Useful as a text for students and a reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory most useful for its application in modern analysis. Coverage includes sets and classes, measures and outer measures, Haar measure and measure and topology in groups. From the reviews: "Will serve the interested student to find his way to active and creative work in the field of Hilbert space theory." --MATHEMATICAL (...) REVIEWS. (shrink)

This book provides an introduction to measurement theory for non-specialists and puts measurement in the social and behavioural sciences on a firm mathematical foundation. Results are applied to such topics as measurement of utility, psychophysical scaling and decision-making about pollution, energy, transportation and health. The results and questions presented should be of interest to both students and practising mathematicians since the author sets forth an area of mathematics unfamiliar to most mathematicians, but which has many potentially significant applications.

The need for quantitative measurement represents a unifying bond that links all the physical, biological, and social sciences. Measurements of such disparate phenomena as subatomic masses, uncertainty, information, and human values share common features whose explication is central to the achievement of foundational work in any particular mathematical science as well as for the development of a coherent philosophy of science. This book presents a theory of measurement, one that is "abstract" in that it is concerned with highly general (...) axiomatizations of empirical and qualitative settings and how these can be represented quantitatively. It was inspired by, and represents a generalization and extension of, the last major research work in this field, Foundations of Measurement Vol. I, by Krantz, Luce, Suppes, and Tversky published in 1971. (shrink)

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric (...) space is countably additive on open sets. (shrink)

This paper presents a novel semantic analysis of unit names (like pound and meter) and gradable adjectives (like tall, short and happy), inspired by measurement theory (Krantz et al. In Foundations of measurement: Additive and Polynomial Representations, 1971). Based on measurement theory’s four-way typology of measures, I claim that different adjectives are associated with different types of measures whose special characteristics, together with features of the relations denoted by unit names, explain the puzzling limited distribution of measure (...) phrases, as well as unit-based comparisons between predicates (as in the table is longer than it is wide). All considered, my analyses support the view that the grammar of natural languages is sensitive to features of measurement theory. (shrink)

Measurement is said to be the basis of exact sciences as the process of assigning numbers to matter (things or their attributes), thus making it possible to apply the mathematically formulated laws of nature to the empirical world. Mathematics and empiria are best accorded to each other in laboratory experiments which function as what Nancy Cartwright calls nomological machine: an arrangement generating (mathematical) regularities. On the basis of accounts of measurement errors and uncertainties, I will argue for two claims: 1) (...) Both fundamental laws of physics, corresponding to ideal nomological machine, and phenomenological laws, corresponding to material nomological machine, lie, being highly idealised relative to the empirical reality; and also laboratory measurement data do not describe properties inherent to the world independently of human understanding of it. 2) Therefore the naive, representational view of measurement and experimentation should be replaced with a more pragmatic or practice-based view. (shrink)

Our primary aim in this paper is to sketch the account of virtue that we think most amenable to virtue measurement. Our account integrates Whole Trait Theory from psychology with a broadly neo-Aristotelian approach to virtue. Our account is ‘ecumenical’ in that it has appeal for a wide range of virtue ethicists. According to WTT, a personality trait is composed of a set of situation-specific trait-appropriate responses, which are produced when certain “social-cognitive” mechanisms are triggered by the perception of (...) trait-relevant stimuli in a person’s external and/or internal environment. Moving from this starting point, we discuss our conception of a virtue and respects in which it both aligns with and diverges from Aristotle’s conception. We discuss roles for practical wisdom and motivation in our conception of virtue, and highlight respects in which WTT provides an amenable empirical framework into which key Aristotelian elements can be integrated. We conclude with brief remarks about our conception as an empirically adequate and measurable account. (shrink)

Our primary aim in this paper is to sketch the account of virtue that we think most amenable to virtue measurement. Our account integrates Whole Trait Theory from psychology with a broadly neo-Aristotelian approach to virtue. Our account is ‘ecumenical’ in that it has appeal for a wide range of virtue ethicists. According to WTT, a personality trait is composed of a set of situation-specific trait-appropriate responses, which are produced when certain “social-cognitive” mechanisms are triggered by the perception of (...) trait-relevant stimuli in a person’s external and/or internal environment. Moving from this starting point, we discuss our conception of a virtue and respects in which it both aligns with and diverges from Aristotle’s conception. We discuss roles for practical wisdom and motivation in our conception of virtue, and highlight respects in which WTT provides an amenable empirical framework into which key Aristotelian elements can be integrated. We conclude with brief remarks about our conception as an empirically adequate and measurable account. (shrink)

Our primary aim in this paper is to sketch the account of virtue that we think most amenable to virtue measurement. Our account integrates Whole Trait Theory from psychology with a broadly neo-Aristotelian approach to virtue. Our account is ‘ecumenical’ in that it has appeal for a wide range of virtue ethicists. According to WTT, a personality trait is composed of a set of situation-specific trait-appropriate responses, which are produced when certain “social-cognitive” mechanisms are triggered by the perception of (...) trait-relevant stimuli in a person’s external and/or internal environment. Moving from this starting point, we discuss our conception of a virtue and respects in which it both aligns with and diverges from Aristotle’s conception. We discuss roles for practical wisdom and motivation in our conception of virtue, and highlight respects in which WTT provides an amenable empirical framework into which key Aristotelian elements can be integrated. We conclude with brief remarks about our conception as an empirically adequate and measurable account. (shrink)

Our primary aim in this paper is to sketch the account of virtue that we think most amenable to virtue measurement. Our account integrates Whole Trait Theory from psychology with a broadly neo-Aristotelian approach to virtue. Our account is ‘ecumenical’ in that it has appeal for a wide range of virtue ethicists. According to WTT, a personality trait is composed of a set of situation-specific trait-appropriate responses, which are produced when certain “social-cognitive” mechanisms are triggered by the perception of (...) trait-relevant stimuli in a person’s external and/or internal environment. Moving from this starting point, we discuss our conception of a virtue and respects in which it both aligns with and diverges from Aristotle’s conception. We discuss roles for practical wisdom and motivation in our conception of virtue, and highlight respects in which WTT provides an amenable empirical framework into which key Aristotelian elements can be integrated. We conclude with brief remarks about our conception as an empirically adequate and measurable account. (shrink)

The book offers an overview of experiences, theories and conceptions of time in the Graeco-Roman world. It presents the results of new research on neglected medical texts, relating to time management, aging and times of life, and the importance of t.

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and (...) the logic of quantum mechanics are given. (shrink)

The paper reconstructs the connections between the evolution of Patrick Suppes’ measurement theory from 1951 to 1958 and the research in utility analysis he conducted between 1953 and 1957 within the Stanford Value Theory Project. In particular, the paper shows that Suppes’ superseding of the classical understanding of measurement, his endorsement of the representational view of measurement, and his conceiving of an axiomatic version of the latter were prompted by his research in utility analysis.

A highly abstracted theory of measurement is synthesized from classical measurement theory, fuzzy set theory, generalized information theory, and predicate calculus. The theory does not require specific truth value concepts, nor does it specify what subsets of the reals can be observed, thus avoiding the usual fundamental difficulties. Problems such as the definition of systems, the significance of observations, numerical scales and observables, etc. are examined. The general logico-algebraic approach to quantum/classical physics is justified as (...) a special case of measurement theory. (shrink)

The Kubovy-Epstein proposal for the formalization of the relation between kinematic geometry and perception of motion has formal problems in itself. Motion phenomena are inadequately captured by the relational structures and the notion of isomorphism taken over from measurement theory. [Kubovy & Epstein].

It is shown that the application of the Lax-Phillips scattering theory to quantum mechanics provides a natural framework for the realization of the ideas of the “Many-Hilbert-Space” theory of Machida and Namiki to describe the development of decoherence in the process of measurement. We show that if the quantum mechanical evolution is pointwise in time, then decoherence occurs only if the Hamiltonian is time-dependent. If the evolution is not pointwise in time (as in Liouville space), then the decoherence (...) may occur even for closed systems. These conclusions apply as well to the general problem of mixing of states. (shrink)

We tested if the newly designed ToMotion task reflects a single construct and if the atypical groups differ in their performance compared to typically developing peers. Furthermore, we were interested if ToMotion maps a developmental sequence in a Theory of Mind performance as exemplified by increasing difficulty of the questions asked in every item. The sample consisted of 13 adolescents that have been diagnosed with a developmental language disorder and 14 adolescents that are deaf or hard of hearing. All (...) of these adolescents were in special schools for secondary vocational education. The control group existed of 34 typical developing adolescents who were in regular intermediate vocational education, ranging from level 2 to 4. The ToMotion, available in a spoken Dutch version and in a version in Sign Language of the Netherlands, was used to map ToM abilities. An attempt has been made to fill the gap of missing studies of ToM in adolescents by developing a new measuring instrument. In conclusion, assessing ToM with the ToMotion results in a picture that DHH adolescents score lower than TD peers. However, their scores are as consistent as those of the TD peers. The picture of DLD adolescents is the reverse. They show no differences in ToM scores, but seem to be somewhat more inconsistent compared to TD peers. We provide a discussion on those results and its implications for future research. What this paper adds? The current study introduces a new visual Theory of Mind task, ToMotion, designed specifically to assess ToM in adolescents in an ecologically valid way and adapted to the needs of adolescents with language and communication difficulties. (shrink)

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that (...) “Ω happens”. Moreover, the Kolmogorov sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers. (shrink)

This is an entry to the Compendium of Quantum Physics, edited by F Weinert, K Hentschel and D Greenberger, to be published by Springer-Verlag.

We prove that for any locally countable $\Sigma _{1}^{1}$ partial order P = 〈2ω,≤P〉, there exists a nonmeasurable antichain in P. Some applications of the result are also presented.

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The “collapsed state-vector” after the “objectification” is simply treated as a random vector of the a posterioristate given by the quantum filtering, i.e., the conditioning of the a prioriinduced state on the corresponding reduced algebra. The nonlinear phenomenological equation of “continuous spontaneous localization” has been (...) derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments. (shrink)

The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in (...) the sense of Holman 1969 and Colonius 1978. The problem of justifying representational conditions is addressed in more detail than is customary in the RTM literature; this continues the study of the foundations of RTM begun in an earlier paper. The most important formal consequence of the present interpretation of physical extensive scales is that the basic existence and uniqueness properties of scales (representation theorem) may be derived without appeal to an Archimedean axiom; this parallels a conclusion drawn by Narens for representations of qualitative probability. It is concluded that there is no physical basis for postulation of an Archimedean axiom. (shrink)

This paper outlines a ‘paradox’ in quantum measurement theory, illustrated with two different types of systems. If this paradox cannot be resolved in ordinary quantum mechanics (as I currently think), it is alarming. If it can be resolved, it can be added to a long list of examples that show the internal consistency of quantum mechanics, and in this case I hope the correct analysis will be an interesting example for students. The immediate paradox involves a failure of Lorentz (...) invariance for measurements on certain types of single particle systems. It seems to show that an absolute frame of reference is required to describe wave function collapse. But if the argument is correct, quantum measurement theory is not merely dependant on the frame of collapse - it is inconsistent, and experimental tests would have to be done to verify whether the apparent non-local effects predicted are real. (shrink)

Theoreticity is closely connected in the (mainstream) philosophy of science to the idea of non-observability. A closer analysis of measurement, however, may give us a deeper perspective into this connection. This was done by Kyburg in his Theory and Measurement, where he argued that theory is much more pervasive then usually thought of -- even the simplest forms of measurement essentially invoke non-observables. In my article I advance Kyburg’s ideas and try to show that theoreticity implicitly invoked by (...) Kyburg’s pervasive theory may be cast in terms of what I call “non-vagueness principle”. Further, I argue that this principle can provide for a natural demarcation between mature science and other more rudimentary forms of science. (shrink)

In the iconic measurements of atomic spin-1/2 or photon polarization, one employs two separate noninteracting detectors. Each detector is binary, registering the presence or absence of the atom or the photon. For measurements on a d-state particle, we recast the standard von Neumann measurement formalism by replacing the familiar pointer variable with an array of such detectors, one for each of the d possible outcomes. We show that the unitary dynamics of the pre-measurement process restricts the detector outputs to the (...) subspace of single outcomes, so that the pointer emerges from the apparatus. We propose a physical extension of this apparatus which replaces each detector with an ancilla qubit coupled to a readout device. This explicitly separates the pointer into distinct quantum and (effectively) classical parts, and delays the quantum to classical transition. As a result, one not only recovers the collapse scenario of an ordinary apparatus, but one can also observe a superposition of the quantum pointer states. (shrink)

State transformations in quantum mechanics are described by completely positive maps which are constructed from quantum channels. We call a finest sharp quantum channel a context. The result of a measurement depends on the context under which it is performed. Each context provides a viewpoint of the quantum system being measured. This gives only a partial picture of the system which may be distorted and in order to obtain a total accurate picture, various contexts need to be employed. We first (...) discuss some basic definitions and results concerning quantum channels. We briefly describe the relationship between this work and ontological models that form the basis for contextuality studies. We then consider properties of channels and contexts. For example, we show that the set of sharp channels can be given a natural partial order in which contexts are the smallest elements. We also study properties of channel maps. The last section considers mutually unbiased contexts. These are related to mutually unbiased bases which have a large current literature. Finally, we connect them to completely random channel maps. (shrink)

According to Suppes, measurement theory, like any scientific theory, should consist of two parts, a set-theoretical defined structure and the empirical interpretation of that structure. An empirical interpretation means the specification – ‘coordinating definitions’ – of a ‘hierarchy of models’ between the theory and the experimental results. But in the case of measurement theory, he defined the relationship between numerical structure and the empirical structure specifically in terms of homomorphism. This is rather a highly restrictive relation (...) between models, and therefore he never succeeded in giving his measurement theory empirical content. This paper discusses what an empirical measurement theory will look like if we would use less restrictive ‘coordinating definitions’ to specify the relationships between the various models. (shrink)

Part II: Suppes and the mature theory. Representation and uniqueness.

The ten contributions in this volume range widely over topics in the philosophy of mathematics. The four papers in Part I (entitled "Set Theory, Inconsistency, and Measuring Theories") take up topics ranging from proposed resolutions to the paradoxes of naïve set theory, paraconsistent logics as applied to the early infinitesimal calculus, the notion of "purity of method" in the proof of mathematical results, and a reconstruction of Peano's axiom that no two distinct numbers have the same successor. Papers (...) in the second part ("The Challenge of Nominalism") concern the nominalistic thesis that there are no abstract objects. The two contributions in Part III ("Historical Background") consider the contributions of Mill, Frege, and Descartes to the philosophy of mathematics. (shrink)

Suppose that fn is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that equation imageT fn < T f0. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that equation imagefn < f0. This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not (...) themselves constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (shrink)

The purpose of this work is to analyse the cognitive process of the domain theories in terms of the measurement theory to develop a computational machine learning approach for implementing it. As a result, the relational data mining approach, the authors proposed in the preceding books, was improved. We present the approach as an implementation of the cognitive process as the measurement theory perceived. We analyse the cognitive process in the first part of the paper and present the (...) theory and method of the logically most powerful empirical theory discovery in the second. The theory is based on the notion of law-like rules, which conform to all the properties of laws of nature, namely generality, simplicity, maximum refutability and minimum number of parameters. This notion is defined for deterministic and probabilistic cases. Based on the method, the discovery system is developed. The system was successfully applied to many practical tasks. (shrink)

In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely.

The role of completely positive mappings in quantum dynamics and measurement theory is reanalyzed in light of the possibility of a generalized dynamics.

At Philebus 23c4-26d10 Socrates makes a division into three kinds: Unbounded (apeiron), Bound (peras), and Mix (meikton). I review problems for the main interpretations of Unbounded and Mix and review kinds of scales defined in abstract measurement theory. Then I take 23c4-26d10 speech by speech, interpreting the Unbounded as a kind containing partial scales, Bound as the kind containing the relations and quantities needed to turn partial scales into appropriate ratio scales, and Mix as the kind containing ratio scales (...) appropriate for the good things that come to be in the world. (shrink)

This work concerns constructive aspects of measure theory. By considering metric completions of Boolean algebras – an approach first suggested by Kolmogorov – one can give a very simple construction of e.g. the Lebesgue measure on the unit interval. The integration spaces of Bishop and Cheng turn out to give examples of such Boolean algebras. We analyse next the notion of Borel subsets. We show that the algebra of such subsets can be characterised in a pointfree and (...) constructive way by an initiality condition. We then use our work to define in a purely inductive way the measure of Borel subsets. (shrink)

It was shown by Frege that four of the five axioms of Peano can be regarded as analytical truths; and it was shown by Russell that the remaining axiom cannot be regarded as being analytically true or even as being analytically false, that this axiom thus is to be regarded as a synthetic statement. In using the concept of apriority in the sense of Reichenbach, it can be shown that this synthetic axiom is to be regarded as an apriorical truth (...) within the usual background theory of measuring theories, which are used not as generalizations of empirical results but as— not moreover provable— preconditions of receiving measuring results and of ordering these results. Furthermore, the systems of numbers, starting with the natural numbers, are developed in a way such that the pre-rational numbers— but not the rational ones— turn out to be those ones which are used in performing measurements according to such theories, while the pre-real numbers— but not the real ones— then turn out to be those ones which are used in using such measuring theories together with their background theories for purely theoretical reasons. (shrink)

The aim of this paper is to reconstruct the historical evolution of the so-called Measurement Theory. MT has two clearly different periods, the formation period and the mature theory, whose borderline coincides with the publication in 1951 of Suppes' foundational work, ‘A set of independent axioms for extensive quantities’. In this paper two previous research traditions on the foundations of measurement, developed during the formation period, come together in the appropriate way. These traditions correspond, on the one hand, (...) to Helmholtz's, Campbell's and Hölder's studies on axiomatics and real morphisms and, on the other, to the work undertaken by Stevens and his school on scale types and transformations. These two lines of research are complementary in the sense that neither of them is enough taken alone, but together they contain all that is necessary to develop the theory, and it is in Suppes that these complementary approaches converge and all the elements of the theory are appropriately integrated for the first time. With Suppes' work, then, begins what may be called the ‘mature’ theory, which was to develop rapidly later on, especially during the 1960s. Our historical reconstruction is divided into two parts, each part devoted to one of the periods mentioned. Part I also contains a conceptual introduction which aims to establish the use of some notions, specifically those of measurement and metrization. Although the reconstruction is not exhaustive, it intends to be quite complete and up to date compared to what is available in measurement literature; in this sense the aim of this paper is mainly historical but, although secondarily, it also attempts to make some conceptual and metascientific clarifications on the subject of the theory. (shrink)

We recall one Tarski's result about the existence of a non-zero additive measure defined on power set of an infinite set vanishing on finite subsets. This rather surprising result goes back to 1930 and it allows to introduce a non-trivial linear functional invariant under changes of finitely many inputs.