We present some results about the burgeoning research area concerning set theory of the “κ‐reals”. We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.

Solovay’s random-real forcing [R.M. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory , Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428] is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay’s model that do not follow from the existence of real-valued measurability.

We generalize Solovay's unfolding technique for infinite games and use an Unfolding Theorem to give a uniform method to prove that all analytic sets are in the $\sigma$ -algebras of measurability connected with well-known forcing notions.

We prove three theorems concerning Laver indestructibility, strong compactness, and measurability. We then state some related open questions.

We show that Σ1 4-Amoeba-absoluteness implies that $\forall a \in \mathbb{R}(\omega^{L\lbrack a \rbrack}_1 < \omega^V_1)$ and, hence, Σ1 3-measurability. This answers a question of Haim Judah (private communication).

There is a common argument form in the metaphysics of natural laws literature: a theory of natural law is attacked by offering a claim L as a law of scientific field F (physics, chemistry, biology, etc.), and from this law metaphysical implications contrary to the theory are drawn. Quite often however, L would not be regarded as a law by a scientist of F. Roberts' "measurability account of laws" offers a new and interesting way to more reliably identify the (...) laws of a field F and to eliminate the L that are not laws of F from the literature. (shrink)

In a recent paper, Yoaav Isaacs, Alan Hájek, and John Hawthorne argue for the rational permissibility of "credal imprecision" by appealing to certain propositions associated with non-measurable spatial regions: for example, the proposition that the pointer of a spinner will come to rest within a certain non-measurable set of points on its circumference. This paper rebuts their argument by showing that its premises lead to implausible consequences in cases where one is trying to learn, by making multiple observations, whether a (...) certain outcome is associated with a non-measurable region or a measurable one. (shrink)

I describe a problem about the relations among symmetries, laws and measurable quantities. I explain why several ways of trying to solve it will not work, and I sketch a solution that might work. I discuss this problem in the context of Newtonian theories, but it also arises for many other physical theories. The problem is that there are two ways of defining the space-time symmetries of a physical theory: as its dynamical symmetries or as its empirical symmetries. The two (...) definitions are not equivalent, yet they pick out the same extension. This coincidence cries out for explanation, and it is not clear what the explanation could be. The Puzzle: Symmetries, Measurability and Invariance 1.1 The symmetries and the measurable quantities of Newtonian mechanics 1.2 The puzzle Two Easy Answers Another Unsuccessful Solution: Appeal to Geometrical Symmetries Locating the Puzzle The Relation between Laws and Measurability A Possible Solution CiteULike Connotea Del.icio.us What's this? (shrink)

A ubiquitous argument against mental-state accounts of well-being is based on the notion that mental states like happiness and satisfaction simply cannot be measured. The purpose of this paper is to articulate and to assess this “argument from measurability.” My main thesis is that the argument fails: on the most charitable interpretation, it relies on the false proposition that measurement requires the existence of an observable ordering satisfying conditions like transitivity. The failure of the argument from measurability, however, (...) does not translate into a defense of mental-state accounts as accounts of well-being or of measures of happiness and satisfaction as measures of well-being. Indeed, I argue, the ubiquity of the argument from measurability may have obscured other, very real problems associated with mental-state accounts of well-being – above all, that happiness and satisfaction fail to track well-being – and with measures of happiness and satisfaction – above all, the tendency toward reification. I conclude that the central problem associated with the measurement of, e.g., happiness as a subjectively experienced mental state is not that it is too hard to measure, but rather that it is too easy to measure. (shrink)

The density matrix ρ describing a decaying system can be expressed in terms of correlations among observables belonging to the subsystems. Due to this structure and to the difficulties in measuring higher rank tensors of decay products for a single decay event, it is found that the mean value of ρ cannot be determined, in general, from measurements on the decay products. We also discuss the consequences of this conclusion as far as tests of quantum mechanics are concerned.

Pollock (2004) argues in favour of Wittgenstein’s (1953) claim that the standard metre bar in Paris has no metric length: Because the standard retains a special status in the system of measurement, it cannot be applied to itself. However, we argue that Pollock is mistaken regarding the feature of the standard metre which supports its special status. While the unit markings were arbitrarily designated, the constitution, preservation and application of the bar have been scientifically developed to optimize stability, and hence (...) predictive accuracy. We argue that it is the ‘hard to improve’ quality of stability that supports the standard’s value in measurement, not any of its arbitrary features. And because the special status of the prototype is tied to its ability to meet this external criterion, the possibility always exists of identifying an alternative, more stable, standard, thereby allowing the original standard to be measured. (shrink)

We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ (...) R such that ω L[ r] 1 = ω 1 and MA holds, then there exists a ▵ 1 3 -selective filter on ω, and from the consistency of ZFC we build a model for ZFC + MA(I) + every ▵ 1 3 -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey. (shrink)

Starting from cardinals κ κ is measurable, we construct a model for the theory “ZF + n < ω[DCn] + ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.

This book considers whether there is a legitimate or even necessary place for the perspective of 'care' when addressing questions of universal justice. To this end, it examines two major frameworks of contemporary moral philosophy_Jürgen Habermas's model of discourse ethics and Jacques Derrida's deconstructive ethics of radical singularity_in which the contrasting standpoints of communicative reciprocation and care for the absolute otherness of the other are respectively prioritized.

The main result of this paper, built on previous work by the author and T. Wilson, is the proof that the theory “ADR+DC + there is an R-complete measure on Θ” is equiconsistent with “ZF+DC+ ADR + there is a supercompact measure on ℘ω1(℘(R))+Θ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of ZF+DC.

We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide.

This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results (...) provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.§1. Complete Boolean algebras and weak distributivity. ABoolean algebrais a setBwith Boolean operationsa˅b,a˄b and −a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b,a˄b and -a, partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S. (shrink)

We examine the problem of the existence (in classical and/or quantum physics) of longitudinal limitations of measurability, defined as limitations preventing the measurement of a given quantity with arbitrarily high accuracy. We consider a measuring device as a generalized communication system, which enables us to use methods of information theory. As a direct consequence of the Shannon theorem on channel capacity, we obtain an inequality which limits the accuracy of a measurement in terms of the average power necessary to (...) transmit the information content of the measurement itself. This inequality holds in a classical as well as in a quantum framework. (shrink)

The measurement of sensory intensity has had a long history, attracting the attention of investigators from many disciplines including physiology, psychology, physics, mathematics, philosophy, and even chemistry. While there has been a continuing doubt by some that sensation has the properties necessary for measurement, experiments designed to obtain estimates of sensory intensity have found that a general rule applies: Equal stimulus ratios produce equal sensory ratios. Theories concerning the basis for this simple psychophysical rule are discussed, with emphasis given to (...) the physical correlate theory, which considers judgments of subjective magnitudes to be based upon estimates of physical dimensions that vary regularly with changes in degree of stimulation. For the most thoroughly investigated sensory scales, brightness and loudness, the physical correlate is considered to be distance. Our “tacit knowledge” of the sensory effects of changing distance plays an essential role in matching motor activities to environmental conditions and in ensuring accurate perceptual evaluations through brightness and loudness constancies. In psychophysical experiments, subjects apparently use this same tacit knowledge when required to estimate relative subjective magnitudes. The evidence related to the physical correlate theory is summarized, and it is concluded that, while under appropriate conditions we demonstrate considerable skill in evaluating environmental relationships, we are quite unable to estimate the neurophysiological nature or quantity of sensory response. A psychophysics devoted to studying conditions required for accuracy and conditions producing errors in the perception of environmental relationships would seem to be more valuable than one devoted to subjective magnitudes. (shrink)

Corporate social responsibility (CSR) is one of the most prominent concepts in the literature and, in short, indicates the positive impacts of businesses on their stakeholders. Despite the growing body of literature on this concept, the measurement of CSR is still problematic. Although the literature provides several methods for measuring corporate social activities, almost all of them have some limitations. The purpose of this study is to provide an original, valid, and reliable measure of CSR reflecting the responsibilities of a (...) business to various stakeholders. Based on a proposed conceptual framework of CSR, a scale was developed through a systematic scale development process. In the study, exploratory factor analysis was conducted to determine the underlying factorial structure of the scale. Data was collected from 269 business professionals working in Turkey. The results of the analysis provided a four-dimensional structure of CSR, including CSR to social and nonsocial stakeholders, employees, customers, and government. (shrink)

Global poverty measurement is important. It is used to allocate scarce resources, evaluate progress, and assess existing projects, policies, and institutional designs. But given the diversity of ways in which poverty is conceived, how can we settle on a conception and measure that can be used for interpersonal and inter-temporal global comparison? -/- This book lays out the key contemporary debates in poverty measurement, and provides a new analytical framework for thinking about poverty conception and measurement. Rather than trying to (...) find some essential meaning of poverty, the author recommends explicitly reflecting on the purposes served by the concept and the values that do and should inform our conceptions and measures. -/- After reviewing the strengths and weaknesses of five competing conceptions of poverty and their corresponding measures, the book concludes with specific recommendations for the future. Poverty measurement should be developed through a process of public reason that gives weight to the voices of those individuals who are most marginalized and deprived. The author suggests new values, desiderata, and candidate indicators that should be used in a pro-poor poverty measure. (shrink)

Conventional quantitative methods for the measurement of organizational legitimacy consider mainly three sources that make judgments about organizations visible: news media, accreditation bodies, and surveys. Over the last decade, however, social media have enabled ordinary citizens to bypass the gatekeeping function of these institutional evaluators and autonomously make individual judgments public. This inclusion of voices beyond functional and formally organized stakeholder groups potentially pluralizes the ongoing discussions about organizations. The individual judgments in blogs, tweets, and Facebook posts give indication about (...) the broader fit between an organization’s perceived behavior and heterogeneous social norms and therefore constitute an indicator of organizational legitimacy that can be accessed and measured. We propose the use of social media data and sentiment analysis to study the affect-based responses to organizational actions by citizens. We critically discuss and compare the method with existing quantitative methods for legitimacy measurement and apply them to a recent case in the banking industry. We discuss the value of the method for studying the process of legitimacy construction as the expression and negotiation of normative judgments about organizations by various evaluators. (shrink)

Measuring the effectiveness of medical interventions faces three epistemological challenges: the choice of good measuring instruments, the use of appropriate analytic measures, and the use of a reliable method of extrapolating measures from an experimental context to a more general context. In practice each of these challenges contributes to overestimating the effectiveness of medical interventions. These challenges suggest the need for corrective normative principles. The instruments employed in clinical research should measure patient-relevant and disease-specific parameters, and should not be sensitive (...) to parameters that are only indirectly relevant. Effectiveness always should be measured and reported in absolute terms (using measures such as 'absolute risk reduction'), and only sometimes should effectiveness also be measured and reported in relative terms (using measures such as 'relative risk reduction')-employment of relative measures promotes an informal fallacy akin to the base-rate fallacy, which can be exploited to exaggerate claims of effectiveness. Finally, extrapolating from research settings to clinical settings should more rigorously take into account possible ways in which the intervention in question can fail to be effective in a target population. (shrink)

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 propose partial measurements as a conceptual tool to understand how to operate with counterfactual claims in quantum physics. Indeed, unlike standard von Neumann measurements, partial measurements can be reversed probabilistically. We first analyze the consequences of this rather unusual feature for the principle of superposition, for the complementarity principle, and for the issue of hidden variables. Then we move on to exploring non-local contexts, by reformulating the EPR paradox, the quantum teleportation experiment, and the entanglement-swapping protocol for the situation (...) in which one uses partial measurements followed by their stochastic reversal. This leads to a number of counter-intuitive results, which are shown to be resolved if we give up the idea of attributing reality to the wavefunction of a single quantum system. (shrink)

One of the most challenging questions surrounding subjective animal welfare is whether these states are measurable: that is, is subjective welfare an appropriately quantifiable target for scientific enquiry and ethical and deliberative calculation? The availability of several different types of measurement scale raises important questions regarding whether subjective experience has the right properties to be meaningfully represented on the types of scale required for different applications. This methodological question has so far received scant attention in the animal welfare literature. In (...) this paper, I address this omission by examining the types of measurement scale we can reasonably expect to apply to animal welfare measurements, and which we will actually need for our applications. I argue that our different applications will require variously ordinal, interval, and ratio scales, and that we have sufficient reason to believe that subjective welfare is a target with the appropriate characteristics to justify the practice of representing it using each of these types of scales. (shrink)

Next SectionAn attempt to resolve the controversy regarding the solution of the Sleeping Beauty Problem in the framework of the Many-Worlds Interpretation led to a new controversy regarding the Quantum Sleeping Beauty Problem. We apply the concept of a measure of existence of a world and reach the solution known as ‘thirder’ solution which differs from Peter Lewis’s ‘halfer’ assertion. We argue that this method provides a simple and powerful tool for analysing rational decision theory problems.

Measurement is a special type of evaluation that is more exact than either opinion or estimation. In the social sciences, in particular, most evaluations are not measures, but rather mixtures of opinion and estimation. Over-measurement represents anchoring to evaluations which are not measures. For an over-measured characteristic, single measures are used when instead a portfolio of possible measures should be used. There are three implications. First, measurements of characteristics which depend on the over-measured characteristic are biased. Secondly, decisions which depend (...) on the over-measured characteristic are biased. Thirdly, over-measurement biases the measurement of uncertainty. (shrink)