**On the Axiomatic Representational Theory Of Measurement**

The Axiomatic Representational Theory of Measurement ('ARTM') is the default tradition of measurement theory in mathematical psychology. It arguably started in a seminal work by the late philosopher of science and polymath Patrick Suppes and the logician Dana Scott [1].

It is 'axiomatic' in that it uses rigorous axiomatic-deductive mathematical methods. It is 'representational' by employing set-theoretical structures to represent roughly both the features of the world world and the information expressed by these features [2].

This kind of theory is contrasted to what is called a *classical* or *realist* theory of measurement [3]. Theories of this sort carry the spirit of ancient Greek sage, philosopher and mathematician Pythagoras (depicted in the painting above, proselytizing about the "marvels" of vegetarianism); it is not merely the case that we can use numbers to represent the information about a given measurand - the values of measurands *are* numbers, and numbers exist *right here*, in the structure of the world.

I've mentioned that in the account of Tiresias, pleasure is a *ratio-scalable* feature of psychological states. What does that mean?

**A bit on the ratio scale-type**

The word 'scale' is ambiguous in contexts of measurement. First, it can mean the choice of what I prefer to call a *representation standard* of a given measurable feature. Some examples include *m*, the meter-standard of the feature *length* and Ω, the Ohm-standard of the feature *electric resistance* . But for our purposes, a scale is the kind of mathematical structure under which the numerical values that we assign to a given feature are constrained. Scales in this second meaning come in different *types* and the modern terminology of *scale types* (of which we find the *ratio scale*) was baptized by psychophysicist Stanley Smith Stevens [4, 5].

Scales in this sense are a kind of mapping, an homomorphism from a structure representing the world to a structure representing the values of features. It's the transformation that allows the relational structure to be preserved, to remain invariant.

Take for example this excerpt from a book by contemporary moral philosopher Fred Feldman [6], a defender of the thesis that pleasure is a quantitative feature (just like Tiresias):

We assume that each episode of pleasure contains a certain ‘amount’ of pleasure, and that this amount is in principle subject to measurement. (We need not assume that these amounts can in practice be precisely determined either by introspection or by any existing technology.) The amount of pleasure in an episode depends upon intensity and duration, with longer-lasting and more intense pleasures being said to contain more total pleasure. For purposes of exposition I will imagine that there is a standard unit of measurement for these amounts. I call one unit of pleasure a “hedon”.

Imagine that Tiresias worked out his own theory of quantitative pleasure and proposed the representation standard *eron* for pleasure.

To say that pleasure is ratio-scalable is to say that we can switch from an *hedon* into an *eron* by a function of the form where '' is a positive real number. If one eron is 2.38 hedons, we can discover how many erons are in 10 hedons by dividing 10 by 2.38. Simple as that - the proportions are constrained and preserved.

There is another common type of scale that is intuitively considered to be "quantitative", the *interval scale*. For the interval scale, the admissable transformation is of the form , again with ' being a positive real number.

**Extensive structures**

Let's go a bit deeper on the proposal of the ARTM. First, we build a set-theoretical structure to represent the province of empirical reality that we're interested - in this case, it is the empirical feature 'pleasure'. *Ex hypothesi*, let's take Tiresias' word of it; a default contender to represent the quantity pleasure is the triple ,,. I like to call this an example of a *material relational structure*, but you'll see it with other names throughout the literature - most likely *empirical relational structure* or *system*.

'' is a nonempty set whose elements are interpreted as pleasure-events or pleasure-states - or '*P-events*', for short.

'' represents an empirical relation that mimics the abstract binary relation of *total order*. This roughly means that there is an empirically verifiable procedure to rank two distinct P-events as being either just as pleasurable or one being more pleasurable than the other.

The last element of our triple, '', is representing an empirical operation that is traditionally called *concatenation*. This is also intuitively graspable; it means that there is an empirical procedure that allows us to "stack" or "pile up" our quantities.

The class of structure with this form most used in science are called *continuous extensive structures* [7]. They need to satisfy a number of axioms.

The pioneering analysis and proposal of a set of axioms describing quantities came from Hermann von Helmholtz in 1887 and Otto Hölder in 1901 [8]. Over the 20th and 21st century, the axiomatic system of Hölder was modified by many philosophers of science and measurement theorists. They include Ernest Nagel in 1931 [9], Patrick Suppes in 1951 [10] and Jean-Claude Falmagne in 1975 [11].

These various adjustments, both minor and major, were motivated by empiricist reasons in an attempt to ground the practice of measurement in a more empirically adequate setting. They involve, for instance, weakening the logical relation of identity '' into a relation of equivalence and getting rid of background assumptions for non-standard numbers (infinitesimals).

Speaking of which, now we need some numbers to build our *numerical relational structure*. This is seemingly more straightforward. For quantities, a common candidate is the triple ,,. Here, '' is the set of positive reals, '' is the standard relation of total order and '' is the standard arithmetic operation of addition. A large share of the ARTM is focused on proving theorems concerning homomorphisms from material relational structures to numerical relational structures.

I hold that there are a series of problems with this particular choice of numerical structure but I'll utter them elsewhere.

Having set this amount of framework, I still have some things to say about quantities, the qualitative-quantitative distinction and I also have bad news for the defenders of the hypothesis that pleasure is quantitative. It will be discussed in a future entry.

[Bibtex]

```
@article{Suppes1958a,
author = {Scott, Dana and Suppes, Patrick},
citeulike-article-id = {13531611},
journal = {Journal of Symbolic Logic},
number = {2},
pages = {113--128},
posted-at = {2015-03-02 22:50:45},
priority = {2},
title = {Foundational Aspects of Theories of Measurement},
volume = {23},
year = {1958}
}
```

[Bibtex]

```
@article{Mari2009a,
author = {Frigerio, Aldo and Giordani, Alessandro and Mari, Luca},
citeulike-article-id = {12534798},
journal = {Synthese},
pages = {123--149},
posted-at = {2013-07-30 07:13:11},
priority = {2},
title = {Outline of a general model of measurement},
volume = {175},
year = {2009}
}
```

[Bibtex]

```
@article{Michell2005a,
author = {Michell, Joel},
citeulike-article-id = {13531643},
journal = {Measurement},
pages = {285--294},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {The logic of measurement: A realist overview},
volume = {38},
year = {2005}
}
```

[Bibtex]

```
@article{Stevens1946a,
author = {Stevens, S. S.},
citeulike-article-id = {13112840},
journal = {Science},
number = {2684},
pages = {677--680},
posted-at = {2014-03-21 00:50:29},
priority = {2},
title = {On the Theory of Scales of Measurement},
volume = {103},
year = {1946}
}
```

[Bibtex]

```
@incollection{Stevens1951a,
author = {Stevens, S. S.},
booktitle = {Handbook of experimental psychology},
citeulike-article-id = {13112820},
pages = {1--49},
posted-at = {2014-03-21 00:50:01},
priority = {2},
publisher = {Wiley},
title = {Mathematics, measurement and psychophysics},
year = {1951}
}
```

[Bibtex]

```
@book{Feldman2004a,
abstract = {Fred Feldman's fascinating new book sets out to defend hedonism as a theory about the Good Life. He tries to show that, when carefully and charitably interpreted, certain forms of hedonism yield plausible evaluations of human lives. Feldman begins by explaining the question about the Good Life. As he understands it, the question is not about the morally good life or about the beneficial life. Rather, the question concerns the general features of the life that is good in itself for the one who lives it. Hedonism says (roughly) that the Good Life is the pleasant life. After showing that received formulations of hedonism are often confused or incoherent, Feldman presents a simple, clear, coherent form of sensory hedonism that provides a starting point for discussion. He then presents a catalogue of classic objections to hedonism, coming from sources as diverse as Plato, Aristotle, Brentano, Ross, Moore, Rawls, Kagan, Nozick, Brandt, and others. One of Feldman's central themes is that there is an important distinction between the forms of hedonism that emphasize sensory pleasure and those that emphasize attitudinal pleasure. Feldman formulates several kinds of hedonism based on the idea that attitudinal pleasure is the Good. He claims that attitudinal forms of hedonism - which have often been ignored in the literature -- are worthy of more careful attention. Another main theme of the book is the plasticity of hedonism. Hedonism comes in many forms. Attitudinal hedonism is especially receptive to variations and modifications. Feldman illustrates this plasticity by formulating several variants of attitudinal hedonism and showing how they evade some of the objections. He also shows how it is possible to develop forms of hedonism that are equivalent to the allegedly anti-hedonistic theory of G. E. Moore and the Aristotelian theory according to which the Good Life is the life of virtue, or flourishing. He also formulates hedonisms relevantly like the ones defended by Aristippus and Mill. Feldman argues that a carefully developed form of attitudinal hedonism is not refuted by objections concerning 'the shape of a life'. He also defends the claim that all of the alleged forms of hedonism discussed in the book genuinely deserve to be called 'hedonism'. Finally, after dealing with the last of the objections, he gives a sketch of his hedonistic vision of the Good Life},
author = {Feldman, Fred},
citeulike-article-id = {13562104},
number = {1},
posted-at = {2015-03-25 19:43:16},
priority = {2},
publisher = {Clarendon Press},
title = {Pleasure and the Good Life: Concerning the Nature, Varieties and Plausibility of Hedonism},
year = {2004}
}
```

```
@book{Narens2002a,
author = {Narens, Louis},
citeulike-article-id = {13531638},
posted-at = {2015-03-02 22:50:46},
priority = {2},
publisher = {Lawrence Erlbaum Associates},
series = {Scientific Psychology Series},
title = {Theories of Meaningfulness},
year = {2002}
}
```

[Bibtex]

```
@article{Holder1901a,
author = {H\"{o}lder, Otto},
citeulike-article-id = {13531626},
journal = {Berichte uber die Verhandlungen der Koeniglich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physikaliche Klasse},
pages = {1--46},
posted-at = {2015-03-02 22:50:46},
priority = {2},
title = {Die Axiome der Quantit\"{a}t und die Lehre vom Mass},
volume = {53},
year = {1901}
}
```

[Bibtex]

```
@article{Nagel1931a,
author = {Nagel, Ernest and Hempel, C. G.},
citeulike-article-id = {13559115},
journal = {Erkenntnis},
number = {1},
pages = {313--335},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {Springer},
title = {Measurement},
volume = {2},
year = {1931}
}
```

[Bibtex]

```
@article{Suppes1951a,
author = {Suppes, Patrick},
citeulike-article-id = {13559116},
journal = {Portugaliae Mathematica},
number = {4},
pages = {163--172},
posted-at = {2015-03-23 22:50:07},
priority = {2},
title = {A Set of Independent Axioms for Extensive Quantities},
volume = {10},
year = {1951}
}
```

[Bibtex]

```
@article{Falmagne1975a,
abstract = {Current axiomatizations for extensive measurement postulate the existence of infinitely small objects. This assumption is neither necessary nor reasonable. This paper develops this theme and presents a more acceptable axiom system. A representation theorem is stated and proved in detail. This work improves some previous results of the author},
author = {Falmagne, Jean-Claude},
citeulike-article-id = {13559114},
journal = {Philosophy of Science},
number = {2},
pages = {137--151},
posted-at = {2015-03-23 22:50:07},
priority = {2},
publisher = {University of Chicago Press},
title = {A Set of Independent Axioms for Positive Holder Systems},
volume = {42},
year = {1975}
}
```