procedure for exploiting valued preference relations and related results

This paper is concerned with procedures which transform valued preference relations on a set of alternatives into crisp relations. We present a simple characterization of a procedure that ranks alternatives in decreasing order of their minimal performance. This is done by means of three axioms that are shown to be independent. Among other results, we characterize in a very similar manner a procedure called ‘leximin’ and investigate two families of procedures whose intersection is the ‘min’ procedure.     

A method of weighing qualitative preference axioms

A method is developed for determining the absolute and relative strengths of qualitative preference axioms in normative Bayesian decision theory. These strengths are calculated for the three most common qualitative axioms; transitivity, the sure-thing principle, and dominance. The relative strength of the latter two axioms with respect to transitivity is calculated for special cases, and a bound is derived which is applicable to a larger class of decision problems. Possible implications of this theoretical work for decision heuristics are discussed.     

社会群体中的偏好逻辑

在许多多主体偏好逻辑系统中,主体之间是没有联系的,因而无法描述主体间的偏好互动。借鉴＂The Logic in the Community＂一文中称为＂群体压力＂的例子对偏好的影响,本文在＂The Logic in the Community＂所提出的系统构架上,加入偏好算子和两个动态算子,通过构造归约公理说明这些算子可以被无偿添加,并借助这些规约公理证明了系统的完全性。     

Internet minimal group paradigm.

Over many years, social psychologists have sought to understand what causes individuals to form themselves into groups. Initially, it was believed that groups were formed when people bonded around a common goal. Later, it was found that, when individuals were divided into groups on a random basis, this in itself was sufficient for them to feel part of a group and show a preference for their own group over others. Since the environment in cyberspace is different from that of the offline world, for example, there is no physical proximity between participants; it may be assumed that it would be difficult to achieve feelings of affiliation among potential or actual group members. This pioneer study seeks to discover which components are requisite to the creation of a group identity among individuals surfing the Internet. For this experiment, 24 people were divided into two Internet chat groups according to their intuitive preference in a decision-making task. It was found that group members perceived their own group performance as superior on a cognitive task as compared with that of the other group. These results demonstrate that for surfers, the Internet experience is very real and even a trivial allocation of people to a group is likely to create a situation of ingroup favoritism.     

Classes of models for selected axiomatic theories of choice

Adding a reversibility axiom to the other axioms of Luce's (1959) probabilistic ranking theory results in an impossibility theorem—that all alternatives in an alternative set are equally likely to be chosen (i.e., that preferences are random). This impossibility theorem is generally avoided by removing the reversibility axiom. Using simple algebraic methods such a modified theory is shown to contain a theorem similiar to the impossibility result. These results are discussed within the framework of mathematical model theory (model theory deals with the relations between sets of sentences (theories) and the structures which satisfy these sentences (models)) to illustrate the applicability of model theory as an analytic tool in theory development.     

Decision theory with complex uncertainties

A case is made for supposing that the total probability accounted for in a decision analysis is less than unity. This is done by constructing a measure on the set of all codes for computable functions in such a way that the measure of every effectively accountable subset is bounded by a number <1. The consistency of these measures with the Savage axioms for rational preference is established. Implications for applied decision theory are outlined.     

My beliefs about your beliefs: a case study in theory of mind and epistemic logic

We model three examples of beliefs that agents may have about other agents’ beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs (‘no preferences’), that agents believe that the beliefs of other agents are in part as their own (‘my preferences’), and the special case that agents believe that the beliefs of other agents are exactly as their own (‘preference refinement’). This multi-agent belief interaction is frame characterizable. We provide examples for introspective agents. We investigate which of these forms of belief interaction are preserved under three common forms of belief revision.     

A Representation Theorem for a Decision Theory With Conditionals

This paper investigates the role of conditionals in hypothetical reasoning and rational decision making. Its main result is a proof of a representation theorem for preferences defined on sets of sentences (and, in particular, conditional sentences), where an agent’s preference for one sentence over another is understood to be a preference for receiving the news conveyed by the former. The theorem shows that a rational preference ordering of conditional sentences determines probability and desirability representations of the agent’s degrees of belief and desire that satisfy, in the case of non-conditional sentences, the axioms of Jeffrey’s decision theory and, in the case of conditional sentences, Adams’ expression for the probabilities of conditionals. Furthermore, the probability representation is shown to be unique and the desirability representation unique up to positive linear transformation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sequent-systems and groupoid models. II

The purpose of this paper is to connect the proof theory and the model theory of a family of prepositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.Below is the sequel to the first part of the paper, which appeared in the previous issue of this journal (vol. 47 (1988), pp. 353–386). The first part contained sections on sequent-systems and Hilbert-formulations, and here is the third section on groupoid models. This second part is meant to be read in conjunction with the first part.     

Linear Measurement Models-Axiomatizations and Axiomatizability

This paper examines, in the scope of representational measurement theory, different axiomatizations and axiomatizability of linear and bilinear representations of ordinal data contexts in real vector spaces. The representation theorems proved in this paper are modifications and generalizations of Scott's characterization of finite linear measurement models. The advantage of these representation theorems is that they use only finitely many axioms, the number of which depends on the size of the given ordinal data context. Concerning the axiomatizability, it is proved by model-theoretic methods that finite linear measurement models cannot be axiomatized by a finite set of first order axioms. Copyright 2000 Academic Press.     

Twenty-five basic theorems in situation and world theory

Conclusion The foregoing set of theorems forms an effective foundation for the theory of situations and worlds. All twenty-five theorems seem to be basic, reasonable principles that structure the domains of properties, relations, states of affairs, situations, and worlds in true and philosophically interesting ways. They resolve 15 of the 19 choice points defined in Barwise (1989) (see Notes 22, 27, 31, 32, 35, 36, 39, 43, and 45). Moreover, important axioms and principles stipulated by situation theorists are derived (see Notes 33, 37, and 38). This is convincing evidence that the foregoing constitutes a theory of situations. Note that worlds are just a special kind of situation, and that the basic theorems of world theory, which were derived in previous work, can still be derived in this situation-theoretic setting. So there seems to be no fundamental incompatibility between situations and worlds — they may peacably coexist in the foundations of metaphysics. The theory may therefore reconcile two research programs that appeared to be heading off in different directions. And we must remind the reader that the general metaphysical principles underlying our theory were not designed with the application to situation theory in mind. This suggests that the general theory and the underlying distinction have explanatory power, for they seem to relate and systematize apparently unrelated phenomena.This research was conducted at the Center for the Study of Language and Information (CSLI). I would like to thank John Perry for generously supporting my research both at CSLI and in the Philosophy Department at Stanford. I would also like to thank Bernard Linsky, Chris Menzel, Harry Deutsch and Tony Anderson for many worthwhile and interesting suggestions for improving the paper. An earlier version of the paper, more narrowly focused on situation theory, has appeared in Zalta (1991).     

A COMMENT ON RCC: FROM RCC TO RCC<Superscript>++</Superscript>

The Region Connection Calculus (RCC theory) is a well-known spatial representation of topological relations between regions. It claims that the connection relation is primitive in the spatial domain. We argue that the connection relation is indeed primitive to the spatial relations, although in RCC theory there is no room for distance relations. We first analyze some aspects of the RCC theory, e.g. the two axioms in the RCC theory are not strong enough to govern the connection relation, regions in the RCC theory cannot be points, the uniqueness of the operation in the theory is not guaranteed, etc. To solve some of the problems, we propose an extension to the RCC theory by introducing the notion of region category and adding a new axiom which governs the characteristic property of the connection relation. The extended theory is named as RCC⁺⁺. We support the claim that the connection relation is primitive to spatial domain by showing how distance relations, size relations are developed in RCC⁺⁺. At last we revisit a sub-family of un-intended models in RCC theory, argue that RCC⁺⁺ is more suitable than RCC with regards to its original intended model, and discuss the representation limitation of the RCC, as well as RCC⁺⁺.     

Color measurement and color theory: II. Opponent-colors theory

Quantitative opponent-colors theory is based on cancellation of redness by admixture of a standard green, of greenness by admixture of a standard red, of yellowness by blue, and of blueness by yellow. The fundamental data are therefore the equilibrium colors: the set A₁ of lights that are in red/green equilibrium and the set A₂ of lights that are in yellow/blue equilibrium. The result that a cancellation function is linearly related to the color-matching functions can be proved from more basic axioms, particularly, the closure of the set of equilibrium colors under linear operations. Measurement analysis treats this as a representation theorem, in which the closure properties are axioms and in which the colorimetric homomorphism has the cancellation functions as two of its coordinates.Consideration of equivalence relations based on opponent cancellation leads to a further step: analysis of equivalence relations based on direct matching of hue attributes. For additive whiteness matching, this yields a simple extension of the representation theorem, in which the third coordinate is luminance. For other attributes, precise representation theorems must await a better qualitative characterization of various nonlinear phenomena, especially the veiling of one hue attribute by another and the various hue shifts.     

A FORMAL CONSTRUCTION OF THE SPACETIME MANIFOLD

The spacetime manifold, the stage on which physics is played, is constructed ab initio in a formal program that resembles the logicist reconstruction of mathematics. Zermelo’s set theory extended by urelemente serves as a framework, to which physically interpretable proper axioms are added. From this basis, a topology and subsequently a Hausdorff manifold are readily constructed which bear the properties of the known spacetime manifold. The present approach takes worldlines rather than spacetime points to be primitive, having them represented by urelemente. Thereby it is demonstrated that an important part of physics is formally reducible to set theory.     

Choice probabilities and choice functions

Choice probabilities are basic to much of the theory of individual choice behavior in mathematical psychology. On the other hand, consumer economics has relied primarily on preference relations and choice functions for its theories of individual choice. Although there are sizable literatures on the connections between choice probabilities and preference relations, and between preference relations and choice functions, little has been done—apart from their common ties to preference relations—to connect choice probabilities and choice functions. The latter connection is studied in this paper. A family of choice functions that depends on a threshold parameter is defined from a choice probability function. It is then shown what must be true of the choice probability function so that the choice functions satisfy three traditional rationality conditions. Conversely, it is shown what must be true of the choice functions so that the choice probability function satisfies a version of Luce's axiom for individual choice probabilities.     

心理学对经济学的涉入——从理性自利到社会偏好的决策框架

理性认知能力与社会偏好存在紧密的关联。文章简要回顾了经济决策理论从理性模型到有限理性模型和社会偏好模型的发展进程,论述了人们理性的局限性及其根源,并进一步探讨理性认知能力与社会偏好的关系。对人类以及灵长目动物的研究显示,有限理性可能是由根源于演化的适应性机制所导致。人类不公平厌恶的起源、个体公平能力的发展规律和表征公平的大脑结构上的证据表明,理性认知能力能让人更好地抑制自私性,实现更高层次的公平。     

Implications of the Dutch Book: Following Ramsey��s axioms

The Dutch Book Argument shows that an agent will lose surely in a gamble (a Dutch Book is made) if his degrees of belief do not satisfy the laws of the probability. Yet a question arises here: What does the Dutch Book imply? This paper firstly argues that there exists a utility function following Ramsey’s axioms. And then, it explicates the properties of the utility function and degree of belief respectively. The properties show that coherence in partial beliefs for Subjective Bayesianism means that the degree of belief, representing a belief ordering, satisfies the laws of probability, and that coherence in preferences means that the preferences are represented by expected utilities. A coherent belief ordering and a utility scale induce a coherent preference ordering; a coherent preference ordering induces a coherent belief ordering which can be uniquely represented by a degree-of-belief function. The preferences (values) and beliefs are both incoherent or disordered if a Dutch Book is made.     

Sequent-systems and groupoid models. I

The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related toBCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hubert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hubert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.     

What is Classical Mereology?

Classical mereology is a formal theory of the part-whole relation, essentially involving a notion of mereological fusion, or sum. There are various different definitions of fusion in the literature, and various axiomatizations for classical mereology. Though the equivalence of the definitions of fusion is provable from axiom sets, the definitions are not logically equivalent, and, hence, are not inter-changeable when laying down the axioms. We examine the relations between the main definitions of fusion and correct some technical errors in prominent discussions of the axiomatization of mereology. We show the equivalence of four different ways to axiomatize classical mereology, using three different notions of fusion. We also clarify the connection between classical mereology and complete Boolean algebra by giving two “neutral” axiom sets which can be supplemented by one or the other of two simple axioms to yield the full theories; one of these uses a notion of “strong complement” that helps explicate the connections between the theories.     

Parametrized preference structures and some geometrical interpretation

Order structures such as linear orders, semiorders and interval orders are often used to model preferences in decision-making problems. In this paper we introduce a family of preference structures where the mutual indifference threshold belongs to a specific family parametrized by extended reals α. This family includes interval orders (α=1), tangent circle orders (α=0) and a new preference structure called ‘diamond order’ (α=−∞). All these preference relations present an asymmetric part which is shown to be always quasi-transitive and to be transitive for α > 1. Diamond orders present ‘forbidden configurations’ which can occur in the case of tangent circle orders. © 1997 John Wiley & Sons, Ltd.