11	Measurement from Empiricist and Realist Points of View

Zoltan Domotor

University of Pennsylvania

The term “measurement” conjures up many meanings. It suggests ways of assigning numbers to extents of quantities, types of interaction between physical systems and sensors, modes of detection, coding, transfer and storage of information, and much more.

In this chapter, I shall concentrate on certain obstructions to empiricist conceptions of measurement and on some realist ways of overcoming them. Furthermore, I will propose a framework for reasoning about measurement theories. The need for such a framework is prompted by the ever-increasing proliferation of representation results in axiomatic measurement theory and the desire for unification. Traditional epistemology holds that measurement is a fundamental means by which we come to know things. No details of this idea seem to be available even within the simplest instance of measurement theories; therefore, I will give an account of how measurement results affect the states of our knowledge.

My main task here is to bring the empiricist and realist viewpoints into sharper focus than customary by showing their specific consequences within the context of measurement theories. The position I will defend takes theoretical quantitative structures to be of primary empirical importance, whereas the spectrum of associated qualitative measurement structures is regarded as secondary. However, unlike some realists, including Friedman (1983), I perceive observable structures not as some sort of partial submodels of certain theoretical models but as appropriate quotient structures of these models, obtained by classifying away some of their theoretical aspects. The idea is that theoretical models are projected onto their observational structures, and being so, some features are understandably lost or forgotten. Formally this is the exact opposite of injecting or embedding observational structures into larger theoretical models, frequently criticized by the realists.

As in Domotor (1981), I continue to recognize the importance of quantitative representations of qualitative structures, but my conclusion will be that representation or injection of the qualitative into a quantitative is fully justified only when coupled with the dual procedure of projection.

In their attempts to understand how science works, some philosophers of science find it important and interesting to single out and contrast the empiricist and realist approaches to scientific theories. These approaches differ primarily over the attitude they take to the nature of theoretical terms and theoretical laws introduced by a scientific theory—in our case, quantities and their lawlike relationships. According to the empiricist approach, the fact that a theory happens to be presented within a language that is blessed with abstract theoretical terms does not mean that these terms are to be taken seriously There is no reason to suppose that the world actually contains anything that corresponds to these special terms. The tasks that the theory has to perform do not in any way require such metaphysical commitments. Similar things are believed about the nature of theoretical laws. Laws are adequate to the extent that they respect the data and lead to good predictions, and the question of their global truth is not for science to answer.

To this, the realist objects on two counts. First, a theory is more than just an instrument for deriving predictions; it actually must satisfy our demand for explanation by telling us how and why things happen. However, for this purpose, the theory must be taken as something that aims at a genuine description of what actually goes on. Further, a theory’s success in yielding predictions would itself be mysterious unless there was in fact some connection between the way the theory worked and the way reality is. Second, the practice of science shows that it is precisely by viewing theories realistically that science progresses. It is by thinking of theories as literal albeit perhaps approximate descriptions of what goes on that we are able to suggest good ways of extending or generalizing them and can see where they might require modification. To this, the empiricist responds by pointing out that realism principally expresses the overall goals of science rather than its actual structure. In addition, what the structure of science suggests is that there is no fixed ontology and everlasting semantics. Instead there are many alternative theoretical ontologies and interpretations, and, in some cases, there is a mathematical approach to how these ontologies are created and how interpretations are conceived.

Unfortunately, in view of their excessive generality and abstractness, these arguments have limited force in the delicate contexts of measurement.

Currently there are two basic approaches to measurement, which I shall call the representational approach and the interactionist approach.

In the representational approach, one deals with numerical (and possibly geometric) representations of certain qualitative empirical structures of measurable attributes. The underlying doctrine of these representations is a modern-day implementation of Ramsey’s empiricist methodology. As is well-known among philosophers of science, for Ramsey (1950), every empirical theory comes in a bifurcated language. Observational facts and their phenomenological generalizations are stated in a primary language L₀, whereas the systematizing, theoretical aspects of these facts are expressed in a conceptually fancier secondary language L. Linkages between the primitives of these languages are made visible by suitable correspondence rules. What is important for us is that the empirical semantics of L₀ is quite different from that of L. Although the ontology and meaning of terms in the primary language are fully determined by the intended experimental procedures, the meaning of theoretical terms in the secondary language is widely open and the metaphysics of these terms is of secondary importance. Theoretical terms are conceptual analogs of fictional characters, and, as such, their job is not so much referring to reality as organizing and systematizing our theoretical reasoning about reality. Calling L a “secondary” language surely expresses this sort of semantic or metaphysical contempt.

Now how does a scientific theory T(L₀,L), erected over a disjoint pair of languages L₀ and L, manage to refer to reality? As Ramsey saw it, the theory succeeds in referring to reality precisely via its body T(L₀,L) ↾ L₀ of observational consequences, and, I would add, its class of validating models Mod T(L₀,L) ↾ L₀. Ramsey also noted that every observable prediction P(L₀) made by the full theory T(L₀,L) can be made equally well, logically speaking, by the ontologically less demanding Ramsey sentence ∃L T(L₀, L) of the theory, obtained by second-order existentially quantifying away all primitives in L as they occur in the formulas of the theory:

However, there is more. The Ramsey sentence is first-order derivable within the observable component T(L₀,L) ↾ L₀ of the theory precisely when the models of this component coincide with the observational reducts of models of the full theory:

Mod(T(L₀,L)↾L₀) = (Mod T(L₀,L))↾ L₀.

When this high-flown logical terminology about the commutativity between models and languages is brought down to earth, it says that a Ramsey sentence is true precisely when the observational fragment of a model of the theory is extendable—by adding appropriate theoretical structure—to its full model. In other words, the L-terms are eliminable from T(L₀,L) precisely when the models of the observational component T(L₀,L) ↾ L₀ are extendible to appropriate models of T(L₀,L).

Applying this view to measurement, the Ramsey-style empiricist point is that a generic measurement theory is built over a pair of disjoint sets of qualitative/quantitative primitives, where only the qualitative part of the theory is of primary empirical importance. Quantitative terms play only a pragmatic role in that they aid the formulation of the theory as a whole and make it conceptually and calculationally manageable.

A representationalist research program, therefore, consists in axiomatic characterizations of the qualitative (observational) components of measurement theories, necessary and/or sufficient for proving the Ramsey sentences of these theories.

According to the interactionist approach, measurement is based on a physical interaction between a measuring instrument by which the measurement is carried out and a physical system on which the measurement is being done. At the end of the interaction, the instrument indicates its result of measurement (i.e., some fragment of information obtained by the instrument about the value of a measured attribute of the system).

The formal background for interactionist theories goes back to the classical distinction between intensive and extensive quantities. Quite specifically, with every ring R of measurable magnitudes (reals being the prime example) and a physical space X, there comes a ring R^x of intensive quantities on X with their possible magnitudes in R, defined by suitable (continuous, measurable) maps from X to R. The central features of intensive quantities, essential in dimensional analysis, are their multiplicativity and contravariance. Thus, in addition to being a linear space, the space R^x is actually a ring under pointwise multiplication

f·g(x) = f(x)·g(x)

for all x in X and f·g in R^x.

Furthermore, intensive quantities transform contravariantly in the sense that every map h : X → Y induces a ring homomorphism R^h : R^Y → R^x in the reversed direction, defined by the usual functional composition R^h(f) = f∘h. Here and below the notation h: X → Y means that h is a function with domain X and codomain or range Y.

Dually with every ring R and a physical space X, there comes a module R(X) of extensive quantities on X with their magnitudes in R, defined by measures on X. The fundamental characteristics of extensive quantities are their linearity or convexity (in the case of probabilities) and covariance. What this means is that R(X) is a module over the ring R^x, and every map h :X → Y induces an unreversed module homomorphism R(h): R(X) → R(Y), defined by the direct image formula

R(h)μ(B) = μ(h^-1(B))

for all measurable subsets B of space Y and measure μ in R(X).

Now these dual pairs of spaces of intensive and extensive quantities are brought together by the fundamental bilinear inner product or evaluation map (|):R^X × R(X) → R, defined by the usual expectation integral

and satisfying the principal substitution law (f/R(h)μ) = (R^h(f)|μ). Finally Radon-Nikodym derivatives produce intensive quantities from pairs of extensive quantities (e.g., density from mass and volume), and, in general, the foregoing inner product generates extensive quantities from pairs of intensive/extensive quantities. By including tensor products of extensive and/or intensive quantities, representing empirical interactions, we obtain virtually every type of quantity used in natural science in general and in physics in particular.

According to the interactionist view, measurement is basically a species of physical interaction that is capable of empirical instantiation of the inner product defined previously. So a result of measurement is nothing more than the numerical value of some empirically realized inner product, involving the quantity measured and some other quantities, characterizing the state of the system and its form of interaction with the measuring instrument.

It is important to point out that the representational measurement theorists have not succeeded thus far in convincing the interactionists to go beyond the inner product conception of measurement. In my own view, this is precisely the place where many realists part company with the empiricists. Inner products lead to numbers in a nice way, so why go beyond?

It is high time we looked at some specific examples, discussed mainly from an empiricist point of view.

OBSTRUCTIONS TO EMPIRICIST INTERPRETATIONS OF MEASUREMENT

How is measurement possible? Empiricists of today hold that measurement hinges on the possibility of a quantitative representation of stereotype bodies of qualitative data regarding objects, events, and actions. The aim of measurement theory, therefore, is to discern and lay bare the species of those data structures that actually admit such representations.

The relevant formal methodology is axiomatic. Data structures are described in terms of mixtures of empirically testable and technically or normatively desirable axioms, and they are reasoned about extensionally in terms of suitable set-theoretic predicates, defined by the axioms, including “is an extensive measurement structure,” “is an intensive measurement structure,” “is an additive conjoint measurement structure,” and many, more. The most important measurement-theoretic task is to find empirically adequate lists of axioms that guarantee the existence of desired quantitative representations and possibly characterize or cast light on the nature of their uniqueness.

I begin, as a way of entering the subject of this section, by characterizing a particular empiricist interpretation of measurement, which, although not representative of the more recent formulations of some writers, is, in my view, the most frequent form encountered in current textbooks and monographs, including Pfanzagl (1968), Krantz, Luce, Suppes, & Tversky (1971), Roberts (1979), and Narens (1985).

Throughout this chapter, I will illustrate the relevant measurement-theoretic concepts with simple examples drawn from the realm of measuring subjective probability. I dwell on these examples in order to emphasize the importance of representational measurement theory in the social sciences as opposed to natural science, where interactionist theories are important. Further, as Adams (chapter 4 in the present volume) shows this circle of examples is worth studying in detail, because it beautifully illustrates and motivates the methodological problems of axiomatic measurement. Finally examples will I hope help lessen the risk of becoming lost in an abyss of formal definitions so typical in modern measurement theory textbooks.

From a logical point of view, much of axiomatic representational measurement theory may be viewed as a series of instantiations of and variations on the following dual pair of Ramsey formulas: Representation

M(E,p) iff ∃P[C(E,ρ,P) & T(E,P)]

Uniqueness

M’(E,ρ) iff ∀P[C(E,ρ,P) ⇒ T’(E,ρ)]

Here M and M’ are suitable set-theoretic predicates, defined by strings of observational (phenomenological) axioms and applicable to various, intended qualitative measurement structures (E,ρ) of a fixed similarity type, consisting of a domain E and some relational structure ρ thereon. In contrast, T and T’ are theoretically motivated set-theoretic predicates, expressed by theoretical axioms and satisfiable by appropriate, real-valued quantitative structures (E,P), closely linked to the presumed qualitative structures (E, ρ). This linkage is captured by a correspondence relation C(E,ρ, P).

As a simple illustration, take the popular qualitative measurement structure (E, ), given by a finite boolean algebra E of events over some sample space and impressed with a de Finetti-style qualitative probability (at most as likely as). Then under the correspondence relation

∀x, y ∈ E [x y⇔ P(x) ≤ P(y)] (1)

linking concepts E, and P, the foregoing Ramsey representation formula is true of (E,), just in case the predicate μ is presented by the familiar Scott–Adams axioms of qualitative probabilities, referred to in Adams (chapter 4 in this volume), and T is the predicate defined by the classical Kolmogorov axioms for finitely additive probability spaces (E,P). Here we use the symbolism T(E, P) as short for the sentence “The set-theoretic structure (E,P) is a finitely additive Kolmogorovian probability space,” and similarly the notation M(E,) abbreviates the subject-predicate nexus “The set-theoretic object (E,) is a qualitative probability structure in the sense of Scott and Adams.”

Under the preceding instantiation, the Ramsey formula for representation says that, granted a qualitative measurement structure (E,) of de Finetti’s probabilistic data, a necessary and sufficient condition for the existence of a Kolmogorovian probability measure on E agreeing with is the truth of Scott–Adams axioms in the measurement structure (E,).

Regarding uniqueness, take μ’ to be the set-theoretic predicate determined by de Finetti’s axioms for subjective probability, and let T’ be a predicate generated by axioms for convex mixing. It is easy to check that, in our context, a convex mixture of agreeing probabilities is again an agreeing probability. We will see later that the uniqueness predicate T’ is fully characterized by the barycentric subdivision of the simplex of all probabilities defined on the boolean algebra E.

As pointed out before, given theoretical frameworks T and T’ together with a version of the correspondence relation (1), the main job of a representational measurement theorist is to identify the bodies of empirically adequate axiomatic systems for μ and M’ in such a way that both Ramsey formulas hold. A plentiful and varied supply of representation results arises in this way; some are trivial, and some are rather difficult.

The Ramsey sentence conception remains valid even in the context of fuzzy or probabilistic relations, where the atomic observations x y are viewed as members of a more general lattice than a boolean algebra, or are replaced by probabilistic data p[x y] requiring an equational correspondence

P[x y]= P[U_X ≤ U_y],

involving a joint probability distribution of a family of real-valued random variables U_x, …, U_y. Here the idea is that the observational probability p of ranking object y higher than object x must coincide with the value of a theoretically given probability distribution P of an agreeing ranking of a corresponding pair of random variables.

If we read the Ramsey formulas from right to left, they in effect tell us how to do science without numbers. Under a related interpretation, they tell us how to define real numbers in terms of their uses. The converse reading is of course what representational measurement is all about: a carefully controlled introduction of quantities into the context of qualities.

Generalizations of Ramsey formulas, involving empirical operations and several types of observational structure, should not detain us here. In any case, they do not lead to qualitatively different situations. Let me just add that the classical extensive measurement structure of von Helmholtz (1887) and Hölder (1901), represented by additive quantities, have won indisputable permanence in the corpus of representation results. Here I want to inject that, in the process of searching the measurement literature, I failed to find references to an equally important early work by Hahn (1907), pertaining to infinitesimal quantities. With due respect to Krantz et al. (1971), a well-known, ex-colleague of mine characterized representational measurement theory as the enterprise of proliferating boring corollaries to Hölder’s theorem. The theorem in question is the foundation-stone of extensive measurement theory, and it concerns the order-preserving embeddability of Archimedean ordered groups into the additive group of reals. Hahn showed a similar embeddability of ordered Abelian groups into lexicographically ordered Cartesian powers of the field of reals. To complete the comment about Holder, my ex-colleague’s point is not to allow a tool become a straightjacket. Although today simple variations on the theme of Hölder’s theorem will no longer do, for a while, nothing was suggested to take its place, except perhaps some separation theorems borrowed from linear programming and the mathematical background for solving functional equations. Be that as it may, in my view, whereas the correct mathematical language in which to speak “measurementese” is not yet available, there are several mathematically and foundationally interesting results that make the business of measurement attractive to both philosophers and applied mathematicians. The reasons why a flexible mathematical theory is demanded by measurement is easy to provide. Consider the measurement of subjective probability in the context of de Finetti’s qualitative data structures (E, ). Suppose we perturb the data by replacing the boolean algebra E with a quantum lattice of events, represented, say, by subspaces of a Hilbert space. In seeking an agreeing representation of in terms of a Gleason probability measure on the quantum lattice E, we find that not only we need some new necessary axioms, including x y ⇔ y x, to supplement the classical list of de Finetti’s axioms, but more importantly we need a new method, because none of the old-known construction techniques work in this context.

Having completed the preliminary remarks on the Ramsey picture of measurement, let me now turn to the major task of this section, namely the discussion of certain methodological obstacles to the development of empiricist measurement theory. To set the tone of my remarks, I will start with some technical questions first.

TROUBLING PROFUSION OF REPRESENTATION RESULTS

One embarrassing problem the empiricist-oriented representational program suffers from is its endless proliferation of measurement representation results. From a logical standpoint, this problem stems from the fact that generally two existential Ramsey-style quantifiers cannot be melted down into one. The following example illustrates my point.

Suppose we have a representation of the qualitative probability relation < on some algebra of events E in terms of a quantitative probability P_t on E under the correspondence rule

x _y ⇔ P₁(x) ≤ P₁(y),

and a representation of the qualitative probabilistic independence relation ⊥ on the same algebra in terms of a probability measure P₂ on E under the correspondence

x ⊥ y ⇔ P₂(x ∩ y) = P₂(x)·P₂(y),

resulting in an appropriate pair of observational axiomatic characterizations M₁(E, ) and M₂(E,⊥). Recall that, for a pair of events x and y, x y means “event y is at least as likely as event x,” and x ⊥ y stands for “events x and y are probabilistically independent.” Now, because is usually representable with more than one agreeing probability measure, there is no reason to suppose that P₁ and P₂ coincide, even if and ⊥ are in some sense compatible. To grant ρ₁ = P₂, a joint representation result M₁₂(E,,⊥) must be found ab ovo! However, usually this is not just the mere conjunction of the earlier established marginal representations. Simply the equivalence

M₁₂(E,,⊥) iff M₁(E,) & M₂(E,⊥),

fails in general! What is needed of course is a brand-new set of empirical axioms (encapsulated in M₁₂) together with another representation result, which may neither be easy to find nor simple to test.

Next time around we may run into yet another species of qualitative data, say, ◊x (symbolizing “event x is more likely than not”), leading to a third marginal correspondence

◊x ⇒ P₃(x) > 1/2.

So we need a new axiomatic system M_l23(E,,⊥,◊), which covers all three species of data with one quantitative representation.

To finish the point, the representation result will have to be reworked from scratch every time a measurer runs into or comes up with a new type of data, closely related to but not definable in terms of the old data. Thus, measurement theories, unlike some nature scientific theories, are not rejected by measurers here and there piecemeal but as a matter of course holistically. We already know, from Ramsey, why this sort of conceptual game with representation results is empiricistically inevitable. For an empiricist, the interpretation of theoretical concepts (such as the probability measures introduced above) is thought to be open in the sense that their empirical meanings are not fully determined by measurements in which they occur. Now since new measurement techniques may be adjoined at any moment of time to the current stock of such techniques, the class of data structures is actually never closed, and thus the need for new representation results never ends.

To remedy this defect, some realists insist on interpreting the quantitative concepts ρ literally in the sense that the observational structures (E,), (E,⊥) and (E, ◊) are presumed to be identical with or plain substructures of the respective (E, _P), (E,⊥_P) and (E,◊_p). Here the P-indexed qualitative concepts are those defined by means of biconditional correspondence rules, involving a single theoretical quantity P. Since under this view correspondence rules are perceived as definitions, the problem of representation disappears.

I do not accept this extreme position in measurement, because it leaves no room for explaining how fundamental quantities are measured, and it leaves out the important fact that measurement theories are projected onto their possible observational consequences, crucial in testing.

What the empiricists seem to need is a kinematics of measurement representation—a theory that characterizes the possible revisions of old representation results in response to new types of qualitative data. In the final subsection, I shall consider what is to be done about the proliferation of representation.

Obstacles to Common Extensions of Qualitative Measurement Structures

Here I want to urge a different point that so far as I know has not been noticed before: Two or more data structures of the same similarity type need not admit aggregation.

It is clear that a single measurement structure, comprehensive as it may be, but isolated from the rest of the universe, is not an appropriate object of study. Measurement specialists measure length, temperature, vacuum, probability, or what have you, in many smaller, possibly overlapping domains and in several compatible ways. The question is how to aggregate the measurement results of these domains and techniques. To understand the problem in question, it will help to compare the qualitative approach with a quantitative treatment. Although what I have to say here applies to many species of extensive measurement structures, I will confine my attention only to the case of measuring subjective probability.

Suppose we are given two qualitative probability structures (E,) and (F, ), defined over the same sample space, but generally with different algebras of events, both satisfying, say, the Scott–Adams axioms for qualitative probabilities. What are the conditions for the existence of a common extension (E ∨ F, ∨ )? Here E ∨ F denotes the smallest boolean algebra of events containing both E and F as subalgebras, and ∨ designates a new qualitative probability, namely the common extension of qualitative probabilities and · to the joint algebra.

Marczewski (1951) showed, among other things, that, for any pair of finitely additive probability spaces (E,P) and (F,Q), conceived over the same sample space, a necessary and sufficient condition for the existence of a common extension (E ∨ F, ρ ∨ Q), which is again a finitely additive probability structure, is their compatibility:

x ⊆ y ⇒ P(x) ≤ Q(y),

for all events x in E and y in F. (Along standard lines ⊆ denotes boolean inclusion.)

This sort of pairwise compatibility of probability spaces enjoys the familiar properties of commutativity of observables arising in quantum mechanics. Unfortunately nobody had previously had the ambition to apply these techniques to the study of aggregating extensive measurement structures.

Observe that, although there are qualitative crossmodality comparisons, as in “Peter is a better Catholic than Paul is a Jew,” nothing nearly that complicated is needed in the quantitative case.

Marczewski showed even a stronger result, stating that a common extension (E ∨ F, ρ ∨ Q) always exists with the mixed-independence property

ρ ∨ Q(x ∩ y) = P(x)·Q(y)

for all x in E and y in F, whenever the constituent subalgebras E and F are algebraically independent:

x∩y = 0 ⇒ (x = 0∨y = 0),

for all x ∈ E and y ∈ F. (Here as well as elsewhere 0 denotes the impossible event and 1 refers to the sure event.)

To see some of the details, let E(x) be the smallest boolean algebra containing the algebraically independent E and {0,x,x,1}. The elements of E(x) have the form of disjoint sums x ∩ y + x ∩ z with y, z ∈ E; thus, we can define the common extension ρ ∨ Q on E(x) of probabilities ρ and Q on the constituent algebras by setting

ρ ∨ Q(x ∩ y + x ∩ z) = P(x)·Q(y) + ρ(x)·Q(z).

Using convex mixing, this definition is quickly extendible to cases where F is generated by a finite partition. Common extension in the context of arbitrary boolean algebras is obtained by passing to limits over directed sets of finitary partitions.

There are serious obstructions to extending these extremely useful modes of aggregation to the qualitative case. We assume as a matter of course that, when two agents measure, say, length in the realm of overlapping domains of rods, there is exactly one attribute measured, namely length, and the magnitudes of this attribute are the same—modulo units and errors—for all rods in the shared part of the domains, irrespective of the agents’ measurement methods.

Unfortunately this is only a small part of a more systematic difficulty besetting the empiricist conception of measurement.

There are no Worthy Categories in Measurement Theory

Most representational measurement theorists readily enlist the help of modern mathematics in general and that of ordered algebraic structures in particular. In the spirit of Klein’s Erlanger Program, and following Maclane (1971), most mathematicians accept the dictum that, whenever new mathematical structures are constructed in a specified way out of given ones, one should regard the construction of the corresponding structure-preserving maps on these new structures as an integral part of their definition. In this manner, one would arrive, quite naturally, at various categories, for example, of qualitative measurement structures and their structure-preserving maps. In particular, the category of qualitative probability structures satisfying the Scott–Adams axioms is defined by the class of structures (E, ) of the familiar sort together with the class of maps h:(E, ) → (F, ), where h is a boolean homomorphism from E to F such that

x y ⇔ h(x) h(y),

for all x,y ∈ E. It is immediately evident that these maps are composable in the usual associative way and that each structure has its unique identity map serving as a unit of composition. Likewise the category of Kolmogorovian probability spaces is given by the class of these spaces and the class of their maps h:(E,P) → (F,Q), where h is a boolean homomorphism from E to F such that Q(h(x)) = P(x). These maps compose in the usual associative way, and the identity maps of probability spaces act as units of the composition operation.

What is particularly upsetting about measurement categories is their lamentably feeble structure. When compared with the category of Kolmogorovian probability spaces, which enjoys several kinds of products, sums, and a host of other extremely important structural constructions, the category of qualitative probability structures lacks these constructs in the worst way. This would not matter, except that their existence is absolutely fundamental in developing a full-fledged probability theory capable of serving as a foundation for statistics and statistical physics. True, in general, qualitative measurement structures do possess lexicographic products, various completions and extensions, but none of these is of much help in attempts at developing the qualitative foundations some empiricists presume to be possible. The traditional view is that, because qualitative measurement structures resist the usual modes of functorial combination, it is better to think of them as being embedded into richer theoretical frameworks and work with the constructions available at a higher conceptual level. Nobody questions this move, except when the higher level theoretical frameworks appeal to concepts completely devoid of empirical meaning.

To illustrate these ideas, let us examine in some detail the case of quantitative versus qualitative probabilities. We begin by noting that the space ℙ(E) of all probabilities on a boolean algebra E is not just any old amorphous set living in Plato’s heaven. It is blessed with an excess of algebraic, topological, geometric, and order-theoretic structure. For example, in addition to Bayesian conditioning everybody uses, ℙ(E) is closed under arithmetic mixing ρ + Q, defined by ρ + Q(x) = a·ρ(x) + (1 - a)·Q(x), where 0 ≤ a ≤ 1. This operation enables us to show that the space of probabilities is actually a simplex, a notion generalizing an important geometric aspect of line segments, equilateral triangles, perfect tetrahedrons, and so on. The space of probabilities is closed under geometric mixing and several species of lattice operations, and it admits strong and weak topologies, and many interesting and important metrics. Last but not least, the simplex ℙ(E) carries the familiar relations of dominance,

ρ ⋖ Q iff Q(y) = 1 ⇒ P(y) = l

for all y ∈ E; and orthogonality,

ρ ⊥ Q iff P(y) = 1 & Q(y) = 0

for some y ∈ E.

In sharp contrast, observe that the space ℚ(E) of all qualitative probability relations on E satisfying the Scott–Adams axioms is much harder to handle conceptually, due to an almost total lack of structure, readily available in ℙ(E). Although we still have dominance and orthogonality even in the qualitative case, the algebraic and metric structures have all but disappeared. Space ℚ(E) is too coarse-grained relative to ℙ(E) to be of foundational use. How is ℚ(E) obtained? Formally it is a collection of abstraction classes of probabilities, obtained by the equivalence

ρ ≡ Q iff P(x) ≤ P(y) ⇔ Q(x) ≤ Q(y)

for all x,y ∈ E. To be more exact, space ℚ(E) is isomorphic to the quotient structure P(E)_/≡ of probability measures, modulo their order indiscernibility ≡ This sort of abstraction explains the importance of the projection map

⊏: ℙ(E)→ℚ(E)

that sends every probability ρ onto its qualitative probability ordering ⊏_P, defined by

x ⊏_p y iff P(x) < P(y).

Thanks to Scott–Adams axioms, there is also a Ramsey embedding or injection

℞: ℚ(E) →ℙ(E),

going in the reversed direction and picking a probability ℞() for every prescribed qualitative probability relation . These fundamental maps are brought together by the following identities:

⊏∘℞ = idℚ_(E) and ℞∘⊏ = idℙ_(E),

where id_D denotes the identity function with domain and codomain being equal to D.

Formally the first identity says that ℞ is a section of the projection ⊏. There may be several of these; therefore, let Ram ℚ(E) be the set of all Ramsey embeddings (injections, sections) of ℚ(E) into ℙ(E). It is clear that, for any pair of Ramsey embeddings ℞¹ and ℞², their arithmetic mixture ℞¹ + ℞², defined coordinatewise, is again a Ramsey embedding. The convex space Ram ℚ(E) provides a neat algebraic characterization of the problem of uniqueness of probabilistic representation.

Geometrically ℙ(E)/≡ represents the barycentric subdivision of the simplex ℙ(E); thus, we have a lemma that characterizes the problem of uniqueness geometrically.

LEMMA 1. The space ℚ(E) of qualitative probability relations on a finite algebra E satisfying the Scott–Adams axioms is in one-to-one correspondence with the set of components of the barycentric subdivision of ℙ(E).

Proof. Ignoring the trivial case with one atom, it is easy to check that algebras generated by two atoms admit exactly five qualitative probability relations, precisely the number of components in a barycentric subdivision of the unit interval.

Now, as the picture (below) of the barycentric subdivision of a 3-simplex (equilateral triangle) indicates, algebras generated by three atoms, x, y, and ζ admit 10 + 21 + 12 = 33 qualitative probability relations.

FIG. 11.1

Indeed direct inspection reveals that 10 such relations are defined by families of equivalences among events, and they correspond to the vertices (0-dimen-sional components) of the barycentric subdivision of ℙ(E). In particular, upon setting x ~ y, iff x y & y x, we get the following list of possible probability relations: (a) x ~ y & z ~ 0, (b) x ~ z & y ~ 0, (c) y ~ z & x ~ 0, (d) x ~ 1 & y ~ z ~ 0, (e) y ~ 1 & x ~ y ~ 0, (f) z ~ 1 & x ~ y ~ 0, (g) x ~ y & x ∪ y ~ z, (h) x ~ z & x ∪ z ~ y, (i) y ~ z & y ∪ z ~ x, and (j) x ~ y ~ z. Another 21 probability relations are determined by mixtures of equivalences and strict comparisons of events, corresponding to open segments (1-dimensional components) of the barycentric partition. Finally the remaining 12 probability relations are specified by possible strict comparisons, corresponding to open faces (2--dimensional components) of the barycentric subdivision.

Straightforward but tedious mathematical induction on the number of atoms in E, coupled with the classification of possible equivalences and strict comparisons of events, proves the validity of the lemma in general.

We see that qualitative probability relations provide a triangulation of the quantitative probability space ℙ(E), which may be refined further by other species of qualitative data. Although qualitative probability relations may determine unique probability measures in some special cases, in no way do they determine their quantitative structure! For example, although every Ramsey embedding satisfies

⊥ · iff ℞() ⊥ ℞(),

and similarly

⊥ · iff ℞() ⊥ ℞(),

it does not define mixing and other important structure in ℙ(E).

The appropriate moral to draw from these brief remarks is that representing quality in terms of quantity is one thing, and matching their structures is yet another.

In conventional treatments of subjective probability, it is customary to study one particular (generic) qualitative probability ordering and one instance of a representing probability measure. With overtones of realism, I am choosing the study of the spaces of these notions, together with their pairs of projective/injective maps.

Some Unexplained Jumps from Qualitative Reality to Quantitative Mathematics

Objects, events, and actions possess myriad properties and stand in countless relations. Of course most of these features are measurement theoretically uninteresting. Measurement focuses primarily on qualities (i.e., empirically detectable aspects in virtue of which things may be identified), qualified or compared. We have seen, for example, that, in considering subjective probabilities of events, the main relations or properties of interest were belief comparison , probabilistic independence ⊥, and epistemic possibility ◊.

It is common to grant axiomatically or as a fait accompli that every quality ϕ comes with its custom-made comparative >_ϕ (is ϕ-er than or is more ϕ than) assumed to be typically transitive and connected. Associated with every (strict) comparative relation >_ϕ, there are what one might call the indifference relation ~_ϕ (is as ϕ as) and the weak comparative relation _ϕ (is at least as ϕ as). The indifference ~_ϕ generates (by virtue of being an equivalence relation) its abstraction classes that define extensionally the notion of ϕ-ness of things. Thus, thing x has the ϕ-ness of y just in case x ~_ϕ y.

In some cases, rational comparisons of the sort “is twice as ϕ as” or “is half as ϕ as” are introduced. In this context, it is particularly common to stipulate a closed empirical operation ⊕_ϕ on things with properties analogous to those of numerical addition. However, often it is not fully realized that, in order to avoid oddities of the sort x ⊕_ϕ x (e.g., placing a straight rod end-to-end in a straight line with itself), one must consider mereological part-whole relations, which will tell us when the combined objects are separated and, hence, composable, and when they share a common part and, thus, are not composable.

What exactly is the logic of passing from qualities ϕ to their comparatives >_ϕ and combinations ⊕_ϕ? To my best knowledge, the empiricists do not have a good answer.

Suppose the quality ϕ we are interested in is television violence. Grammar enables us to form a comparative >_ϕ (movie x is more violent than movie y). This is hardly worth much, but measurement theory will tell us something about the order-theoretic properties of >_ϕ. However, how will this knowledge alone help the measurer in measuring television violence without knowing beforehand a great deal about television violence per se?

It is important to note that empirically it is far clearer whether one movie is more violent than another, than whether a given movie is violent. In view of this, some philosophers tend to analyze qualities in terms of their comparatives rather than the other way around. For example, a violent television movie is on this view a movie more violent than most movies of its kind. If this view were correct, then it should explain how different species of comparative relations (including interval ordering, semiordering, and weak ordering—one of these for each measurement method) manages to pick out always the same quality, in our case television violence. However, as we have seen in the case of measuring subjective probability, it is far from trivial to show that two comparatives of a given species must refer to the same quality.

Transformation grammarians tell us that “x is more violent than y” is actually derived by a suitable transformation from “x is violent” and “y is violent.” The problem with this suggestion is that there are no interesting comparatives that are definable classically in terms of properties. The following lemma explains why this must be so.

LEMMA 2. Given a binary relational structure (E,ρ), the merging condition

(x ρ y & z ρ w)⇒xρ w

is both necessary and sufficient for the existence of unary relations U and V on E such that the decomposition

x ρ y iff Ux & Vy,

holds for all x, y ∈ E.

Proof. Given U and V as wanted, evidently we have

(Ux & Vy) & (Uz & Vw) ⇒ (Ux & Vw).

Now, granted a binary relation ρ with the merging property, take the direct U = ρ(E) and inverse V = ρ-1(E) images of the domain E under the relation ρ, and you will see at once that the decomposition holds.

Furthermore, it is not hard to see that the boolean dual of the margining condition in Lemma 2 (reverse the implication, and replace conjunction with disjunction) is necessary and sufficient for the existence of a disjunctive decomposition x ρ y iff Ux ∨ Vy. Indeed upon selecting U and V in such a way that their complements satisfy Ü = ρ(E) and V = ρ^-1(E), we find that the dual result also holds.

It is natural to ask whether there are similar results for some other logical decompositions. A truth table for binary logical operations shows that the only remaining meaningful decomposition we have not considered so far is the following: x ρ y iff Ux ⇔ Vy. This decomposition works precisely when (x ρ y ⇔ z ρ w) iff (z ρ y ⇔ x ρ w). In verifying this result, we can set U = ρ(a) and V = ρ^-1(b) for some fixed a,b ∈ E. Here as well as previously ρ denotes the complement of ρ, and ρ(a) is the image of element a.

Getting back to our problem, we see that there cannot be any logically interesting transformations that will take us from qualities to comparatives.

An equally objectionable move is to derive comparatives by quantifying over degrees or extents, as Seuren (1973) suggested. Seuren proposed analyzing “x is longer than y” in terms of “x is long to an extent to which y is not.” Here “x is long to extent e” means not that x is exactly that long but that x is at least that long. Although the truth conditions for the sentences of the form

∃e[x is long to extent e & y is not long to extent e]

come out correctly under this interpretation, note that Seuren quantified into an intensional context. A far more troubling point is that the ontology of extents and degrees is quite unclear. Extents and degrees fall prey to objections raised against abstract objects: What is the nature of their domains, and what are the rules that generate them? We need to know what is meant by claiming that a given quality is present in each object to some extent or another.

There are a variety of other ways that extents or degrees might be built into comparative relations, but none of them is satisfactory as far as I can tell. For example, “x is brighter than y” may be analyzed in terms of a quantified conjunction

∃d₁∃d₂[x is bright to degree d₁ & y is bright to degree d₂ & d₁ is higher than d₂].

The inherent circularity is obvious: The relation “is higher than” is one of the comparatives we are trying to define!

Often differences of degree can themselves be compared, and the comparative constructions may be iterated. For example, we can pass from “x is longer than y” to “x is more longer than y than z is longer than w,” and in turn to “x is more (more longer than y than z is longer than w) than (u is more longer than ν than ν is longer than t).” Never mind whether anybody can take in by ear this comparative. The question follows: How do we know that it makes sense for some given qualities and not for others? Comparatives of this sort are important in measuring the rate of change of a rate of change (e.g., in accelerated motion). Clearly these comparatives thrive in the riches of a given scientific theory.

The production of double comparatives is even more mysterious. Think about the relationship between the following comparatives and the underlying pairs of qualities:

1.   Agent x has read more books in more languages than agent y.

2.   Policy x makes more people happier than policy y.

3.   Agent x types more words per minute with less number of mistakes than agent y.

In the last example, the description “the greatest number of words per minute with fewest mistakes” does not pick out a unique agent in general. It is not clear to me how these comparatives ought to be numerically represented.

Nozick (1981, pp. 490–494) proposed to measure moral weight by considering a simultaneous comparison of wrong-making features and right-making features of pairs of human actions. With considerable hand waving, Nozick assumed that the underlying measurement structures will lead to feasible numerical representations. Unless he meant the usual variety of conjoint measurement that ignores the crucial factor of interaction between right and wrong, which, therefore, is rather uninteresting empirically, one must remain skeptical about the merit of Nozick’s measurement-theoretic proposal.

Finally we have the intensional comparatives of the sort “Jones is heavier than he was.” It seems that this type of comparative might be reduced to the usual treatment of “heavier than” by stipulating the relation on the set of temporal slices of Jones. Unfortunately no such treatment will avail for “Jones looks heavier than he is” or “Jones is heavier than he looks.” Some philosophers seek to refer to possible worlds or situations in which Jones might live or in which there is a counterpart of Jones. Evidently the domain of possible worlds in this case is quite arbitrary, and any relational or topological structure one wishes to impute to it is beyond proper justification.

It is unclear to me how measurement theory ought to reckon with more complex intensional comparatives, such as “Object x is heavier than S thinks y is” or “Agent S thinks x is heavier than x in fact is.”

So how big a leap do we in fact make when we pass from real-world qualities to mathematical quantities? We are aware that a mathematical structure is somehow associated with a stereotype domain of objects and that this structure is then studied by purely mathematical means in order to obtain empirically important and theoretically interesting theorems. The knowledge of these structures is presumed to constitute a form of understanding of the nature of measurement.

The gap between qualitative reality and quantitative mathematics is measured by the extent to which measurement results developed internally within measurement theory reflect our external practice. The search for a smooth passage from the qualitative to a quantitative is in my view misguided, because every step in conceptualization is in effect a form of abstraction that denies or imputes degrees of freedom to reality and thereby creates a gap between what really is and what our concepts say there is.

A Plea for More Realism in Measurement

Often order relations are introduced not because there are some operational procedures to instantiate them, but because there is a need for characterizing approximation, one wants to solve extremal problems, and so on. For example, depending on the choice of a meet-closed subset S of E, the simplex of probability measures ℙ(E) may be ordered in a number of ways for purposes of approximation by setting

P Q iff P(x) ≤ Q(x),

for all x ∈ S. This sort of stochastic order relation may be refined further by defining a series of conditional stochastic orderings

ρ _W Q iff P_y Q_y,

for all y ∈ W, where ρ_y denotes the classical Bayesian conditional “P given y,” and W is a suitable subset of events in E.

For example, if the underlying sample space of E is partially ordered, we can take S to be the set of monotonic events in E (i.e., events containing with every sample all the other samples ranked higher).

These order relations are introduced primarily on theoretical grounds with no particular regards to possible operationalization; thus, there is neither a need nor a hope for a testable representation result. Additional examples of order relations on ℙ(E) that are introduced in this measurement-like manner may be found in Stoyan (1983). Some might suggest that this sort of methodology ought not to be part of measurement theory.

As I see it, representation of qualitative measurement structures is only half of the measurement-theoretic program, the embedding or injective part. The other half of the program—the projective part—concerns the development of qualitative structures, which serve as forgetful images of quantitative structures.

The nonexistence of a common extension of qualitative probabilities is seen in a different light when we realize that these probability orderings are coarsegrained projections of powerful quantitative concepts. In the passage to the qualitative, these structures have lost their capability to aggregate. Representation results proliferate because of the representationalists’ perception that every qualitative structure must somehow be quantitatively represented. We have seen that, although joint representations cause serious problems, projections are free. In answer to the question raised at the end of the section about proliferating representation results, I suggest to project first and represent later.

REASONING ABOUT MEASUREMENT THEORIES

Whoever browses the recent theoretical books on measurement is struck by the type of reasoning used in their texts. Besides the extensive use of set-theoretic predicates (such as “is an extensive measurement structure” or “is a concatenation measurement structure”), there are repeated references to and justifications of various representation results.

The variety of measurement representation results is best seen as a repeated instantiation of a small number of simple Ramsey sentences of the following form:

∃P[T(P) & C(ρ, P)],

whereas uniqueness results may be viewed as realizations of the dual Ramsey sentence

∀P[C(ρ, P) ⇒ T’(P)].

Here T and T’ are suitable set-theoretic predicates, representing quantitative theories, and C is a set-theoretic correspondence between qualitative relations and numerical functions or quantities P. From now on, we shall adopt the habit of dropping explicit references to domains E, serving as parameters in Ramsey sentences, on which the relations and quantities ρ are presumed to be defined.

Let us now see what more can be said about the relationships between Ramsey sentences and measurement theories, as understood by the empiricists. We begin by associating with every correspondence C a conjugate pair of modal (closure) operators [C] and (C). In particular, for every set-theoretic predicate M(ρ), we define a new set-theoretic predicate [C]M(ρ) by putting

[C]M(ρ) iff ∃P{∀ς[C(ς,P) ϕ μ(ς)] & C(ρ,P)},

and, in a dual fashion, we define (C)M(ρ) by setting

(C)M(ρ), iff VP{∀ς[M(ς) ⇒ C*(ς, P)] ⇒ C*(ρ, P)},

where C* denotes the logical dual of C.

Therefore, in the case of C(ρ,P) being interpreted as “P is a Kolmogorovian probability satisfying x ρ y ⇔ P(x) ≤ P(y), the predicate formula [C]M(ρ) captures the existence of a probability measure together with the necessity of the qualitative axioms, characterizing the predicate M(ρ). The meaning of (C)M(ρ) is related to uniqueness statements in the same manner. It would require a longer detour to give a full discussion of these modal ideas here. I confine myself to a simple lemma, which justifies the importance of previously given definitions.

LEMMA 3. For any correspondence C the necessity operator [C] satisfies the following axioms of modal logic:

1.   [C]M(ρ) ⇒ M(ρ).

2.   [C][C]M(ρ) ⇔ [C]M(ρ).

3.   [C]M,(ρ) ⇒ [C]M₂(ρ), if M,(ρ) ⇒ M₂(ρ).

4.   [C,]M(ρ) ⇒ [C₂]M(ρ), if C,(ρ, P) ⇒ C₂(ρ, P).

5.   [C]C(ρ, P) ⇔ C(ρ,P).

6.   Solutions of the fixed-point equation

[C]M(ρ) ⇔ M(p)

with predicate unknown μ (ρ) have precisely the form of a Ramsey sentence: M(p) ⇔ ∃ρ[T(ρ) & C(ρ, P)] for some predicate T.

7.   Solutions of the fixed-point equation

(C)M(ρ) ⇔ M(ρ)

with predicate unknown M(ρ) have precisely the form of a dual Ramsey sentence: M(ρ) ⇔ ∀P[C(ρ,P) ⇒ T(P)]for some predicate T.

Proof. The clauses of this lemma are entirely trivial, except perhaps clauses (6) and (7), and these we proceed to prove.

Let M(ρ) iff ∃P[T(P) & C(ρ,P)], and suppose M(ρ) is true. Then T(P₀) and C(ρ, P₀) must hold for some P₀. Now C(ς,P₀) ⇒ [T(P₀) & C(ς, P₀)] gives C(ς, P₀) ⇒ ∃Q[T(Q) & C(ς,Q] for all ς, and, hence, after substitution, ∀ς[C(ς,P₀) ⇒ μ(ς)] & C(ρ,P₀). Consequently we have the truth of ∃P{∀ς[C(ς, P) ⇒ μ(ς)] & C(ρ,P)} and, hence, that of [C]M(ρ). Thus, Ramsey sentences of the foregoing type are indeed solutions of the fixed-point equation in clause (6).

Now suppose S(ρ) is a solution of the fixed-point equation in (6). Then by definition the modal closure [C]S(ρ) is automatically a special case of a Ramsey sentence, and, therefore, so is S(ρ), in view of (2). The proof of clause (7) is a formal dual of that of (6).

Therefore, every qualitative measurement predicate M(ρ) is logically approximated from below by its associated Ramsey sentence [C]M(ρ) in the sense of clause (1), where the degree of approximation is contingent upon the choice of the correspondence relation C. Empiricists tend to select C parsimoniously with special regards to the truth of M(ρ), whereas some realists grant C a priori with special emphasis on its explanatory power.

The obvious failure of the equivalence

[C ∩ D](M(ρ) & ν(ς)) iff [C]M(ρ) & [D]N(ς),

explains the phenomenon of proliferation of representation results. The kinematics of representation is captured by iterating the correspondence-driven modalities, as in [C][D]M(ρ). Note, however, that iteration collapses to joint representation in the sense of [C][D]M(ρ) = [C ∩ D]M(ρ) precisely when the correspondence relations are independent.

The preceding modal reasoning is a special case of a more general approach, involving categories. In what follows, I will present a convenient category Meas of measurement predicates and relations between them. In this setting, the modal operators will act as abstract maps. Specifically an object of this category is an equivalence class of set-theoretic predicate formulas, modulo equivalence relation ~, where

M(ρ) ~ ν(ς) iff ν is obtained from M by a type-preserving renaming of the variables of M.

To make things as simple as possible, we shall write formulas (playing the role of objects in Meas) with disjoint sets of variables. As our next step, we turn correspondence relations into abstractly conceived maps. To define a map C(ρ,ς): M(ρ) → N(ς) in Meas, we first grant that the domain M and codomain ν have no variables in common and then proceed to define C(ρ,ς) as an equivalence class, modulo ~, of formulas satisfying the following.

C(ρ,ς) ⇒ M(ρ) & ν(ς).

For example, the identity map of M(ρ) has the form [ρ = ς] & M(ρ): M(ρ) → μ(ς). The composition of maps

is given by existential quantification

DoC(p, P) iff 3[C(p, a) & O(ς,ρ)],

modulo equivalence ~. It is simple to check that these definitions make Meas into a category.

What is Meas good for? It enables us to formulate many of the representation problems in terms of simple categorical constructions. For example, the pullback diagram in the following

FIG. 11.2

captures the idea of joint representation. Indeed a joint axiomatization defines a predicate

which is precisely the pullback of the marginal representations M(ρ) and ν(ς). Needless to add, none of this says anything about how to find such joint representations! Along similar lines, Ramsey sentences come out as maps C ∘ T(ρ): 1→ M(ρ) obtained from composing a correspondence rule with the theoretical predicate determining M(ρ). Here 1 denotes the terminal object of Meas, corresponding to a universally true, set-theoretic predicate formula, such as [ρ = ρ].

Looking at measurement theories this way gives us a nice, global picture of how representation results are put together. The price paid, however, is a total lack of computational power in Meas. To provide a more realistic framework, we need a category that captures the full import of the domains of measurement structures. For example, in the case of measuring subjective probability, we can remain entirely within the category of measurable spaces and measurable maps. Indeed, if we let sets of the form

[x ≥ a] = {ρ ∈ ℙ(E) | P(x) ≥ a}

with x ∈ E band together so as to form a new boolean algebra, the simplex ℙ(E) becomes an important measurable space that carries probabilities of probabilities μ together with their averages ⊔μ—special probabilities on E—defined by the integral

Likewise the space of qualitative probability relations ℚ(E) is converted into a measurable space by taking the boolean algebra of its subsets generated by collections of the form {p ∈ ℚ(E) | x ρ y} for all x,y ∈ E. Under this arrangement, the projection/injection pairs of maps become measurable, and the problem of measurement can be developed entirely within the category of measurable spaces.

Our basic observation in this context concerns ℙ and ℚ. These are functors on measurable spaces, and the projection ⊏ is a natural transformation between them. What this means is that it does not matter whether we apply a boolean homomorphism first to the probability measure and then to the corresponding qualitative probability ordering or the other way around:

x⊏_Phy iff h(x) ⊏_p h(y),

for all x,y ∈ E, where h is a boolean homomorphism from E to F. In contrast, the Ramsey embeddings ℞; are not natural transformations in general. Furthermore, whereas ℙ is actually a monad (triple), the measurement functor ℚ is most assuredly not! What this means in a less technical language is that, whereas ℙ has the fundamental averaging operation ⊔ defined previously, there is no similar operation in the qualitative case. Although it is possible to define a higher-order comparison of lower-level beliefs, it is impossible to define an average of higher-order comparisons. The reasons for this are simple to state: Averaging in ℙ(E) does not commute with the order indiscernibility ≡ defined by the barycentric partition of ℙ(E). For this reason, there are no composable transition qualitative probabilities, corresponding to Markovian kernels, and there are no product qualitative probabilities either.

The overall situation in the context of measuring subjective probability is as follows. The category of measurable spaces is furnished with a powerful functor ℙ that carries the structural information of probability calculus. Next we have a longer list of measurement functors (e.g., one for comparative probabilities, one for probabilistic independence relations, and so on) that interact with one another in rather weak ways. In general each of them carries only a fragment of the total structure available in ℙ. We already know what this structure is in the case of comparative probabilities. Upon introducing the equivalence

ρ ℙ Q iff P(x ∩ y) = P(x)·P(y) ⇔ Q(x ∩ y) = Q(x)·Q(y),

for all x,y ∈ E on probabilities, we get a new quotient set ℙ(E)/≡, which is isomorphic to the set ℚ(E) of all qualitative probabilistic independence relations. It is easy to see that, in this case, the partitioning lumps together the Dirac probabilities (0-dimensional faces), the open faces of dimension 1,2. …, and so on. The functor ℚ* carries only the dominance structure, while ℚ has in addition the orthogonality structure as well. Now the joint measurement of comparatives and independence amounts to having a pullback ℚ ∨ ℚ* in the diagram below:

The pullback object is obtained by factoring ℙ(E) with the common refinement of the respective indiscernibility relations for ℚ and ℚ*. What I am advocating here is of course that the theoretical functor ℙ is always primary, and all the other functors are derived from it by a quotienting process. As we emphasized before, a good deal of important theoretical structure present in ℙ is destroyed in this process of abstraction.

This type of reasoning about measurement theories is still in a preliminary state, but when more is known about the measurement functors involved, non-trivial applications in extensive measurement may be expected. This optimism is nourished by the success of this framework in clarifying the problem of uniqueness of (probabilistic) representations.

One virtue of categorical reasoning is that it helps get rid of ad hoc structures and reasoning about them, and retains only the natural ones. It seems to me that some ad hoccery is often present in various representation results in which suitable technical assumptions are added solely for the purposes of being able to obtain their proofs. The apparent lack of harmony in these accidental structures shows up at once at a functorial level, where certain, desirable natural transformations are simply not available.

I believe that the inherent unifying power and accompanying realist ontology sufficiently justify this general approach.

REVISING KNOWLEDGE IN RESPONSE TO
MEASUREMENT RESULTS

Although we all recognize that measurement is a fundamental means by which we come to know the values of quantities, traditional measurement theories do not bring this fact out in the open.

The purpose of this section is to show how an observer’s extant state of knowledge or belief may be revised in response to new measurement results.

Observational statements regarding a qualitative measurement structure (E, <) may be viewed as propositions

[x y] = {ω | x _ω y}

given by the amount of samples ω in a sample space ω. In a subjective self-parametrizing situation, we can put [x y] ={p∈ ℚ(E) | x ρ y}. In either case, we get a boolean algebra B (E,) of observational propositions generated by those mentioned previously. This opens up the road to defining probabilities on B(E,). For example, if μ₀ is such a probability, then μ₀[x y] may be thought of as the probability that x is not ranked higher than y in some experiment.

On the quantitative side, let (E,P) be a corresponding quantitative structure satisfying a set-theoretic predicate T. Along similar lines as above, we have a boolean algebra B(E,P) of propositions of the form [P(x) ≥ a].

Now let us assume that the observer’s beliefs are representable by suitable probability measures on B(E,P). Let μ be the probability representing the observer’s current state of belief about the values of quantity P. Thus, μ [P(x) ≥ a] denotes the observer’s degree of belief that the quantity ρ exceeds or is equal to a on object x.

As is well known, a change of belief from the current state μ to a new state μ* regarding the values of quantity P, brought about by observing μ₀[x y], is given by the mixture of the following Bayesian conditionals:

where C(x y) = [P(x) ≤ P(y)] denotes the translation of observation [x y] into the theoretical algebra by a given correspondence rule C.

In the case of knowledge, we may assume with the empiricists that the states of knowledge are representable by deductively closed bodies (filters) of propositions. In analogy with Bayesian conditionals, we can associate with every deductive system K its conditional K_c(xy) It represents the body of knowledge, enriched by additional information C(x y), and it is defined by the smallest deductive system containing A and C(x y).

Let K₀ be a body of observational propositions, obtained during observing the structure (E,). Then the passage from the current state of knowledge K to a revised state of knowledge K* regarding quantity P, brought about by receiving proposition [x y], is given by the following boolean conditional:

In these rather simple examples, we showed how knowledge and belief states are affected by qualitative measurement results. It should be quite clear after a little thought that these revision rules can be generalized to more complicated species of data in a routine fashion. What is much more interesting, however, is the fact that this formalism is a special case of interactionist measurement theory in which the instrument is identified with the observer.

REFERENCES

Domotor, Z. (1981). Review of Roberts’s measurement theory. Journal of Mathematical Psychology, 24, 88–92.

Friedman, M. (1983). Foundations of space time theories. Princeton: Princeton University Press.

Hahn, H. (1907). Über die nichtarchimedischen Grössensysteme. Sitzungsberichte der Acad. Wiss., Wien, mathem. phys. Klasse, 116, 601–655.

von Helmholtz, H. (1887). Zählen und Messen, erkenntnistheoretisch betrachtet. Philosophische Aufsätze, Fues Verlag, Leipzig, 17–52.

Hölder, O. (1901). Die axiome der quantität und die lehre vom mass. Berichte der Sächsischen Gesellschaft der Wissenschaften, mathem. phys. Klasse, 53, 1–64.

Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of Measurement (Vol. I). New York: Academic Press.

Maclane, S. (1971). Categories for the working mathematician. New York: Springer.

Marczewski, E. (1957). Measures in almost independent fields. Fundamenta Mathematicae, 38, 217–229.

Narens, L. (1985). Abstract measurement theory. Cambridge, MA: MIT Press.

Nozick, R. (1981). Philosophical explanations. Cambridge, MA: Harvard University Press.

Pfanzagl, J. (1958). Theory of measurement. New York: Wiley.

Ramsey, P. F. (1950). The foundations of mathematics. London: Routledge & Kegan Paul. Original work published 1929

Roberts, F. S. (1979). Measurement theory. Reading, MA: Addison-Wesley.

Seuren, P. (1973). The comparative. In F. Kiefer & N. Ruwet (Eds.), Generative grammar in europe (pp. 528–564). Dordrecht: Reidel.

Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: Wiley.