On the empirical status of axioms in theories of fundamental measurement

JOURNAL

OF MATHEMATICAL

PSYCHOLOGY

On the Empirical

7, 379-409

Status

of Axioms

of Fundamental W.

of California,

University

in Theories

Measurement1

ERNEST University

(1970)

ADAMS

Berkeley,

ROBERT

F.

of Oregon,

Eugene,

California

94720

FAGOT Oregon

97403

AND RICHARD University

of British

E.

Columbia,

ROBINSON Vancouver,

British

Columbia

If one regards the purpose of axiomatization in fundamental measurement theory as being that of exhibiting directly testable sets of conditions (the axioms) that are necessary and/or sufficient to guarantee the existence of numerical representations (numerical measurements) of certain kinds, then it should be clear that most of the axiom systems that have been given in measurement theory do not realize this objective. This raises the question of precisely what the empirical content is of axioms in theories of fundamental measurement. The basic concept in terms of which our formal results are stated is that of data equivalence between two theories. Roughly, two theories are data equivalent if and only if they are consistent with exactly the same finite sets of basic “observation” statements. The significance of data equivalence is that if two theories are data equivalent and their primitive notions are given the same interpretation, then no experimental test can refute one without refuting the other. We are able to show, for example, that certain axioms (Archimedean and continuity conditions) are purely “technical” in the sense that the theory with the axiom in question is data equivalent to the theory without the axiom, whereas certain other axioms, previously treated as technical, are indirectly testable; i.e., have testable consequences when considered in combination with other directly testable axioms. The concept of data equivalence is used to analyze the Lute-Tukey axioms for conjoint measurement, axiom systems for interval measurement, for additive and bisection measurement (Pfanzagl), and for extensive measurement (Suppes). r This research was supported in part by National Science Foundation Grant GS-824, in part by the Advanced Research Projects Agency of the Department of Defense, and monitored by the Air Force Office of Scientific Research under Contract No. F44620-67-C-0099. Preparation of the manuscript was facilitated by a Leave Fellowship awarded to Robinson Canada Council,

379 0 1970 by Academic 480/7/3-1

Press, Inc.

and was by

380

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FAGOT,

AND

ROBINSON

In this paper, we discuss certain elementary considerations from model theory that bear on the “empirical status” of axioms in various theories of fundamental measurement, and on the function of these axioms in empirical tests of the theories. In particular, we are concerned with the vague but practically important intuition shared by many to the effect that only certain axioms may contain the “empirical content” and may be ignored in of the theory being axiomatized, while others are “technical” assessing the empirical adequacy of the theory as a whole.2 We shall attempt to give a precise, if limited, sense to the notion of the empirical content of a theory of fundamental measurement. This will make it possible to ascertain clearly which axioms in certain well-known theories of fundamental measurement contain a part of the empirical content of the theories, and so can not be ignored in empirical tests, and those that do not, and so can (with certain qualifications) be ignored as “technical.” To illustrate the methodological issues which concern us, consider the Lute and Tukey (1964) axioms for conjoint measurement. These apply to ordered pairs (a, X) from a domain that is a Cartesian product A x --I-, and an observationally determined binary relation < over this domain. The axioms are sufficient conditions for the existence of a pair of numerical conjoint measures m, and m2 over the sets A and X, respectively, that satisfy the conjoint scale condition: (CS)

For all (a, X) and (b, y) in A x iii, (a, X) < (b, y) if and only if

ml(a) + 44

G ml(b) + ma(y).

The assumption that there exists a pair of conjoint scales satisfying condition (CS) will be called the conjoint scaling hypothesis, CSH. The Lute-Tukey axioms are trivially equivalent to the following four: If (a, X) < (6, y) and (b, y) < (c, z) then (a, X) < (c, z); (WO) WEAK ORDERING. and either (a, x) < (b, y) or (b, y) < (a, x). If (a, x) Q (b, y) and (b, x) < (c, .x) then (a, 2) < (c, y).

(C) CANCELLATION. (S)

SOLUTIONS.

X such that (a, x) generated by <).

For all a and b in A and x and y in X there exists c in A and z in (c,y) and (a, x) - (b, z) (where - is the indifference relation

(A) An ARCHIMEDEAN Although

condition

(see Def. 7.3, Sec. III below).

the four axioms above are axiomsfor

conjoint

scaling theory,

in the sense

’ For instance, Lute and Galanter (1963, p. 259) say in discussing the empirical reasonableness of certain axioms in a theory of measurement due to Pfanzagl (1959): “Axioms 1 and 3 are largely technical and need not be discussed.”

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AXIOMS

that they entail the validity of the underlying scaling hypothesis CSH, their exact relation to this theoretical assumption is somewhat complicated. Of the four axioms, only the first two, WO and C, are actually necessary conditions for the validity of CSH, but these two alone are not suficient for CSH to hold, and so experimentally verifying just WO and C would not verify CSH. The four axioms together are sufficient for CSH, but as the last two are not necessary, rejection of either of them would not show that CSH had to be rejected. The Lute-Tukey axioms give assumptions justifying not just CSH but the use of a particular scaling procedure (involving dual standard sequences). This scaling procedure can be used in determining numerical scale values satisfying condition CS, but the applicability of this particular scaling procedure is not entailed by the fundamental hypothesis CSH itself. The four axioms together are therefore necessary and sufficient conditions for the validity of CSH together with the assumptions of the particular scaling procedure. Although one cannot object to simultaneously axiomatizing both CSH and the special scaling procedure, it would clearly be desirable to formulate axioms just for CSH alone, so that failure of the axioms would entail rejection of CSH. This is particularly true in view of the fact that of the two theoretical hypotheses the Lute-Tukey theory axiomatizes, CSH is clearly the more important, and there are many empirical realizations in which there are strong a priori reasons to think that the Lute-Tukey special scaling procedure is unrealistic,3 even though CSH alone might be expected to hold. Unfortunately, recent results of Titiev’s (1969), based on results of Scott and Suppes (1958), show that it is impossible to give a finite list of simple elementary conditions applicable to < that are necessary and sufficient for CSH to hold, even in finite domains. More exactly, what has been shown is that no finite list of purely universal laws, of which WO and C are examples, are necessary and sufficient for CSH to hold in any finite domain. And this trivially entails that no finite list of purely universal laws plus universal existential laws, exemplified by S, are necessary and sufficient for CSH in finite domains. Since CSH alone does not entail the existence of members of the domain A x X having any specific properties, it is hard to imagine what kinds of elementary empirical laws other than purely universal ones could be entailed by it, and therefore Titiev’s result at least strongly suggests that the desirable objective of formulating elementary empirical laws that are necessary and sufficient for the validity of CSH alone, even in finite domains, may be impossible to realize.q Some of the force of the objection to the Lute-Tukey axiom set as a formuzation of the “empirical content” of the underlying theoretical hypothesis CSH could be met if it a We are referring to the consequence that the numerical scales are unbounded, which follows from Axioms S and A. However, Lute (1966) has recognized that Axiom S is too strong, and has provided a more plausible alternative axiomatization which weakens S. a Scott (1964) does give an infinite list of purely universal laws that are jointly necessary and sufficient for the validity of CSH to hold in any finite domain, but as infinite sets of independent empirical

laws

are

not

empirically

testable

in any

obvious

way,

we shall

not

consider

them

here.

382

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FAGOT,

AND

ROBINSON

could be shown that the nonnecessary Axioms, S and A, included in the set were “technical,” hence, all the “empirical content” of CSH was contained in just the two axioms WO and C, and S and A could therefore be ignored in experimental tests of the theory. In that case testing WO and C alone would be an “adequate” test of CSH, since if either of these were found not to hold, CSH would have to be rejected, and if both were found to hold, CSH could be regarded as empirically established simply because all of its empirically significant consequences were verified. Unfortunately, it is easy to show that the Lute-Tukey axioms are not “empirically adequate” as an axiomatization of CSH even in this weaker sense. It is easy to describe empirical data that would be logically consistent with axioms WO and C, but would be inconsistent with CSH (and also to imagine situations in which WO and C would be confirmed to hold for all members of an infinite domain, but in which CSH would be inconsistent with simple empirical observations in the domain). A more limited objective would be to formulate axioms for CSH in which all of the empirical content of the axioms was contained in some subset of the axioms that were actually necessary conditions for the validity of CSH. However, it appears impossible to realize even this more limited objective, and this holds for certain other systems of measurement, as well as the Lute-Tukey system. Although the foregoing conclusion as to the possibility of getting “nice” systems of axioms for CSH and similar theoretical hypotheses may be somewhat discouraging, nonetheless we feel that the intuition to the effect that among axioms of a theory, certain ones are to be distinguished as having “empirical content,” while others are lacking in it, is an important one deserving of some analysis. One outcome of such an analysis would in fact be to make more precise the vague conclusion just stated. Another would be to show that, even though certain of the nonnecessary axioms for theoretical hypotheses like CSH did contribute to the “empirical content” of the theory, and so could not be ignored in empirical tests, others did not and therefore could safely be ignored. As a matter of fact, one result of our investigation will be to show that in the admittedly limited sense of “empirical content” we shall define, the LuceTukey axiom A (the Archimedean axiom) is technical; hence, of the two nonnecessary axioms of their theory, only one-the Solutions axiom-needs to be given special attention in empirical applications of the theory. Similar results will be found to hold in a number of related theories of fundamental measurement. In clarifying our intuitions as to the “technical” or “substantive” status of axioms in theories of measurement, the key concept in need of explication is that of the empirical content of such a theory. We shall not attempt a careful “analysis” of this fuzzy but highly important notion, but shall make what is hopefully a plausible assumption concerning it, and then investigate the consequences of that assumption. The assumption is that the empirical content of a fundamental hypothesis like CSH is equivalent to the set of all purely universal laws that it entails. It seems hardly deniable that any purely universal law entailed by CSH expresses part of the empirical content of CSH,

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since universal laws are empirically testable par excellence-not only as to whether they are logically consistent with immediate observational data, but also for how well they are satisfied by data not perfectly consistent with them. They are also inductively con$rmabZe in a straightforward way. In the present case, however, it is at least plausible that the universal laws entailed by CSH and similar theoretical hypotheses actually exhaust the empirical content of these hypotheses, simply because (as previously noted) these hypotheses do not imply the existence of any particular objects in the domains to which they apply, and so it is hard to imagine what kinds of directly testable empirical laws other than purely universal ones might be entailed by CSH. Admittedly, this argument is inconclusive, and to that extent it suggests the desirability of investigating alternative precizations of “empirical content” that would throw further light on the empirical status of axioms. We shall not attempt any further justification for our basic assumption here, however, and shall turn instead to the investigation of its consequences. Even if alternative approaches should lead to conclusions relative to the empirical status of axioms that differ from those we arrive at, it seems likely that such results would not so much contradict as complement our conclusions. Given the assumption that the empirical content of a theory like CSH is equivalent to that of the purely universal laws it entails, we shall be concerned with the question: Which axioms of a particular theory contain all the empirical content of the basic theoretical hypothesis in the sense that they entail all of the purely universal laws that are entailed by the basic hypothesis itself? This question is trivially equivalent to the following, which is the form that we shall consider in what follows: Which subsets ZJ of the axioms of a theory have the property that any finite set of observational data that are logically consistent with axioms ~2 must also be logically consistent with the underlying theoretical hypothesis? In the next section, this question is made more precise by our formulating a precise notion of an observational datum (relevant to a given theory or set of axioms), and of what it is for such a datum or set of data to be consistent with a theory. We shall then be able to give a precise answer to the question whether any finite set of data could possibly be consistent with certain axioms of the theory, although inconsistent with the fundamental hypothesis being axiomatized. More exactly, we shall ask whether data of particular hinds could be consistent with some subset of the axioms, but inconsistent with the fundamental hypothesis. It will be shown that in some cases certain subsets of axioms in fundamental measurement theories are “empirically equivalent” to basic measurement hypotheses relative to one kind of data (in that any finite set of data of that kind is consistent with this subset if and only if consistent with the basic hypothesis) but not relative to other sorts of data. In particular, it proves necessary in some cases to consider data of two kinds: one to the effect that certain observable entities stand in some ordinal relation and one to the effect that certain of these entities are or are not identical. It is important to note that the observable entities referred to in data expressions might be ordered pairs or

384

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concatenations of objects. Thus, in order to avoid misinterpreting “ordinal” in the narrow sense, we shall call data of the first kind generalized ordinal data as such data would be relevant to theories of additive, extensive, conjoint, and interval measurement, as well as to theories of ordinal measurement. It will be shown that axioms and theories differ in systematic ways with respect to their implications for data of the different kinds. The next section presents some fundamental definitions and states a main theorem, essentially due to Goldman (1956), that is the basis of most of the proofs that follow. Section II then gives specific results concerning theories of ordered algebras, including, in particular, theories of additive and bisection measurement due to Pfanzagl (1959, 1968) and extensive measurement due to Suppes (1951). Section III gives results on systems of conjoint measurement, (Lute and Tukey, 1964), and interval measurement due to Davidson and Suppes (1956) and Suppes and Winet (1955). Modifications of the above axiom systems have appeared in the literature and investigations of the empirical status of the axioms that appear in such modifications could be made. However, since the purpose of this paper is to introduce a basic methodological approach to questions regarding empirical status of axioms, we shall restrict our treatment to those mentioned. Certain of the results to be stated here were originally presented in Adams, Fagot and Robinson (1965), where the present approach to assessing the empirical status of axioms was first proposed. The 1965 results are somewhat restricted versions of those given here concerning ordered algebras, conjoint and interval measurement (the generalization in the present version includes data involving identity). Pfanzagl (1968), following the approach of Adams et al. (1965), and using similar methods, has proved theorems generalizing some of their 1965 results in a different direction. We give here a much shortened version of the proof of the basic theorem concerning the existence of solutions to systems of linear inequalities which is fundamental to all of the other theorems. The presentation below is rather informal, and proofs are given only if they are not routine, and if the method of proof has some interest of its own. Although the bulk of our discussion is technical, our concern is primarily with the methodological issues these technicalities may throw light on-i.e., with the problem of assessing the empirical status of axioms-and only incidentally with the technicalities themselves.

I.

GENERAL

CONCEPTS

AND

METHODS

In this section we formulate definitions of classes of ordered structures, theories of ordered structures, data (relevant to a given kind of ordered structure) and what it is for a set of data to be consistent with a theory (of a kind of ordered structure), and, finally, for different kinds of data equivalences that can hold between two or more

EMPIRICAL

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theories of ordered structures of a given kind. We also describe some general features of the methods used to prove that certain theories stand in particular data equivalence relations. All of the theories considered are theories of ordered structures (0%). An ordered structure is here defined to be a set-theoretical system GT = (A,

‘1~~,..., o,,

R, ,..., R,,

<>,

such that: (1) A is a set called the domain of &, (2) or ,..., 13, are set-theoretical operations, called the basic operations of a, (3) R, ,..., R, are set-theoretical properties and relations defined over A, which, together with <, will be called the basic relations of a; and (4) < is a weak ordering of A (see axiom WO in the introduction), which will be called the basic ordering of GY. The domain A may be a set of individuals or the members of A may have some structure-e.g., they may be ordered pairs. We allow the possibility that either or both of p and q may be 0, so that OSs with no basic operations and only one basic relation are permitted. We do not require that the basic operations always be closed; i.e., that they be defined as applied to all objects in A. When we wish to impose closure as a requirement, we shall do so explicitly. The kinds of ordered structures that will be considered in detail are: (1) ordered aZgebras (OAs), which are OSs of the form (A, cm,<> with just one basic operation, which is binary; (2) conjoint structures (CSs), which are OSs of the form (A, x -iz, , <>; i.e., whose domain is a Cartesian product, and which have a single basic relation and no operations; (3) deference structures (DSs), which are OSs of the form (A x A, <>; i.e., DSs are CSs whose domains are “squares.” Within each of the three categories of OSs, we shall be able to define appropriate concepts (1) of one such structure being a substructure of another (or, equivalently, that the second is an extension of the first); (2) of a function mapping one structure homomorphically onto another or of its homomorphically embedding one structure in another. The details of the definitions of these notions vary with the kind of structure to which they are applied, but the following will hold in general. First, a substructure of a given structure will be another structure of the same kind, which is obtained from the first by restricting the domain of the first, and then restricting the basic operations and relations of the first to this sudomain. Second, a homomorphism of one structure onto another is a function mapping the domain of the first structure onto the domain of the second, preserving all the basic operations and relations of the first structure. Third, data that are consistent with a given structure (in the sense that they could hold in or be true when suitably interpreted in the structure) are also consistent with any extension of the structure; and data formulated just in terms of the basic relations of the structure are consistent with the structure if and only if they are consistent with another structure to which it is homomorphic. Finally, most of the basic measurement hypotheses to be considered, such as the CSH already discussed, can be formulated

386

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FAGOT,

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as requirements to the effect that structures of particular kinds be homomovphically embeddable in a fixed numerical ordered structure or a “canonical class” of such structures. [This way of formulating basic measurement hypotheses was first proposed by Scott and Suppes (1958).] For formal purposes we may identify a theory of ordered systems of a given kind with the class of all ordered structures that satisfy its basic assumptions. However, we shall also speak occasionally of the various assumptions themselves as the assumptions of the theory. We must now define a datum relevant to a particular kind of theory; and then what it is for a set of such data to be consistent with the theory. Data relevant to a theory of a given kind will, in general, be expressions that assert or deny that certain observable objects stand in one of the basic relations of the theory, or else that these objects are identical. We need not be perfectly formal and specify particular formal languages, but we indicate in a general way, for the given theories, what sorts of expressions are terms and what sorts of expressions are data. For theories of OAs, CSs, and DSs the basic ordering is the only basic relation of the theory, and therefore basic data for these theories can be taken to be expressions of the form t, < t, and their denials (to be called generalized ordinal data as noted in the Introduction), or t, = t, and their denials (identity data) where t, and t, are terms capable of denoting objects in the domains of the structures to which the theory applies. The kinds of terms that may enter into data expressions vary, depending on the kind of theory involved, and it is necessary to say something about their structure. The terms entering into OA data (OA-terms) are variabZes (letters from “u” through “2,” possibly with subscripts), together with all expressions formable from them by combining them through the use of the binary operation symbol “0.” It is convenient for us to use the notation “nt” as an abbreviation for “(f-+- (t G t) -** in t) Cl f)” (n-times). This can be defined recursively, as follows: 1 t = t; (n + 1) t = (nt 2 t). In the context of expressions such as “tl o nt, ” it is occasionally helpful to let n be any nonnegative integer. We can also define this recursively: t, n Ot, = t, ; t, o (n + 1) t, = ((tl o nt,) o tz). (These notations are well-defined even when u is not associative, but they clearly are useful only when it is.) Terms entering into CS data (CS-terms) are expressions of the form (01,/3), where ~1 is a variable in the sense already described, and /3 is a variable of the second kind, which is a lower case greek letter, possibly with subscripts. DS-terms are simply expressions of the form (OL,/I) in which 01and /3 are both variables. Now, given a theory T of OSs of a particular kind and a set D of data of the same kind, we want to say in general what it is for D to be consistent, first with a particular ordered structure GZ belonging to T, and second with T itself. To define these notions we need the concept of a variable valuation in a structure. For OAs, a variable valuation in GY is simply a function v mapping the set of variables into the domain of GE For a CS 6Y = (A, x A 2 , <), a variable valuation in GZ is a function whose domain is

EMPIRICAL

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the union of the set of variables and the set of variables of the second kind which maps variables into A, and which maps variables of the second kind into A, . A variable valuation in a DS (A x A, <) is simply a function mapping the set of variables into A. Next, consider an OS @of one of the three kinds, a term t of that kind, and a variable valuation v of the appropriate kind in GZ We want to define what it is for t to have a value under v in 02, and then, if t has a value under v in (2, what the value, v(t), is. These definitions vary, depending on what kind of structure is involved, though in all cases where a term t has a value in an OS 6l? under a variable valuation v in GY,the value v(t) will be a member of the domain of 6Y. The simplest cases are CSs and DSs, where terms are all of the form (LX, fl) such that 01and /3 are variables or variables of the second kind. In this case all terms (OL,8) have values in GYunder v; namely, the ordered pair (v(g), v(/3)), which is clearly a member of the domain of 0Z. For an OA-term t, a corresponding OS GZ, and a variable valuation v in a, the class of terms having values in 02 under v is defined recursively together with their values. In these cases all variables 01are terms, and all have values in GZ under v; namely v(a). For compound terms Q(t, ,..., tn) formed from terms t, ,..., t, and operation symbol Q, the compound term has a value in 02 under v if and only if all of t, ,..., t, have values v(Q,..., v(tn) in @ under v, and the operation o denoted by the operation symbol Q is defined as applied to these values. In this case v(o(tl ,..., t,)) is defined as o(v(t,),..., v(Q). In this special case we have taken care to distinguish between the operation symbol Q, and the set-theoretical operation o which it denotes; in what follows we shall not preserve this distinction, and so shall use symbols like “0” ambiguously as names for operations and as names of names of operations. Now, consider a datum of the form t, < t, or t, = t, , an OS GZof the corresponding type, and a variable valuation v in GZ. We shall say that the datum is de$ned in 02 under v if and only if the terms t, and t, both have values in 6Y under v and that t, < t, holds in GZ under v if and only if the formula is defined, and the pair (v(tJ, v(Q) belongs to the basic ordering of 02, and t, = t, holds if it is defined and v(tJ = v(&). The denial of any basic datum holds in an OS GYunder variable valuation v if and only if the datum is defined in fZ under v (i.e., both of its terms are defined), but it does not hold in 0L under v. We are now in a position to define consistency of a set D of data of a given kind (e.g., OA-data, CS-data, etc.) with an OS Q? of that kind or a theory T of OSs of that kind. D is consistent with 02 if and only if there is some variable valuation v in 62 under which all data in D hold in 02, and D is consistent with T if and only if D is consistent with some OS belonging to T. Intuitively, a set of data of a kind appropriate to theory T is consistent with the theory if there is some structure to which the theory applies and an interpretation of the terms occurring in the data as objects in the domain of the structure (which is represented formally by the variable valuation involved) such that if the data asserts that two terms stand in a given relation (ordering or identity), then the corresponding objects have that relation in the structure; and if the data deny

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that these terms have this relation, then the objects corresponding to the terms do not have that relation in the structure. Data equivalence is now very simply definable. Two theories of the same kind are data equivalent if it is the case that any finite set of data (of that kind) is consistent with one if and only if it is consistent with the other. The notion of data equivalence is relativized to particular kinds of data by replacing the word “data” in the foregoing definition by the appropriate kind of data. Thus, theories are generalized ordinal data equivalent in case any finite set of generalized ordinal data is consistent with one if and only if it is consistent with the other. Identity data equivalence is defined similarly. Before we describe in some detail the general methods we shall use in establishing data equivalence relations among theories defined by various well-known axiom systems for fundamental measurement, we first note some trivial facts that wiI1 be used repeatedly in these demonstrations. As already noted, any set of data that is consistent with an OS is also consistent with any extension of that OS. It therefore follows trivially that if T1 and T, are two theories such that any structure in TI is a substructure of a structure in T, , then all data consistent with TI must be consistent with T, . For any two homomorphic structures, generalized ordinal data are consistent with one if and only if they are consistent with the other. Hence, if any structure belonging to a theory TI is homomorphically embeddable in a structure belonging to theory T, , then any generalized ordinal data consistent with T, must also be consistent with T, . In what follows we shall usually be concerned with the following kind of question: What is the relation between the “empirical content” of a fundamental scaling hypothesis, FSH, and of a system of axioms -91, which are generally sufficient but not necessary conditions for FSH to hold ? We shall be interested in two things in particular: (1) Do certain subsets of ~4 contain all of the empirical content of FSH in the sense that any finite set of data consistent with the subset must also be consistent with FSH, and (2) does the entire set have more empirical content than FSH alone in that there might be data consistent with FSH but not consistent with JZ?? As it happens, it is useful to relativize the above questions to particular kinds of data, because, as has already been mentioned, identity data play roles with respect to the theories we shall consider that are very different from that played by generalized ordinal data. Hence, we shall usually begin by asking whether certain subsets of d exhaust the empirical content of FSH relative to generalized ordinal data, and whether FSH is generalized ordinal data equivalent to the entire set JZJ’.The question, what role, if any, identity data may play, will then be considered separately. In determining generalized ordinal data ralations between axiom systems LX! and their subsets and fundamental scaling hypotheses FSH, we shall follow essentially the same procedures in all cases, which are most easily described in their application to theories of ordered algebras LZ = (A, 3,
EMPIRICAL

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(2) FSH can be expressed as the requirement that the OA 6T be homomorphically embeddable in a fixed, “canonical” numerical ordered algebra N = (R, L?‘, t, (i.e., the denial oft, < tz) occurs in D, then v(tl) > v(t,). Assuming that data D were inconsistent with FSH would therefore entail that there did not exist such a valuation a. In the theories we consider, the operation o’ is a linear function of its operands, and therefore the values v(t) of compound terms t occurring in data D are linear functions of the values V(X) of the variables occurring in t-in fact in the simplest case v(t) is simply the sum of these values. Therefore, to say that data D are inconsistent with an FSH is to say that there do not exist variable valuations v(x) for the variables x occurring in D such that if the linear functions v(t) for terms t in D are computed from them, the inequality v(tJ < v(te) holds if t, < t, is in D, and the inequality v(tl) < v(tJ holds if t, < t, is in D. And this in turn means simply that this set of linear inequalities in unknowns V(X) (for variables x in D) has no solution in R. To recapitulate: the fact that a set of data D is inconsistent with an FSH entails that there is no solution in R to the unknown variable values V(X) for variables occurring in D to the linear inequalities v(tl) < v(tJ or v(ti) > v(tJ corresponding to data

390

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t, < t, or their denials occurring in D. But a theorem of Goldman (1956, Theorem 2) gives us very useful information concerning finite sets of strong and weak linear inequalities that have no solution; according to that theorem, if these linear inequalities have no solution, then certain relations must hold among the coe@cienfs of the unknowns that occur. Since the coefficient of an unknown variable value, V(X) in a compound term v(t) that contains X, is determined by the syntactical structure of t; the fact that the inequalities have no solution and consequently certain relations must hold among the coefficients of the unknowns entails, in turn, that the syntactic structures of the terms in the data must satisfy corresponding conditions. The final step in the argument to show that data D, inconsistent with FSH, hence with N, must be inconsistent with 3’ also, consists in showing that any data whose terms exhibit the syntactic features entailed by the unsolvability of the linear inequalities to which D corresponds, must be inconsistent with $8. We shall use the fundamental argument to show that certain subsets 3? of axioms for FSHs contain all of the “ordinal” empirical content of the FSH in only two cases: namely in the theory of additive scales, due to Pfanzagl, and the theory of extensive measurement, due to Suppes. Although it is possible in principle to use this direct method in other theories we consider, it proves simpler in these cases to use one or another device to “reduce” such theories to additive or extensive systems, or to systems that are in turn reducible to such systems. We do this first in the case of bisection measurement, due to Pfanzagl, then with the Lute-Tukey version of conjoint mesaurement, and then with the Suppes-Winet theory of interval measurement. We conclude this section by stating Goldman’s theorem in a form that will be most directly applicable to the sorts of problems we are considering, together with some immediate corollaries. The version given here is essentially a translation of Goldman’s formulation into terms of linear inequalities, whereas Goldman’s formulation was in terms of convex polyhedra and polyhedral cones. Inasmuch as this formulation is no more than a translation, we shall simply give it without proof. Likewise, the corollaries, some of which are not used in this paper but are used in related investigations to be reported elsewhere, are almost immediate consequences of the main theorem, and their proofs also will be omitted. Goldman’s Consider

Theorem

and Corollaries

a set of inequalities (IS) i

> 0,

i = I,..., p

bijxj > 0,

i = I,..., q,

aijxj

j=l

(1) (IW)

i i=l

EMPIRICAL

STATUS

OF MEASUREMENT

391

AXIOMS

where for i = I,..., p, j = l,..., n, aij is a real number and for i = l,..., q and j = I,..., n, bij is real (and the possibility that p and/or q is 0 is allowed), and x1 ,..,, X, are real variables. Then: (1) There exist real numbers x1 ,..., x, satisfying (I) if and only if there do not exist nonnegative real numbers rr ,..., r, and sr ,..., sp such that at least one of rl ,..., rv is positive, and all of the sums

Sj = i

riaij + i

i=l

are 0 forj

sibfj

j=l

= l,..., 72.

(2) There exist nonnegative real numbers xl ,..., x, satisfying (I) if and only if there do not exist nonnegative real numbers y1 ,..., rp , at least one of which is positive and nonnegative real numbers sr ,..., s, such that the sums Sj are nonpositive for j = I,..., n. (3) If p > 0 then there exist nonnegative x1 ,..., x, which sum to 1 satisfying (I) if and only if there exist nonnegative xi ,..., x, satisfying (I). If p = 0, then there exist nonnegative x1 ,..., x, summing to 1 satisfying (I) if and only if there do not exist nonnegative real numbers sr ,..., s, such that the sums S, are negative for j = I ,.,,, n. (4) There exist positive real numbers xi ,..., x, satisfying (I) if and only if there do not exist nonnegative real numbers rr ,..., Y, , at least one of which is positive and nonnegative real numbers s1 , .. ., sa such that the sums S, are nonpositive forj = I,..., n, and there do not exist nonnegative real numbers ri ,..., yD and sr ,..., sq such that all the sums Sj are nonpositive and at least one is negative. (5) If the coefficients aij and bij are all rational then the term “real” in (l)-(4) above. rr ,..., rp and sr ,..., s, can be replaced by “integer”

qualifying

Clause (1) of the above theorem is a direct translation of Goldman’s theorem into terms of solutions of linear inequalities, and clauses (2)-(4), giving necessary and sufficient conditions for the existence of special kinds of solutions, are immediate corollaries of the basic result, which are derived from the fact that a solution of a special kind exists to (I) if and only if there exists some solution to inequalities (I) together with a finite number of additional linear inequalities in xi ,..., x, . The final clause, (5), is also a trivial modification of the basic results. It will be the clause that is most useful in what follows. II. SOME

DATA

EQUIVALENCES

IN

THEORIES

OF ORDERED

ALGEBRAS

The results stated in this section are concerned with systems of axioms for three kinds of fundamental scaling hypotheses that may be satisfied by OAs: namely, they may be representable by additive measures, by extensive measures (which are a special

392

ADAMS,

FAGOT,

AND

ROBINSON

kind of additive measure), or by bisection measures. Intuitively, these fundamental scaling hypotheses are that the operation :Cacts like numerical addition over the domain of all numbers, like numerical addition over the domain of all positive numbers, or like the operation of taking the arithmetical mean of (“bisecting”) two numbers. We shall consider here axiom systems due to Pfanzagl (additive and bisection representation) and Suppes (extensive representation). The problem is to determine which subsets of the axioms of these systems contain all of the “empirical content” of the basic scaling hypothesis in the sense that data consistent with the axioms of these subsets must be consistent with the scaling hypothesis itself. Throughout most of this section we consider only generalized ordinal data, and our results demonstrate generalized ordinal data equivalences. At the end of this section we state a model-theoretic result that helps to make clear the special roles of data including identities, and of certain axioms involving identities in the theories of Pfanzagl. We begin by listing various groups of conditions or assumptions that Suppes and Pfanzagl combine in different ways in their axiom systems. The exact form of the assumptions given here differ in inessential ways from their original formulations. DEFINITION

1.

Let GZ = (A, 0, <)

be an ordered

algebra,

1.1. cl is closed b A if and only if for all x and y in A there is a z in A such that o(x,y) = z; I .2. GE is closed if and only if o is closed in ‘4; I .3. if x and y belong such that c$x,y) = Z; 1.4. if x andy

to A, then x c y is dejined if and only if there is a z in A

belong to A and x 3 y is defined, then x C:y = <1(x, y).

Definition 1.3 is inelegant and creates some problems for the formal logician, but it provides a methods of expression that is intuitively clear and it avoids other methods of expression that are lengthy and obscure to the nonlogician. Given Definition 1.3, if Q? = (A, O, <) is an ordered algebra, then o is closed in A if and only if x 1~)y is defined for all x and y in A. It is important, however, that the closure condition can be stated in first order logic. Thus, if “2” is a 3-place predicate (which is customarily variables in interpreted by binary operations), and “x,” “y,” and “a” are individual a given formal language, then the closure condition can be written (with the quantifiers written out) as follows: for all x, for all y, there is a x such that ~(x, y, z). (The condition of being a binary operation can also be stated in first order logic with identity by the statement for all X, if 0(x, y, 4

for all y, and

for all Z, i?(x, y, 4,

for all w, then

z = w.)

EMPIRICAL

STATUS

OF MEASUREMENT

AXIOMS

393

In the following definitions if the operation symbol is used, e.g., in “x o y,” it is assumed that the definition is hypothetical, the hypothesis being that x 5 y is defined. Thus, the definition for commutativity, given below, would be, more formally, “02 is commutative if and only if for all x and y in A such that x o y and y cl x are defined, x 0 y zz=y I:, x.” We shall also eliminate, in the customary mathematical manner, universal quantifiers. The definitions in Definition 2 are of algebraic conditions that can be stated formally in first order logic with identity. The definitions are stated using the identity symbol; corresponding definitions using the symbol for the indifference relation generated by the basic ordering relation are for corresponding set-theoretic predicates preceded by the word “weakly.” DEFINITION 2. Let 02 = (A, ‘18,<> mutative, (ii) associative, (iii) idempotent, solvable, as the following hold:

be an ordered algebra (OA) (iv) bisymmetric, (v) solvable,

Q? is (i) comor (vi) positive

(i) x 1 y = y ;I x; (ii) x ‘3 (y 0 z) = (x 0 y) 0 z; (iii)

x I: x = x;

(iv) (x C y) C (z 0 w) = (x 0 x) 0 (y 0 w); (v) thereisauinAsuchthatxou (vi)

=y;

if y > x, then there is a u in A such that x ~1 u = y,

The next set of definitions is also formulable in first order logic, but do not involve identities (or indifferences) as did those of Definition 2. 3.

DEFINITION

monotonic, (iii) positive, or (vii)

Let GY = (A, c, <) be an OA. 02 is (i) left monotonic, (ii) right left cancellable, (iv) right cancellable, (v) Suppes monotonic, (vi) dense, as the following conditions hold:

(i) if x < y, then x o z < y c z; (ii) if x < y, then z o x < z o y; (iii)

if .x 12 z < y 1~ z, then x < y;

(iv) if z o x < z c y, then x < y; (v) x:~z x; (vii)

if x > y, then there is a u in A such that x > u and u > y.

394

ADAMS,

The following order logic.

set of definitions

FAGOT,

AND

ROBINSON

consists of conditions

that can not be stated in first

DEFINITION 4. Let 02 = (A, o, <) be an OA. 02 is (i) additive Archimedean, (iii) left continuous, or (iv) right continuous, as the (ii) bisection Archimedean, following conditions hold:

(i) if x cl x > x and y o y > y, then there exists a positive

integer

n such that

nx>y;

(ii) if x > y > .a, then there exists a positive

integer

n such that y > x ~1 nz;

(iii) both the following are topologically open sets: the set of all u in A such that u o y is defined and u I; y > x; and the set of all t in A such that t ti y is defined and toy
(iv) both the following are topologically open sets: the set of all u in A such that y o u is defined and y c u > x; and the set of all t in A such that y o t is defined and yot
DEFINITION

concern

Let Q! = (A, o, <)

5.1. An additive representation such that for all x and y in A,

the various

be an ordered

representations

and FSH

algebra.

for GZ is a real valued function

m with

domain

A

(i) m(x) < m(y) if and only if x < y, (ii) if x c) y is defined,

then m(x o y) = m(x) + m(y);

5.2. An extensive representation that m(x) > 0, for all x in A; 5.3. A bisection representation that for all x and y in A, (i) m(x)

for GI is an additive

representation

for CPG is a real valued function

m for 02 such

m with domain A such

< m(y) if and only if x < y,

(ii) if x o y is defined, 5.4. O! is additive for 02.

then m(x o y) = $(m(x) + m(y));

representable

if and only if there is an additive

representation

5.5. GT is uniquely additive representable if and only if for any two additive representations m, and m2 for @, there exists a real number k > 0 such that m,(x) = km,(x), for all x in A. 5.6. for 6Y.

6Y is extensive representable

if and only if there is an extensive representation

EMPIRICAL

STATUS

OF

MEASUREMENT

395

AXIOMS

5.7. CZ is uniquely extensive representable if and only if for any two extensive representations m, and rns for 02, there exists a real number k > 0 such that ml(x) = Km,(x), for all x in A. 5.8. 02 is bisection representable for @.

if and only if there is a bisection

representation

5.9. 02 is uniquely bisection representable if and only if for any two bisection representations m1 and m2 for GY, there exist real numbers K > 0 and g such that ml(x) = km,(x) + g, for all x in A. In our considerations of data equivalences for ordered algebras we have dealt with only the original sets of axioms given by Pfanzagl (1959), for additive representations and bisection representations, and by Suppes (1951) f or extensive representations. DEFINITION

6.

Let cZ’ = (A,

o, <)

be an ordered

algebra.

6.1. GT is an additive structure if and only if it is closed, weakly weakly associative, and left and right cancellable; 6.2. GZ is an extensive structure if and only if it is closed, weakly positive solvable, positive, and Suppes monotonic;

commutative,

weakly

6.3. G? is a bisection structure if and only if it is closed, weakly weakly idempotent, weakly bisymmetric, and left and right cancellable.

associative,

commutative,

Pfanzagl, extending the work of AczCl, and Suppes, extending the work of Helmholtz and Hijlder, provided sets of conditions that were sufficient for the existence of additive (Pfanzagl), extensive (Suppes), and bisection (Pfanzagl) representations. Pfanzagl has shown that additive structures that are left and right continuous are additive representable and uniquely additive representable; Suppes has shown that extensive structures that are additive Archimedean are extensive representable and uniquely extensive representable; and Pfanzagl has shown that bisection structures that are left and right continuous are bisection representable and uniquely bisection representable. The only nonnecessary axioms contained in Pfanzagl’s theories of additive and bisection representations are those of closure and continuity. In Suppes’s theory of extensive representation the nonnecessary axioms are those of closure, positive solvability, and the additive Archimedean condition. Our task is to determine which among these nonnecessary axioms is technical. First, however, some comment on the above definitions is appropriate. The definitions of various “algebraic conditions” in Definition 2 are considered in two forms, one using identity and the other using the indifference relation generated by the basic ordering. Suppes uses the indifference versions and Pfanzagl, in his earlier versions, did so as well. However, in his recent book Pfanzagl (1968) states most of the axioms for the quotient-systems generated by the indifference relation 4w7/3-2

396

ADAMS,

FAGOT,

AND

ROBINSON

(which is a congruence relation relative to the various OAs). At the end of this section we shall show that axiom systems using identities and those using the indifference relation are empirically equivalent relative to generalized ordinal data, but that when identity data are considered the identity versions are stronger than the indifference versions and some care must be taken in determining the exact role played by identities. The various monotonicity and cancellation conditions of Definition 3 are given because they are stated differently by different authors. Suppes uses the version named for him (but essentially due to Holder) to combine monotonicity and commutativity in a single axiom. We distinguish the special class of “nonelementary axioms” stated in Definition 4 on the grounds that they are not formulable in first order logic. The question how such nonelementary propositions might be tested empirically is, in general, a very difficult one, but we shall be able to show that in the cases we consider such axioms do not add to the empirical content of the theories of which they are parts. This will follow directly from our next comment. The conditions that there exist additive, extensive, and bisection representations are equivalent to the conditions that there exist homomorphic embeddings in the systems, respectively, Add = (Re, +, ~1, Ext = (Re, , +, <>, and Bisect = (Re, Mean, is homomorphically embeddable in, for example, Add, if and only if there is a functionf whose domain is A and whose range is a subset of Re, the set of real numbers, such that (i) if x CIy is defined,

thenf(x)

(ii) x < y if and only if.f(x)

+f(y)

is defined

andf(x

c y) =f(~)

+f(y);


This definition can be extended to other numerical OAs in an obvious way. Assume that a set of generalized ordinal data D holds in an OA @ under a variable valuation v and that a functionf homomorphically embeds Cptin G!‘. It will follow, by definitions, that D holds in GY’ under the variable valuationf(v), that is the composition off with U. It also follows that if D is consistent with Q? and @ is homomorphically embeddable in GY’, then D is consistent with @‘. From this we immediately infer that any purely generalized ordinal data consistent with the theory of additive (extensive, bisection) representable OAs must be consistent with Add (Ext, Bisect). Furthermore, since the entire system of axioms of Pfanzagl and Suppes entail the existence of such representations, it follows that the hypotheses to the effect that these representations exist are generalized ordinal data equivalent to the entire axiom sets. This last remark also throws light on the significance of uniqueness assumptions. It follows from our observations in the last paragraph that hypotheses to the effect that representations are unique up to appropriate classes of transformations do not add empirical content to bare hypotheses that the representations exist. We now state and prove the main result of this section; namely, a theorem that

EMPIRICAL

STATUS

OF MEASUREMENT

certain subsystems of axioms are generalized hypotheses for ordered algebras.

ordinal

397

AXIOMS

data equivalent

to basic scaling

THEOREM 1. (a) The theory of additive representable OAs is generalized ordinal data equivalent to the theory of additive structures (OAs that are closed, weakly commutative, weakly associative, and left and right cancellable); (b) the theory of extensive representable OAs is generalized ordinal data equivalent to the theory of extemive Suppes structures (OAs that are closed, weakly associative, weakly positive-solvable, monotone, and positive); (c) the theory of bisection representable OAs is generalized ordinal data equivalent to the theory of bisection structures (OAs that are closed, weakly commutative, weakly idempotent, weakly bisymmetric, and left and right cancellable).

Proof. Let D be a finite set of generalized ordinal data partitioned into two subsets DS and D W such that DS consists of negations of sentences of the form hi

For convenience,

<

we shall regard

t,i

7

i=l

DS as consisting

hi < t,i 7

,..., P. of sentences of the form

i = l,..., p.

DW consists of sentences of the form t;t

<

t;i

)

i = I,.... q.

All terms occurring in the sentences of D are OA-terms. We shall assume that n different variables, xi ,..., x, , occur in these terms. We shall regard terms tli and tli , occurring in DS and D W, respectively, as occurring on the left-hand side of the inequality; similarly, we regard tzi and t& as occurring on the right-hand side of their inequalities. From our previous remarks it follows that if D is consistent with the theory of additive (extensive, bisection) representable OAs, then D is consistent with the theory of additive (extensive, bisection) structures. It remains to show that if D is inconsistent with the theory of additive (extensive, bisection) representable OAs, then it is also inconsistent with the theory of additive (extensive, bisection) structures. We follow the the additive and extensive cases general procedure outlined in Sec. I, considering first. Thus, the assumption that D is inconsistent with the theory of additive (extensive) representable OAs is equivalent to D’s being inconsistent with Add (Ext); that is, that there is no numerical valued variable valuation v, defined in Add (Ext) for all terms t occurring in D such that: (i) if a term t, CCt, occurs in D, then ordinal data in DS, v(tIi) < v(t,,), v(t, c, t2) = v(tl) + v(t,); (ii) f or all generalized i = l,..., p; and (iii) for all generalized ordinal data in DW, c(t;J < v(tkJ, i = l,..., q.

398

ADAMS,

FAGOT,

AND

ROBINSON

This entails that there do not exist real numbers (positive real numbers case) zI(xi),..., V(XJ assigned to the variables xi ,..., J, such that

in the extensive

and

WV

j=l

j=l

where N,(t) is the number of occurrences of variable xj in term t. From Clause (5) of Goldman’s theorem it follows, given that there is no solution to the above system of linear inequalities, that there exist nonnegative integers ri ,..., rD , sr ,..., sQ , where at least one ri is positive such that all of the sums

Sj = i

j = l,..., n,

rj[Nj(t2j) - Nj(lli)l + f si[Nj(tii) - Nj(tii)13

i=l

i=l

are zero, and if there do not exist positive I,..., z)(xJ satisfying (IS) and (IW), then there exist integers ri ,..., Ye , s1 ,..., s, , with at least one of yi positive, such that all the sums Sj are nonpositive. We have assumed that D is inconsistent with the theory of additive (extensive) representable OAs; let us now assume, contrary to what we want to show, that D is consistent with the theory of additive (extensive) structures. This entails that the datum rltl,

CJ(.** Y&1,, $3 (&, < r,t,,

0 ... i‘ S&J

.*.)

(2 (... r,tzp s (S&, ‘-, ‘.- -1 &)

. ..)

(9

is also consistent with the theory of additive (extensive) structures. On the other hand, in the case in which D is inconsistent with Add, all of the sums Sj are zero, and thus the number of occurrences of each variable xj in the right-hand side of(S) is equal to the number of occurrences of that variable on the left-hand side of (S). Hence, (S) would be an expression of the form t, ( t, , where, for each j, Nj(tl) = Nj(t2). However, it is easy to show that no such datum could be consistent with the theory of additive structures. It follows, in the additive case, that D is not consistent with the theory of additive structures. In the case in which D is inconsistent with Ext, all of the sums Sj are nonpositive. It would follow from this that all variables xi occurred at least as many times on the left-hand side of (S) as on the right, and this is easily seen to be inconsistent with the theory of extensive structures. It follows then that D is inconsistent with the class of extensive structures if D is inconsistent with the class of extensive representable OAs.

EMPIRICAL

STATUS

OF

MEASUREMENT

AXIOMS

399

The proof of part (c) of the theorem is based on reducing the bisection case to the additive case by considering what we shall call “uniform terms.” It will be remembered that if t, and t, are terms, then (ti o ts) is also a term; it is important to note how parentheses are added in forming terms. An occurrence of a variable x in a term t is said to be order n if n parentheses-pairs surround that occurrence of x in t, and the order oft itself is defined to be the highest order of variable-occurrences in t. t will be said to be zmiform of order n if all variable-occurrences in t are of order n, and to be uniform if it is uniform of some order. For any variable x there is a uniquely constructable term n(x) that is uniform of order n and in which only the variable x occurs. Any term t of order n can be reducedin a unique way to a uniform term of order p 3 n by replacing an occurrence of a variable x of order m < p in t by the term (p - m)(x). Let us call this “reduced term” U,(t). Finally, generalized ordinal data D can be reduced to uniform data of any order p, which is at least as great as the order of any term in D, by replacing each term t in it by U,(t). Call the uniform data U,(D). We can establish the following (proofs are omitted, as they are straightforward): (1) D is consistent with the theory of bisection representable OAs if and only if U,(D) is; (2) D is consistent with the theory of bisection structures if and only if U,(D) is; and (3) U,(D) is consistent with the theory of bisection representable OAs if and only if it is consistent with the theory of additive representable OAs. The first two of these show that we can consider without loss of generality that D itself is uniform, hence that D = U,(D) for some p. If D = U,(D) is inconsistent with the theory of bisection representable OAs, it is also inconsistent with the theory of additive representable OAs, and, therefore, by our previous argument, there are integers ri ,..., yp, si ,..., sa, with at least one ri positive, such that all the sums Sj are zero. Again, if data D were consistent with the theory of bisection structures, then datum S constructed as before would also be. However, each variable occurring in (S) would occur the same number of times on the left-hand side as on the right-hand side of the symbol for the strict ordering. It is possible that the terms occurring in (S) are not uniform, but reducing them to uniform terms of the same order preserves the property that each variable occurs the same number of times in the left- and right-hand terms in this case. But a uniform datum of form S in which all the variables occur the same number of times in the left hand terms and the right hand terms is easily seen to be inconsistent with the class of bisection structures. Hence, D itself is inconsistent with the class of bisection structures. This concludes the proof of Theorem 1. From Theorem 1 we conclude that for both the additive and bisection cases the continuity axioms given by Pfanzagl can be regarded as “technical” relative to generalized ordinal data. It is a trivial consequence of the Scott-Suppes-Titiev results that the closure axiom is not technical, for if it were this would mean that the empirical content of these scaling hypotheses could be axiomatized in a finite number

400

ADAMS,

FAGOT,

AND

ROBINSON

of universal sentences. Of the Suppes axioms for extensive measurement it follows directly from Theorem 1 that the additive Archimedean condition is technical relative to generalized ordinal data, but it appears that neither the closure nor the positive solvability condition is. We now wish to consider the special roles that identity data and axioms involving algebraic identities play in empirical realizations in determining the validity of axiom systems containing algebraic identities (e.g., that the operation 0 is commutative). As already noted, the fact, for instance, that o should be commutative is not logically entailed by any of the basic scaling hypotheses considered here, and therefore it is at least a logical possibility that data consistent with the existence of any of these scaling hypotheses might be inconsistent with the axiom systems given for them. We now wish to consider subsets A of the set of axioms given in Def. 24. Let A be partitioned into AI (the algebraic identities included in A, or the identity version of the solutions axioms contained in A) and A0 (the “generalized ordinal axioms” contained in A). Note that if 6Yi and @Z are two homomorphic OAs then axioms in A0 hold in GZ1 if and only if they hold in G!, . Also, let O(AI) be the “generalized ordinal versions” of the identity axioms in AI. It is trivial that any OA satisfying AI also satisfies O(AI). Next, let D be a finite set of OA-data, which may be partitioned into three subsets DO (generalized ordinal data contained in D), DI+ (data of the form t, = t, in D) and DI(data of the form t, f t2 in D). Let O(DI+) be the set of data t, - t, for each identity t, = t, in DI+. W e now wish to prove the following: THEOREM 2. D is consistent with A if and only if DO u DI+ is consistent with A and DI+ u DIis consistent with AI. If A0 u O(AI) entails the monotonicity axioms, then DO u DI+ is consistent with A if and only if DO u O(DI+) is consistent with A0 u O(AI).

Proof. In proving the first part, clearly if D is consistent with A then DO u DI + is consistent with A and DI+ u DIis consistent with A hence with AI, which is a subset of A. Conversely, suppose that DO u DI+ holds in an OA

that satisfies A, and DI+

U

DI-

that satisfies AI. Now construct

holds in an OA

the “quasi-product”

where < is the binary relation over K, x K2 such that for all (x1 , ~a) and (yi , ya) in K, x K, , (~1, x,) < (yi , yz) holds if and only if x1
EMPIRICAL

STATUS

OF MEASUREMENT

AXIOMS

401

following are all easily shown: (1) the entire set D is consistent with @i/Q?s ; (2) G!!i/flZ is homomorphic to @r , and therefore satisfies all axioms in AO; and (3) because axioms in AI are satisfied in both G& and G?s , they are satisfied in the quasi-product ar/eC, . The foregoing three facts then entail directly that data D are consistent with the entire set .il since they are consistent with G&/0& which satisfies all axioms in A. To prove the second part, suppose that A0 u O(AI) entails the monotonicity conditions. Clearly if DO u DI+ is consistent with A, DO u O(DO+) is also consistent with A, and therefore is also consistent with A0 u O(AI). Conversely, suppose that DO u O(DI+) is consistent with A0 u O(‘41): i.e., DO u O(DI+) holds in some OA GZ = (K, r,, <) that satisfies the axioms of A0 u O(AI), and therefore satisfies the monotonicity axioms. Since the monotonicity conditions are satisfied we can construct the quotient system a-, = (K, , ;,, , 2) in which K, is the set of equivalence classes of members of K under the equivalence relation N, 2 is the quotient ordering on K, , and for any two equivalence classes [x] and [y] in K, generated from members Y and y of K, [x] cc- [y] is the class of all x’ I.:,y’ for all r’ and y’ in K such that x’ N x and y’ N y. The following three facts are easily demonstrable: (1) 02 is homomorphic to @- , hence GF?, satisfies axioms A0 because ~7 does; (2) 02- satisfies axioms AI; and (3) DO u DI+- is consistent with 02, . Thus, there is an OA, Q!- that satisfies all axioms of il and data DO u DIf is consistent with a-, . This concludes the proof. The point of the foregoing is this. When we are considering if “mixed data” D, including both generalized ordinal data and identities, are consistent with any subset of the axioms listed in Def. 2-4, we can reduce this to the following question: Are the identity data included in D consistent with the algebraic identities among the axioms and are the generalized ordinal data included in D together with the “positive identities” consistent with these axioms? This in turn reduces to the question of determining whether the “generalized ordinal equivalents” of the identity data plus the original generalized ordinal data are consistent with the “generalized ordinal equivalents” of the axioms in question, in the case of systems of axioms entailing monotonicity.

III.

DATA

EQUIVALENCES

IN

THEORIES

OF CONJOINT

AND

DIFFERENCE

STRUCTURES

In this section results are obtained concerning axiom systems given by Lute and Tukey (1964) for conjoint measurement, by Suppes and Winet (1955) and by Davidson and Suppes (1956) for difference measurement. Conjoint structures (CSs) were defined in Sec. I as ordered structures with a single ordering relation, whose domain is a Cartesian product; i.e., structures of the form . Difference structures (DSs) were defined as conjoint structures whose domain is the Cartesian product of a set with itself, i.e., structures of the form
402

ADAMS,

FAGOT,

AND

ROBINSON

The techniques used in proving the results for these structures are essentially those described in Sec. I: (1) We begin by assuming that fundamental scaling hypotheses (FSHs) can be formulated as requirements of homomorphic embeddability (suitably defined for such structures) in certain “canonical systems.” From this it follows that data consistent with the FSH must be consistent with all laws satisfied by the corresponding canonical systems. (2) To establish that data consistent with certain subsets of the axiom systems must be consistent with the relevant FSH we “reduce” the structures satisfying the axioms to ordered algebras (OAs) satisfying axioms that are generalized ordinal data equivalent to an FSH for the existence of a corresponding OA representation. We are then in a position to use the results of the last section to complete the proofs. For conjoint and difference structures we do not need to consider identity data in establishing data equivalences because for each subset of the axiom systems we consider the class of structures satisfying those axioms is data equivalent to the corresponding FSH if and only if they are generalized ordinal data equivalent. This fact will not be proved, but it follows easily from the fact that all the axioms to be considered are formulated solely in terms of the basic ordering relation and do not involve any identities. A pair of functions (f, g) is a CS homomorphic embedding of a CS structure (A x X, <) in a CS structure (A’ x X’, <‘) if and only if f is a function whose domain is A and whose range is a subset of A’; g is a function whose domain is X and whose range is a subset of X’; and for all (a, x) and (b, y) in A x X, (a, x) < (b, y) if and only if (f(a), g(x)) <,’ (f(b), g(y)). A function f is a DS homomorphic embedding of a DS structure (K x K, <> in a DS structure (K’ x K’, <‘) if and only if f is a function whose domain is K and whose range is a subset of K’, and for all x, y, z and w in K, (x, y) < (x, w) if and only if (f(x),f(y)) <’ (f(z),f(w)). It trivially follows, as before, that generalized ordinal data of the appropriate kind hold in a structure Q! under a variable valuation if the data hold in a structure that is homomorphically embeddable in GZ. We begin by listing various definitions for CSs. We shall not separate the various kinds of definitions as was done in Sec. II. DEFINITION

7.

Let 02 = (A

7.1. CY (i) is solvable, as the following conditions

x X, Q)

be a conjoint

(ii) satisfies cancellation, hold:

structure

(CS).

(iii) is a Lute-Tukey

structure,

(i) for all a and b in A and all x and y in X there exist c in A and .z in X such that (a, x) - (c, y) and (a, x) - (b, z); (4 (iii)

if (a, 4 < (6 Y) and (b, 4 < Cc, xl, then (a, 4 < Cc,Y); GZ is solvable and satisfies cancellation;

EMPIRICAL

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7.2. A dual standard sequence for 6Y is a pair of doubly infinite sequences ai, xi , such that if i +i = p + q, then i = 0, &l, f2 ,..., from A and X respectively, (ai , xi) - (a, , x0); the dual standard sequence is trivia2 if (ai, x0) N (aO, x0), for all i; 7.3. Gl is conjoint Archimedean if and only if for any nontrivial dual standard sequence ai , xi , i = 0, &I,..., and a in A and x in X, there exist integers n and m such that (a, , x,) < (a, x) and (a, x) < (a,, x,). 7.4. A pair of functions (mr , m,) mapping A and X, respectively, into the set of real numbers is a conjoint representation for C!! if and only if for all a, b in A and all x, y in X, (a, x) < (b, y) if and only if ml(a) + m,(x) < m,(b) + mz(y); 7.5. GZ is (i) conjoint representable, (ii) uniquely structure CON J, as the following conditions hold: (i) there exists a conjoint

representation

conjoint

representable,

(iii) the

for @,

(ii) if (m, , m,) and (m,‘, m,‘) are two conjoint representations for GY, then there exists a positive real number K and real numbers g and h such that for all a in A and x in X, m,‘(a) = km,(a) + g and m,‘(x) = km,(x) + h; (iii) A = X = Re and < is the relation <+ such that for all real numbers x, y, Z, and w, (x, y) <+ (x, w) if and only if x + y < z + w (i.e., CONJ

= (Re

x Re, <+>).

From the definitions it follows that a CS structure @ is conjoint representable if and only if it is homomorphically embeddable in CONJ. It follows from this that generalized ordinal data are consistent with the theory of conjoint representable structures if and only if consistent with CON J. Given any law satisfied by CON J, generalized ordinal data consistent with the theory of conjoint representable structures are consistent with the theories consisting of models of that law. We now state and prove our main result concerning conjoint measurement. THEOREM 3. The theory of conjoint equivalent to the theory of Lute-Tukey cancellation).

representable CSs is generalized ordinal data structures (CSs that are solvable and satisfy

Proof. That generalized ordinal data consistent with the theory of conjoint representable CSs are consistent with the theory of Lute-Tukey structures follows from the remarks made in the last paragraph. It remains to show that generalized ordinal data consistent with the theory of Lute-Tukey structures are consistent with the theory of conjoint representable CSs. Let @ = (A x X, <) be a Lute-Tukey structure. For any pair (a,, , x0) in A x X there is an OA system Ad(@, a,, x0) associated with the triple (02, a,, x0) such that Ad(& a,, , x0) = (K, o, <‘) where: (1) K is the set of all classes [a] and [x] defined as follows, for all a in A and x in X:

404 [a] [x] (2) x in

ADAMS,

FAGOT,

AND

ROBINSON

= the set of all b in il and y in X such that (a, x0) y (6, x0) and (a, x0) b (u,, , y); = the set of all b in A andy in X such that (uU , X) N (b, x0) and (a,, , X) N (a0 , y); For all c1and /3 in K, 01 3 /3 is the unique element [c] in K such that for some a in =1, X, a E 01,x E p, (a, x) - (c, x0);

(3) For all 0: and /? in K, cy <’ /3 if and only if for some a and b in A, a E 01, b Ej3,

and(a, x0) =G(6 x0). The existence and uniqueness of the element [c] in K satisfying condition (2) above follows from the assumption that 0Z is a Lute-Tukey structure. This assumption also entails the following, the proofs of which we omit since they are straightforward:

(4) Ad(@, a0, x0) is an additive associative,

and left and right

structure cancellable);

(5) For all (a, X) and (b, y) in il

x

(it is closed, weakly commutative,

weakly

X, (a, X) < (b,y) if and only if [a] 8:) [x] <’

VI cl [YI. It is clearly the case that Ad(CON J, 0,O) = ADD/ = which is isomorphic to ADD. CS data D can also be made to correspond to generalized ordinal data ad(D) as follows. First, to each variable of the second kind, ,6, occurring in D, let c(p) be a variable (of the first kind) not occurring in D (we assume that D is finite). Next, if Q = (in., p) Q (y, 8) is a CS datum, then let ad(Q) be the OA datum OL1:;c(p) 6 y o c(6) and let ud(-9) = --ad(Q). (If D is any set of DS data, then ad(D) is the set of OA data obtained in the above fashion by ad.) It is trivial to show that if D is any set of CS data, then D is consistent with a Lute-Tukey structure 0! = !A x X, < if and only if ad(D) is consistent with Ad(@, a, x), for all a in A and x in ,ri. The proof of our theorem is simple given these preliminaries. Let D be a set of generalized ordinal CS data consistent with the theory of Lute-Tukey structures. Hence D is consistent with some member 02 of that theory. By our remark in the last paragraph, ad(D), then, is consistent with any associated OA, Ad(@, a, x). Since such an OA Ad(02, a, X) is an additive structure, by Theorem 1, ad(D) must be consistent with ADD. But ad(D) is consistent with ADD if and only if D is consistent with to ADD. CONJ, since Ad(CONJ, 0,O ) = ADD/ = and ADD/ = is isomorphic Hence we conclude that D must be consistent with CONJ and must, then, be consistent with the theory of conjoint representable CSs if it is consistent with the theory of Lute-Tukey structures. This concludes the proof. Since the empirical content of the conjoint representable hypothesis is contained in in the axioms for solvability and cancellation (and well-ordering), as shown b! Theorem 3, it follows that the conjoint Archimedean condition is technical in the sense we have employed that term. It also follows from Theorem 3 that solvability is the only nontechnical, nonnecessary axiom in the axiom system given by Lute and Tukey. Our next results concern interval measurement, including the Suppes-Winet theory of difference structures. In this case our results are less neat as we have not

EMPIRICAL

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AXIOMS

isolated a single nonnecessary, nontechnical axiom. The results of Scott-SuppesTitiev show that it is impossible to formulate interval measurement, just as it is impossible to formulate conjoint measurement, using only universal axioms. However, the Suppes-Winet axiomatization uses no less than three axioms that are not universal. We begin by listing some definitions. DEFINITION

8.

Let 02 = (/I

8.1. LE is a basic interval

x A, <)

be a difference

structure

(DS).

structure if and only if

(i) (x, y) < (x, w) if and only if (x, z) < (y, w) and (ii)

(x, y) < (z, w) if and only if (w, x) < (y, x);

8.2. A real-valued function m whose domain is A is an interval repcesentation for @ if and only if for all X, y, z, and w in A: (x, y) < (z, w) if and only if m(x) - m(y) < m(z) - m(w); 8.3. @ is (i) interval representable, (ii) uniquely the structure INT, as the following conditions hold: (i) there exists an interval

representation

interval

representable,

(iii)

for m,

(ii) if m, and ma are two interval representations for 0Z, then there exists a real number k > 0 and a real number h such that for all x in A: ml(x) = km,(x) + h; (iii) A = Re and < is the relation <- such that for all real numbers x, y, z, and w, (~,y) Q- (z, w) if and only if .X - y < z - w (i.e., INT = (Re x Re, <-)). DEFINITION

9.

Let a = (A

x A, =Q be a basic interval

structure.

9.1. The weak ordering of A associated with GZ is the unique weak ordering such that for all x and y in d4, x < y if and only if (x, y) < (y, y);

of A

9.2. A finite sequence xi ,..., x, of elements of A is equally spaced if and only if xi+i > zii , i = l,..., n -- 1 and (xi+i , xi) N (xi , xi-i), i = 2 ,..., n ~ 1. 9.3. 67 is (i) bisectable, (ii) interval-bisymmetric, (iii) interval Archimedean, (iv) continuous, (v) a Suppes-Winet infinite difference structure, (vi) a finite equally spaced difference structure, as the following conditions hold: (i) for all x and y in A there exists a x in A such that (x, z) N (z, y); (ii) if (2, a> - (a, y), (x, b) - (6 ~1, (y, c) - (c, w), and (x, 4 - (4 w), then (a, 4 - Cc, 4; (iii) if x > y and z > w, then there exists a positive integer n and an equally spaced sequence yi ,..., y,, , with yi = y and yn = X, such that (y% , yl) < (z, w); (iv) if x > y and (s, y) > (a, w), then th ere exists a u in A such that x > u ,r y and (a, w) < (x, u);

406

ADAMS,

(v) G! is bisectable,

FAGOT,

continuous,

AND

ROBINSON

and interval

(vi) there exists an equally spaced sequence for all x in A, x - xi , for some i = l,..., 71.

Archimedean; xi ,..., X, of elements

of A such that

The interval-bisymmetry condition has not appeared before in the literature. What we show is that the empirical content of the FSH of interval representability is contained in the set of conditions defining a basic interval structure together with the conditions of bisectability and interval-bisymmetry. Three of the axioms given by Suppes and Winet-bisectability, the interval Archimedean condition, and continuityare nonnecessary for interval representability. If it could be shown that bisectability together with the conditions defining a basic interval structure entailed intervalbisymmetry, then our result would show that continuity and the interval Archimedean condition are technical. We have not, however, taken up this question, and, therefore, the status of the interval Archimedean condition and continuity remain in doubt. Condition 9.3 (iv) is called “continuity,” but it should be noticed that, unlike the usual continuity axioms, it is expressable in first order logic. From the definitions it follows that a DS structure @ is interval representable if and only if it is homomorphically embeddable in INT. Hence, generalized ordinal data consistent with the theory of interval representable structures are consistent with INT. It follows from this that generalized ordinal data consistent with interval representability are consistent of Defiwith all the laws satisfied by INT. INT satisfies the first five conditions nition 9.3, but it does not satisfy the finite equal spacing condition. We shall consider the status of this law separately. We now state and prove the main result concerning interval measurement. THEOREM 4. The theory of interval representable DSs is generalized equivalent to the theory of basic interval structures that are bisectable bisymmetric.

ordinal data and intevval-

Proof. That generalized ordinal data consistent with the theory of interval representable DSs are consistent with the theory of basic interval structures that are bisectable and interval-bisymmetric follows from the remarks made in the last paragraph. Let 6Y = (A x A, <> be a basic interval structure that is bisectable and interval-bisymmetric. We construct an associated OA, Bi(@) = (K, G, <‘) in which: (I) K is the set of all classes [x], for all x in A, where [x] = the set of ally in A such that y N x; (2) For all LYand p in K, (Y s @ is the unique andzinA,xisinor,yisinfi,zisiny,and(x,z)m(Z,y); (3) For all OLand /3 in K, 01 <’ in B.

element y in K such that for some X, y,

/3 if and only if x < y, for some x in (Y and some y

EMPIRICAL

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AXIOMS

The existence and uniqueness of the element y in K satisfying (2) above follows from the assuption that a is a basic interval structure that is bisectable and intervalbisymmetric. This assumption also entails the following, the proofs of which are omitted; (4) Bi(02) is a bisection structure (it is closed, weakly commutative, potent, weakly bisymmetric, and left and right cancellable);

weakly

idem-

(5) For all X, y, x, and w in A, (x, y) < (z, w) if and only if [x] 0 [w] <’ [y] o [z]. It should be clear that Bi(INT) = INT/=, which is isomorphic to INT. Similar to our procedure for conjoint measurement we associate to any generalized ordinal DS datum da corresponding OA datum h(d). If d is of the form (01, /3) < (y, S), then k(d) is the OA datum 01 o 6 < /3 o y; we also let h-d) = A(d). If D is a set of generalized ordinal DS data, then h(D) is the set of OA data, each corresponding to a member of D by the above-defined function bi. Given this correspondence it easily follows that a set of generalized ordinal DS data D is consistent with a bisectable interval-bisymmetric, basic interval structure 02’ if and only if h(D) is consistent with the associated OA, &(a). Let D be a set of generalized ordinal DS data consistent with a bisectable, intervalbisymmetric, basic interval structure 02. Then h(D) is consistent with the associated OA, Bi(02). By Theorem 1, part (c), b(D), then, is also consistent with BISECT. But h(D) is consistent with BISECT if and only if D is consistent with INT, since Bi(INT) = BISECT/= and BISECT/= is isomorphic to BISECT. Hence, D is consistent with INT. From our remarks following Def. 9 it follows that D is consistent with the theory of interval representable DSs. This concludes the proof. Regarding the status of the finite equal spacing condition, it can easily be shown that any finite set of generalized ordinal DS data is consistent with INT if and only if consistent with a subsystem INT, = ([n] x [n], Q-i of INT, for some positive integer n, where [n] is the set of the first n positive integers. This follows immediately from the fact that a finite system of inequalities involving a finite set of unknowns, and having integral coefficients, has a solution if and only if it has a solution in the integers. Now, all the systemsINT,satisfy the finite equal-spacing axiom, and therefore any finite set of DS data consistent with INT is also consistent with the finite equalspacing axiom by itself. On the other hand, the finite equal-spacing axiom is not satisfied by any but trivial DSs that are basic interval structures which are also bisectable and interval-bisymmetric. This shows what is no doubt intuitively obvious; namely, that though the Suppes-Winet and the Davidson-Suppes axiom systems are both equivalent so far as concerns their logical compatibility with finite quantities of generalized ordinal data, each, in a sense, contradicts the other. It follows, then, that if these two competing theories are to be compared in terms of their validity or consistency with data, this comparison will have to be made on grounds other then logical consistency with finite amounts of generalized ordinal data.

408

ADAMS,

FAGOT,

AND

ROBINSON

For both the Lute-Tukey and Suppes-Winet axiom systems we showed that in each case one nonnecessary axiom added empirical content to the system (i.e., was not technical): the solutions axiom in the Lute-Tukey system and the bisectability axiom in the Suppes-Winet system. These two axioms were the keys to the possibility of reducing, on the one hand, conjoint structures to additive structures, and, on the other, interval structures to bisection structures. With certain qualifications, the solutions axiom can be interpreted as saying that for any conjoint pairs, another pair can be found whose composite conjoint measure is the sum of the measures of the two original pairs; similarly, the bisectability axiom can be interpreted as saying that for any two members of the domain it is possible to find another member of the domain that “bisects” the interval between them. In a sense, then, these two axioms say that we can always perform particular kinds of operations, in one case an “empirical addition, ” in the other, a “bisection.” Thus, these axioms play the role of the closure axiom for systems designed for ordered algebras. We showed in Sec. II that in many cases the closure axiom was the one nonnecessary axiom, among the empirical axioms for certain kinds of numerical representations, that added substantive content to the system. The fact that conjoint and interval measurement can be axiomatized with their only nonnecessary axioms representing closures of operations-of an additive operation in the conjoint case and of a bisection operation in the interval case-suggests that it would be worthwhile to investigate axiom systems for these same basic representations that postulate that different kinds of operations might always be carried out.5 Studies of the empirical status of axioms could obviously be extended to systems other than those reported here. In this paper we have attempted to suggest a basic methodological approach and have used that approach for some of the standard theories of measurement. In a follow-up study, one of the authors will report on further studies, using the methodological approach introduced here, concerning subjective probability theories, cast in the form of ordered Boolean algebras, and certain theories of utility measurement based on probability mixtures.

REFERENCES ADAMS, E. W., FAGOT, R. F., AND ROBINSON, R. E. On the empirical status of axioms in theories of fundamental measurement. Technical Report No. 3, Measurement Theory and Mathematical Models Reports. University of Oregon, September 1.5, 1965. DAVIDSON, D., AND SUPPES, P. A finitistic axiomatization of subjective probability and utility. Econometrica, 1956, 24, 264-275. GOLDMAN, A. J. Resolution and separation and A. W. Tucker (Eds.), Linear Studies

No.

38,

1956.

Pp.

theorems

for

inequalities and

polyhedral convex sets. In H. W. Kuhn related systems. Annals of Mathematics

41-51.

5 For a treatment of conjoint measurement assuming an interval

measurement reproduction

assuming operation,

a bisection operation see Adams et al.,

and 1965,

of interval pp. 61-73.

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LLJCE, R. D. Two extensions of conjoint measurement. Jowlral of Mathematical Psychology, 1966, 3, 348-370. LUCE, R. D., AND GALANTER, E. Psychophysical scaling. In R. D. Lute, R. R. Bush, and E. Galanter (Eds.), Handbook of mathematical psychology. Vol. 1, Ch. 5. New York: Wiley, 1963. Lute, R. D., AND TUKEY, J. W. Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1964, 1, l-27. PFANZAGL, J. Die axiomatischen Grundlagen einer allgemeinen Theorie des Messens. Schriftenreihe des Statistischen Instituts der Universitiit Wien New Folge Nr 1, Physica-Verlag. Wtirzburg, 1959. (a) PFANZAGL, J. A general theory of measurement-applications to utility. Naval Research Logistics Quarterly, 1959, 6, 283-294. (b) PFANZAGI., J. Theory of Measurement. In cooperation with V. Baumann and H. Huber. New York: Wiley, 1968. SCOTT, D. Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1964, 1, 233-247. SCOTT, D., AND SC’PPES, P. Foundational aspects of theories of measurement. jortrnal of Symbolic Logic, 1958, 23, 113-128. SUPPES, P. A set of independent axioms for extensive quantities. Portugal& Mathematics, 195 I, IO, 163-172. SUPPES, P., AND WINET, M. An axiomatization of utility based on the notion of utility differences. Manugement Science, 1955, 1, 259-270. TITIEV, R. J. Some model-theoretic results in measurement theory. Technical Report No. 146, Psychology Series, Institute for Mathematical Studies in the Social Sciences, Stanford University, May 22, 1969. RECEIVED:

August 1, 1966