The purpose of this work is to analyse the cognitive process of the domain theories in terms of the measurement theory to develop a computational machine learning approach for implementing it. As a result, the relational data mining approach, the authors proposed in the preceding books, was improved. We present the approach as an implementation of the cognitive process as the measurement theory perceived. We analyse the cognitive process in the first part of the paper and present the theory and (...) method of the logically most powerful empirical theory discovery in the second. The theory is based on the notion of law-like rules, which conform to all the properties of laws of nature, namely generality, simplicity, maximum refutability and minimum number of parameters. This notion is defined for deterministic and probabilistic cases. Based on the method, the discovery system is developed. The system was successfully applied to many practical tasks. (shrink)

It is argued that a number of important, and seemingly disparate, types of representation are species of a single relation, here called structural representation, that can be described in detail and studied in a way that is of considerable philosophical interest. A structural representation depends on the existence of a common structure between a representation and that which it represents, and it is important because it allows us to reason directly about the representation in order to draw conclusions about the (...) phenomenon that it depicts. The present goal is to give a general and precise account of structural representation, then to use that account to illuminate several problems of current philosophical interest — including some that do not initially seem to involve representation at all. In particular, it is argued that ontological reductions (like that of the natural numbers to sets), compositional accounts of semantics, several important sorts of mental representation, and (perhaps) possible worlds semantics for intensional logics are all species of structural representation and are fruitfully studied in the framework developed here. (shrink)

We revisit classical Utilitarianism by connecting and generalizing two ideas. The first is that there is a representation theorem possible for hedonic value similar to, but also importantly different from, the one provided by von Neumann and Morgenstern to measure decision utility. The idea is to use objective time, in place of objective chance, to measure hedonic value. This representation for hedonic value delivers a stronger kind of scale than von Neumann–Morgenstern utility, a ratio scale rather than merely an interval (...) scale. The second idea is that measurement on a ratio scale allows the meaningful aggregation of utilities over a group. This is aggregation by product rather than sum. Aggregation by product is known to have interesting Prioritarian consequences. Aggregation becomes complicated when the two approaches are mixed, when hedonic value is mixed with uncertainly. It becomes problematic when pain as well as pleasure is taken into account. (shrink)

In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection (...) of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group R\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

The formal methods of the representational theory of measurement (RTM) are applied to the extensive scales of physical science, with some modifications of interpretation and of formalism. The interpretative modification is in the direction of theoretical realism rather than the narrow empiricism which is characteristic of RTM. The formal issues concern the formal representational conditions which extensive scales should be assumed to satisfy; I argue in the physical case for conditions related to weak rather than strong extensive measurement, in the (...) sense of Holman 1969 and Colonius 1978. The problem of justifying representational conditions is addressed in more detail than is customary in the RTM literature; this continues the study of the foundations of RTM begun in an earlier paper. The most important formal consequence of the present interpretation of physical extensive scales is that the basic existence and uniqueness properties of scales (representation theorem) may be derived without appeal to an Archimedean axiom; this parallels a conclusion drawn by Narens for representations of qualitative probability. It is concluded that there is no physical basis for postulation of an Archimedean axiom. (shrink)

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as involving (...) arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)

The thesis of the empirical underdetermination of theories (U-thesis) maintains that there are incompatible theories which are empirically equivalent. Whether this is an interesting thesis depends on how the term incompatible is understood. In this paper a structural explication is proposed. More precisely, the U-thesis is studied in the framework of the model theoretic or emantic approach according to which theories are not to be taken as linguistic entities, but rather as families of mathematical structures. Theories of similarity structures are (...) studied as a paradigmatic case. The structural approach further reveals that the U-thesis is related to problems of uniqueness in the representational theory of measurement, questions of geometric conventionalism, and problems of structural underdetermination in mathematics. (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)

A serious flaw in Hartry Field’s instrumental account of applied mathematics, namely that Field must overestimate the extent to which many of the structures of our mathematical theories are reflected in the physical world, underlies much of the criticism of this account. After reviewing some of this criticism, I illustrate through an examination of the prospects for extending Field’s account to classical equilibrium statistical mechanics how this flaw will prevent any significant extension of this account beyond field theories. I note (...) in the conclusion that this diagnosis of Field’s program also points the way to modifications that may work. (shrink)

Three of Zeno's objections to motion are answered by utilizing a version of nonstandard analysis, internal set theory, interpreted within an empirical context. Two of the objections are without force because they rely upon infinite sets, which always contain nonstandard real numbers. These numbers are devoid of numerical meaning, and thus one cannot render the judgment that an object is, in fact, located at a point in spacetime for which they would serve as coordinates. The third objection, an arrow never (...) appears to be moving, is answered by showing that it only applies to a finite number of instants of time. A theory of motion is also advanced; it consists of a finite series of contiguous infinitesimal steps. The theory is immune to Zeno's first two objections because the number of steps is finite and each lies outside the domain of observation. Present motion is hypothesized to be an unobservable process taking place within each step. The fact of motion is apparent through a summing (Riemann integration) of the steps. (shrink)

In the late 1970s and early 1980s a number of philosophers, notably Churchland, Field, Stalnaker, Dennett, and Davidson, began to argue that propositional attitude predicates (such as believes that it’s sunny outside) are a species of measure predicate, analogous in important ways to numerical predicates by which we attribute physical magnitudes (such as mass, length, and temperature). Other philosophers, including myself, have subsequently developed the idea in greater detail. In this paper I sketch the general outlines of measurement‐theoretic accounts of (...) propositional attitudes, explaining in the briefest terms the basic idea of such accounts, why some have thought such accounts plausible, how these accounts might go, what their implications might be both for our conception of propositional attitudes and for their role in cognitive scientific theorizing, and where the potential problems with such accounts might lie. (shrink)

This discussion centers on two desiderata: the role of measurement in information-gathering and physical interaction in scientific practice. By taking inspiration from van Fraassen’s view, we present a methodological account of perspectival measurement that addresses empirical practice where there is complex intervention, disagreeing results, and limited theory. The specific aim of our account is to provide a methodological prescription for developing measurement processes in the context of limited theory. The account should be useful to philosophers of science, who are interested (...) in the intersection between representation and intervention; scientists, who are interested in methodological suggestions for theory-development and reliability; and interdisciplinary researchers, who are interested in the intersections between the pandemic, built environments, and social processes. We apply the process-view of measurement to COVID-19, specifically, measuring replication in the SARS-CoV-2 virus. The aim is to show that our account tracks key elements—manipulation/intervention, independence, invariance, and theory-development—by organizing unfolding measurement processes. Additionally, we use our account to make prescriptive suggestions for measurement practice in the COVID-19 context by discussing the need to broaden measurement perspective on interaction, manipulation, and production. (shrink)

In experimental settings, scientists often “make” new things, in which case the aim is to intervene in order to produce experimental objects and processes—characterized as ‘effects’. In this discussion, I illuminate an important performative function in measurement and experimentation in general: intervention-based experimental production (IEP). I argue that even though the goal of IEP is the production of new effects, it can be informative for causal details in scientific representations. Specifically, IEP can be informative about causal relations in: regularities under (...) study; ‘intervention systems’, which are measurement/experimental systems; and new technological systems. (shrink)

Though he maintained a significant interest in theoretical aspects of measurement, Henry E. Kyburg, Jr. was critical of the representational theory that in many ways has come to dominate discussions concerning the foundations of measurement. In particular, Kyburg (in Savage and Ehrlich (eds) Philosophical and foundational issues in measurement theory, 1992 ) asserts that the representational theory of measurement, as introduced in (Scott and Suppes, Journal of Symbolic Logic, 23:113–128, 1958 ) and developed in (Krantz et al., Foundations of measurment: (...) additive and polynomial representations. Academic Press, 1971 ), cannot account for the measurement of error. The present work examines and responds to this charge. (shrink)

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. (shrink)

In the first section of this paper I review Measurement Theoretic Semantics – an approach to formal semantics modeled after the application of numbers in measurement, e.g., of length. In the second section it is argued that the measurement theoretic approach to semantics yields a novel, useful conception of propositions. In the third section the measurement theoretic view of propositions is compared with major other accounts of propositional content.