We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak (...) models. (shrink)

Erik Angner has argued that simultaneous endorsement of the representational theory of measurement and psychometrics leads to inconsistency. His claim rests on an implicit assumption: RTM and psychometrics are full-fledged approaches to measurement. I argue that RTM and psychometrics are only partial approaches that deal with different aspects of measurement, and that therefore simultaneous endorsement of the two is not inconsistent. The argument has implications for the improvement of measurement practices.

In 1957, Feyerabend delivered a paper titled ‘On the Quantum-Theory of Measurement’ at the Colston Research Symposium in Bristol to sketch a completion of von Neumann's measurement scheme without collapse, using only unitary quantum dynamics and well-motivated statistical assumptions about macroscopic quantum systems. Feyerabend's paper has been recognised as an early contribution to quantum measurement, anticipating certain aspects of decoherence. Our paper reassesses the physical and philosophical content of Feyerabend's contribution, detailing the technical steps as well (...) as its overall philosophical motivations and consequences. Summarising our results, Feyerabend interpreted collapse as a positivist assumption in quantum mechanics leading to a strict distinction between the uninterpreted formalism of unitary evolution in quantum mechanics and the classically interpreted observational language describing post-measurement outcomes. Thus Feyerabend took the no-collapse completion of the von Neumann measurement scheme to show the dispensability of the positivist assumption, leading the way to a realistic interpretation of quantum theory. We note, however, that there are substantial problems with his account of measurement that bring into question its viability as a legitimate foil to the orthodox view. We further argue that his dissatisfaction with the von Neumann measurement scheme is indicative of early views on theoretical pluralism. (shrink)

The notion of fuzzy event is introduced in the theory of measurement in quantum mechanics by indicating in which sense measurements can be considered to yield fuzzy sets. The concept of probability measure on fuzzy events is defined, and its general properties are deduced from the operational meaning assigned to it. It is pointed out that such probabilities can be derived from the formalism of quantum mechanics. Any such probability on a given fuzzy set is related to the (...) frequency of occurrence within that set of points in a random sample, where the sample points are themselves fuzzy sets obtained as outcomes of measurements of, in general, incompatible observables on replicas of the system in the same prepared state. (shrink)

In this paper we motivate and develop the analytic theory of measurement, in which autonomously specified algebras of quantities (together with the resources of mathematical analysis) are used as a unified mathematical framework for modeling (a) the time-dependent behavior of natural systems, (b) interactions between natural systems and measuring instruments, (c) error and uncertainty in measurement, and (d) the formal propositional language for describing and reasoning about measurement results. We also discuss how a celebrated theorem in (...) analysis, known as Gelfand representation, guarantees that autonomously specified algebras of quantities can be interpreted as algebras of observables on a suitable state space. Such an interpretation is then used to support (i) a realist conception of quantities as objective characteristics of natural systems, and (ii) a realist conception of measurement results (evaluations of quantities) as determined by and descriptive of the states of a target natural system. As a way of motivating the analytic approach to measurement, we begin with a discussion of some serious philosophical and theoretical problems facing the well-known representational theory of measurement. We then explain why we consider the analytic approach, which avoids all these problems, to be far more attractive on both philosophical and theoretical grounds. (shrink)

This paper discusses a relational modeling of measurement which is complementary to the standard representational point of view: by focusing on the experimental character of the measurand-related comparison between objects, this modeling emphasizes the role of the measuring systems as the devices which operatively perform such a comparison. The non-idealities of the operation are formalized in terms of non-transitivity of the substitutability relation between measured objects, due to the uncertainty on the measurand value remaining after the measurement. The (...) metrological structure of traceability is shown to be an effective solution to cope with the problem of the general non-transitivity of measurement results. A preliminary theory is introduced as a possible formalization for the presented model. (shrink)

This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules (...) for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics. (shrink)

This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly classical observables of Newtonian physics to their quasi-classical counterparts the known rules (...) for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics. (shrink)

This paper is concerned with the description of the process of measurement within the context of a quantum theory of the physical world. It is noted that quantum mechanics permits a quasi-classical description (classical in the limited sense implied by the correspondence principle of Bohr) of those macroscopic phenomena in terms of which the observer forms his perceptions. Thus, the process of measurement in quantum mechanics can be understood on the quasi-classical level by transcribing from the strictly (...) classical observables of Newtonian physics to their quasi-classical counterparts the known rules for the measurement of the former. The remaining physical problem is the delineation of the circumstances in which the correlation of a peculiarly quantum mechanical observable A with a classically measurable observable B can result in a significant measurement of A. This is undertaken within the context of quantum theory. The resulting clarification of the process of measurement has important implications relative to the philosophic interpretation of quantum mechanics. (shrink)

"Proceedings of an International Symposium on Measuring Economic Freedom, held July 28-30, 1988, in Vancouver, British Columbia"--T.p. verso. Includes bibliographical references (p. [174]-175).

The new techniques and ideas in quantum interferometry with neutrons, photons, atoms, electrons, and Bose condensates that fluorished in the last two decades have influenced in a decisive way the thinking and the research in the foundations and interpretation of quantum mechanics. The controversies existing among different schools on the reality of matter waves of quantum theory, the postulates of quantum measurement theory, and the (in)completeness of quantum mechanics have to be approached now in a new way. (...) Our argumentation follows the spirit of the Paris school. (shrink)

We discuss the criteria for deriving new information from coordinate transformations, focusing on the property of implementability, or measurability in practice. We contrast the role of coordinate transformations in classical and quantum physics, and demonstrate that many well-known applications fail to meet the criteria for new information. Finally, we discuss some mathematical properties of the coordinate transformations, and then relate these properties to a practical measurement scheme.

Psychometrics and the representational theory of measurement are widely used in social scientific measurement. They are currently pursued largely in isolation from one another. I argue that despite their separation in practice, RTM and psychometrics are complementary approaches, because they can contribute in complementary ways to the establishment of what I argue is a crucial measurement property, namely, representational interpretability. Because RTM and psychometrics are complementary in the establishment of representational interpretability, the current separation of (...) class='Hi'>measurement approaches is unfounded. 1Introduction2Two Approaches to Measurement 2.1Representational theory of measurement2.2Psychometrics2.3Representational interpretability3Complementarity, Conceptually 3.1Representational theory of measurement: Conditions of representational interpretability3.2Psychometrics: Evidence of representational interpretability4Complementarity in Action 4.1What is the Rasch model?4.2Rasch and conjoint measurement5Conclusion: Critics and Fruits of Complementarity. (shrink)

On the received view, the Representational Theory of Measurement reduces measurement to the numerical representation of empirical relations. This account of measurement has been widely criticized. In this article, I provide a new interpretation of the Representational Theory of Measurement that sidesteps these debates. I propose to view the Representational Theory of Measurement as a library of theorems that investigate the numerical representability of qualitative relations. Such theorems are useful tools for concept (...) formation that, in turn, is one crucial aspect of measurement for a broad range of cases in linguistics, rational choice, metaphysics, and the social sciences. (shrink)

It has become customary to locate the origins of modern measurement theory in the works of Helmholtz and Hölder. If by ‘modern measurement theory’ is meant the representational theory, then this may not be an accurate assessment. Both Helmholtz and Hölder present theories of measurement which are closely related to the classical conception of measurement. Indeed, Hölder can be interpreted as bringing this conception to fulfilment in a synthesis of Euclid, Newton, and Dedekind. (...) The first explicitly representational theory appears to have been Russell's. He rejected the traditional concept of quantity and its connection to number, making the use of number in quantitative science problematical. He solved the problem via representationalism. (shrink)

We propose a logic to reason about data collected by a num- ber of measurement systems. The semantic of this logic is grounded on the epistemic theory of measurement that gives a central role to measure- ment devices and calibration. In this perspective, the lack of evidences (in the available data) for the truth or falsehood of a proposition requires the introduction of a third truth-value (the undetermined). Moreover, the data collected by a given source are here (...) represented by means of a possible world, which provide a contextual view on the objects in the domain. We approach (possibly) conflicting data coming from different sources in a social choice theoretic fashion: we investigate viable opera- tors to aggregate data and we represent them in our logic by means of suitable (minimal) modal operators. (shrink)

The present paper is concerned with Helmholtz’s theory of measurement. It will be argued that an adequate understanding of this theory depends on how Helmholtz’s application of the concepts of perception and coincidence is interpreted. In contrast both to conventionalist and Kantian readings of Helmholtz’s theory, a more realistic interpretation will be suggested.

The thesis that numbers are ratios of quantities has recently been advanced by a number of philosophers. While adequate as a definition of the natural numbers, it is not clear that this view suffices for our understanding of the reals. These require continuous quantity and relative to any such quantity an infinite number of additive relations exist. Hence, for any two magnitudes of a continuous quantity there exists no unique ratio. This problem is overcome by defining ratios, and hence real (...) numbers, as binary relations between infinite standard sequences. This definition leads smoothly into a new definition of measurement consonant with the traditional view of measurement as the discovery or estimation of numerical relations. The traditional view is further strengthened by allowing that the additive relations internal to a quantity are distinct from relations observed in the behaviour of objects manifesting quantities. In this way the traditional theory can accommodate the theory of conjoint measurement. This is worth doing because the traditional theory has one great strength lacked by its rivals: measurement statements and quantitative laws are able to be understood literally. 1 This paper was completed in 1990-1. while the author was a visiting scholar at the Irvine Research Unit in Mathematical Behavioral Sciences. University of California. Irvine. The author wishes to thank the Director. Professor R. D. Luce, for the generous provision of space and facilities within the Unit and for his critical comments upon some of the ideas expressed herein: Professor L. Narens. for his trenchant criticisms: and the University of Sydney, for granting Special Study Leave and financial assistance to make the visit possible. (shrink)

A theory of the joint measurement of quantum mechanical observables is generalized in order to make it applicable to the measurement of the local observables of field theory. Subsequently, the property of local commutativity, which is usually introduced as a postulate, is derived by means of the theory of measurement from a requirement of mutual nondisturbance, which, for local observables performed at a spacelike distance from each other, is interpreted as a requirement of macrocausality. (...) Alternative attempts at establishing a deductive relationship between relativistic causality and local commutativity are reviewed, but found wanting, either because of the assumption of an unwarranted objectivity of the object system (algebraic approach) or because of the use of a projection postulate (operational approach). Finally, the quantum mechanical nonobjectivity is related to certain features of nonlocality which are present in the formalism of quantum mechanics. (shrink)

The Representational Theory of Measurement (RTM), especially the canonical three volume Foundations of Measurement by Krantz et al., is a landmark accomplishment in our understanding of measurement. Despite this, it has been far from easy to pinpoint what exactly we can learn about measurement from RTM, and who the target audience for RTM’s formal results should be. In what sense does RTM provide foundations of measurement, and what is the philosophical significance of such foundations? (...) I argue that RTM provides semantic foundations of measurement, and that their philosophical significance lies in a shift towards structural representation. (shrink)

The transition from the traditional to the representational theory of measurement around the turn of the century was accompanied by little sustained criticism of the former. The most forceful critique was Bertrand Russell''s 1897 Mind paper, On the relations of number and quantity. The traditional theory has it that real numbers unfold from the concept of continuous quantity. Russell''s critique identified two serious problems for this theory: (1) can magnitudes of a continuous quantity be defined without (...) infinite regress; and (2) can additive relations between such magnitudes exist if magnitudes are not divisible? The present paper shows how the traditional theory answers these questions and compares the traditional and representational theories as contributions to our understanding of the logic of application. (shrink)

Von Neumann's theory of measurement in quantum mechanics is reinterpreted so that the experimental arrangement specifies the location of the “cut” by calling for the separate observation of the object and the measuring apparatus after the initial measurement interaction. The measurement ascertains which element of the mixture describing the final state of the apparatus is actually present. The relevance and feasibility of observing the final coherent state of the object plus apparatus is criticized and the paradoxes (...) of “Schrödinger's cat” and “Wigner's friend” are discussed. (shrink)

It has been suggested that we ought to idealize the apparatus used to measure quantum systems as consisting of an infinite number of particles. Various authors have claimed that if we do so we do not need to take seriously the limitations on measurement incorporated into the Wigner-Araki-Yanase quantum theory of measurement. Bub and claims we can solve the measurement problem if we make this assumption. I argue against both claims on the basis of differences between (...) the role of such idealizations in classical and quantum mechanics. (shrink)

The problem of measurement can be reformulated as the problem of measurability: What are the conditions under which measurement becomes possible at all? And what is the ontological status of concrete measurement outcomes? It will be shown in the course of this article that what Joel Michell deemed the ‘representational theory’ of measurement provides an adequate framework for answering these questions. However, contrary to Michell, I will point out that Hermann von Helmholtz should be seen (...) as an important forerunner of the representational view. Furthermore, it will be argued that the Finnish logical empiricist Eino Kaila carried on Helmholtz’s approach by combining it with a certain form of ontological invariantism. On the whole, a ‘structural realist’ account of measurement will be suggested. (shrink)

It has been suggested that the measuring apparatus used to measure quantum systems ought to be idealized as consisting of an infinite number of quantum systems. Let us call this the infinity assumption. The suggestion that we ought to make the infinity assumption has been made in connection with two closely related but distinct problems. One is the problem of determining the importance of the limitations on measurement incorporated into the Wigner-Araki-Yanase quantum theory of measurement. The other (...) is the measurement problem. These problems are not internal to the quantum mechanical formalism. They arise when we obtain particular formal results in applying the quantum mechanical formalism to measurement interactions. The formal results are problematic because they resist explanation. Various authors have claimed that by making the infinity assumption we can neglect the limitations on measurement. Bub (1988) and (1989) claims it allows us to solve the measurement problem. I will argue that both claims are unjustified. (shrink)

The central problem in the quantum theory of measurement, how to describe the process of state reduction in terms of the quantum mechanical formalism, is solved on the basis of the relativity of quantal states, which implies that once the apparatus is detected in a well-defined state, the object state must reduce to a corresponding one. This is a process termed by Schrödinger disentanglement. Here, it is essential to observe that Renninger's negative result does constitute an actual (...) class='Hi'>measurement process. From this point of view, Heisenberg's interpretation of his microscope experiment and the Einstein-Podolsky-Rosen arguments are reinvestigated. Satisfactory discussions are given to various experimental situations, such as the Stern-Gerlach-type experiment, successive measurements, macroscopic measurements, and Schrödinger's cat. Finally it is proposed to regard a state vector in quantum mechanics as an irreducible physical construct, in Margenau's sense, that is not further analyzable both mathematically and conceptually. (shrink)

We argue that inductive analysis and operational assessment of the scientific process can be justifiably and fruitfully brought together, whereby the citation metrics used in the operational analysis can effectively track the inductive dynamics and measure the research efficiency. We specify the conditions for the use of such inductive streamlining, demonstrate it in the cases of high energy physics experimentation and phylogenetic research, and propose a test of the method’s applicability.

Max Albert has recently argued that the theory of power indices “should not ... be considered as part of political science” and that “[v]iewed as a scientific theory, it is a branch of probability theory and can safely be ignored by political scientists”. Albert’s argument rests on a particular claim concerning the theoretical status of power indices, namely that the theory of power indices is not a positive theory, i.e. not one that has falsifiable implications. (...) I re-examine the theoretical status of power indices and argue that it would be unwise for political scientists to ignore such indices. Although I agree with Albert that the theory of power indices is not a positive theory, I suggest that it is a theory of measurement that can usefully supplement other positive and normative socialscientific theories. (shrink)

An account of the contributions of Louis de Broglie to the quantum theory of measurement processes is presented. It being impracticable to cover all research work done by de Broglie and his School about this matter, stress is put on the refutation of von Neumann's theorem, on the alternative description arising from the double solution theory, and on some critical analysis of the usually accepted formalism.

The concept of retroactive apparent occurrence, the main ingredient of Wheeler's delayed-choice thought experiments, is systematically incorporated into the quantum theory of measurement (in the framework of the recent review of Busch, Lahti, and Mittelstaedt). Besides, the (general) notion of individual-system measurement is introduced, and, due to it, premeasurement is defined by a truly minimal condition. Finally, retroactive apparent occurrence is made use of to derive apparent objectification in measurement. The derivation is discussed in the framework (...) of the quantum mechanical approach to the objectification problem. (shrink)

It is demonstrated that neither the arguments leading to inconsistencies in the description of quantum-mechanical measurement nor those “explaining” the process of measurement by means of thermodynamical statistics are valid. Instead, it is argued that the probability interpretation is compatible with an objective interpretation of the wave function.

Modifications of current theories of ordinal, interval and extensive measurement are presented, which aim to accomodate the empirical fact that perfectly exact measurement is not possible (which is inconsistent with current theories). The modification consists in dropping the assumption that equality (in measure) is observable, but continuing to assume that inequality (greater or lesser) can be observed. The modifications are formulated mathematically, and the central problems of formal measurement theory--the existence and uniqueness of numerical measures consistent (...) with data--are re-examined. Some results also are given on a problem which does not arise in current theories: namely that of determining limits of accuracy attainable on the basis of observations. (shrink)

Three- to six-year-olds were given Heyes's proposed task and theory of mind tasks. Although they correlated, Heyes's was harder; only 50% of participants with a theory of mind reached a criterion of 75% correct. Because of the complex series of inferences involved in Heyes's task, it is possible that one could have a theory of mind and fail Heyes's version.

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)

This chapter contains sections titled: Introduction Qualitative Hedonism Objections to Mill's Qualitative Hedonism: Internal Inconsistency and Value Pluralism The Judgment of Competent Agents: Self‐Development and Value Measurement Self‐Development and Virtue Ethics Further Reading.