1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References A Defence of the Representational Theory of Measurement Conrad Heilmann Email: heilmann@fwb.eur.nl Erasmus Institute for Philosophy and Economics (EIPE) and Faculty of Philosophy Erasmus University Rotterdam Arctic Workshop on Measurement in Economics Rovaniemi, 14-15 December 2012 A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Claim Starting point: The Representational Theory of Measurement (RTM) is not a complete / comprehensive account of measurement. (Luca is right!) Claim: RTM can be highly useful as a tool to structuring conceptual reflections. I argue for this in two steps: Offer a more general interpretation of RTM 1 Discuss hypothetical measurement and backwards 2 engineering of foundations as ways to use RTM in conceptual reflections A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Claim Starting point: The Representational Theory of Measurement (RTM) is not a complete / comprehensive account of measurement. (Luca is right!) Claim: RTM can be highly useful as a tool to structuring conceptual reflections. I argue for this in two steps: Offer a more general interpretation of RTM 1 Discuss hypothetical measurement and backwards 2 engineering of foundations as ways to use RTM in conceptual reflections A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Claim Starting point: The Representational Theory of Measurement (RTM) is not a complete / comprehensive account of measurement. (Luca is right!) Claim: RTM can be highly useful as a tool to structuring conceptual reflections. I argue for this in two steps: Offer a more general interpretation of RTM 1 Discuss hypothetical measurement and backwards 2 engineering of foundations as ways to use RTM in conceptual reflections A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Agenda 1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata Hypothetical Measurement Backwards Engineering of Foundations Disentangling RTM from Operationalism 5 Conclusions A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Agenda 1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata Hypothetical Measurement Backwards Engineering of Foundations Disentangling RTM from Operationalism 5 Conclusions A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Classic RTM The authoritative statement of RTM can be found in the three monographs Krantz et al. (1971), Suppes et al. (1971) and Luce et al. (1971). A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Classic RTM Accordingly, representation theorems establish homomorphisms between empirical and numerical structures that allow to characterise properties of numerical assignment. We assume an empirical relation R on a set of objects A and a numerical relation S on R . A homomorphism is established by a function that assigns real numbers to elements in A in a way that numerically captures their empirical relation. More formally, ‘. . . if � A , R 1 , . . . , R m � is an empirical relational structure and � R , S 1 , . . . , S m � is a numerical relational structure, a real valued function φ on A is a homomorphism if it takes each R i into S i , i = 1 , . . . , m .’ (Krantz et al. , 1971, 8ff.) A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Classic RTM Such homomorphisms can be characterised formally to render explicit what kinds of transformations are possible which is captured by the concept of scales: ‘A homomorphism into the real numbers is often referred to as a scale in the psychological measurement literature. From this standpoint measurement may be regarded as the construction of homomorphisms (scales) from empirical relational structures of interest into numerical relational structures that are useful.’ (Krantz et al. , 1971, 9) A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Classic RTM The exact characterisation of what kind of scale a given measurement procedure yields is given by uniqueness theorems which specify the permissible transformations of the numbers. More formally, uniqueness theorems assert that ‘. . . a transformation φ �→ φ ′ is permissible if and only if φ and φ ′ are both homomorphisms of � A , R 1 , . . . , R m � into the same numerical structure � R , S 1 , . . . , S m � .’ (Krantz et al. , 1971, 12) A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Criticism of RTM RTM is much maligned: . . . it allegedly reduces measurement to scale construction, without specifying problems associated with the actual process of measuring something, for instance measurement error and the construction of reliable measurement instruments. . . . it is closely associated with operationalism, the idea that measurement is equal to and nothing more than actually perform a measurement operation (most notorious in revealed preference theory) . . . A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Agenda 1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata Hypothetical Measurement Backwards Engineering of Foundations Disentangling RTM from Operationalism 5 Conclusions A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Reinterpreting RTM In classic RTM, we speak of a homomorphism between an empirical relational structure (ERS) and a numerical relational structure (NRS). For example, for simple length measurement, we might want to specify the ERS as � X , ◦ , � � , where X is a set of rods, ◦ is a concatenation operation, and � is a comparison of length of rods. All going well, there is a homomorphism into a NRS that we can specify as � R , + , ≥� . A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Reinterpreting RTM From a formal point of view, the representation and uniqueness theorems simply characterise mappings between two kinds of structures, with one of these structures being associated with properties of numbers, and the other with qualitative relations. Since the theorem just concerns the conditions under which the concatenation operation and the ordering relation can be represented numerically, it is possible to furnish a more abstract interpretation of what hitherto has been called the empirical relational structure . A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Reinterpreting RTM Reinterpreting the empirical relational structure � X , ◦ , � � as a qualitative relational structure (QRS) does not require any change, addition or reconsideration of the measurement and uniqueness theorems in RTM. All what is needed in order to apply the latter is that there is: a set of well specified objects in the mathematical sense: that we have clear membership conditions for the set X . well-defined qualitative relations, such as ◦ and � . RTM theorems do not require that these objects and relations are interpreted empirically. A Defence of the Representational Theory of Measurement Conrad Heilmann
1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References Reinterpreting RTM Reinterpreting the empirical relational structure � X , ◦ , � � as a qualitative relational structure (QRS) does not require any change, addition or reconsideration of the measurement and uniqueness theorems in RTM. All what is needed in order to apply the latter is that there is: a set of well specified objects in the mathematical sense: that we have clear membership conditions for the set X . well-defined qualitative relations, such as ◦ and � . RTM theorems do not require that these objects and relations are interpreted empirically. A Defence of the Representational Theory of Measurement Conrad Heilmann
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