A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all the advantages of (...) calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper topology. However, this condition fails to be sufficient. Instead of search (...) for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements. (shrink)

A ubiquitous argument against mental-state accounts of well-being is based on the notion that mental states like happiness and satisfaction simply cannot be measured. The purpose of this paper is to articulate and to assess this “argument from measurability.” My main thesis is that the argument fails: on the most charitable interpretation, it relies on the false proposition that measurement requires the existence of an observable ordering satisfying conditions like transitivity. The failure of the argument from measurability, however, does not (...) translate into a defense of mental-state accounts as accounts of well-being or of measures of happiness and satisfaction as measures of well-being. Indeed, I argue, the ubiquity of the argument from measurability may have obscured other, very real problems associated with mental-state accounts of well-being – above all, that happiness and satisfaction fail to track well-being – and with measures of happiness and satisfaction – above all, the tendency toward reification. I conclude that the central problem associated with the measurement of, e.g., happiness as a subjectively experienced mental state is not that it is too hard to measure, but rather that it is too easy to measure. (shrink)

We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it (...) contains neither 2 \oplus 2 nor 3 \oplus 1. (shrink)

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 (...) results also are given on a problem which does not arise in current theories: namely that of determining limits of accuracy attainable on the basis of observations. (shrink)

The modern discussion on the concept of truthlikeness was started in 1960. In his influential Word and Object, W. V. O. Quine argued that Charles Peirce's definition of truth as the limit of inquiry is faulty for the reason that the notion 'nearer than' is only "defined for numbers and not for theories". In his contribution to the 1960 International Congress for Logic, Methodology, and Philosophy of Science at Stan­ ford, Karl Popper defended the opposite view by defining a compara­tive (...) notion of verisimilitude for theories. The concept of verisimilitude was originally introduced by the Ancient sceptics to moderate their radical thesis of the inaccessibility of truth. But soon verisimilitudo, indicating likeness to the truth, was confused with probabilitas, which expresses an opiniotative attitude weaker than full certainty. The idea of truthlikeness fell in disrepute also as a result of the careless, often confused and metaphysically loaded way in which many philosophers used - and still use - such concepts as 'degree of truth', 'approximate truth', 'partial truth', and 'approach to the truth'. Popper's great achievement was his insight that the criticism against truthlikeness - by those who urge that it is meaningless to speak about 'closeness to truth' - is more based on prejudice than argument. (shrink)

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)

The interpretation of the calculus of probability as a logic of partial belief has at least two advantages: it makes the assignment of probabilities plausible in cases where classical frequentist interpretations must find such assignments meaningless, and it gives a clear meaning to partial belief and to consistency of partial belief.

Axiomatic measurement theories are commonly interpreted as claiming that, in order to quantify an empirical domain, the qualitative structure of data about that domain must be mapped to a numerical structure. Such mapping is supposed to be established independently, i.e., without presupposing that the domain can be quantified. This interpretation is based on two myths: that it is possible to independently infer the qualitative structure of objects from empirical data, and that the adequacy of numerical representations can only be justified (...) by mapping such qualitative structures to numerical ones. I dispel the myths and show that axiomatic measurement theories provide an inadequate characterization of the kind of evidence required to detect quantities. (shrink)

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving (...) operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics. (shrink)

In a previous paper it was proposed that theories of empirical phenomena should satisfy conditions of finite and irrevocable testability. Roughly speaking, a theory is finitely testable if, for every set of observations that falsifies the theory, there exists a finite subset of observations that falsifies it. A theory is irrevocably testable if, for any finite set of observations that falsifies the theory, any superset of observations that includes that set also falsifies it—a theory falsified by observations cannot be resuscitated (...) by adding new observations to the original set. (shrink)

In this article I give an overview of some recent work in philosophy of science dedicated to analysing the scientific process in terms of (conceptual) mathematical models of theories and the various semantic relations between such models, scientific theories, and aspects of reality. In current philosophy of science, the most interesting questions centre around the ways in which writers distinguish between theories and the mathematical structures that interpret them and in which they are true, i.e. between scientific theories as linguistic (...) systems and their non-linguistic models. In philosophy of science literature there are two main approaches to the structure of scientific theories, the statement or syntactic approach -- advocated by Carnap, Hempel and Nagel -- and the non- statement or semantic approach --advocated, among others, by Suppes, the structuralists, Beth, Van Fraassen, Giere, Wojcicki. In conclusion, I briefly review some of the usual realist inspired questions about the possibility and character of relations between scientific theories and reality as implied by the various approaches I discuss in the course of the article. The models of a scientific theory should indeed be adequate to the phenomena, but if the theory is 'adequate' to (true in) its conceptual (mathematical) models as well, we have a model-theoretic realism that addresses the possible meaning and reference of 'theoretical entities' without relapsing into the metaphysics typical of the usual scientific realist approaches. (shrink)

This paper studies the theory of uniqueness of scales of measurement, and in particular, the theory of meaningfulness of statements using scales. The paper comments on the general theory of meaningfulness adopted by Luce in connection with his work on dimensionally invariant numerical laws. It comments on Luce's generalization of the concept of meaningfulness of a statement involving scales to a concept of meaningfulness of an arbitrary relation relative to the defining relations in a relational structure. It is argued that (...) in studying the concept of meaningfulness, it is necessary to consider invariance under endomorphisms, not just automorphisms. The difference between the endomorphism and automorphism concepts of meaningfulness is studied. Luce's primary result, that automorphism meaningfulness is preserved under isomorphism, is extended to the result that endomorphism meaningfulness is preserved under homomorphism. (shrink)

Suppose it is possible to make approximate interpersonal comparisons of welfare gains and losses. Thus, if w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w$$\end{document}, x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}, y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y$$\end{document} and z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z$$\end{document} are personal states, then it is sometimes possible to say “The welfare gain of the state change w⇝x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\leadsto x$$\end{document} is greater than the welfare gain of the state change y⇝z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\leadsto z$$\end{document}”. We can represent this by the formula “⪰\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\succeq $$\end{document}”, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} is a difference preorder: an incomplete preorder on the space of all possible personal state changes. A social state change is a bundle of personal state changes. A social difference preorder is an incomplete preorder on the space of social state changes, which satisfies Pareto and Anonymity axioms. We characterize a family of SDPs which are roughly utilitarian in nature. We also apply the model to redistributive taxation. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.

In the literature, there are many axiomatizations of qualitative probability. They all suffer certain defects: either they are too nonspecific and allow nonunique quantitative interpretations or are overspecific and rule out cases with unique quantitative interpretations. In this paper, it is shown that the class of qualitative probability structures with nonunique quantitative interpretations is not first order axiomatizable and that the class of qualitative probability structures with a unique quantitative interpretation is not a finite, first order extension of the theory (...) of qualitative probability. The idea behind the method of proof is quite general and can be used in other measurement situations. (shrink)

A formal theory of quantity T Q is presented which is realist, Platonist, and syntactically second-order (while logically elementary), in contrast with the existing formal theories of quantity developed within the theory of measurement, which are empiricist, nominalist, and syntactically first-order (while logically non-elementary). T Q is shown to be formally and empirically adequate as a theory of quantity, and is argued to be scientifically superior to the existing first-order theories of quantity in that it does not depend upon empirically (...) unsupported assumptions concerning existence of physical objects (e.g. that any two actual objects have an actual sum). The theory T Q supports and illustrates a form of naturalistic Platonism, for which claims concerning the existence and properties of universals form part of natural science, and the distinction between accidental generalizations and laws of nature has a basis in the second-order structure of the world. (shrink)

The numerical representations of measurement, geometry and kinematics are here subsumed under a general theory of representation. The standard theories of meaningfulness of representational propositions in these three areas are shown to be special cases of two theories of meaningfulness for arbitrary representational propositions: the theories based on unstructured and on structured representation respectively. The foundations of the standard theories of meaningfulness are critically analyzed and two basic assumptions are isolated which do not seem to have received adequate justification: the (...) assumption that a proposition invariant under the appropriate group is therefore meaningful, and the assumption that representations should be unique up to a transformation of the appropriate group. A general theory of representational meaningfulness is offered, based on a semantic and syntactic analysis of representational propositions. Two neglected features of representational propositions are formalized and made use of: (a) that such propositions are induced by more general propositions defined for other structures than the one being represented, and (b) that the true purpose of representation is the application of the theory of the representing system to the represented system. On the basis of these developments, justifications are offered for the two problematic assumptions made by the existing theories. (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)

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. This (...) is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields real-valued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense. (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)

Shepard has supposed that the mind is stocked with innate knowledge of the world and that this knowledge figures prominently in the way we see the world. According to him, this internal knowledge is the legacy of a process of internalization; a process of natural selection over the evolutionary history of the species. Shepard has developed his proposal most fully in his analysis of the relation between kinematic geometry and the shape of the motion path in apparent motion displays. We (...) argue that Shepard has made a case for applying the principles of kinematic geometry to the perception of motion, but that he has not made the case for injecting these principles into the mind of the percipient. We offer a more modest interpretation of his important findings: that kinematic geometry may be a model of apparent motion. Inasmuch as our recommended interpretation does not lodge geometry in the mind of the percipient, the motivation of positing internalization, a process that moves kinematic geometry into the mind, is obviated. In our conclusion, we suggest that cognitive psychologists, in their embrace of internal mental universals and internalization may have been seduced by the siren call of metaphor. Key Words: apparent motion; imagery; internalization; inverse projection problem; kinematic geometry; measurement; metaphors of mind. (shrink)

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable (...) us to treat formally errors arising in nominal and ordinal measurements. (shrink)

Die Wissenschaftstheorie hat sich in der Vergangenheit hauptsächlich mit dem Aufbau und der Analyse wissenschaftlicher Theorien und den logischen Problemen in ihrem eigenen Gebiet beschäftigt, während Probleme der Wissenschaftspraxis, hier vor allem die theoretischen Grundlagen des Messens, nur am Rande oder gar nicht behandelt wurden. Dies ist insofern bemerkenswert, weil die Messung das wichtigste erfahrungswissenschaftliche Hilfsmittel zur Gewinnung von Erkenntnis darstellt. Beim Messen erfolgt der wichtige Übergang vom Empirischen zum Formalen, indem die empirisch vorliegende Intensität einer Meßgröße durch eine mathematische (...) Größe beschrieben und damit überhaupt erst die Voraussetzung für eine erfahrungswissenschaftliche Theorie geschaffen wird. Die vorliegende Meßtheorie ist in Meßprozeßtheorie, Metrisierungstheorie und Fehlertheorie gegliedert. Die Meßprozeßtheorie behandelt die Vorgänge zwischen Meßobjekt und Meßgerät, die Metrisierungstheorie die Darstellung der empirischen Größe als formale und die Fehlertheorie die Schätzung des gesuchten wahren Wertes aus mehreren fehlerbehafteten gleichwertigen Meßwerten. Der Schätzwert ist das Endergebnis einer Messung. Ziel der Arbeit ist es, die Meßtheorie aus dem engen Kreis der Metrisierungs- bzw. Skalentheorie herauszulösen und auf die häufig übersehenen notwendigen Bedingungen einer Messung aufmerksam zu machen. (shrink)

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this holistic conception does (...) not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori. (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)

Generalizations of partial meet contraction are introduced that start out from the observation that only some of the logically closed subsets of the original belief set are at all viable as contraction outcomes. Belief contraction should proceed by selection among these viable options. Several contraction operators that are based on such selection mechanisms are introduced and then axiomatically characterized. These constructions are more general than the belief base approach. It is shown that partial meet contraction is exactly characterized by adding (...) to one of these constructions the condition that all logically closed subsets of the belief set can be obtained as the outcome of a single (multiple) contraction. Examples are provided showing the counter-intuitive consequences of that condition, thus confirming the credibility of the proposed generalization of the AGM framework. (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)

a–b* c–d is taken to mean that your degree of preference for a over b is less than your degree of preference for c over d. Various properties of the strength-of-preference comparison relation * are examined along with properties of simple preferences defined from *. The investigation recognizes an individual's limited ability to make precise judgments. Several utility theorems relating a–b * c–d to u(a)–u(b) are included.