Luca Mari

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 ways in which the interactions of such systems with the entities under measurement provide a structure to an empirical domain. In particular it is claimed that (1) it is on the ground of the interaction with a measurement system that a partition can be induced on the domain of entities under measurement and that relations among such entities can be established, and that (2) it is the usage of measurement systems that guarantees a degree of objectivity and intersubjectivity to measurement results. As modeled in this paper, measurement systems link the abstract theory of measuring, as developed in representational terms, and the practice of measuring, as coded in standard documents such as the International Vocabulary of Metrology.

The common account of the analog vs digital distinction is based on features of physical systems, being related to the usage of continuous vs discrete supports respectively. It is proposed here to alternatively characterize the concepts of analog and digital as related to coding systems, of which a formal definition is given, by suggesting that the distinction refers to the strategy adopted to define the coding function: extensional in digital systems, isomorphic intensional in analog systems. This thesis is supported by examples, in particular of analog systems exploiting discrete supports, and is discussed to explain why digital coding is currently so widespread in technological and social practice

Measurement results are stated in terms of sentences ascribing measured values, as obtained via measurement processes, to measurands, as defined by measuring agents. Since both the definition of the measurands and the characterization of the processes depend on models constructed on the basis of relevant theories, the issue arises of the theory dependence of the truth of those sentences. This paper aims at assessing the question by introducing suitable distinctions about the sense and reference of the terms used to refer to individual and general quantities, in a framework where measurement is construed as an agent-dependent, model-based, purpose-driven activity.

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.

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 another, as sometimes claimed nowadays. The crucial problem is whether assuming this standpoint implies definitely renouncing to maintain a role for truth and the related concepts, particularly accuracy, in measurement. It is argued here that the well known objections against true values in measurement, which would lead to refuse the concept of accuracy as non-operational, or to maintain it as only qualitative, derive from a not clear distinction between three distinct processes: the metrological characterization of measuring systems, their calibration, and finally measurement. Under the hypotheses that (1) the concept of true value is related to the model of a measurement process, (2) the concept of uncertainty is related to the connection between such model and the world, and (3) accuracy is a property of measuring systems (and not of measurement results) and uncertainty is a property of measurement results (and not of measuring systems), not only the compatibility but actually the conjoint need of error-based and uncertainty-based modeling emerges.

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 analysis, performed in systematic comparison with the paradigm of measurement of physical quantities. Some background questions arising here are: – Are the chosen indicators appropriate performance indicators? – Do such indicators convey complete and non-redundant information on performance? – Does the chosen combination rule generate results suitably interpretable as performance values? And enlarging the focus: – Does the obtained value specifically convey information on the system under analysis, instead of some different entity (typically including the subject who is evaluating)? Operatively: would different subjects evaluate the same system in the same way? i.e., is the obtained information objective? – Does the obtained value convey information that is interpretable in the same way by different subjects? Operatively: would different subjects who have agreed on a decision procedure make the same decision from the same performance information? i.e., is the obtained information intersubjective? Any well founded positive answers to these questions significantly support a structural interpretation of measurement encompassing both physical and soft measurement.

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 (VIM3).

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.

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.

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 viewpoint. Finally, from the analysis of the inter-relations between calibration and measurement the fundamental reasons of the claimed objectivity and intersubjectivity of measurement are highlighted, a valuable epistemological result to characterize measurement as a particular kind of evaluation.

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 sketched, which gives the context to highlight the unavoidable, although sometimes implicit, presence of models in measurement and, finally, to propose some remarks on the relations between models and measurement uncertainty, complementarily classified as due to the idealization implied in the models and their realization in the experimental setup.

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 obtained, which is then generalized to 'property' and 'property value'. This analysis also throws some light on the elusive concept of magnitude.