2.1 Preliminary Concepts

I presume familiarity with predicate logic, and with the elementary concepts and notation of set theory. Much of what follows will revolve around properties of binary relations and operations, though, so the following are worth stating:

Preorders – especially weak orders – will be important. Throughout, I’ll use $\succsim$ to represent a number of qualitative preorder relations, and I’ll use $\sim$ and $\succ$ for the symmetric and asymmetric parts of $\succsim$ respectively. That is, I’ll henceforth take it as read that

• $x\sim y$ if and only if $x\succsim y$ and $y\succsim x$ , and

• $x\succ y$ if and only if $x\succsim y$ and $y̸\succsim x$ .

Note that properties are just the special case of $n$ -ary relations where $n=1$ , and every $n$ -ary operation can be recast as an $\left(n+1\right)$ -ary relation. For example, addition is a total binary operation on the set of real numbers $R$ , since it maps $R\times R$ back into $R$ ; it is also the ternary relation $R$ on $R$ such that $(x,y,z)\in R$ if and only if $x+y=z$ . As such, for what follows I’ll usually just write ‘relations’ rather than ‘properties and relations’ or ‘relations and operations’ – but wherever I intend to refer to operations in particular, this will be explicitly marked.

Next we need the generic notion of a relational system. This is a system comprising a set, one or more distinguished relations on that set, and zero or more distinguished binary operations:

The relations and operations used to characterise a relational system are known as the primitives of that system. Note the semi-colon, used to explicitly separate the primitive relations from the primitive operations.Footnote ⁵

An example of a simple relational system is $⟨R,\ge ⟩$ , comprising the set $R$ and the primitive at least as great relation $\ge$ on $R$ . A richer relational system would be $⟨R,\ge ;+⟩$ , which includes also the primitive binary operation $+$ . These are what we’ll call numerical systems – they’re comprised of a set of numbers and one or more relations thereupon. More generally, we take a numerical system to be any relational system constructed from numerical stuff. (There’s no need to be very precise here – some relational systems have a numerical feel about them, and that’ll suffice for referring to them as numerical systems.) In contrast are qualitative systems, or systems constructed from qualitative stuff. For example, if $L$ is the set of determinate length properties (as described at the beginning of the section), and $\succsim$ is the at least as long relation between them, then $⟨L,\succsim ⟩$ will count as a qualitative system. Likewise, for any two lengths $L$ and $L^{\prime }$ , we let their end-to-end concatenation, $L\circ L^{\prime }$ , be the length $L^{\prime \prime }$ of any object that’s as long as what you get when you take two disjoint rigid objects of length $L$ and $L^{\prime }$ and attach them end-to-end. (See Figure 1.) Then $\circ$ will be a binary operation on $L$ , and $⟨L,\succsim ;\circ ⟩$ will also be a qualitative system.

Figure 1 $L^{\prime \prime }$ is the end-to-end concatenation of $L$ and $L^{\prime }$ (i.e., $L\circ L^{\prime }=L^{\prime \prime }$ ).

Henceforth, I’ll use $N$ for numerical systems and $Q$ for qualitative systems. We need then a way of expressing when a qualitative relational system possesses a similar structure to that of some numerical system, such that the latter might be exploited to represent the former. For this we make use of structure-preserving mappings, or homomorphisms:

Corresponding to the distinction between weak homomorphisms and strong homomorphisms, we can say that $\phi$ weakly maps $\bullet$ into $*$ whenever

$\begin{array}{c}x\bullet y=z\text{ implies }\phi \left(x\right)*\phi \left(y\right)=\phi \left(z\right).\end{array}$

and strongly maps $\bullet$ into $*$ whenever the converse also holds.

An example will help to make this clearer. Start first with the simple qualitative system $⟨L,\succsim ⟩$ . A function $\phi :L\mapsto R$ is a homomorphism from $⟨L,\succsim ⟩$ into $⟨R,\ge ⟩$ when

$\begin{array}{c}L\succsim L^{\prime }\text{ if and only if }\phi \left(L\right)\ge \phi \left(L^{\prime }\right).\end{array}$

Since there are no primitive operations in $⟨L,\succsim ⟩$ , conditions 3 and 4 are trivially satisfied and so we don’t bother with the weak/strong distinction. Next, consider the richer system $⟨L,\succsim ;\circ ⟩$ , this time endowed with a primitive concatenation operation. This time, then, a function $\phi :L\mapsto R$ counts as a weak homomorphism from $⟨L,\succsim ;\circ ⟩$ into $⟨R,\ge ;+⟩$ whenever, in addition to the preceding, it weakly maps $\circ$ into $+$ :

$\begin{array}{c}\phi (L\circ L^{\prime })=\phi \left(L\right)+\phi \left(L^{\prime }\right).\end{array}$

And $\phi$ is a strong homomorphism if it strongly maps $\circ$ into $+$ :

$\begin{array}{c}\phi \left(L\right)+\phi \left(L^{\prime }\right)=\phi \left(L^{\prime \prime }\right)\text{ if and only if }L\circ L=L^{\prime \prime }\end{array}$

If $\circ$ is a total operation and $\succsim$ is antisymmetric, then every weak homomorphism from $⟨L,\succsim ;\circ ⟩$ into $⟨R,\ge ;+⟩$ will be a strong homomorphism – but otherwise this needn’t be the case.

2.2 Representation Theorems and Uniqueness

A homomorphism maps the primitive relations and operations of one relational system into the primitive relations and operations of another. When at least a weak homomorphism from $Q$ into $N$ exists, we can say that $N$ has – or otherwise includes as a proper part – a structure similar to that of $Q$ . A strong homomorphism establishes a slightly stronger similarity of structure. In either case, it is this similarity that justifies representing $Q$ using (or ‘in’) $N$ . Because of this, the central theoretical objects of the RTM are results that establish precise conditions for when an arbitrary qualitative system $Q$ can be represented in some specific numerical system $N$ . These are known as representation theorems.

Let $\Phi (Q,N)$ denote the set of all weak homomorphisms from $Q$ into $N$ . Then, for a prespecified $N$ , a representation theorem supplies (at least) sufficient conditions on $Q$ to guarantee that some such homomorphism exists. The conditions are usually called the axioms of that theorem. Typically, the axioms will be chosen such that (at least) most of them are individually necessary for representability – that is, they’re direct consequences of the assumption that $\Phi (Q,N)$ is non-empty. Axioms that are not necessary for representability are usually known as structural axioms.Footnote ⁶ For example:

1.	$X$ is finite	(finitude)
2.	$\succsim$ is a weak order	(weak order)

The weak order axiom is necessary: since $\ge$ is a weak order on $R$ , if $X$ is to be mapped into $R$ then $\succsim$ must itself be a weak order if it’s to be mapped into $\ge$ . The finitude axiom is structural – it’s possible to represent $⟨X,\succsim ⟩$ in $⟨R,\ge ⟩$ even if $X$ is infinite, though in that case additional axioms are needed to ensure representability. (See Reference Krantz, Luce, Suppes and TverskyKrantz et al. (1971), 40–1, for details.)

A representation theorem will also usually include or otherwise be associated with a uniqueness result. In the ideal case, the uniqueness result tells us about the relationship between homomorphisms belonging to $\Phi (Q,N)$ for all $Q$ satisfying the axioms of the associated representation theorem. Continuing the example, it’s plain to see that if $\phi$ is any homomorphism from $⟨X,\succsim ⟩$ into $⟨R,\ge ⟩$ , then so too is $\psi :X\mapsto R$ if and only if

$\begin{array}{c}\psi \left(x\right)\ge \psi \left(y\right)\text{ if and only if }\phi \left(x\right)\ge \phi \left(y\right).\end{array}$

Any $\psi$ satisfying this condition is related to $\phi$ by a strictly increasing (or order-preserving) transformation. So the kind of uniqueness result we’d expect to find attached to Theorem 5 would say that given weak order and finitude, the homomorphisms in $\Phi (⟨X,\succsim ⟩,⟨R,\ge ⟩)$ are unique up to an order-preserving transformation. The ‘unique up to’ phrasing is another way to say that the homomorphism set is constrained by the specified transformation – hence it designates a property shared by all and only the functions in the set.

Two points of caution. First: a uniqueness result applies to all systems satisfying the axioms of the associated representation theorem – not necessarily to all systems that are representable in the specified numerical system simpliciter. This is important if the representation theorem includes structural axioms, which are sometimes used to strengthen the uniqueness result. As a rule of thumb, the more structural constraints imposed on $Q$ , the more restricted the potential homomorphisms from $Q$ into $N$ , leading to a stronger uniqueness result. Second: many uniqueness results apply only to a proper subset of the possible homomorphisms in $\Phi (Q,N)$ . For example, the uniqueness result may assert that there is only one homomorphism from $⟨X,\succsim ⟩$ into $⟨R,\ge ⟩$ which satisfies such-and-such properties (e.g., is a probability measure), even while there are infinitely many homomorphisms in $\Phi (Q,N)$ that do not. For these reasons, one must be careful when interpreting a uniqueness result – some results that on first glance appear rather impressive may end up only really reflecting the strength of the structural conditions employed in the representation theorem and/or arbitrary restrictions to a particular representational format.

Moving on – the final thing to do in this section is outline the major scale types. (See Table 1.) In the preceding example, the $\phi$ in $\Phi (Q,N)$ are unique up to order-preserving transformations. In that case, the set $\Phi (Q,N)$ is said to be an ordinal scale of $Q$ , and the $\phi$ in $\Phi (Q,N)$ are also called ordinal scales of $Q$ . (The ambiguity is unfortunate, but context usually suffices for disambiguation.) Three other scale types are also important. The next is an interval scale: $\Phi (Q,N)$ is an interval scale when the $\phi \in \Phi (Q,N)$ are unique up to a positive affine (or interval-preserving) transformation – that is, if $\phi \in \Phi (Q,N)$ then so is $\psi$ , for any $\psi$ defined such that for some real values $r$ and $s$ , with $r>0$ ,

$\begin{array}{c}\psi \left(x\right)=r\phi \left(x\right)+s.\end{array}$

Whereas order-preserving transformations merely preserve orderings, interval-preserving transformations preserve ratios of differences (and thus also orderings). So, if $\phi$ and $\psi$ are related by an interval-preserving transformation, then

$\begin{array}{c}\frac{\phi \left(x\right)-\phi \left(y\right)}{\phi \left(z\right)-\phi \left(w\right)}=\frac{\psi \left(x\right)-\psi \left(y\right)}{\psi \left(z\right)-\psi \left(w\right)}.\end{array}$

Next are ratio scales: $\Phi (Q,N)$ is a ratio scale when the $\phi \in \Phi (Q,N)$ are unique up to a positive similarity (or ratio-preserving) transformation – that is, if $\phi$ is in $\Phi (Q,N)$ , then so is $\psi$ , for any $\psi$ defined such that for some real value $r>0$ ,

$\begin{array}{c}\psi \left(x\right)=r\phi \left(x\right).\end{array}$

Ratio-preserving transformations preserve ratios (and thus also ratios of differences, and thus also orderings). So, if $\phi$ and $\psi$ are related by a ratio-preserving transformation, then

$\begin{array}{c}\frac{\phi \left(x\right)}{\phi \left(y\right)}=\frac{\psi \left(x\right)}{\psi \left(y\right)}.\end{array}$

Finally, there are absolute scales. This is just the case where $\Phi (Q,N)$ contains exactly one homomorphism.

The foregoing classification scheme originates with Reference StevensStevens (1946). It’s the most widely known means of classifying scale types by a wide margin. It works well for most purposes, and it’ll suffice for ours, though it’s not the only classification scheme nor is it the most general. (A more general classification scheme, though also more complicated, can be found in Reference NarensNarens (1981).)

2.3 Extensive and Conjoint Measurement

Of special interest to the theory of measurement are ‘additive’ representations. Roughly, these are representations that make use of addition in some important way. It can be a little hard to define precisely, though, as what it takes for a representation to count as ‘additive’ can vary across measurement structures. The simplest case is that of extensive measurement. Here, we can say that a homomorphism from $Q$ into $N$ is weakly additive when it weakly maps one of $Q$ ’s primitives into addition; strong additivity can then be defined in the obvious parallel way. The qualitative operation that gets mapped into addition is usually referred to as a concatenation operation.

Let’s discuss one example of an extensive measurement structure in more detail – a positive concatenation structure. Since it doesn’t make sense to speak of lengths shorter than no length at all, we conventionally measure length using additive homomorphisms from $⟨L,\succsim ;\circ ⟩$ into $⟨R^{\ge 0},\ge ;+⟩$ , where $R^{\ge 0}$ is the set of real numbers not smaller than zero. The metre scale is one such homomorphism. Let $L_m$ be the metre length, defined as the length of the path light travels in a vacuum in 1/ $299,792,458$ of a second. Then the metre scale, ${\phi }_m$ , corresponds to the (unique) strong homomorphism from $⟨L,\succsim ;\circ ⟩$ into $⟨R^{\ge 0},\ge ;+⟩$ that assigns the unit value to $L_m$ . In other words,

1. 1. ${\phi }_m\left(L\right)\ge 0$ and ${\phi }_m\left(L_m\right)=1$

2. 2. $L\ge L^{\prime }$ if and only if ${\phi }_m\left(L\right)\ge {\phi }_m\left(L^{\prime }\right)$

3. 3. $L\circ L^{\prime }=L^{\prime \prime }$ if and only if ${\phi }_m\left(L\right)+{\phi }_m\left(L^{\prime }\right)={\phi }_m\left(L^{\prime \prime }\right)$

This method of measuring length is possible precisely because the behaviour of $\succsim$ and $\circ$ is mirrored by the behaviour of $+$ and $\ge$ over the non-negative reals. The most important conditions are as follows:

1.	$\succsim$ is a weak order	(weak order)
2.	$L\circ (L^{\prime }\circ L^{\prime \prime })=(L\circ L^{\prime })\circ L^{\prime \prime }$	(associativity)
3.	$L\circ L^{\prime }=L^{\prime }\circ L$	(commutativity)
4.	$L\succsim L^{\prime }\text{ if and only if }L\circ L^{\prime \prime }\succsim L^{\prime }\circ L^{\prime \prime }$	(monotonicity)
5.	$L\circ L^{\prime }\succsim L$	(weak positivity)
6.	$L\circ L^{\prime }\sim L$ only if $L^{\prime }$ is minimal in $\succsim$	(identity element)

Compare, for $x,y,z\in R^{\ge 0}$ :

1.	$\ge$ is a weak order	(weak order)
2.	$x+(y+z)=(x+y)+z$	(associativity)
3.	$x+y=y+x$	(commutativity)
4.	$x\ge y\text{ if and only if }x+z\ge y+z$	(monotonicity)
5.	$x+y\ge x$	(weak positivity)
6.	$x+y=x$ only if $y=0$	(identity element)

Epistemic approaches to the measurement of belief focus on representing the internal structure of the belief system, and typically posit systems that look a great deal like positive concatenation structures. However, not all ‘additive’ representations follow the same model – they do not all require a primitive concatenation operation that gets mapped into addition. An alternative way to generate ‘additive’ representations employs conjoint measurement structures, wherein multiple quantities are represented simultaneously and the additive structure of the representation is derived from the nature of their lawlike relationships. Since conjoint measurement is important for decision-theoretic approaches to the measurement of belief, it’s worth considering an example in a bit of detail. The procedure is more complicated than the case of extensive measurement (see Figure 2).Footnote ⁷

Figure 2 Conjoint measurement structure

We start with a single weak ordering, $\succsim$ , defined for some quantity $C$ that’s determined by two independent factors $A$ and $B$ . For example, suppose $C$ is discomfort as determined by temperature $A$ and humidity $B$ (Reference Krantz, Luce, Suppes and TverskyKrantz et al. 1971, 17–18), momentum as determined by mass and velocity (Reference Luce and TukeyLuce & Tukey 1964, 4–5), or overall value as determined by monetary and sentimental value.

In any case, we suppose $\succsim$ on $C$ is determined by these two factors $A$ and $B$ , whatever they may all be. Formally we can represent this by reconstructing $\succsim$ as an ordering not over $C$ directly but instead over $A\times B$ . So, for example,

$\begin{array}{c}(A_1,B_1)\succsim (A_2,B_2)\end{array}$

is understood to mean that the level of $C$ determined by the combination of $A_1$ and $B_1$ is at least as great as the level of $C$ determined by $A_2$ and $B_2$ , where the $A_1,A_2$ and $B_1,B_2$ are levels of $A$ and $B$ respectively.

The next step is to extract from $\succsim$ two extensive ‘subsystems’ for $A$ and $B$ separately. We start by defining an ordering ${\succsim }_a$ over $A$ , by comparing the levels of $C$ that result from varying the $A$ factor while holding the $B$ factor fixed. That is,

$\begin{array}{c}{A}_1{\succsim }_a{A}_2\text{ iff }(A_1,B_i)\succsim (A_2,B_i)\text{ for all }{B}_i\in B.\end{array}$

So $A_1$ is greater than $A_2$ when $A_1$ contributes more to $C$ than $A_2$ does, holding the level of $B$ fixed. Note, of course, that the definition alone doesn’t guarantee ${\succsim }_a$ will be a weak order – for that we need to suppose that if changing from $A_1$ to $A_2$ increases the level of $C$ while holding the level of $B$ fixed for any particular level of $B$ , then the same should hold for all levels of $B$ . Essentially this amounts to saying that the contribution $A$ makes to $C$ is independent of the contribution made by $B$ . This is established by the independence axiom, which will be explained in a later paragraph. An exactly parallel definition gets us an ordering ${\succsim }_b$ over $B$ .

At this point we’ve got two very simple subsystems, $⟨A,{\succsim }_a⟩$ and $⟨B,{\succsim }_b⟩$ . But we should like to construct extensive structures so as to enable a richer numerical representation. Thus we will need to define a concatenation operation as well. Assume that $A$ and $B$ combine in an intuitively ‘additive’ fashion. (This will be qualitatively expressed by means of the independence and double cancellation axioms.) Then, it will be possible to draw meaningful correlations in size between intervals in ${\succsim }_a$ and in ${\succsim }_b$ by comparing the effects on the level of $C$ that result from varying one factor while holding the other fixed. For suppose there are $A_1,A_2,B_1,B_2$ such that

$\begin{array}{c}(A_1,B_2)\sim (A_2,B_1)\succ (A_1,B_1).\end{array}$

We can read this as saying that changing from $A_1$ to $A_2$ (while holding the $B$ -level fixed) has the same effect on $C$ as changing from $B_1$ to $B_2$ (while holding the $A$ -level fixed). If we let $A_i\to A_j$ designate the interval between $A_i$ and $A_j$ as observed in the effect on $C$ , and likewise for $B_i\to B_j$ mutatis mutandis, then what we’ve said is that $A_1\to A_2$ is equal to $B_1\to B_2$ , and thus we compare the size of intervals in one factor to intervals in the other. Given that, if there also are minimal levels $A_0$ and $B_0$ of $A$ and $B$ , then we can define concatenation operations ${\oplus }_a$ and ${\oplus }_b$ for each of $A$ and $B$ . Starting with ${\oplus }_a$ , we say

$\begin{array}{c}{A}_1{\oplus }_a{A}_2=A_3\end{array}$

just in case the effect on $C$ that results from increasing $A_0$ to $A_3$ while holding the level of $B$ fixed at $B_0$ is equal to the effect on $C$ that results from increasing the level of $A$ from $A_0$ to $A_1$ and increasing the level of $B$ from $B_0$ to some level $B_x$ such that the result is equal in effect on $C$ as observed from an increase from $A_0$ to $A_2$ . That is, if

$\begin{array}{c}(A_3,B_0)\sim (A_1,B_x),\end{array}$

then $A_0\to A_3$ is equal to $A_0\to A_1$ plus $B_0\to B_x$ , where the latter is known to be equal to $A_0\to A_2$ . Treating $A_0$ and $B_0$ as ‘zero’ points, then, the ‘size’ of the interval $A_0\to A_i$ gives the absolute ‘size’ of $A_i$ alone, and so this essentially amounts to saying that $A_3$ equals $A_1$ plus $A_2$ .

The upshot is that, with the appropriate axioms on $\succsim$ , we can extract extensive subsystems $⟨A,{\succsim }_a;{\oplus }_a⟩$ and $⟨B,{\succsim }_b;{\oplus }_a⟩$ out of the initial system $⟨A\times B,\succsim ⟩$ , which will admit of separate additive representations ${\phi }_a$ and ${\phi }_b$ . The final step is to then show that there exists some numerical operation, $f$ , that combines ${\phi }_a$ and ${\phi }_b$ so as to represent $\succsim$ on $A\times B$ ; that is,

$\begin{array}{c}(A_1,B_1)\succsim (A_2,B_2)\text{ if and only if }f({\phi }_a\left(A_1\right),{\phi }_b\left(B_1\right))\ge f({\phi }_a\left(A_2\right),{\phi }_b\left(B_2\right)).\end{array}$

The function $f$ may take a wide variety of forms depending on the shape of $\succsim$ , but one simple case is when ${\phi }_a$ and ${\phi }_b$ combine additively to determine a final value that represents $C$ :

$\begin{array}{c}(A_1,B_1)\succsim (A_2,B_2)\text{ if and only if }{\phi }_a\left(A_1\right)+{\phi }_a\left(B_1\right)\ge {\phi }_a\left(A_2\right)+{\phi }_b\left(B_2\right)\end{array}$

The result is a conjoint representation of all three quantities $A$ , $B$ , and $C$ simultaneously, achieved via a two-component vector homomorphism $\phi$ from $⟨A\times B,\succsim ⟩$ into $⟨R\times R,\ge ⟩$ that ‘decomposes’ into ${\phi }_a$ and ${\phi }_b$ via $f$ .

All of this obviously requires that $\succsim$ will satisfy the axioms required for the existence of such a representation. These axioms will together essentially assert that $\succsim$ behaves in the manner one would expect if levels of $C$ were determined by the sum of two independent factors $A$ and $B$ . For instance, the following are very typical necessary axioms for additive conjoint measurement structures:

1.	For all $A_i,A_j,A_k,A_l\in A$ and $B_i,B_j,B_k,B_l\in B$ , $(A_i,B_k)\succsim (A_j,B_k)$ if and only if $(A_i,B_l)\succsim (A_j,B_l)$ , and $(A_k,B_i)\succsim (A_k,B_j)$ if and only if $(A_l,B_i)\succsim (A_l,B_j)$	(independence)
2.	For all $A_i,A_j,A_k\in A$ and $B_i,B_j,B_k\in B$ , $(A_i,B_j)\succsim (A_j,B_k)$ and $(A_j,B_i)\succsim (A_k,B_j)$ implies $(A_i,B_i)\succsim (A_k,B_k)$	(double cancellation)

Again, it’s helpful to compare the qualitative axiom with the intended numerical representation. The independence axiom is straightforward:

$x+z\ge y+z$ for some $z$

$\downarrow$

$x+z\ge y+z$ for all $z$ The double cancellation axiom is a little less obvious; it concerns cases in which the common terms of two inequalities cancel out to determine a third:

$x+m\ge y+o$

$y+n\ge z+m$

$\downarrow$

$x+n\ge z+o$

Let’s sum up. In the example of a conjoint measurement structure I’ve just outlined, the numerical representations of $A$ , $B$ , and $C$ are a package deal. Or, more accurately, they’re three parts of a single representational system comprising several functions and an operation that ties them together. Note in particular – and this will be important – that the two constructed subsystems are defined such that they only make sense as parts of the larger system. The primitives of $⟨A,{\succsim }_a;{\oplus }_a⟩$ , for instance, are characterised in terms of how $A$ relates to $B$ in the determination of $C$ . Likewise, the operation ${\oplus }_a$ needn’t correspond to any ‘natural’ concatenation operation that can be readily defined in terms of $A$ alone, without reference to how $A$ interacts with $B$ and $C$ . To the extent that $A$ is represented as having an ‘additive’ structure in this manner, then, that structure is manifest in its relationship with $B$ and $C$ . This is all to say that the meaning of the representation ${\phi }_a$ in this context can only be fully grasped by reference to its relation to ${\phi }_b$ as specified by the rule $f$ by which they combine to represent $C$ . The three numerical representations are, in that sense, inseparable.

Contrast this with the extensive measurement of $⟨L,\succsim ;\circ ⟩$ , where the primitives of that system can be characterised without any direct reference to other quantities. One can appreciate what it is for the system of lengths to have an ‘additive’ structure just by considering how determinate length attributes relate to other determinate length attributes. One needn’t embed the system of lengths into a larger relational structure involving multiple quantities in order to comprehend what it is for one length to be twice as long as another, for example, since one can just see it directly by placing the lengths alongside one another. An intuitive way to characterise the difference between the two kinds of measurement structure, then, is to say extensive measurement is geared towards representing the internal relational structure of a single determinable attribute, whereas the conjoint measurement is geared more towards representing the relationships between several attributes.

2.4 Conventionality

One of the more important lessons of the RTM concerns the extent to which our use of numbers to represent the world is grounded in convention. (By ‘conventional’, I mean unforced from a purely mathematical point of view, and so setting aside pragmatic considerations.) It’s useful to divide it up into three distinct grades of conventionality.

Most will be plenty familiar already with conventionality of the first grade, choice of scale – that is, in the choice of homomorphism from $\Phi (Q,N)$ , for a fixed choice of $Q$ and $N$ . This arises, for instance, when we are free to choose between metres, inches, light-years, or beard-seconds (the amount a typical beard grows in one second) as our units for measuring length.

A rather deeper and not as widely appreciated form of conventionality arises in the choice of numerical system. A very simple example is the choice to use $\ge$ to represent a weak order $\succsim$ , rather than $\le$ . Either would obviously work just as well as the other – and just as well as any other weak order on the reals. But a more complicated example is also worth mentioning. As I noted in the previous section, conventional measures of length are almost always additive homomorphisms from $⟨L,\succsim ;\circ ⟩$ to $⟨R,\ge ;+⟩$ . On any such measure, the value assigned to $L\circ L$ will always be twice the value assigned to $L$ . Our overwhelming familiarity with these additive measures can lead to the sense that there’s something uniquely correct about this representation – that the qualitative relation holding between $L$ and $L\circ L$ is an essentially twice-ish relation. As Brian Reference EllisEllis (1968, 83) put it, there’s a common sense that the ‘twice’ in ‘twice as long’ has a significance independent of the conventions of measurement. (‘Clearly, 2 meters is twice as long as long as 1 meter – that is the natural and obvious way to describe their relation!’) However, the axioms that justify the additive measurement of length – associativity, commutativity, monotonicity, and so on – are consistent with a multitude of non-additive representations whereby $L\circ L$ need not be assigned a value twice that which is assigned to $L$ .

Consider multiplicative measures, which map $⟨L,\succsim ;\circ ⟩$ not into $⟨R^{\ge 0},\ge ;+⟩$ but into $⟨R^{\ge 1},\ge ;\times ⟩$ instead (see Reference HölderHölder 1901; Reference Krantz, Luce, Suppes and TverskyKrantz et al. 1971, 11–12, 99ff; Reference NarensNarens 1985, 27–31). Let the multiplicative (base 2) version of the metre scale be called the schmetre scale; it corresponds to a homomorphism ${\phi }_{sch}$ that maps $⟨L,\succsim ;\circ ⟩$ onto $⟨R^{\ge 1},\ge ;\times ⟩$ such that

1. 1. ${\phi }_{sch}\left(L\right)\ge 1$ and ${\phi }_{sch}\left(L_m\right)=2$

2. 2. $L\ge L^{\prime }$ if and only if ${\phi }_{sch}\left(L\right)\ge {\phi }_{sch}\left(L^{\prime }\right)$

3. 3. $L\circ L^{\prime }=L^{\prime \prime }$ if and only if ${\phi }_{sch}\left(L\right)\times {\phi }_{sch}\left(L^{\prime }\right)={\phi }_{sch}\left(L^{\prime \prime }\right)$

On the schmetre scale, the value assigned to $L\circ L$ will always be equal to the square of the value assigned to $L$ . Since 1 metre is 2 schmetres, then, 2 metres is 4 schmetres and 4 metres is 16 schmetres. Hence, if 4 metres is twice as long as 2 metres, it follows that 16 schmetres is twice as long as 4 schmetres. The point is not that there’s some sort of contradiction here – there isn’t. Rather, it’s that the qualitative relation between $L\circ L$ and $L$ is no more a twice-ish relation than it is a square-ish relation. Our use of ‘twice as long’ to refer to and describe the relation between $L\circ L$ and $L$ reflects only a conventional preference for additive representations over an infinite variety of alternative representational formats that are, from a mathematical point of view, equally adequate to the task.

But the conventionality runs deeper still, for it arises also in the choice of qualitative system. Again, length supplies a useful example. Earlier I characterised $\circ$ on $L$ in terms of laying objects end-to-end. However, there are other ways of concatenating lengths that we might have employed as primitives instead. One alternative (also discussed by Reference EllisEllis 1968, 80–1) is right-angled concatenation. Say that $L\odot L^{\prime }=L^{\prime \prime }$ just when $L^{\prime \prime }$ is the length of the hypotenuse of the right-angled triangle with catheti of lengths $L$ and $L^{\prime }$ . (See Figure 3; I did have to look that word up.) Right-angled concatenation has all the same key properties as end-to-end concatenation which permit additive measurement. In an alternative history, then, we might have chosen to measure length by mapping $⟨L,\succsim ;\odot ⟩$ into $⟨R^{\ge 0},\ge ;+⟩$ . (Or $⟨R^{\ge 1},\ge ;\times ⟩$ , or $⟨R,\ge ;*⟩$ , or …) This would have simplified how we express the relationships between sides of a right-angled triangle, though it would also have made calculating end-to-end concatenations of distances slightly more difficult.

Figure 3 $L^{\prime \prime }$ is the right-angled concatenation of $L$ and $L^{\prime }$ (i.e., $L\odot L^{\prime }=L^{\prime \prime }$ ).

With that said, I don’t want to give the impression that anything goes. A minimal constraint on the choice of qualitative system is that the primitives must be natural. Without some such constraint, we trivialise the whole endeavour. For instance, assuming no more than that $L$ can be mapped into $R$ , we know already that there exists a binary relation $R$ on $L$ that maps into $\ge$ , in the sense that

$\begin{array}{c}(L,L^{\prime })\in R\text{ if and only if }\phi \left(L\right)\ge \phi \left(L^{\prime }\right).\end{array}$

Likewise, supposing only that $⟨L,\succsim ⟩$ maps into $⟨R,\ge ⟩$ , we know already that there will exist at least one ternary relation $R$ such that

$\begin{array}{c}(L,L^{\prime },L^{\prime \prime })\in R\text{ if and only if }\phi \left(L\right)+\phi \left(L^{\prime }\right)=\phi \left(L^{\prime \prime }\right).\end{array}$

So the fact that we can then find some relations on $L$ corresponding to $\ge$ and $+$ is thoroughly uninteresting. We can derive such relations from any mapping of $L$ into $R$ , so long as we’re permissive enough about what counts as a relation. It’s a matter of convention what we take the primitive relations in our qualitative systems to be, that is true, but measurement is only interesting when those relations are natural.

2.5 Meaningfulness

A focus of the discussion to follow involves differentiating what is meaningful from what is not in the measurement of belief. The most common strategy for drawing such distinctions goes via invariance. Essentially, the idea is that a numerical property or relation is meaningful only if it’s invariant across alternative numerical representations; otherwise, it’s a mere artefact of convention.

Compare the case of temperature. When represented in ${}^{\circ }$ C, water freezes at 0 and boils at 100, the hottest temperature recorded in Australia is almost exactly half way between these values (50.7) and more than double the hottest temperature recorded in Antarctica (19.8). But not all of the numerical properties and relations just mentioned are meaningful. Measured in ${}^{\circ }$ F, water freezes at 32 and boils at 212, the hottest recorded temperature in Australia (123) is less than twice the hottest temperature in Antarctica (68), though it’ll still be just over halfway between the freezing and boiling points of water. Celsius and Fahrenheit are equally legitimate interval scale measures of temperature – they’re numerically distinct but they’re not meaningfully distinct. The particular values associated with each temperature and the ratios between those values vary between alternative scales, and they are therefore not meaningful. Ratios of differences are invariant, on the other hand, and so we call them meaningful.

So far, so good. But consider again the additive measures of $⟨L,\succsim ;\circ ⟩$ . If $\phi$ and $\psi$ are any two additive measures of that system, then

$\begin{array}{c}\phi \left(L\right)=2\phi \left(L^{\prime }\right)\text{ if and only if }\psi \left(L\right)=2\psi \left(L^{\prime }\right).\end{array}$

But this, too, is an artefact of conventions – in the choice of numerical system. As we’ve just discussed, the qualitative relation that holds between $L$ and $L^{\prime }$ whenever $L^{\prime }$ is twice as long as $L$ isn’t itself a ratio relation in any deep sense, and if $\theta$ is a multiplicative measure of length, then

$\begin{array}{c}\phi \left(L\right)=2\phi \left(L^{\prime }\right)\text{ if and only if }\theta \left(L\right)=\theta (L^{\prime }{)}^2.\end{array}$

In that broader sense, almost all the information in any numerical representation of $⟨L,\succsim ;\circ ⟩$ is an artefact of convention. There’s approximately nothing that’s invariant across all numerical representations of a qualitative system, and what is preserved is far too little to be of much interest.

The upshot is that meaningfulness needs to be understood relative to a fixed choice of numerical system. A more precise account of meaningfulness, and one that incorporates this lesson, originates with Reference PfanzaglPfanzagl (1968). I present it here in lightly modified form:

Definition 6. Suppose that $\Phi (Q,N)$ is non-empty, where $Q=⟨X,R_i;{\bullet }_j⟩$ and $N=⟨Y,S_i;{*}_j⟩$ . For any $\phi \in \Phi (Q,N)$ and any $n$ -ary relation $S$ on $Y$ , $R(S,\phi )$ is the relation induced on $X$ by $S$ and $\phi$ if and only if

$\begin{array}{c}(x_1,\dots ,x_n)\in R(S,\phi )\text{ if and only if }(\phi \left(x_1\right),\dots ,\phi \left(x_n\right))\in S.\end{array}$

$S$ is $Q$ -meaningful relative to $N$ when $R(S,\phi )$ doesn’t depend on the choice of $\phi$ in $\Phi (Q,N)$ .

Where the $Q$ and $N$ are obvious given context, we simply say that $S$ is meaningful. Note one: if $S$ is among the primitive relations $S_i$ of $N$ , then $R(S,\phi )$ is just the corresponding primitive relation $R_i$ in $Q$ and thus $R(S,\phi )$ is automatically $Q$ -meaningful relative to $N$ . So we’re only interested in the case where $S$ isn’t among the primitives relations of $N$ . Note two: if $S$ is one of the primitive operations of $N$ , then it doesn’t automatically follow that $S$ is meaningful, except in the special case where every homomorphism in $\Phi (Q,N)$ is a strong homomorphism. So being a primitive relation in $N$ suffices for being meaningful, being a primitive operation doesn’t.

To get a grip on Definition 6, it’s helpful to compare cases where a numerical relation isn’t meaningful. Observe first of all that every numerical property or relation $S$ induces a corresponding property or relation $R(S,\phi )$ on the qualitative system relative to each $\phi \in \Phi (Q,N)$ . So, if some ordinal scale $\phi$ maps $Q=⟨L,\succsim ⟩$ into $N=⟨R,\ge ⟩$ , the 2:1 ratio relation induces a corresponding relation on $L$ that holds for $L,L^{\prime }$ whenever $\phi \left(L\right)=2\phi \left(L^{\prime }\right)$ . But that relation isn’t $Q$ -meaningful relative to $N$ , precisely because $R(2:1,\phi )$ needn’t equal $R(2:1,\psi )$ for every other $\psi$ in $\Phi (Q,N)$ . By contrast, the 2:1 ratio is meaningful relative to the additive measures of $⟨L,\succsim ;\circ ⟩$ , since $R(2:1,\phi )$ equals $R(2:1,\psi )$ for any two additive measures $\phi$ and $\psi$ . (Why does that matter? Because if the 2:1 ratio is meaningful with respect to the additive measurement of length, then we can draw generalisations and formulate laws involving that ratio without worrying that it all depends on an arbitrary choice of scale.)

In general, the idea is that a numerical relation is meaningful inasmuch as it always corresponds to the same qualitative relation regardless of what homomorphism we care to use, given a fixed choice of numerical system. That’s just what ‘meaningful’ means in this context: always picks out the same thing independent of the choice of scale. So to make some headway on the matter of what should be considered meaningful in our numerical representations of belief, we need to say more about the kinds of qualitative structures these representations are supposed to be representations of.

3.1 Quantitation, Not Elicitation

In the classic presentations of the RTM, qualitative systems are understood to be empirical relational systems. These systems are built around primitives that are directly and publicly observable in the context of some experimental procedure. For example, rather than characterising the length system $⟨L,\succsim ;\circ ⟩$ as a set of attributes and higher-order relations between them, if I were doing things in the classical manner, then I’d have characterised it as a set of rigid physical objects, the observable at least as long relation between them, and a concatenation operation interpreted as the physical process of taking two rigid objects and joining them to form a new composite object. Essentially, empirical relational systems are systems in which nothing is hidden from view – the relations should be open to observation, the relata should be things we can touch and see and poke and prod, and the operations are physical procedures on or processes observed in the entities being measured.

There are some obvious problems that arise when measurement is understood this way. (These problems were not unknown to the founders of the representational theory; cf. Reference Krantz, Luce, Suppes and TverskyKrantz et al. 1971, 27–31.) In violation of ubiquitous transitivity axioms, for example, one might have a series of objects each longer than the preceding by an imperceptible amount, such that adjacent objects will be observed to be of the same length even while the last is much longer than the first. Similar problems arise for all empirical relational systems, and will be familiar from the history of operationalism. They all point to the same basic issue: quantities cannot be perfectly characterised in terms of the experimental procedures by which they’re measured, since no such procedure is ever perfect. Instead, measurement procedures are developed on the basis of what our best theories imply about the conditions under which observable experimental outcomes will reliably (albeit imperfectly) correlate with variations in some limited range of magnitudes of the quantity we desire to measure.Footnote ⁸