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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 (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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  2. Joseph Agassi (1968). Precision in Theory and in Measurement. Philosophy of Science 35 (3):287-290.
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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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.
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  9. Zoltan Domotor (1972). Species of Measurement Structures. Theoria 38 (1-2):64-81.
  10. 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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  11. 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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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. Stephan Hartmann & Luc Bovens, The Variety-of-Evidence Thesis and the Reliability of Instruments: A Bayesian-Network Approach.
    The variety of evidence thesis in confirmation theory states that more varied supporting evidence confirms a hypothesis to a greater degree than less varied evidence. Under a very plausible interpretation of this thesis, positive test results from multiple independent instruments confirm a hypothesis to a greater degree than positive test results from a single instrument. We invoke Bayesian Networks to model confirmation on grounds of evidence that is obtained from less than fully reliable instruments and show that the variety of (...)
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  18. Stig Kanger (1972). Measurement: An Essay in Philosophy of Science. Theoria 38 (1-2):1-44.
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  19. Kareem Khalifa (2004). Erotetic Contextualism, Data-Generating Procedures, and Sociological Explanations of Social Mobility. Philosophy of the Social Sciences 34 (1):38-54.
    Critics of the erotetic model of explanation question its ability to discriminate significant from spurious explanations. One response to these criticisms has been to impose contextual restrictions on a case-by-case basis. In this article, the author argues that these approaches have overestimated the role of interests at the expense of other contextual aspects characteristic of social-scientific explanation. For this reason, he shows how procedures of measuring occupational status and social mobility affected different aspects of one explanation that Peter Blau and (...)
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  20. Arnold Koslow (1982). Quantity and Quality: Some Aspects of Measurement. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1982:183 - 198.
    A description is given of the quantitative-qualitative distinction for terms in theories of measurable attributes, and, adjoined to that account, a suggestion is made concerning the sense in which empirical relational systems have an empirical attribute as their topic or focus. Since this characterization of quantitative terms, relative to a partition, makes no explicit reference to numbers, concatenation operations, or ordering relations, we show how our results are related to some standard theorems in the literature. Analogs of representation and uniqueness (...)
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  21. Alexandre Koyré (1968/1992). Metaphysics and Measurement. Gordon and Breach Science Publishers.
    This collection of six essays centers on Professor Koyre;'s great theme: the relative importance of metaphysics and observation, with controlled experiment a kind of marriage between the two. Professor Koyre;'s thesis might be summed up as a claim that when one is seeking to explain the scientific revolution, attention must be concentrated on the philosophical outlook of the scientist and away from speculative theories. At the time of his death, Alexandre Koyre; was a professor at the Ecole Pratique des Hautes (...)
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  22. David Krantz, Duncan Luce, Patrick Suppes & Amos Tversky (eds.) (1971). Foundations of Measurement, Vol. I: Additive and Polynomial Representations. New York Academic Press.
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  23. Ulrich Krohs, A Priori Measurable Worlds.
    Part of the scientific enterprise is to measure the material world and to explain its dynamics by means of models. However, not only is measurability of the world limited, analyzability of models is so, too. Most often, computer simulations offer a way out of this epistemic bottleneck. They instantiate the model and may help to analyze it. In relation to the material world a simulation may be regarded as a kind of a “non-material scale model”. Like any other scale model, (...)
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  24. Thomas S. Kuhn (1961). The Function of Measurement in Modern Physical Sciences. Isis 52:161-193.
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  25. Henry E. Kyburg Jr (1997). Quantities, Magnitudes, and Numbers. Philosophy of Science 64 (3):377-410.
    Quantities are naturally viewed as functions, whose arguments may be construed as situations, events, objects, etc. We explore the question of the range of these functions: should it be construed as the real numbers (or some subset thereof)? This is Carnap's view. It has attractive features, specifically, what Carnap views as ontological economy. Or should the range of a quantity be a set of magnitudes? This may have been Helmholtz's view, and it, too, has attractive features. It reveals the close (...)
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  26. Henry E. Kyburg Jr (1969). Measurement and Mathematics. Journal of Philosophy 66 (2):29-42.
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  27. Henry E. Kyburg (ed.) (1984). Theory and Measurement. Cambridge University Press.
    Measurement is fundamental to all the sciences, the behavioural and social as well as the physical and in the latter its results provide our paradigms of 'objective fact'. But the basis and justification of measurement is not well understood and is often simply taken for granted. Henry Kyburg Jr proposes here an original, carefully worked out theory of the foundations of measurement, to show how quantities can be defined, why certain mathematical structures are appropriate to them and what meaning attaches (...)
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  28. Benedikt Löwe & Thomas Müller (2011). Data and Phenomena in Conceptual Modelling. Synthese 182 (1):131-148.
    The distinction between data and phenomena introduced by Bogen and Woodward (Philosophical Review 97(3):303–352, 1988) was meant to help accounting for scientific practice, especially in relation with scientific theory testing. Their article and the subsequent discussion is primarily viewed as internal to philosophy of science. We shall argue that the data/phenomena distinction can be used much more broadly in modelling processes in philosophy.
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  29. Duncan Luce, David Krantz, Patrick Suppes & Amos Tversky (eds.) (1990). Foundations of Measurement, Vol. III: Representation, Axiomatization, and Invariance. New York Academic Press.
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  30. R. Duncan Luce (1965). A "Fundamental" Axiomatization of Multiplicative Power Relations Among Three Variables. Philosophy of Science 32 (3/4):301-309.
    Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. (...)
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  31. Luca Mari (2005). The Problem of Foundations of Measurement. Measurement 38 (4):259-266.
    Given the common assumption that measurement plays an important role in the foundation of science, the paper analyzes the possibility that Measurement Science, and therefore measurement itself, can be properly founded. The realist and the representational positions are analyzed at this regards: the conclusion, that such positions unavoidably lead to paradoxical situations, opens the discussion for a new epistemology of measurement, whose characteristics and interpretation are sketched here but are still largely matter of investigation.
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  32. Luca Mari (2003). Epistemology of Measurement. Measurement 34 (1):17-30.
    The paper introduces what is deemed as the general epistemological problem of measurement: what characterizes measurement with respect to generic evaluation? It also analyzes the fundamental positions that have been maintained about this issue, thus presenting some sketches for a conceptual history of measurement. This characterization, in which three distinct standpoints are recognized, corresponding to a metaphysical, an anti-metaphysical, and relativistic period, allows us to introduce and briefly discuss some general issues on the current epistemological status of measurement science.
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  33. Luca Mari (2000). Beyond the Representational Viewpoint: A New Formalization of Measurement. Measurement 27 (2):71-84.
    The paper introduces and formally defines a functional concept of a measuring system, on this basis characterizing the measurement as an evaluation performed by means of a calibrated measuring system. The distinction between exact and uncertain measurement is formalized in terms of the properties of the traceability chain joining the measuring system to the primary standard. The consequence is drawn that uncertain measurements lose the property of relation-preservation, on which the very concept of measurement is founded according to the representational (...)
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  34. Luca Mari & Alessandro Giordani (2012). Quantity and Quantity Value. Metrologia 49 (6):756-764.
    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 relation. This paper argues that quantity values are in fact individual quantities, and that a complementarity exists between measurands and quantity values. This proposal is grounded on the analysis of three basic 'equality' relations: (i) between quantities, (ii) between quantity values and (iii) between quantities and quantity values. A consistent characterization of such concepts is (...)
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  35. Luca Mari, Valentina Lazzarotti & Raffaella Manzini (2009). Measurement in Soft Systems: Epistemological Framework and a Case Study. Measurement 42 (2):241-253.
    Measurement in soft systems generally cannot exploit physical sensors as data acquisition devices. The emphasis in this case is instead on how to choose the appropriate indicators and to combine their values so to obtain an overall result, interpreted as the value of a property, i.e., the measurand, for the system under analysis. This paper aims at discussing the epistemological conditions of the claim that such a process is a measurement, and performance evaluation is the case introduced to support the (...)
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  36. Luca Mari & Sergio Sartori (2007). A Relational Theory of Measurement: Traceability as a Solution to the Non-Transitivity of Measurement Results. Measurement 40 (2):233-242.
    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 (...)
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  37. James W. McAllister (1997). Phenomena and Patterns in Data Sets. Erkenntnis 47 (2):217-228.
    Bogen and Woodward claim that the function of scientific theories is to account for 'phenomena', which they describe both as investigator-independent constituents of the world and as corresponding to patterns in data sets. I argue that, if phenomena are considered to correspond to patterns in data, it is inadmissible to regard them as investigator-independent entities. Bogen and Woodward's account of phenomena is thus incoherent. I offer an alternative account, according to which phenomena are investigator-relative entities. All the infinitely many patterns (...)
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  38. Ernest Nagel & C. G. Hempel (1931). Measurement. Erkenntnis 2 (1):313-335.
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  39. Louis Narens (ed.) (1985). Abstract Measurement Theory. MIT Press.
    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 (...)
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  40. Louis Narens (1974). Measurement Without Archimedean Axioms. Philosophy of Science 41 (4):374-393.
    Axiomatizations of measurement systems usually require an axiom--called an Archimedean axiom--that allows quantities to be compared. This type of axiom has a different form from the other measurement axioms, and cannot--except in the most trivial cases--be empirically verified. In this paper, representation theorems for extensive measurement structures without Archimedean axioms are given. Such structures are represented in measurement spaces that are generalizations of the real number system. Furthermore, a precise description of "Archimedean axioms" is given and it is shown that (...)
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  41. Omar W. Nasim (2013). Observing by Hand: Sketching the Nebulae in the Nineteenth Century. University of Chicago Press.
    Today we are all familiar with the iconic pictures of the nebulae produced by the Hubble Space Telescope’s digital cameras. But there was a time, before the successful application of photography to the heavens, in which scientists had to rely on handmade drawings of these mysterious phenomena. Observing by Hand sheds entirely new light on the ways in which the production and reception of handdrawn images of the nebulae in the nineteenth century contributed to astronomical observation. Omar W. Nasim investigates (...)
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  42. Isabelle Peschard, Is Simulation a Substitute for Experimentation?
    It is sometimes said that simulation can serve as epistemic substitute for experimentation. Such a claim might be suggested by the fast-spreading use of computer simulation to investigate phenomena not accessible to experimentation (in astrophysics, ecology, economics, climatology, etc.). But what does that mean? The paper starts with a clarification of the terms of the issue and then focuses on two powerful arguments for the view that simulation and experimentation are ‘epistemically on a par’. One is based on the claim (...)
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  43. Fred S. Roberts (ed.) (1985). Measurement Theory. Cambridge University Press.
    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.
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  44. Kim Sawyer, Howard Sankey & Ric Lombardo (2013). Measurability Invariance, Continuity and a Portfolio Representation. Measurement 46 (1):89-96.
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  45. Dana Scott & Patrick Suppes (1958). Foundational Aspects of Theories of Measurement. Journal of Symbolic Logic 23 (2):113-128.
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  46. Michael J. Shaffer (2013). E Does Not Equal K. The Reasoner 7:30-31.
    This paper challenges Williamson's "E = K" thesis on the basis of evidential practice. The main point is that most evidence is only approximately true and so cannot be known if knowledge is factive.
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  47. David Sherry (2011). Thermoscopes, Thermometers, and the Foundations of Measurement. Studies in History and Philosophy of Science Part A 42 (4):509-524.
    Psychologists debate whether mental attributes can be quantified or whether they admit only qualitative comparisons of more and less. Their disagreement is not merely terminological, for it bears upon the permissibility of various statistical techniques. This article contributes to the discussion in two stages. First it explains how temperature, which was originally a qualitative concept, came to occupy its position as an unquestionably quantitative concept (§§1–4). Specifically, it lays out the circumstances in which thermometers, which register quantitative (or cardinal) differences, (...)
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  48. Barry Smith (2012). Classifying Processes: An Essay in Applied Ontology. Ratio 25 (4):463-488.
    We begin by describing recent developments in the burgeoning discipline of applied ontology, focusing especially on the ways ontologies are providing a means for the consistent representation of scientific data. We then introduce Basic Formal Ontology (BFO), a top-level ontology that is serving as domain-neutral framework for the development of lower level ontologies in many specialist disciplines, above all in biology and medicine. BFO is a bicategorial ontology, embracing both three-dimensionalist (continuant) and four-dimensionalist (occurrent) perspectives within a single framework. We (...)
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  49. Patrick Suppes (2006). Transitive Indistinguishability and Approximate Measurement with Standard Finite Ratio-Scale Representations. Journal of Mathematical Psychology 50:329-336.
    Ordinary measurement using a standard scale, such as a ruler or a standard set of weights, has two fundamental properties. First, the results are approximate, for example, within 0.1 g. Second, the resulting indistinguishability is transitive, rather than nontransitive, as in the standard psychological comparative judgments without a scale. Qualitative axioms are given for structures having the two properties mentioned. A representation theorem is then proved in terms of upper and lower measures.
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  50. Patrick Suppes (2002). Representational Measurement Theory. In J. Wixted & H. Pashler (eds.), Stevens' Handbook of Experimental Psychology. Wiley.
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