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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:

1

Offer a more general interpretation of RTM

2

Discuss hypothetical measurement and backwards 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:

1

Offer a more general interpretation of RTM

2

Discuss hypothetical measurement and backwards 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:

1

Offer a more general interpretation of RTM

2

Discuss hypothetical measurement and backwards 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, R1, . . . , Rm is an empirical relational structure and R, S1, . . . , Sm is a numerical relational structure, a real valued function φ on A is a homomorphism if it takes each Ri into Si, 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

• nly if φ and φ′ are both homomorphisms of

A, R1, . . . , Rm into the same numerical structure R, S1, . . . , Sm.’ (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

1 Claim 2 Classic RTM 3 Reinterpreting RTM 4 Desiderata of the Reinterpretation 5 Conclusions References

Reinterpreting RTM

We can just use the theorems provided in RTM and apply them to any concept that we might care to investigate with regards to its potential for numerical representation. This is much more general than viewing RTM theorems

• nly as investigating possibilities of representing empirical

relations with numbers. The general advantage of this strategy: Krantz et al. (1971), Suppes et al. (1971) and Luce et al. (1971) becomes a library of theorems that we can use as tools to structure any conceptual exercise.

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

Hypothetical Measurement

Interpreting measurement theorems as specifying conditions of mappings between QRS and NRS, we can use them to speculate about possible numerical representations of abstract properties of abstract concepts. What is required of this are simply concepts that specify a well-defined set of objects and qualitative relations.

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

Example: Personal Identity over Time

Consider the theory of personal identity over time offered by Parfit (1984). He maintains that persons can be viewed as sets of temporal selves, and that personal identity consists in connectedness, which in turn is determined by an appropriate degree of psychological continuity between selves. In hypothetical measurement, we might understand this as specifying a set of temporal selves X and a conception of what it means for those selves to be psychologically continuous with one another, for instance by comparing them with each other, or by comparing them all with regards to how continuous they are with a temporal self that is associated with a certain point in time (say, the present). If that is the case, then we have some reason to believe

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

Example: Personal Identity over Time

Consider the theory of personal identity over time offered by Parfit (1984). He maintains that persons can be viewed as sets of temporal selves, and that personal identity consists in connectedness, which in turn is determined by an appropriate degree of psychological continuity between selves. In hypothetical measurement, we might understand this as specifying a set of temporal selves X and a conception of what it means for those selves to be psychologically continuous with one another, for instance by comparing them with each other, or by comparing them all with regards to how continuous they are with a temporal self that is associated with a certain point in time (say, the present). If that is the case, then we have some reason to believe

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

Example: Personal Identity over Time

Consider two temporal selves, Si and Sj. One way to characterise their psychological continuity is with regards to certain attitudes they might take to small number propositions P = {a, b, c, x, y, z}. Now suppose Si has the following preference ordering: {a ≻P b ≻P c ≻P x ≻P y ≻P z} and Sj has the preference

• rdering {a ≻P b ≻P c ≻P x ∼P y ∼P z}. These two
• rderings determine 15 binary relations and 3 of those are

different. We can now compare the similarity between the two selves Si and Sj with other pairs, such that (Si, Sj) (Si, Sk)

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

Example: Personal Identity over Time

It is natural that these kinds of comparisons can be investigated with measurement theorems: do the comparisons satisfy certain conditions such that the QRS

• f temporal selves and comparisons can be represented

by some NRS? If so, we have a hypothetical numerical representation of psychological continuity. The above example of comparisons of attitudes has taken a very narrow view of psychological continuity that might not be in light with the rather rich idea of psychological continuity in Parfit (1984) that includes hopes, ideals, and plans, which are hard to depict in the kind of QRS that I have adopted in the example above. This is exactly the kind of insight that a hypothetical use of RTM measurement theorems can achieve.

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

More examples along those lines

Happiness measurement Social Preferences Preference aggregation frameworks Utility maximisation failures etc.

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

Backwards Engineering of Foundations

Ideally, measurement of new concepts would be introduced ‘bottom up’: first specifying what kind of properties, relations, and objects one is interested in, and secondly investigating what kinds of ERS could be used to capture them, and thirdly, proving a representation theorem. Yet, especially in economics and the social sciences, things do not always work like that. There is a need to improve models in this respect, such that the parameters used capture meaningful concepts.

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

Example: Time Discounting

Time Discounting Functions 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

Example: Time Discounting

Definition (Time discounting function) A time discounting function D is a decreasing mapping D : T → (0, 1], from a set of clock-time points T to the real interval (0, 1], such that D(0) = 1. What concept is represented by the values?

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

Backwards Engineering of Foundations

We need ‘backwards engineering of foundations’.

We already have time discounting functions that are used in all areas of economic theory and policy-making, but we neither do we have agreement nor do we have clarity over the conceptual motivation for time discounting. Using measurement theorems from RTM ‘backwards’, first specifying what properties we are looking for in the NRS, and then asking what kinds of concepts would be able to sustain the required interpretation of the QRS seems a fruitful way to investigate the foundations of time discounting.

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

Backwards Engineering of Foundations

This, then, is another advantage of the more general interpretation of RTM:

for cases in which numbers are in search of foundations, we can explore different possibilities to provide them, while at the same time not falling into the trap of being bound by what is currently done empirically in any given research domain.

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

More Examples

Time discounting:

different discounting functions competing motivations elements in the social discount rate

Happiness measurement Concept formation for any context in which numbers are used ambiguously etc.

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

Disentangling RTM from Operationalism

In economics and the social sciences, RTM has since long been associated with operationalism, the idea that measurement is equal to and nothing more than actually perform a measurement operation.

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

Disentangling RTM from Operationalism

In the more general interpretation of RTM as specifying conditions for mapping qualitative relational structures into numerical ones, any such link simply goes away. Just as we can discuss the numerical representation of preferences as observed choice behaviour, we can discuss preferences as mental states, and ask what kinds of conditions are plausible in a hypothetical measurement exercise. Indeed, ‘the measurement question’ is no longer one that goes straight to the empirical level, but rather is one that allows us – and one might even say: encourages us – to consider conceptual questions in greater detail.

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

Conclusions

I have proposed to generalise the interpretation of empirical relational structures in RTM to qualitative relational structures. I have argued that this interpretation allows us to use RTM theorems for:

1

Hypothetical measurement

2

Backwards engineering of foundations, and

3

Disentangling RTM from operationalism.

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

How is this a defence of RTM?

The three uses of RTM offered in this paper mitigate or sidestep well-rehearsed criticisms against RTM:

1 RTM does not take into account problems of measuring,

such as measurement error or the construction of measurement devices.

The strengths of RTM might just lie elsewhere: in providing a formal framework with which processes of concept formation can be structured.

2 RTM places too much an emphasis on foundations.

RTM could be used much more than it is now to explore possible foundations for numbers, such as parameters in economic models that lack foundations.

3 RTM is linked to operationalism.

In the general interpretation it is not.

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

Thank you! Thank you!

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

References

Krantz, D. H., Luce, R. D., Tversky, A., and Suppes, P . (1971). Foundations of Measurement Volume I: Additive and Polynomial Representations. Mineola: Dover Publications. Luce, R. D., Krantz, D. H., Tversky, A., and Suppes, P . (1971). Foundations of Measurement Volume III: Representation, Axiomatization, and Invariance. Mineola: Dover Publications. Parfit, D. (1984). Reasons and Persons. Clarendon. Suppes, P ., Krantz, D. H., Luce, R. D., and Tversky, A. (1971). Foundations of Measurement Volume II: Geometrical, Threshold, and Probabilistic Representations. Mineola: Dover Publications.

A Defence of the Representational Theory of Measurement Conrad Heilmann