Search

We provide an axiomatic approach to general premium principles in a probability-free setting that allows for Knightian uncertainty. Every premium principle is the sum of a risk measure, as a generalization of the expected value, and a deviation measure, as a generalization of the variance. One can uniquely identify a maximal risk measure and a minimal deviation measure in such decompositions. We show how previous axiomatizations of premium principles can be embedded into our more general framework. We discuss dual representations of convex premium principles, and study the consistency of premium principles with a financial market in which insurance contracts are traded. (C) 2021 Elsevier B.V. All rights reserved.

Extending classical algorithms for ordinary weighted or string-to-string automata to automata with underlying more general algebraic structures is of significant practical and theoretical interest. However, the generalization of classical algorithms sets certain assumptions on the underlying structure. In this respect the maximal factorization turns out to be a sufficient condition for many practical problems, e.g. minimization and canonization. Recently, an axiomatic approach on monoid structures suggested that monoids with most general equalizer (mge-monoids) provide an alternative framework to achieve similar results. In this paper, we study the fundamental relation between monoids admitting a maximal factorization and mge-monoids. We describe necessary conditions for the existence of a maximal factorization and provide sufficient conditions for an mge-monoid to admit a maximal factorization.

This chapter deals with the fundamentals of mechanics of beam, plate and shell structures. The asymptotic and axiomatic approaches in structural mechanics are thoroughly discussed. The main strengths and weaknesses, of the above mentioned methodologies, are highlighted. The most known classical models for beam, plate and shell structures are reviewed. Various examples involving different structural elements are proposed.

To improve comparisons of journals, which are typically based on single-value indicators, such as the journal impact factor (JIF), this paper proposes a functional approach. We discuss interpretatively three progressively finer dominance relations. The first one corresponds to a comparison between the quantile functions of the citation distributions. The second one consists in comparing the integrals of the quantile functions—namely, the generalized Lorenz curves (GLCs). The third one consists in comparing the integrals of the GLCs, where the integration is designed to emphasize the role of the “central body” of the articles of the journal. Although dominance relations are generally not complete orders, we demonstrate with an empirical analysis that it is possible to increase significantly the proportion of pairs of journals that are comparable by moving from the first to the second criterion, and then from the second to the third. Because, in practical applications, it may be convenient to reduce such a functional comparison to a scalar comparison between indicators, we follow an axiomatic approach to identify classes of indicators that are isotonic with the criteria introduced. We demonstrate that the established JIF may be usefully improved if it is corrected simply by multiplying it by one minus the Gini coefficient. The resulting index, defined as stabilized-JIF , has many attractive features and it is isotonic with all the dominance relations introduced.

Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.

Guilt aversion has been shown to play an important role in economic decision-making. In this paper, we take an axiomatic approach to guilt by deducing a utility representation from a list of easily interpretable assumptions on an agent's preferences. It turns out that our logarithmic representation can mitigate the problem of multiplicity of equilibria to which psychological games are prone. We apply the model in three well-known games and show that its predictions are consistent with experimental observations.

This note axiomatically justifies a simple scoring rule for multiplechoice\ud tests. The rule permits choosing any number, k, of available options and\ud grants 1/k-th of the maximum score if one of the chosen options is correct, and\ud zero otherwise. This rule satisfies a few desirable properties: simplicity of implementation,\ud non-negative scores, discouragement of random guessing, and rewards\ud for partial answers. This is a novel rule that has not been discussed or empirically\ud tested in the literature.

Kinship involves social interactions and forms of social organization based on systems of culturally constructed social relations expressed linguistically through the kin terms constituting a kinship terminology. The organization and structure for these social relations expressed in a kinship terminology is formally modeled here as being part of an axiomatic theory in which a few, primary kinship concepts are the equivalent of axioms in a mathematical theory. The cultural knowledge embedded in a kinship terminology therefore is part of a cultural theory – analogous to a mathematical theory – that expresses the culturally salient kinship properties derived from the primary kinship concepts.

L. J. Savage was born in Detroit in 1917 and obtained a mathematics Ph.D at the University of Michigan. After a series of posts, he went, as a statistician, to the University of Chicago in 1946, staying until 1960. He then returned to Michigan, before going in 1964 to Yale, where he died in 1971. He is today principally known for his 1954 book The Foundations of Statistics, the first seven chapters of which develop an axiomatic approach to decision-making under uncertainty. The main result is that the uncertainties should be described by probabilities, the worth of the outcomes by utilities, and that the best decision is that obtained by maximizing expected utility. This approach is today usually described as Bayesian. Although Savage developed it as a justification for accepted statistical practice, it effectively destroyed that practice, instead producing a system based on probability as a degree of belief. The remainder of his life was devoted to exploring the consequences of his approach. In this he worked with the great Italian probabilist, de Finetti and, with Hewitt, proved an important extension of a major result of the Italian. He also wrote a book on gambling with Dubins. For social scientists, the main importance of his work lies in the construction of a belief system about the world as appreciated by a single individual.

This chapter examines different approaches to the measurement of multidimensional inequality and poverty. It first outlines three aspects preliminary to any multidimensional study: the selection of the relevant dimensions, the indicators used to measure them, and the procedures for their weighting. It then considers the counting approach and the axiomatic treatment in poverty measurement. Finally, it reviews the axiomatic approach to inequality analysis. The chapter also provides a selective review of the rapidly growing theoretical literature with the twofold aim of highlighting areas for future research and offering some guidance on how to use multidimensional methods in empirical and policy-oriented applications.

This chapter deals with the philosophy and sociology of science and the different epistemological and methodological approaches discussed in economics. The guiding question is what benefit the student may yield from the knowledge on complexity economics, gained through this textbook, when applying it to “reality,” “empiricism,” “action,” and “policy.” “Theory,” “model,” and “approach” are defined. The epistemological conceptions introduced include “pattern modeling,” “logical positivism,” Popper’s “critical rationalism,” “model Platonism,” “axiomatic method,” Friedman’s methodological instrumentalism, McCloskey’s “rhetoric of economics,” measurement theory, the Duhem–Quine thesis, and Kuhn’s and Lakatos’ “paradigms” approach, Bhaskar’s and Lawson’s “critical realism,” “constructivism,” discourse theory, and “epistemological pluralism.” Furthermore, econometric testing and “meta-regressions” are discussed. Finally, cutting-edge discussions in economics are presented such as publication biases, replicability of results, and data transparency of journal publications, theoretical pluralism, and scientific-output evaluation through journal rankings. A broad understanding of professionally applying theoretical knowledge through “reality,” action, and policy is developed.

We survey the literature on coalitional games, focusing on recent developments and on the axiomatic approach to the subject.

Publisher Summary This chapter discusses independence results in set theory. The chapter explains of the proof that the axiom of choice (AC) and the continuum hypothesis (CH) are independent of the axioms of Zermelo-Fraenkel (Z-F) set theory. Complete proofs will not be presented, but all essential points will be treated and only those details which can be handled by standard methods will be omitted. The latter is an infinite collection of axioms generated by a simply stated rule. In contrast, AC has been accepted by most mathematicians as an axiom not requiring further proof and thus the independence of AC is primarily of technical interest. AC was first explicitly stated by Zermelo in the course of attempting to prove the well-ordering theorem of Cantor. This “naive” attitude came to an end as these attempts proved fruitless and with the axiomatization of set theory the way was opened to discuss their logical relation to the other axioms.

In this chapter the reader will find, a rigorous, axiomatic, development of the fundamental equations of motion. The main focus is on the flow of incompressible fluids, but the equations are developed for any Newtonian fluid. The material in this chapter is intended as a resource, when rigor is required, readers may wish to skip this sections of this chapter on their first reading, especially if they are from a non-engineering background. The material in this chapter was developed for a first year graduate course that I have taught at various institutions, in Australia, the US, Italy and Germany.

Rational choice constitutes a field in which philosophers share interests particularly with economists but also with political theorists, sociologists, psychologists. Theory, over the past century, has been dominated by the axiomatic approach which offers a distinctive contribution to our understanding of deliberation and choice and one that contains some heavy-weight theories that have been highly influential. Within both the programme of research and its intellectual disaspora , two theories stand out as demanding our attention — expected utility theory, previously the main workhorse for the analysis of decision-making and Arrow's impossibility theorem — a theorem that suggests reasonable social choice mechanisms are destined to being undemocratic. Both theories are axiomatic and share the fact that they were taken to identify reasonable behaviour in individual or group settings.

This paper reexamines key results from the measurement of opportunity freedom, or the extent to which a set of options offers a decision maker real opportunities to achieve. Three cases are investigated: no preferences, a single preference, and plural preferences. The three corresponding evaluation methods—the cardinality relation, the indirect utility relation, and the effective freedom relation—and their variations are considered within a common axiomatic framework. Special attention is given to representations of freedom rankings, with the goal of providing practical approaches for measuring opportunity freedom and the extent of people's capabilities.

Statecharts is a behavioural specification language proposed for specifying large real-time, event-driven reactive systems. It is a graphical language based on state-transition diagrams for finite state machines extended with many features like hierarchy, concurrency, broadcast communication and time-out. We supply Statecharts with a compositional axiomatization for both safety and liveness properties. By generating external events symbolically, Statecharts can be executed, thereby turning it into a programming language for real-time concurrency (as well as enabling rapid prototyping). As such it is well suited for compositional program verification. In addition to our compositional axiomatic system, we give a denotational semantics and prove that the axiomatization is sound and relatively complete with respect to this semantics.

This chapter introduces the most important notion in probability theory - mathematical expectation or (in mathematical language) a Lebesgue Integral taken with respect to a probabilistic measure (see, for example, Chapters 15 and 16 in Poznyak (2008)). Physically, this operator presents some sort of ‘averaging’, or a probabilistic version of ‘the center of gravity of a physical body’. In fact, it compresses the information on a random variable in to a single number. There exist two possibilities to introduce the operator of the mathematical expectation: using the so-called axiomatic approach, suggested by Whittle ( Whittle, 1984), which is based on some evident properties of ‘an averaging operation’, and showing then that it is exactly the Lebesgue-Stieltjes integration; introducing directly the mathematical expectation operator as the Lebesgue-Stieltjes integration with respect to a distribution function or probability measure. Here we will present both of these approaches and demonstrate their internal interconnection.

Publisher Summary In economic analysis, decision theory is developed with a purely axiomatic method. The theory proceeds by first defining a set of choices that a subject (the decision-maker, DM) faces. A choice is a finite set of options that are offered to the DM; a decision is the selection of one of these options. The observed data are pairs of choices offered and decisions taken: it is possible to collect these data experimentally asking a real DM to pick one out of two options, under the condition that the object selected is actually delivered to him or her. For neuroeconomics and any research program that tries to determine how decisions are implemented, the utility function is the most interesting object. This function ties observed behavior with a simple one-dimensional quantity, the utility of the option, and predicts that the decision between two options is taken by selecting the option with the highest utility. However, if interested in determining the neural correspondents of the objects one has introduced, one must first know whether these objects are unique. For example, one may formulate the hypothesis that the decision is taken depending on some statistics of the firing rate of a group of neurons associated with each of the options.

Publisher Summary This chapter describes the axiomatic approach to neuroeconomics. The axiomatic approach to neuroeconomics forms part of a wider agenda for the incorporation of nonstandard data into economics. Recent advances in experimental techniques have led to an explosion in the range of data available to those interested in decision-making. This has caused something of a backlash within economics against the use of nonstandard data in general and neuroscientific data in particular. In their impassioned defense of “mindless economics” Gul and Pesendorfer claimed that nonchoice data cannot be used as evidence for or against economic models, as these models are not designed to explain such observations. By design, our axiomatic approach is immune to such criticisms as it produces models, which formally characterize whatever data is under consideration. Ideally, an expanded conception of the reach of the axiomatic methodology will not only open new directions for neuroeconomic research, but also connect the discipline more firmly with other advances in the understanding of the process of choice, and the behaviors that result.

Publisher Summary This chapter looks at the embodiment of stable inferential patterns created and sustained by communities of individuals, which exist beyond the individuals themselves. The approach taken here is comparable to the study of, say, speech accents, in that although they are created, manifested, and sustained by individuals, they—the speech accents themselves—constitute distinctions we make at a supra-individual level. The chapter analyzes the embodied cognitive mechanisms that make human abstraction and their supra-individual crystallization possible, and sees how they bring mathematics, its concepts, and inferential organization, into being. The chapter aims to provide a brief overview of what is the nature of mathematics from the perspective of embodied cognitive science and conceptual systems. It intends to show how the inferential organization of mathematics emerges from everyday cognitive mechanisms of human imagination realized via embodied conceptual mappings such as metaphor, metonymy, and conceptual blends.

Graphical User Interfaces (GUIs) have been extensively applied to the user interfaces of almost every computer system. An important feature of GUIs is the icon. While it is not difficult to find relevant information for the design of icons from handbooks, industry standards or guidelines, it is surprising to find many inconsistencies in the information. Hence, it is necessary to find a new and improved method for the design of icons. An axiomatic method was applied to icon design. This method was based on the two axioms of the Axiomatic Design (AD) principles – The Independence Axiom and the Information Axiom. From the viewpoints of semiotics and human information processing, visual distinctiveness and the appropriateness of representation are two key factors in icon design. It was proposed that the discriminability of icons could be analyzed through the Independence Axiom, and the meaningfulness of icons could be evaluated through the Information Axiom. From a previous study, a set of icons used in a Distributed Control System (DCS) product for ASEAN market was reviewed. The review was used as an example to show how the axiomatic method could be applied as a framework for the design of icons in process control displays.

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.

In recent years a discussion could be followed where the pros and cons of the applicability of the Cosserat continuum model to granular materials were debated [Bardet, J.P., Vardoulakis, I., 2001. The asymmetry of stress in granular media. Int. J. Solids Struct. 38, 353–367; Kruyt, N.P., 2003. Static and kinematics of discrete Cosserat-type granular materials. Int. J. Solids Struct. 40, 511–534; Bagi, K., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1329–1331; Bardet, J.P., Vardoulakis, I. 2003a. Reply to discussion by Dr. Katalin Bagi. Int. J. Solids Struct. 40, 1035; Kuhn, M., 2003. Discussion on “The asymmetry of stress in granular media”. Int. J. Solids Struct. 40, 1805–1807; Bardet, J.P., Vardoulakis, I., 2003b. Reply to Dr. Kuhn’s discussion. Int. J. Solids Struct. 40, 1809; Ehlers, W., Ramm, E., Diebels, S., D’Addetta, G.A., 2003. From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses. Int. J. Solids Struct. 40, 6681–6702; Chang, C.S., Kuhn, M.R., 2005. On virtual work and stress in granular media. Int. J. Solids Struct. 42, 3773–3793]. The authors follow closely this debate and try, with this paper, to provide a platform where the various viewpoints could find their position. We consider an ensemble of rigid, arbitrarily shaped grains as a set with structure. We establish a basic mathematical framework which allows to express the balance laws and the action–reaction laws for the discrete system in a “global” form, through the concepts of “part”, “granular surface”, “separately additive function” and “flux”. The independent variable in the balance laws is then the arbitrary part of the assembly rather than the single grain. A parallel framework is constructed for Cosserat continua, by applying the axiomatics established by [Noll, W., 1959. The foundation of classical mechanics in the light of recent advances in continuum mechanics. In: The axiomatic method, with special reference to Geometry and Physics, North-Holland Publishing Co., Amsterdam pp. 266–281, Gurtin, M.E., Williams, W.O., 1967. An axiomatic foundation of continuum thermodynamics. Arch. Rat. Mech. Anal. 26, 83–117, Gurtin, M.E., Martins, L.C., 1976. Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324]. The comparison between the two realisations suggests the microscopic interpretation for some features of Cosserat Mechanics, among which the asymmetry of the Cauchy-stress tensor and the couple-stress.

This paper studies axiomatically the cognitive process of reference-dependent preferences (RDP). We show that the effect of a shifting reference point is two sided: first, modifying the criteria deemed relevant for a given choice and second, modifying the desirability of an option, i.e. how these criteria combine in order to yield decision. We also relate this notion of desirability and the strength of the status quo bias that allows for a reference-independent definition of the desirability of an option.

The idea of topological invariants defined via path integrals was introduced by A.S. Schwartz (1977) in a special case and by E. Witten (1988) in its full power. To formalize this idea, Witten [Wi] introduced a notion of a Topological Quantum Field Theory (TQFT). Such theories, independent of Riemannian metrics, are rather rare in quantum physics. On the other hand, they admit a simple axiomatic description first suggested by M. Atiyah [At]. This description was

This article is about notational conventions in formal logic, which is (in the view of most mathematicians) that branch of mathematics (logicians, by contrast, tend to think of mathematics as a branch of logic; both metaphors are correct, in the appropriate formal axiomatic system) most concerned with many questions that arise in natural language, e.g., questions of meaning, syntax, predication, well formedness, and – for our purposes, the most important such – detailed, precise specification. Specification is the purpose of notation, both in mathematics and in science, but such precise conventions are unavoidably context sensitive. Thus, the use of logical notation is different in logic and in linguistics.

Publisher Summary In mathematics David Hilbert's Grundlagen der Geometrie was an influential work that led by its axiomatic method to a new thinking in all mathematical fields in the 20th century. In addition, following in the traces of Euclid, it became the classical textbook for geometry in educating mathematicians and mathematics teachers for nearly the whole century. Towards the end of the 19th, century a remarkable change came about in the field of the foundations of geometry. For the first time David Hilbert constructed the axioms in what was subsequently to be their usual sequence. In this manuscript the arrangement of the later Festschrift is already apparent: axioms, proofs of independences, segment arithmetic, Desargues's theorem, Pascal's theorem, and problems concerning constructability. The manuscript Grundlagen der Euklidischen Geometrie (EG) contains an exhaustive discussion of those areas that were mostly treated in brief in the vacation course. The logical meaning of the axioms was studied by construction of arithmetical models. Amongst these were proofs of independence for axioms of the first two groups. The book had some consequences for physics, which gained Hilbert's attention from the 1900s onwards. For during the 1910s, there was a remarkable connection between Einstein and Hilbert.

Publisher Summary This chapter describes Kurt Godel's paper on the incompleteness theorems. Godel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Godel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Godel used is applicable in a wide variety of circumstances. Godel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations.

This chapter examines the history of measure theory. At the beginning of civilization, mathematics could be differentiated from science and technology as rational art of solving abstract problems with numbers and geometrical figures. It means that the solution has to be obtained from initial data with some kind of rational, logical mental process. Greek philosophers and mathematicians made the great discovery that mathematical rules could be proved and that the whole of mathematics can be organized in axiomatic theory. Pappos proves that the spiral and quarter of the circumference that connects its starting and end point bound the surface, which has the same ratio to the hemisphere as the sector of the quadrant to the quadrant. Translational invariance of Lebesgue integral has generalization with important application in the representation theory of groups. It is suggested that much of character and representation theory of finite groups remains valid if suitable integration is applied over the compact manifold formed by the elements of the rotation group.

Within the framework of the axiomatic approach three types of voting schemes are investigated according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision. These types of rules are Social Decision Rules, Functional Voting Rules, and Social Choice Correspondences. Consideration is given to local rules, i.e., to the rules which satisfy some analogue of Arrow's Independence of Irrelevant Alternatives condition. A general description of the problem of axiomatic synthesis of local rules, and various formalizations of voting schemes are given. The notion of “rationality” of individual opinions and social decision is described. Various types of binary relations (preferences) are introduced. The characteristic conditions (Expansion-Contraction Axioms) on choice functions are defined, and the interrelations between them are established. Two types of Social Decision Rules (transforming individual preferences to social ones) are studied. The explicit forms of those rules are investigated. The rules restricted by rationality constraints, i.e., by the constraints on domains and ranges of the rules, are studied as well. Functional Voting Rules are investigated which transform individual opinions defined as choice functions into a social choice function. In doing so, a rationalizability of those choice functions is not assumed. The explicit form of these rules is obtained, and the rules which satisfy different rationality constraints are studied. Social Choice Correspondences deal with the case when the individual opinions are formalized as binary relations, and the collective decision that we look for is a choice function. The explicit form of rules is studied. The obtained classes comprise the rules such as the generalized Pareto rules. Several new classes of the rules are introduced and analyzed. The explicit form of the Nash-implementable rules is found. The analysis of publications on the axiomatic synthesis of the local aggregation rules is made.

Theories of utility and subjective probability focus on quantitative representations of people's preferences and beliefs that facilitate analyses of choice behavior and decision making under risk or uncertainty. A utility theory consists of assumptions about a person's preferences over decision options that lead to numerical values or utilities for options whose ordering reflects the subjective preference ordering. Different option formulations lead to specialized properties of utilities. A theory of subjective probability consists of assumptions about a person's judgments of relative likelihoods of uncertain events that lead to numerical probabilities for events that reflect the underlying judgments. A joint theory of utility and subjective probability sets forth axioms for an individual's preferences over a set of options whose outcomes depend on realizations of uncertain events. The axioms imply an integrative numerical model for options that combines outcome utilities and event probabilities. Investigators have proposed a variety of axiomatic systems that lead to different numerical models of utilities and subjective probabilities.

Publisher Summary This chapter focuses on visual confrontation naming and explores different models of naming. Visual confrontation naming is more than a standard clinical tool and research task. It has many virtues that help make it a critical instrument for understanding many aspects of human cognition and their neural basis. The chapter presents an empirical axiomatic approach toward constructing and validating theories in cognitive neuroscience. The reasons to aim for an empirical axiomatic approach are rooted in both the fundamental problem of understanding mental operations and their neural substrates, and in the practical problems of evaluating competing theories. Naming is similar to many other mental processes in that it must be a multistage, dynamic process. Multistage dynamic processes are almost certain to have extremely complex behaviors that are very sensitive to the properties of their internal processes. Naming is known to be the product of relatively specific brain regions, and relatively small ones, so the neural mechanisms involved must be comparatively circumscribed. It is possible to make detailed computational models of a wide range of levels involved in naming, from neural activity to overt behaviors, and determine what consequences such models have for observable behavior and neuroscience measures.

Publisher Summary This chapter explores that semantic theory plays a central role in the normative and descriptive theory of deductive inference, but its role in the study of inductive inference has been much less prominent. The disparity is both odd and understandable. It is odd because deductive and inductive reasoning both rely heavily on linguistic representations and semantic theory is the natural tool for investigating inference within propositional structures. The axiomatic theory of subjective probability specifies further properties of belief strength that characterize so-called coherent beliefs; if strength of belief is coherent in this sense, then there exist numerical probabilities that represent the belief strength ordering and satisfy the mathematical laws of probability. A theory of semantics relates three aspects of language: (1) the syntactic structure of propositions, that is, a specification of how complex propositions are built from simpler parts, (2) the semantic structure of propositions, that is, a specification of the relation between propositional structure, reference, and truth values, and (3) inference rules that define inferential relations in terms of syntactic and semantic structure. The chapter considers variety of empirical results that illustrate concretely the relationship between strength of belief, propositional structure, and the structure of evidence. Many of the phenomena discussed in the chapter are characterized as compositional anomalies-they are cases in which observed relations in belief strength conflict with the semantic structure of propositions, or at least with well-established theories of this semantic structure.

Publisher Summary This chapter describes apportionment. Apportionment determines the number of seats each state will receive. Redistricting refers to the drawing of congressional district boundaries within a state, one for each representative. The districts should have approximately ‘equal populations'. These considerations place severe restrictions on the districting plans, and make districting a quite difficult computational problem. All of the major historical methods except Hamilton's fall into the specific computational framework implied by a common divisor. The approach to measure bias suggests a mathematical model for theoretical analysis. Mathematics, through the axiomatic method, gives a precise statement about the consistency or inconsistency and thus provides a sensible basis for choosing the method that seems right for a given application. Population monotonicity implies house monotonicity. Not only do the Hamilton and Quota methods violate population monotonicity, so does any method that satisfies quota.

Publisher Summary This chapter discusses cooperative models of bargaining. The axiomatic theory of bargaining originated in a fundamental paper by J.F. Nash. Nash introduced an idealized representation of the bargaining problem and developed a methodology that gave the hope that the undeterminateness of the terms of bargaining could be resolved. The canonical bargaining problem is that faced by management and labor in the division of a firm's profit. Another example concerns the specification of the terms of trade among trading partners. The chapter presents the classic characterizations of the three solutions that occupy center stage in the theory as it stands today: the Nash solution, the Kalai–Smorodinsky solution, and the Egalitarian solution. Solutions to abstract bargaining problems, most notably the Nash solution, have been used to solve concrete economic problems, such as management–labor conflicts, on numerous occasions.

We offer a system to capture the relationship between knowledge and belief, which also sheds new light on each of them in isolation. In the case of knowledge, we strongly reject the property of negative introspection. In the case of belief, we propose a distinction between belief (whose defeasibility is recognized by the agent) and certainty (whose defeasibility is not). The relationship between the three notions - knowledge, certainty, and belief - goes far beyond mere hierarchy. In particular, knowledge has the flavor of belief that is stable under incorporation of correct facts. We explore these first through a model theory, which is based on the notions of the agent's subconscious biases and its conscious preferences (or plausibility measure). We then provide a sound and complete axiomatic system, and point to some of its illuminating properties. We compare our construction to previous ones in AI and philosophy, and in particular point to connections with recent work in AI based on conditionals. (Proofs of our theorems are omitted from this version of the article, hence the subtitle.)

This chapter discusses the characteristics of the algorithms, the future of the art of optimization, and approaches to the problems. Although many managers might wish it otherwise, the writing of efficient computer programs will always be an art, not a science. Clarke's Law states that if a distinguished elderly scientist says that something is possible, he is almost always right; if he says that something is impossible, he is almost always wrong. As demonstrated by Goedel's Proof, no automatic procedure can discover all of the true statements in a consistent axiomatic system such as the behavior of a programmable computer. This implies that the development of new algorithms will always be an adventure. Software engineers, however, have a fundamentally different situation: Once solved a given problem, one should never need to solve it again, as one can reuse the solution already developed. In the development projects, one is always venturing into unexplored territory and one knows how to identify pioneers. On the other hand, the thought of going into the office to face the same problems every day would greatly reduce interest in this field.

An EBL-based system for acquiring conceptual design knowledge in physical systems was proposed and implemented based on Value Engineering Methodologies and Axiomatic Design Approaches. The key idea behind this system is that such knowledge can be acquired by analyzing existing designed objects which can be regarded as the result of rational decisions and actions. In this system, the structural features of designed objects are analyzed by domain specific knowledge to yield a systematic explanation of how they function and attain their design goals and why they are used for attaining the goals. The “how” explanation results in a generalized version of the Functional Diagram used in Value Engineering from which the object in question can be interpreted via two kinds of design rationalities, i.e., teleological and causal ones. Namely, a designed object is regarded as being consisted of a hierarchy of design goals (primary functions), subgoals (subfunctions), structures and substructures toward attaining those goals. We applied the EBL method to this hierarchical model to acquire general design knowledge which is operational at various phases and fields in the conceptual design. Through organizing domain-specific knowledge of this EBL system according to the above hierarchical model, various levels of knowledge can be extracted by a single positive instance. The quality of the extracted knowledge is then discussed with reference to its level in the hierarchy of acquisition. The “why” explanation gives us a deeper understanding of the designed objects from which we can then extract meta-planning or strategic knowledge for selecting rational plans from among other possible alternatives. This deep explanation is obtained by regarding the object in question as being the result of a sequence of strategically rational decisions and actions which are subject to the principles of “good” design that are formalized as an axiomatic system in the Axiomatic Design Approach.

Publisher Summary This chapter focuses on the concept of probability. Probability is a concept that is used in conversation. The problems surrounding the definition of probability itself can be similarly circumscribed by summarizing its properties in purely mathematical terms. This is called the axiomatic theory of probability. In brief, if a number conforms to the axioms of probability, it is a probability. The axioms are deceptively simple but they are sufficient to underpin the entire structure of mathematical probability and its offspring, statistics. The frequency definition of probability has been adapted to address (and evade) this problem by defining probability as the relative frequency of the event in the long run. No meaning can he attached to a statement that literally assigns a frequency based probability to a single event or individual. And that, according to the opponents of the frequency view of probability, is exactly what is wrong with the whole frequency approach. An event is a particular group of outcomes having some common property with respect to the random variable.

Publisher Summary This chapter presents a number of ideas that originated an evolution of programming from arts and crafts to a science. The chapter describes computer arithmetic in two stages. In the first stage, axioms are given for arithmetic operations on natural numbers, which are valid independently of their computer representation, and choices of supplementary axioms are proposed for characterizing various possible implementations. In the second part, an axiomatic definition of program execution is introduced. An axiomatic approach is indispensable for achieving program reliability. The usefulness of program proving is advocated in view of the cost of programming errors and program testing. The chapter discusses the definition of formal language. The axioms and rules of inference can be understood as the ultimate definitive specification of the meaning of the language.

Publisher Summary A recent development in category theory has been the description and study of a useful class of categories called “elementary topoi.” This class includes the ordinary category of sets: diagrams of sets and sheaves of sets on a topological space. The axiomatic description of these categories provides a formulation of axiomatic set theory wholly different from the usual set-theoretic axioms on the membership relation and the further study casts considerable light on a number of problems of foundations. Systematic developments of the properties of a topos from axioms are discussed in this chapter. Other categories with internal logic and quantifiers and their relations to standard logical systems are also described in the chapter. Given the basic geometric character of sheaf theory, this common development of ideas from geometry and concepts from set theory embodies exciting prospects.

Publisher Summary This chapter discusses logical aspects of the axiomatic method and focuses on the diversity of uses for the option; cf. [M 1] for a sustained use of it. The chapter explains that the option is cheap for specialists in m.l. and may help consolidate better some of the shifts in the 60's. As a matter of experience, the particular tool kit allows those shifts to be articulated in civilized terms. The modest word hygiene contains a warning against a two-fold mistake assuming that insights or errors involved in l.f. are dramatic and if they are not, then they are of no consequence. The second assumption is a corollary to high-minded ideas of depth.

This chapter discusses the axiomatic theory of sets. It adds universal quantification denoted by “∀ x ” (read as “for all x”) and existential quantification, denoted by “∃#” (read as “there is an x”) to the logical operations. Thus, given some property Px of a real number x it writes (∀x) Px to denote the false sentence “for all x , x is an even prime” and it writes (∃x) P ( x ) to denote the true sentence “there is an x such that x is an even prime.” Clearly, (∀ x ) Px→ Py is true for every property P , and any y and so is the sentence P y →(∃ x )P x . Primitive sentences are of the form x ∈ y, and every sentence is formed from primitive sentences by quantification, negation, disjunction, conjunction, and equivalence.

Publisher Summary This chapter discusses the basic ideas and results in discrete probability theory. In setting up a probability theory, there are various approaches that could be used. The chapter presents a modern axiomatic approach. To the probabilist, a random experiment, an experiment whose outcome cannot be determined in advance, could be reduced to an abstract formulation in terms of three mathematical entities: (1) a set of all possible outcomes of the experiment, called the sample space; (2) a collection of subsets of the sample space, which is called the family of events; and (3) a numerical function that assigns to each event a number between 0 and 1, called the probability of the event. The function defined on the family of events is called the probability measure. In a simplistic way, a random variable is essentially a quantitative measure that is attached to each possible outcome of the random experiment.

The development of probability theory in the beginning of the 20 th century necessitated a reevaluation and refinement of its logical foundations. This was required by the progress in statistical physics and for the development of probability theory. The Laplace-type classical foundations were insufficient and inadequate. To establish a logical order and consistency for any kind of inference, it is necessary to single out the initial concepts, establish the rules for inference, and show the absence of contradiction in all the results obtained. This can be achieved using an axiomatic method that makes it possible to encompass the totality of objects studied by a given mathematical theory. The essence of this method is that certain propositions—called axioms—are postulated as the basis of the theory, and all the subsequent propositions are deduced from these axioms. The rules of deduction should be distinctly formulated. A variety of mathematical theories then appeared, which were constructed using axiomatization. Toward the beginning of the 20 th century, the axiomatic method penetrated various branches of mathematics.

Publisher Summary The successor function s(x) = xtl, where x is a variable ranging through the natural numbers, is a basic function in ordinary recursion theory. All the recursive functions can be generated from s and some other simple functions (projections and constant functions) by a few schemes. It is also chosen as a basic function in several settings of abstract recursion theory. In [31 Moschovakis defines the functions on an abstract domain by schemes, and is needed already to formulate the scheme of primitive recursion. For the same reason it is needed in the axiomatic approach in [41. [I] is also an axiomatic approach to recursion theory, and there Fenstad has proved some simple results from the axioms alone without using s (in chapter 1 , section 21, but s is brought in in the proofs of more advanced results, for instance the simple representation theorem 1.6.3: Let 0 be a precomputation theory. Then there exists a 0 - computable function f such that 0 is equivalent to PR[fl. Here PR[f] is a computation theory which is inductively defined by an operator r . The n-th level of r , denoted by rn, is defined from ,,n-l. The following result is an important part of the proof.

Publisher Summary This chapter presents an overview of the theory and applications of superconvex spaces. It discusses about Rode who developed an axiomatic theory on the formation of countable convex combinations named “superconvex analysis” and several applications of it to different parts of analysis. The chapter describes the internal aspects of the superconvex theory. The remarkable theorem on superconvex subsets by Kuhn demonstrates the fundamental distinction between convex and superconvex structures. The chapter presents a complete proof of the James theorem for Banach spaces; and to achieve this, two appendices are added on the Pryce selection theorem and the double limit criterion. The chapter describes the application of the superconvex theory to obtain a central result in Choquet theory; it presents an essential fortification and simplification by Kremp.