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Summary Measurement is a fundamental empirical process aimed at acquiring and codifying information about an entity, the object or system under measurement. This process is commonly interpreted in functional terms as a production process, accomplished by means of a measurement system, whose input is the system under measurement and whose output is a piece of information, the property value, about a certain instance of a general property of that system, the measurand. As a consequence, the central problem concerning the definition of measurement turns into the one of characterizing the just mentioned process. When an empirical general property is specified, any system under measurement can be viewed as a member of a class of systems characterized by that property. When provided with a set of relations between its elements, this class is called an empirical relational system and measurement can be conceived of as a mapping assigning numbers to elements of this system in such a way that the relations between these elements are preserved by relations between numbers in a numerical relational system. This is the model underlying the so-called representational theory of measurement, considered nowadays the standard measurement theory. According to this model to measure is to construct a representation of an empirical system to a numerical system, under the hypothesis that relations in the empirical system are somehow observable. The model has many merits, but it is also subject to many problems. In particular, the crucial drawback is given by the difficulty of linking the proposed conception of measurement with the way in which measurement is accounted for from a metrological point of view, specifically the point of view underlying the International Vocabulary of Metrology. Hence, the debate concerning the characterization of measurement is still open, where the principal task consists in defining a general model aiming at (i) providing a sound interpretation of measurement as structured process; (ii) identifying the ontological conditions to be fulfilled for measurement to be possible; (iii) identifying the epistemic conditions to be fulfilled for measurement results to be able to justify empirical assertions.
Key works The representational theory of measurement has its roots in the work of Scott and Suppes 1958 and has found its more extensive exposition in the three volumes of the Foundations of Measurement (1971, 1989, 1990), but see also Roberts 1985, for a more friendly presentation, and Narens 1985. The metrological standpoint is summarized in the International Vocabulary of Metrology (VIM). For a problematization of the representational theory see Domotor et al. 2008, where an analytical approach to measurement is developed, and Frigerio et al. 2010, where a synthesis between the representional approach and the metrological approach is proposed.
Introductions See Suppes 2002 for a general introduction to the representational standpoint.
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  1. Ernest W. Adams (1982). Approximate Generalizations and Their Idealization. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982:199 - 207.
    Aspects of a formal theory of approximate generalizations, according to which they have degrees of truth measurable by the proportions of their instances for which they are true, are discussed. The idealizability of laws in theories of fundamental measurement is considered: given that the laws of these theories are only approximately true "in the real world", does it follow that slight changes in the extensions of their predicates would make them exactly true?
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  2. Ernest W. Adams (1966). On the Nature and Purpose of Measurement. Synthese 16 (2):125 - 169.
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  3. Ernest W. Adams (1965). Elements of a Theory of Inexact Measurement. Philosophy of Science 32 (3/4):205-228.
    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 (...)
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  4. Joseph Agassi (1968). Precision in Theory and in Measurement. Philosophy of Science 35 (3):287-290.
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  5. P. Alexander (1969). ELLIS, B. - "Basic Concept of Measurement". [REVIEW] Mind 78:627.
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  6. Holger Andreas (2008). Ontological Aspects of Measurement. Axiomathes 18 (3):379-394.
    The concept of measurement is fundamental to a whole range of different disciplines, including not only the natural and engineering sciences, but also laboratory medicine and certain branches of the social sciences. This being the case, the concept of measurement has a particular relevance to the development of top-level ontologies in the area of knowledge engineering. For this reason, the present paper is concerned with ontological aspects of measurement. We are searching for a list of concepts that are apt to (...)
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  7. William H. Angoff (1987). Philosophical Issues of Current Interest to Measurement Theorists. Theoretical and Philosophical Psychology 7 (2):112-122.
    The research interests of measurement theorists in psychology are, expectedly, largely methodological, entailing a search for improved ways of applying statistical principles and methods to problems of measurement. However, these theorists are fundamentally psychologists, and their interests are, also expectedly, rooted in the substantive areas in psychology and education in which their methods are applied. Several such areas are of particular importance today, provoking continuing discussion at a broad range of conceptual and methodological levels. Among the most perplexing of these (...)
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  8. Gavin Ardley (1969). Metaphysics and Measurement. Philosophical Studies 18:227-227.
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  9. W. Balzer (1983). Theory and Measurement. Erkenntnis 19 (1-3):2 - 25.
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  10. W. Balzer & C. M. Dawe (1986). KYBURG Jr, H. E. [1984]: Theory and Measurement. Cambridge University Press. British Journal for the Philosophy of Science 37 (4):506-510.
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  11. Vadim Batitsky (1998). Empiricism and the Myth of Fundamental Measurement. Synthese 116 (1):51 - 73.
  12. Michael Baumgartner (2010). Measuring and Governing, Review of "The Law-Governed Universe" by John T. Roberts. [REVIEW] Metascience 19 (3):409-412.
  13. José A. Benardete (1968). Continuity and the Theory of Measurement. Journal of Philosophy 65 (14):411-430.
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  14. A. Cornelius Benjamin (1933). The Logic of Measurement. Journal of Philosophy 30 (26):701-710.
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  15. Karel Berka (1984). Measurement. Its Concepts, Theories and Problems. Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 15 (2):354-363.
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  16. Karel Berka (1974). D. H. Krantz, R. D. Luce, P. Suppes and A. Tversky, "Foundations of Measurement", Vol. I. [REVIEW] Theory and Decision 5 (4):461.
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  17. Marcel J. Boumans, Invariance and Calibration.
    The Representational Theory of Measurement conceives measurement as establishing homomorphisms from empirical relational structures into numerical relation structures, called models. Models function as measuring instruments by transferring observations of an economic system into quantitative facts about that system. These facts are evaluated by their accuracy. Accuracy is achieved by calibration. For calibration standards are needed. Then two strategies can be distinguished. One aims at estimating the invariant (structural) equations of the system. The other is to use known stable facts about (...)
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  18. Dragana Bozin (1998). Alternative Combining Operations in Extensive Measurement. Philosophy of Science 65 (1):136-150.
    This paper concerns the ways in which one can/cannot combine extensive quantities. Given a particular theory of extensive measurement, there can be no alternative ways of combining extensive quantities, where 'alternative' means that one combining operation can be used instead of another causing only a change in the number assigned to the quantity. As a consequence, rectangular concatenation cannot be an alternative combining operation for length as was suggested by Ellis and agreed by Krantz, Luce, Suppes, and Tversky. I argue (...)
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  19. Dragana Bozin (1993). Alternative Scales for Extensive Measurement: Combining Operations and Conventionalism. Dissertation, Rice University
    This thesis concerns alternative concatenating operations in extensive measurements and the degree to which concatenating operations are matter of convention. My arguments are directed against Ellis' claim that what prevents us from choosing alternative ways of combining extensive quantities is only convenience and simplicity and that the choice is not based on empirical reasons. ;My first argument is that, given certain relational theories of measurement, there can be no more than one concatenating operation per quantity; because combining operations are the (...)
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  20. William Brown (1911). The Essentials of Mental Measurement.
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  21. Henry C. Byerly & Vincent A. Lazara (1973). Realist Foundations of Measurement. Philosophy of Science 40 (1):10-28.
    This paper defends a realist interpretation of theories and a modest realism concerning the existence of quantities as providing the best account both of the logic of quantity concepts and of scientific measurement practices. Various operationist analyses of measurement are shown to be inadequate accounts of measurement practices used by scientists. We argue, furthermore, that appeals to implicit definitions to provide meaning for theoretical terms over and above operational definitions fail because implicit definitions cannot generate the requisite descriptive content. The (...)
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  22. N. R. Campbell & H. Jeffreys (1938). Measurement and Its Importance for Philosophy. Aristotelian Society Supplementary Volume 17:121-151.
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  23. N. R. Campbell & H. Jeffreys (1938). Symposium: Measurement and Its Importance for Philosophy. Aristotelian Society Supplementary Volume 17:121 - 151.
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  24. Robert L. Causey (1969). Derived Measurement, Dimensions, and Dimensional Analysis. Philosophy of Science 36 (3):252-270.
    This paper presents a representational theory of derived physical measurements. The theory proceeds from a formal definition of a class of similar systems. It is shown that such a class of systems possesses a natural proportionality structure. A derived measure of a class of systems is defined to be a proportionality-preserving representation whose values are n-tuples of positive real numbers. Therefore, the derived measures are measures of entire physical systems. The theory provides an interpretation of the dimensional parameters in a (...)
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  25. C. West Churchman & Philburn Ratoosh (1960). Measurement: Definitions and Theories. Journal of Philosophy 57 (15):513-514.
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  26. Aaron Cicourel (2003). Method and Measurement*(1964). In Gerard Delanty & Piet Strydom (eds.), Philosophies of Social Science: The Classic and Contemporary Readings. Open University 191.
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  27. Hans Colonius (1978). On Weak Extensive Measurement. Philosophy of Science 45 (2):303-308.
    Extensive measurement is called weak if the axioms allow two objects to have the same scale value without being indifferent with respect to the order. Necessary and/or sufficient conditions for such representations are given. The Archimedean and the non-Archimedean case are dealt with separately.
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  28. Gordon Cooper & Stephen M. Humphry (2012). The Ontological Distinction Between Units and Entities. Synthese 187 (2):393-401.
    The base units of the SI include six units of continuous quantities and the mole, which is defined as proportional to the number of specified elementary entities in a sample. The existence of the mole as a unit has prompted comment in Metrologia that units of all enumerable entities should be defined though not listed as base units. In a similar vein, the BIPM defines numbers of entities as quantities of dimension one, although without admitting these entities as base units. (...)
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  29. O. Darrigol (2003). Number and Measure: Hermann Von Helmholtz at the Crossroads of Mathematics, Physics, and Psychology. Studies in History and Philosophy of Science Part A 34 (3):515-573.
    In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics (Grassmann / Du Bois), on the possibility of quantitative psychology (Fechner / Kries, Wundt / Zeller), and on the meaning of temperature measurement (Maxwell, Mach). Late nineteenth-century scrutinisers of the foundations of mathematics (Dedekind, Cantor, Frege, Russell) made little of Helmholtz's essay. Yet it inspired two mathematicians with an eye on (...)
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  30. Elton Ray Davis (1980). Fundamental Measurement: Some Lessons From Classical Physics. Dissertation, University of California, Riverside
    One can conclude then that in some instances theories are invoked in introducing quantitative concepts into science, even when the concept is treated as though it had been introduced by fundamental measurement. On the other hand, one sees in Maxwell an instance of fundamental measurement that does not invoke theory. It is not the case either that all our metric concepts are theory-laden or that they are theory-free. We must look to individual cases to discover how measurement functions in science. (...)
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  31. J. A. Diez (1993). Comentario a Foundations of Measurement 2 y 3'. Theoria 19:163-168.
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  32. JoséA Díez (1997). A Hundred Years of Numbers. An Historical Introduction to Measurement Theory 1887–1990. Studies in History and Philosophy of Science Part A 28 (1):167-185.
    Part II: Suppes and the mature theory. Representation and uniqueness.
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  33. Herbert Dingle (1950). A Theory of Measurement. British Journal for the Philosophy of Science 1 (1):5-26.
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  34. Zoltan Domotor (1972). Species of Measurement Structures. Theoria 38 (1-2):64-81.
  35. Zoltan Domotor & Vadim Batitsky (2008). The Analytic Versus Representational Theory of Measurement: A Philosophy of Science Perspective. Measurement Science Review 8 (6):129-146.
    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 (...)
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  36. Brian Ellis (1961). Some Fundamental Problems of Indirect Measurement. Australasian Journal of Philosophy 39 (1):13 – 29.
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  37. Brian Ellis (1960). Some Fundamental Problems of Direct Measurement. Australasian Journal of Philosophy 38 (1):37 – 47.
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  38. Tyler Estler (1999). Measurement as Inference: FundamentalIdeas. CIRP Annals - Manufacturing Technology 48 (2):611-631.
    We review the logical basis of inference as distinct from deduction, and show that measurements in general, and dimensional metrology in particular, are best viewed as exercises in probable inference: reasoning from incomplete information. The result of a measurement is a probability distribution that provides an unambiguous encoding of one's state of knowledge about the measured quantity. Such states of knowledge provide the basis for rational decisions in the face of uncertainty. We show how simple requirements for rationality, consistency, and (...)
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  39. Jean-Claude Falmagne (1980). A Probabilistic Theory of Extensive Measurement. Philosophy of Science 47 (2):277-296.
    Algebraic theories for extensive measurement are traditionally framed in terms of a binary relation $\lesssim $ and a concatenation (x,y)→ xy. For situations in which the data is "noisy," it is proposed here to consider each expression $y\lesssim x$ as symbolizing an event in a probability space. Denoting P(x,y) the probability of such an event, two theories are discussed corresponding to the two representing relations: p(x,y)=F[m(x)-m(y)], p(x,y)=F[m(x)/m(y)] with m(xy)=m(x)+m(y). Axiomatic analyses are given, and representation theorems are proven in detail.
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  40. Ludwik Finkelstein (2009). Widely-Defined Measurement. An Analysis of Challenges. Measurement 42 (9):1270–1277.
    The paper examines fundamental problems of widely-defined measurement that lie outside the representation concerns of measurement theory. It is intended as a starting point of a research agenda. It shows that measurement is applied in a wide range of diverse domains of knowledge and enquiry for which a wide-sense definition of measurement is necessary. It examines philosophical objections to the application of measurement. It considers in particular problems of measurand concept formation, validity, verifiability and of theories for the measurand. It (...)
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  41. Ludwik Finkelstein (2003). Widely, Strongly and Weakly Defined Measurement. Measurement 34 (1):39-48.
    The paper discusses the concept of measurement. Measurement, in the wide sense, is defined as a process of empirical, objective assignment of symbols to attributes of objects and events of the real world, in such a way as to describe them. Strongly defined measurement is measurement that conforms to the paradigm of the physical sciences. Weakly defined measurement is measurement in the wide sense, but which is not strongly defined. Strongly and weakly defined measurements are analysed and compared. Other forms (...)
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  42. Ludwik Finkelstein (1994). Measurement and Instrumentation Science. An Analytical Review. Measurement 14 (1):3-14.
    The paper reviews the state and trends of Measurement and Instrumentation Science viewed broadly as the generic transferable principles of measurement, and of the instrumentation by which it is performed. It regards measurement as an information process and instruments as information machines. It treats Measurement and Instrumentation Science as a coherent part of the sciences of information, systems, and computing, with distinctive features which it discusses. The paper provides an analytical review of the development of Measurement and Instrumentation Science and (...)
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  43. Ludwik Finkelstein (1984). A Review of the Fundamental Concepts of Measurement. [REVIEW] Measurement 2 (1):25-34.
    The paper surveys the current state of the theory of the fundamentalconcepts of measurement which is based on the model theory of logic. A brief review is given of the historical development of measurement theory. The model-theoretic definition of measurement is presented, together with a discussion of representation and uniqueness conditions. Nominal, ordinal, extensive and interval measurement structures are outlined. The classification of scale types and the problem of meaningfulness are considered. A survey is given of conjoint and derived measurement. (...)
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  44. Aldo Frigerio, Alessandro Giordani & Luca Mari (2010). Outline of a General Model of Measurement. Synthese 175 (2):123-149.
    Measurement is a process aimed at acquiring and codifying information about properties of empirical entities. In this paper we provide an interpretation of such a process comparing it with what is nowadays considered the standard measurement theory, i.e., representational theory of measurement. It is maintained here that this theory has its own merits but it is incomplete and too abstract, its main weakness being the scant attention reserved to the empirical side of measurement, i.e., to measurement systems and to the (...)
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  45. Alessandro Giordani & Luca Mari (2014). Modeling Measurement: Error and Uncertainty. In Marcel Boumans, Giora Hon & Arthur Petersen (eds.), Error and Uncertainty in Scientific Practice. Pickering & Chatto 79-96.
    In the last few decades the role played by models and modeling activities has become a central topic in the scientific enterprise. In particular, it has been highlighted both that the development of models constitutes a crucial step for understanding the world and that the developed models operate as mediators between theories and the world. Such perspective is exploited here to cope with the issue as to whether error-based and uncertainty-based modeling of measurement are incompatible, and thus alternative with one (...)
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  46. Alessandro Giordani & Luca Mari (2012). Measurement, Models, and Uncertainty. IEEE Transactions on Instrumentation and Measurement 61 (8):2144 - 2152.
    Against the tradition, which has considered measurement able to produce pure data on physical systems, the unavoidable role played by the modeling activity in measurement is increasingly acknowledged, particularly with respect to the evaluation of measurement uncertainty. This paper characterizes measurement as a knowledge-based process and proposes a framework to understand the function of models in measurement and to systematically analyze their influence in the production of measurement results and their interpretation. To this aim, a general model of measurement is (...)
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  47. Alessandro Giordani & Luca Mari (2012). Property Evaluation Types. Measurement 45 (3):437-452.
    An appropriate characterization of property types is an important topic for measurement science. On the basis of a set-theoretic model of evaluation and measurement processes, the paper introduces the operative concept of property evaluation type, and discusses how property types are related to, and in fact can be derived from, property evaluation types, by finally analyzing the consequences of these distinctions for the concepts of ‘property’ used in the International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (...)
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  48. Alessandro Giordani & Luca Mari, Quantity and Quantity Value. Proc. TC1-TC7-TC13 14th IMEKO Joint Symposium.
    The concept system around ‘quantity’ and ‘quantity value’ is fundamental for measurement science, but some very basic issues are still open on such concepts and their relations. This paper proposes a duality between quantities and quantity values, a proposal that simplifies their characterization and makes it consistent.
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  49. Alessandro Giordani & Luca Mari (2010). Towards a Concept of Property Evaluation Type. Journal of Physics CS 238 (1):1-6.
    An appropriate characterization of property types is an important topic for measurement science. This paper proposes to derive them from evaluation types, and analyzes the consequences of this position for the VIM3.
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  50. Alvin I. Goldman (1974). On the Measurement of Power. Journal of Philosophy 71 (8):231-252.
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