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Journal of Physics Conference Series

Probability as a logic for measurement representations

Giovanni Battista Rossi, Francesco Crenna

Published Sep 6, 2013• 3 citations
Publisher

Abstract

Measurement can be defined on the basis of the notion of empirical relation, which appears as a primitive term in the representational theory. Here a language is introduced for dealing with empirical relations and a probabilistic logic is applied to statements concerning them.
DOI
10.1088/1742-6596/459/1/012005

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In the representational approach, measurement is defined on the basis of primitive (in the context) notion of empirical relation. To account for measurement uncertainty, it is quite natural to consider probabilistic (or, more generally, non deterministic) empirical relations. Here the notion of probabilistic relation is investigated. After a brief historical outline, we point out that probabilistic relations appear in everyday measurement situation. To demonstrate this, a simple illustrative experiment has been developed, whose results are presented and discussed.
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We take the view that physical quantities are values generated by processes in measurement, not pre-existent objective quantities, and that a measurement result is strictly a product of the apparatus and the subject of the measurement. We habitually make an inaccurate statements when we speak of the measurement of a quantity by an apparatus. These statements can be formalised as a many valued logic with the structure of a vector space with a hermitian form in such a way as to generate probabilities in the results of measurement. The difference between this and classical probability theory is that we are not finding probabilities generated by unknown variables, but probabilities generated by unknown structure. We thus interpret quantum logic as the application of complex truth values to statements in an inaccurate language, and find that the properties of vector space hold for approximate measurement as well as for measurement of optimal accuracy, suggesting that Planck's constant governs the scale of the fundamental structures of matter.
The history of philosophy teaches us that metaphysical claims of sweeping generality are neither verifiable nor conclusively falsifiable. One can only aim at assessing their plausibility on the basis of the best available evidence provided by the sciences – both formal and empirical sciences. This is what has been done in this book in connection with the Principle of the Common Cause. In Chapter 2, Reichenbach's notion of common cause and the related Common Cause Principle was formulated explicitly in terms of classical, Kolmogorovian probability measure spaces. The Definition 2.4 of common cause followed Reichenbach's original definition closely, and insisting on the quite obvious methodological principle that probabilistic concepts and statements (in particular claims about random events being correlated or probabilistically independent) are only meaningful within the context of a fixed probability measure space in terms of which some segment of reality is modeled, we specified the notions of common cause incomplete and common cause complete probability theories: a theory was defined to be common cause complete if it contains a proper common cause of every correlation it predicts, common cause incomplete otherwise. (There is a strong version of common cause completeness as well: a probability space was defined to be strongly common cause closed if it contains common causes of all admissible types – Definition 4.4.) These explicitly defined notions form the basis on which one can start assessing the status of the Common Cause Principle in the spirit of empirical philosophy. Suppose we have an empirically confirmed scientific theory T describing some segment of reality using (possibly among other mathematical structures) a probability measure space ( X, S, p ).
According to the Kolmogorovian Censorship Hypothesis, everything that quantum theory says about the world in the language of the quantum mechanical Hilbert space formalism is actually about relationships between ordinary relative frequencies expressible in operational terms using classical Kolmogorovian probability theory. In other words, a quantum theoretical description of a system should in principle be translatable into a purely operational-probabilistic description. However, our goal in this paper is different; we do not want to deal with the problem how to translate the known theory of quantum mechanics into operational terms, or to reconstruct the theory from postulates which can be interpreted in operational terms. Our aim is somewhat broader and points in the opposite direction. We start with a general scheme for the operational description of an arbitrary physical system. The description is based solely on the notion of observable events (measurement operations and measurement results) and on general, empirically established simple laws concerning their relative frequency. These laws are so simple and fundamental that they apply equally to any physical system. In the first part of the paper, we outline the basic elements of such an operational-probabilistic theory. In the second part, we discuss how this operational-probabilistic description compares to the quantum mechanical description and to what extent the standard Hilbert space quantum mechanics can be regarded as a reformulation of the general operational-probabilistic theory.