On the possible scientific laws

19

Mathematical Social Sciences 20 (1990) 19-36 North-Holland

ON THE POSSIBLE

SCIENTIFIC

LAWS

Suh-Ryung KIM * Division

of Mathematics

and Science,

St. John’s

University,

Siaten Island,

NY 10301, U.S.A.

Communicated by F.W. Roush Received 11 January 1990

In a foundational paper, Lute (1959) observes that the scale type of the dependent and independent variables in a possible psychophysical or other scientific law determines the general form of the law through the solution of a certain functional equation. Lute (1964) extends the earlier results to the situation where there may be more than one independent variable. In a subsequent paper, Osborne (1970) extends Lute’s early results to other scale types. Osborne (1970) discusses the possible psychophysical laws in the situation where the independent variables are independent ratio, interval, log-interval or ordinal scales and the dependent variable is a ratio, interval, loginterval or ordinal scale. In this paper we generalize Osborne’s results in the cases that involve ordinal scales and in the case where the independent variables are independent ratio scales and the dependent variable is a log-interval scale and present new, more detailed proofs of his theorems. In addition, for the cases where the independent variables are ratio scales and the dependent variable is a log-interval or ordinal scale, we correct small errors in Osborne’s results. Key words:

Scale type; functional equation.

1. Introduction In a foundational paper, Lute (1959) discusses the general forms of the possible psychophysical or other scientific laws and argues that knowledge of the scale types of the dependent and independent variables can lead to the determination of this general form through the solution of a certain functional equation. Here, we briefly trace the subsequent literature on this subject and then discuss some improvements on some results of Osborne (1970) which extend Lute’s early results. To give a brief history, suppose x1,x2, . . . ,x,,x,, + 1 are n + 1 variables, xi has domain Ri c R, and Ti denotes the set of admissible transformations T under the measurement theory for the ith variable. Thus, q is a function from Ri to R. Suppose that u is a function from fly=, Ri to R,+,, and x,,,, =u(x~, . . ..x.). The funcThis work forms part of the author’s doctoral dissertation in Mathematics at Rutgers University. The author expresses her deepest gratitude to her advisor, Dr. Fred S. Roberts for his tireless guidance and support from the beginning to the end of this work. Also, the author thanks Professor Janos Acztl and Dr. C.T. Ng for their helpful comments on earlier drafts of this paper. This work has been supported by NSF grant number IST-86-04530 to Rutgers University. l

0165-4896/90/%3.50 0 1990-Elsevier

Science Publishers B.V. (North-Holland)

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tion u is thought of as an unknown scientific (psychophysical) law. We say that xi defines a ratioscale if T, consists of all functions T : R; -t R, of the form 7;(xi)= y;Xi for some yi>O; Xi defines an interval scale if Ti consists of all functions 7;:: Ri + R of the form q(Xi) = yiXi+ 6i for some yi> 0 and 6i in R; X; is called a fogintehal scale if Ti consists of all ‘T : R; -+ R, SO that T(Xi) = Y~x,!~for some yi, Si> 0 in R; and x, defines an ordinal scale if Ti consists of all strictly increasing #i on Ri. Moreover, we say that xl, . . . . x, are independent ratio scales if there is no connection among the y, in IT;(Xi)= YiXifor different i;xl,....x, are independent interval scales if there is no connection among yi and 6i in 7;,(~,)= yiXi+ai for different i; xl, . . . , x, are independent log-interval scales if there is no connection among yi and 6i in ~(Xi)=yiXG' for different i; and xt, ....x. are independent ordinal scales if there is no connection among @i in q(Xi)=@i(Xi) for different i. Nowweassumeforeachsetof7;~Ti,i=l,...,n,thereisD(T,,...,T,)suchthat for all+E Ri, i= 1, . . ..n. u(T,x,,...,T,x,,)=D(T,,...,T,)[u(x,,...,x,,)]. This is written by Lute as the ‘principle of theory’. Lute (1962) points out that if there are no ‘dimensional parameters’ which enter the relation and cancel out the effects of the transformation, then there is a D(T,, . . . , T,) such that for all XiE Ri 9 i = 1, . . . , n, x,)]. Lute (1959) considers the special case u(T,x,, **., T,x,)=D(T,,...,T,)[u(x,,..., n = 1 where the independent variable and the dependent variable are a ratio, interval or log-interval scale and he assumes continuity of u. Lute (1964) considers the case of the n independent variables where (i) the independent ratio scales map into a ratio scale; (ii) the independent ratio scales map into an interval scale; and (iii) the independent ratio or interval scales with at least one interval scale map into a ratio scale. Osborne (1970) adds the cases of ordinal scales and log-interval scales to the results of Lute (1964). Aczel, Roberts and Rosenbaum (1986) retrieve Lute’s solutions, replacing the continuity conditions on u by quite weak regularity conditions, where ‘regular’ means bounded on an (arbitrarily small, open) n-dimensional interval I. They also permit partial independence or no independence at all of different transformations on ratio or interval scales. They specify both the general solution and the regular solutions. The results of Aczel, Roberts and Rosenbaum are applied by AczCl and Roberts (1989), Roberts (lQQO),and Aczel(lQ88) to study the solutions under different assumptions about u such as agreement, symmetry and linear homogeneity. The results are also applied by AczCl and Roberts (1989) to extend some of Osborne’s results, namely those where the independent variables are independent log-interval scales and the dependent variable is a ratio, interval, or loginterval scale, those where the independent variables are independent ratio or interval scales and the dependent variable is a log-interval scale and those where the independent variables are independent ordinal scales and the dependent variable is a ratio, interval or log-interval scale. The extensions include eliminating Osborne’s assumption that n> 1, dropping his continuity assumption, and dropping the assumption that when n> 1, the function U, which maps the independent variables to the dependent variable depends upon each of its arguments. (Following Lute, 1964, a function u of n variables is said to depend upon each of its arguments if,

S.-R. Kim / Scientific laws

21

for every choice of n- 1 variables, there are at least two values of the nth variable for which the function has different values.) In this paper we present new proofs of Osborne’s results in the cases that involve ordinal scales and in the ratio-log-interval case. The new proofs are much more detailed than Osborne’s very sketchy proofs. Moreover, the new proofs lead us to extend and correct some of Osborne’s results in small but useful ways. As do Aczel and Roberts (1989), we eliminate the Osborne assumption that n> 1 and also the assumption that the function u which maps the independent variables to dependent variables depends upon each of its arguments. However, Osborne’s restriction that u be continuous still remains. Together with the results in Lute (1959, 1964) and Aczel and Roberts (1989). the results for the case n= 1 complete the derivation of the general forms of the possible psychophysical laws in all situations where independent and dependent scales are combinations of ratio, interval, log-interval, and ordinal scales. To begin the analysis, we introduce several notational conventions. We shall use a bold-roman letter to stand for a vector. For example, x symbolizes the vector (x 1, . . . ,x,_ ,,x,,). Exponentiation of a vector over the base e is pointwise, as are addition and multiplication: eX = (eX’,. . ..e”$ x+y

=(x,+Y,,...,x”+Ynh

XY =(x,Y,,...,&Y”h

Ri denotes the projection

of ny=, R” onto the ith component

and

RI = {(x,, . . . . x,,)ER” 1Xi>0 for any i, 1 li
2. Preliminaries AczCl, Roberts and Rosenbaum (1986) provide the definitions of the terminology we need here and also the conventions about the domain and range of different types of scales. Before we verify the main theorems, we present the lemmas which we will often use. Lemma 1. If u : R2 -) R is continuous, (uPY~), so that u(-~Jz)=~YI,Y~).

then there are (x1,x2) and (~1, ~21, 61, ~2) f

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Proof.

This is a close analogue of Theorem 17, Set 2.4 in Buck (1978) and the same proof can be applied. Q.E.D.

Lemma 2. Suppose n 12,

(*)

u : R” + R is continuous,

and satisfies

For any x, y in R”, and any vector b, u(x)> u(y) if and only if u(x + b) > u(y + b).

Then there exist x,y in R”

that xi =yj for at most one j, 15 j I n, and

SO

Proof. Apply induction on n. The case where without using the condition (*). Now assume Fix x,” in R, and consider u(x ,, . . . ,x,, _ ,, x,f). (x: ,..., x,*-,.x,*) and (y: ,..., y,*_ ,,x,*) in R” lsjln-1 and

u
= U(X:v

=

u(y).

n=2 has been verified in Lemma 1 Lemma 2 holds for k, 25kln - 1. By inductive hypothesis, there exist so that x7 = y; for at most one j,

= u
Since if x,?‘#yj* as compared assume xi* = y,?. Let h(Xj,Xn)

U(X)

(1)

to xi* = y,?‘, then the lemma follows, and we may

. . ..Xi*_!.Xj,Xi*+I,...rXn*-I,Xn)

for fixed x: ,..., x,?r,xj*+r ,..., x:-r. Again, applying the inductive hypothesis, we can find x,T* and x,T** in Rj, and x,** and xz** in R, so that either x,?*#x,?** or x,** #x;** and

u = u. ,...)

(2)

Then since y,F=x,F, (l), (2), and (*) imply U(Y:,

. . ..Yi*-l.xj*,Yi*+l,...(Yn*-l,xn*)

= u(xl* ,..., x,?I,x,:x,f++ I,...,

x,*-*,x,*>

= U(X;I,...,Xj*_*,Xj***-Xj**+Xj*,...,Xn*__1,Xn***-Xn**+Xn*).

We know that either x~#x~**-xr*+xF or x:#xz**-x:*+x: ,,?** or xn**fxn*** and Lemma 2 follows. Q.E.D.

since either xF*#

From Lemma 2, the following corollary immediately

follows.

Corollary 1. Suppose nr2,

and satisfies

(**)

u : R:-+ R is continuous,

For any x,y in R: and any positive diagonal matrix A, u(x)>u(y) and onfy if u(Ax)>u(Ay).

Then there exist x,y in Ry so that Xj=yj for at most one j, 15 jr n, and

U(X)

if

= U(Y).

S.-R.

Proof.

Kim / Scientific

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Iaws

Introduce a function U: R” + R by U(X) = u(eX). Then for any a’s, the in-

equality, U(X) > U(Y)

e

u(eX) > u(e’)

0

u(e”‘exl , . . . . e”~exn)>u(e’le~, . . . . e”neY”)

0

U(X,+a,,...,

X,+a,)>U(Y]+a

,,..., Y,+a,).

Therefore U satisfies (*) in Lemma 2. Hence there exist X,Y in R” so that U(X)= U(Y) with Xj# rj for at most onej, 1 ~j
conditions: (i) u is any continuous function; (ii) for any x, y in R:, and linear transformation L such that Lx=ax where a > 0, u(x) > u(y) if and only if u(Lx) > u(Ly); and (iii) there are two distinct points (x,, . . ..x.) and (y,, . . ..y.,) in RT so that U(XI, *a*,x/l) = u(Y,* . . . ,Y,).

Then

u((;)r,(;):(;)r,...,(2)‘)=u((~)r,(~):($)r ,...,(zy) = u(1, . . . . 1) for any reR. Proof.

Case 1. r is a natural number. Then apply induction on r. First, consider the case where r = 1. Since u(xt, . . .,x,)= U(Y*, **.,Yn), U

by condition assumption,

x1

*2

*3

-,--9 YI Y2’Y3

=u(l,...,

1)

-“’

(ii). Suppose Lemma 3 holds for r, 25rrm

- 1. By the inductive

u((;>“-‘,(~+t,(;>“-‘,...,(;~-‘) = u(1, . . . . 1) =

u((;)“-‘, ($)“-I, (z-‘, ....(g-I).

By condition (ii) and the first equality of (3),

u((;y,(;>“,(~y ,...,(27) =u($z,; ,...,;)

=u(1, . . ..l).

(3)

S.-R.

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laws

Also, by condition (ii) and the second equality of (3),

= U(l,...,l). Case This Case Then

2. r=O. is trivial. 3. r is a negative integer. r= -m for a natural number m. Then

u((~):(~):(;y,...,(z)‘> =u(($)“@y~>”,...,(g) = u(1, . . . . 1)

and

u((~):(;):($)‘,..., (E)? =u((g,(g,(~y ,...,(z>“) = u(l,...,l). Case 4. r is a rational number. Then r =p/q, where p and q are integers and q # 0. We have just shown that

From this, we can deduce

u((yyyy;~ by condition

,...,(~@pu(l,...,

1)

(ii). By symmetry,

u((yyyy)“” Case 5. r is an irrational number.

(...,

(gy=u(l,..., 1).

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laws

There is a sequence of rational numbers, (r,,,}, which converges to r. Then

converges to

((~>:(~)‘,(~>‘,..., (E>?

as m-+Qo.

Hence,

= lim u(l,...,l) n-0 by the continuity of u and Case 4. Similarly, we can obtain u((;):(?):(?)’

,..., (f)?=~(l,...,

1).

Q.E.D.

3. Generalizations and alternative proofs for Osborne’s theorems In this section we state generalizations of several of Osborne’s theorems and provide proofs of all. The results are summarized in Table 1. We begin by generalizing the ‘Osborne cases’ in which the independent variables are independent interval or log-interval scales and the dependent variable is an ordinal scale. We use the same numbering as does Osborne, both for theorems and for the four ‘conditions’ he studies. Thus, we start with his Theorem 1 and Theorem 2. We allow n = 1 in all of the theorems and we do not assume that the function u depends on each of its arguments. Condition 1. For any x,y in R”, and any affine transformation Ax = ax + b, where a > 0, u(x) > u(y) if and only if u(Ax) > u(Ay).

A such that

Theorem 1. Suppose u : R” * R, nl 1, is any continuous nonconstant function. Then u satisfies Condition 1 if and only if u(x) = g(Xj) for some j E { 1, . . . , n} , where g is a continuous and strictly monotonic function. Proof. c=) Recall that an affine transformation is a linear transformation followed by a translation. Let u(x) =g(xi), where g : Rj -+ R is a strictly increasing (decreasing) function. Then, for any affine transformation A such that Ax=ax+ b, where a>O,

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Table 1 Scale type of independent variables

Scale type of dependent variable

General continuous solution to functional equation (n 2 1)

Comparison to Osborne (n> I)

Independent interval scales

Ordinal scales

u is constant or u(x) =g(xj), for some j, I ajl n, where g is continuous and strictly monotonic.

Nonea

Independent log-interval scales

Ordinal scales

u is constant or u(x)=g(xj), for some j, I sj< n, where g is continuous and strictly monotonic.

Independent ordinal scales

Ordinal scales

u is constant or u(x)=g(xj), for some j, I sjsn, where g is continuous and strictly monotonic.

Nonea

Independent ordinal scales

Ratio, interval or loginterval scales

u is constant.

Nonea

Independent ratio scales

Ordinal scales

u is constant or u(x) =g[P(x)], where g is a continuous and strictly increasing function and P(x) = nF=, xp’, pj # 0, for some j.

u(x)=g[P(x)], where g is a continuous and strictly increasing function and P(x)=Kfl;=‘=,

xp’, K>O,

pjt0.c Independent ratio scales

u is constant, u(x) = K#=, xpi, K>O, pj+O, for some j, or u(x)=Cexp(a#=, xp), C>O, a#O, pj#O* for some j.

Log-interval scales

u(x) = Kfl;= , xpi, K>O. pi+0 or

u(x)=Cexp(a#S, xp). C>O, a>O, pi#O.cd

a Osborne assumes that u depends on each of its arguments. b Osborne assumes that when Ti(xi) =yixp, yi. 6i>O, then at least one 6i is not I. c Osborne’s conclusion that all pi+0 requires the assumption that u depends on each of its arguments. d In the second solution, Osborne omits the possibility that a
u(x) > MY)

*

g(Xj)>g(Yj)

c)

xj>Yj(xj
w

ajXj+bj>ajyj+bj (ajXj+ bj
0

g(ajxj+bj)>g(ajYj+bj)

0

u(Ax)> u(Ay).

Hence u satisfies Condition

1.

S.-R.

Kim / Scientific

=) Now suppose u satisfies Condition

27

laws

1.

Claim. u is independent of xi for aN i, 1 sir i=j.

n, except for at most one value of i,

Proof of Claim. If n = 1, then it is obvious since u(x) = u(xI). If n L 2, then we know by Lemma 2 that there exist (x:, . . . , x,*> and CV:, . . . ,_Y,*>which belong to R” so that 24(x?,. . . ,x,*> = u<_Y:,. . . ,r,*) and x,F=y,? for at most one j,

(4) Then by Condition

1s jln.

1, for any x =

(qr . . ..X”). 24(x:--_Y:+x,, . . . . x,*-u,*+x,> Again, by Condition

= u(x,, . . . . x,).

1, for i#j,

u(x:--y;c+xI,..., Xi*_l-_Yi*_l +Xi-I9O,Xj?+I-_Y,*,, +xi+l,***,x,*-U,*+xn>

= u(x*9. . ..Xi_l.y,*-xi+,xi+1,

***sxn)

(3

and U(X[,***,xi-_l,O*xi+l9 ***Sxn) = u(xT-y:+x,

,...,Xi*-l-~i*-l +Xi-l,X:-vi*,Xi*,r-ri*,I

x,*--y,*+x,>. +xi+1, ***,

(6)

(6) leads to u(x,+y:-x:,

. . . . Xi_~+~i*-~-Xi*_~,O,Xj+~+yi*,~-xi*,~,...,x~+Y,*-x,*)

= 24(x, 9 ****Xi-_IrXi*-_Yi*,xi+Ir *ea,x~)e

We may assume $ -JJ: > 0 by symmetry.

(7)

NOW fix XI, . . . , Xi- l,Xi+ 1, . . . ,x, and let

k(z) = ucq ,... ,~i-I,z,xj+l,...,xn). Then, for any z>O,

= 24(x,

+y;c-x:, ....xj_~+~i*-,-xj*_,,o,xi+,+vi*,,-~i*,,~...~x,+u,*-x,+)

from (7) and Condition k(Z)=U

1.

Also, for any z
X],...,Xi-lr ~(y:-*:)~X,'I, ( 1-I

***9X" >

=u(x~-y:+X,,...,xi*_,-yi*_,+x~_~,o,xi*+*-Yi*+~+Xi+,,....xn*-Yn*+xn~

28

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from (5) and Condition k(z) =

Kim

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laws

1. Hence,

Cl 9

if z>O,

c29

if z
But c,=c,=k(O) by the continuity of k. Thus, U(XI,...,Xi_lrX,,Xi+l,...,X,) dependent of xi. This proves the claim. Hence, u(x, ,...,Xj_l,Xi,Xj+I,...,X,)

is in-

= Il(Xl,...,Xj_I,Xi+]r...,X,)

for some CJ: RR-’ +R. We now show by induction on n that u(x,, . . ..x.)=g(xi) for some j and some function g. When n = 1, this is trivial. Assume it is true for n _Yj and g(xj)Zg(Yj)v and Wj>Zj and g(Wj)sg(Zj). If g(Wj+Xj-Zj)Ig(Xj), then g(wj)Zg(Zj) by Condition 1; hence, g(wj) = g(Zj)s

which implies g(0) = g(Zj- Wj) = g(wj-Zj)Thus, g(O) = g(a(Zj- wj)) = g(b(wj-Zj))9 for all a and b>O, from which it follows that g is constant. assumption that u is nonconstant. Hence,

This contradicts

the

g(Wj+Xj-Zj)
Since Wj>Zj, Wj+Xj-Zj>Xj>Yj. Case (i). g(Wj+Xi-Zj)
Wj+Xj-Zj,

= g(b(Yjmuj))*

for any a and b> 0. Again, g is constant since Case (ii). g(Yj)Ig(Wj+Xj-Zj)
Uj#_Yj.

g(Uj) =S(Wj+Xj-Zj),

which implies that g is constant by repeating the argument in Case (i). The results Q.E.D. in both cases are contradictory to the hypothesis. Theorem

1 can be restated as follows:

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Corollary 1.1. If a continuous function

u maps independent interval scales into an ordinal scale, then either u is a constant function or u(x) =g(Xj) for some jE{l,..., n}, where g is a continuous and strictly monotonic function. Corollary 1.2. If a continuous function

u maps independent ordinal scales into an ordinal scale, then either u is a constant function or u(x) =g(Xj) for some n}, where g is a continuous and strictly monotonic function. jE{l,..., Proof.

Every affine transformation A such that Ax = ax + b,a > 0, is admissible for ordinal scales. Hence, Condition 1 applies and by Theorem 1, if u is nonconstant, then U(X)=g(xj) for some jE l,...,n, where g is a continuous and strictly monotonic function. Conversely, if u has this form, it can be easily checked that independent ordinal scales map into an ordinal scale. The proof is similar to the t) part of Theorem 1. Q.E.D. Corollary 1.3.’ If a continuous function u maps independent ordinal scales into a ratio, interval, or log-interval scale, then u is a constant function. Proof.

In each of these situations, Condition 1 holds. Hence, if u is nonconstant, u has the form u(x) =&Xi) as in Theorem 1. But now if #j is any strictly monotone increasing transformation of R, we have

where Fej is the transformation appropriate for a ratio, interval, or log-interval scale. For instance, if the dependent variable is an interval scale, we get the equation

g(@ji(xj)) = Y(@j’i)gtxj) + S(@ji)In particular,

this must hold for

@j(Xj>

= kXj+C,

k>O*

Thus,

g(kXj+C)= YCk* c)g(xj)+ dCks C)* This case tion pose

so

is the functional equation for the Lute (1959) interval scale to interval scale when n = 1. By Lute (1959), the continuous solutions to this functional equaare g(x) =a~+& if rz=O, then g is constant, and so the corollary holds. Sup(r #O. Then for every strictly monotone increasing ~j,

a@ji(xj>+8=Y(@j)(axj+8)+~(@j’i)s

’ Corollary 1.3 and its proof are due to Fred S. Roberts.

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But not every strictly monotone increasing @j has this form, which is a contradiction. A similar contradiction is obtained in the other two cases, using the results of Lute (1959) for the n = 1 interval to ratio maps and the n = 1 interval to log-interval maps. Q.E.D. Note: Aczel and Roberts (1989) prove this result without the hypothesis of continuity. Condition 2. For any x, y in RI and any transformation T such that TX= (alx? , . . . , a,x$) with positive a’s and b’s, U(X)> u(y) if and only if u(Tx) > u(7’y). Osborne’s hypothesis that at least one b # 1 in the transformation T in Condition 2 is not used in the following proof. Hence it is omitted from the theorem. Theorem 2. If u : RT+ R, nz 1, is any continuous nonconstant function, then u satisfies Condition 2 if and only if u(x) = g(Xj) for some j E { 1, . . . , n} , where g is a continuous and strictly monotonic function. Proof. This is directly deduced from Theorem 1 by defining a function U: R” -, R by U(X) = u(eX). Then for any a’s and positive b’s,

0

u(eX) > u(eY)

0

u(eUl(exl)bl , . . . . eUn(eXn)bn)>u(eUk(eK)bl, . . . . e”n(eh)bn)

++

U(b,X,+a,,...,

b,X,+a,)>U(blY~+al,...,b,Y,+a,).

Accordingly, U satisfies Condition 1. Hence U(X) = h(Xj) for some j, where h is continuous and strictly monotonic. If we let ex =x, then we have the equality, U(X) =g(Xj), where g(Xj) = h(log xj). Since log is continuous on R, and strictly increasing, g is continuous and strictly monotonic. Q.E.D. Corollary 2.1. If a continuous function u maps independent log-interval scales into an ordinal scale, then either u is a constant function or u(x) =g(Xj) for some jE{l,..., n), where g is a continuous and strictly monotonic function. Suppose n> 1 and every admissible transformation of n variables depends upon each of its arguments. Then Theorems 1 and 2 imply that we cannot find any continuous and nonconstant functions obeying Condition 1, or Condition 2, which are the results of Osborne (1970) (except that he needs’one additional hypothesis for Theorem 2).

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In his Theorems 3 and 4, Osborne does not assume that f depends on each of its arguments. Nevertheless, he concludes that each exponent pi+O. However, iffhas the same form as in his Theorem 3 or Theorem 4 with the exception that just one of pi#O, then f is a continuous nonconstant function and satisfies Condition 3 or Condition 4, respectively. Also, in his Theorem 3, Osborne omits the case where f(x) = C exp(anl,, xi”), C>O, a< 0, pj#O for some i, and it can be easily checked that thisfis a continuous nonconstant function and satisfies Condition 3. We now correct Osborne’s Theorems 3 and 4 and provide proofs for the corrected theorems. Condition 3. For all x,y,w in RP and any linear transformation Lx = ax, where a > 0,

logf(x) 1wfW

logf(W -

lWf(Y)

- lOa-

= logf(Lw)

logf(Ly)

- logf(Ly)

*

Theorem 3. If f: RT -+ R, n 2 1, is a continuous nonconstant function, fies Condition 3 if and only if (9

f(x) = $I,

(ii)

f(x)=Cexp

K> 0,

x?,

L such that

then f satis-

pj # 0, for some j,

Or *fix? (

i=l

,

C>O,

Cf#O,

pj*O,

for some j.

>

Proof. Under the assumption that f is not a constant function, Condition 3 is equivalent to the functional equation for the case where independent ratio scales map into a log-interval scale and AczCl and Roberts (1989) have proven the latter without the hypothesis of continuity. (We note that the former does not include the case where f is a constant function while the latter does.) Q.E.D. Corollary 3.1. If a continuous nonconstant function u maps independent ratio scales into a log-interval scale, then either u is Kny= 1 x,p’, K> 0, pi + 0, for some j, or u is exp(any=, xip+p), a#O, pj#O, for some j. Condition 4. For any x,y in R:, and any linear transformation Lx = ax, where a > 0, u(x) > u(y) if and only if u(Lx) > u(Ly).

L such that

The following theorem omits the constant K in Osborne’s theorem, subsuming it in the function g. Theorem 4. Let u : R:-+ R, nz 1, be any continuous function. Then u satisfies Condition 4 if and only if (i) there exists a function P(x)= ny=, xy, pj+O, for

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some j, and a continuous and strictly increasing function g so that u(x) = g[P(x)] (ii) u is constant. Proof.

or

-1 x,,) is either constant or one-to-one for fixed

Claim. h(z)=u(xI,...rXi_l,Z,Xi+Ir..., x,, . . . ,x, for any i.

Proof of Claim. If h is not one-to-one, h(z,) = h(z2). Then by Condition 4,

x,) = u(x1 ,...,xj-IrZ2,xi+*r...rX,)

u(x, ,...,xi-l,ZI,xi+I,..., =

u

(l,...,l,z’ z2’

then there are z1,z2,zt 7522 so that

I,...)

1 =u

1

>

‘...’

(

I,%,1,...,1 ZI

=u(l

,...I

1).

>

By Lemma 3, U(l)..., 1,(;Y,1,...,

1) +l,...,

l,@l,...)

1)

= u(l,...,l). Hence,

SO

h((;y)=h((;>‘> =h(l),

(8)

for any r in R. For any z in (0, oo), let r = log z. LI& Then h(z)= h(l) from (8). Hence h is constant. This proves the claim. A well-known result shows that if u is continuous and one-to-one in is strictly monotonic in xi. By redefining u as follows: r&t, ***,X”) = ucx;, . . ..x.>, where Xi’ =

{

xi,

if u is constant or strictly increasing in

l/Xi,

if u is strictly decreasing in Xi,

we may regard u as an increasing function in Xi for any i.

Xi,

Xi,

then u

S.-R. Kim / Scientific laws

33

If n = 1, then clearly u(x) is a strictly increasing function of xt or constant. For nz2, the theorem will follow if we can show that for any (x,, . . ..x.,) in R:, there are ri, rj # 0 for some i, SO that u(x,, . . . . X”) = u(l, . . . . l,x,“-*x$,l,

. . . . l),

(9)

where the term x{’ --.x2 appears in some component. To prove this, use induction on n. (i) We prove (9) for n = 2. By Lemma 1, there are (x:,x:) and (Y,*,Y:) so that u(x:,x~*)=MY:,Y~*) and (x:,xZ*)f(~:,~:). Case (a). x:=y:,x2*fy:. Then r&:,x2*) = u(Y:,Y2*). By Lemma 3,

xz*r

(( 1)

u 1,

7

=

U(l,l).

Y2

For any r, and x, and x2 in (0, oo), let

Then u(l,x2)=u(l, 1) follows, Hence, u(x,,x~)=u(x,, I). Case (b). x:fy:,xz*=yz*. By symmetry, u(x,,x2) is a strictly increasing function in x2 or constant. Case (c). x:#y:,x2*fy2+. Then by Lemma 3, U(($>:

($>?

= U(l,l),

for all r in R. Hence,

u(l,@) =u(($l) by Condition 4. For any (x,,x2), let r= logx;,ri x2. Then

Hence,

34

S.-R. Kim / Scientific laws

where

Hence, u(xI,x2) = u(x,xi, 1). (ii) Now assume (9) is true for n - 1. We prove it for n. By Lemma 1, we can find cx:. . . . . x,*) and (y:, . . . . y,*) so that xT+yt for i#tl and ucx:, . . . ,x,*) = u(y;c, . ..) y,*>. By Lemma 3,

u((~):(~):($)r ,...,($1)=U(l)...,

l),

for any r in R. For any (x ,, . . . ,x,,) in R",, choose i # I and let r = loo,,,; long as xk*/y:# 1,

= x!‘(‘Og,i/,; ,

(10) Xi. Then as

(x:'Y~))

Now (10) implies that U($‘, . . . ,X~i-‘,Xi,X~+‘,

. . ..X~)

=

U(l, *a*)1),

(11)

where

for k#l and if xr*=y:,

From (1 l), it follows that u(x, 9 **.,Xi-_l,Xi*xi+lr

..*sXn)

= u(x,x,FS’ , . . ..x. r-1 xFS’ 1 ‘-‘*l,Xi+lX,‘St+‘r

by

Condition

4.

Now

U(Y ,,...,Yn-*)=u(XI,...,Xi-Itl,Xi+l we find that

applying

the

e*m,XpjX~““)

inductive hypothesis to the function which clearly satisfies Condition 4,

, . . ..x.),

S.-R. Kim / Scientific laws

3s

WI, . . ..x.) --I,

= u(x,x,-‘, . . . . x,-lx; = U(x@

l,x;+lx,-S~*‘, . . . . X,X,7’“)

,..., X;_,X,~SI-‘,Xi+,X,T3f’*‘, . ..) x,x;‘“)

= U(l, . . . . l,(XIX,~~~)‘~...(X;_,X,~~~-~)‘~-~(X;+,X;~~~~)~~+~...(X,X;~”)~: 1, . . . . 1) rj+O for somej#i, = u(l, . ..( l,(x,x,~~~)‘~...(~i_~x,~~~-~)~~-~(xi+,x,~~~~~)’~+~...(x”x,~~~)’~, 1, . . . . l), where u has one more 1 in its ith argument than U. Hence we can complete the proof by taking r, = tk for any k # i and ri = -si ti sztz- . . . -Si_ltj_l-Si+lti+l-...-S~t~. I) Let u(x) =g(P(x)) for some continuous and strictly increasing function g and some function P(x) = ny=, x,?, pj+O for some i. Then

u(x) > U(Y) *

M(x))

> g(P(y))

*

P(x) > P(y) since g is strictly increasing

*

P(f.x)>P(Ly)

e

g(P(Lx))>g(P(Ly))

c)

u(Lx) > u(Ly).

by the structure of P since g is strictly increasing Q.E.D.

Corollary 4.1. If a continuous function u maps independent ratio scales into an ordinal scale, then either u is a constant function or u(x) =g[P(x)], where g is a continuous and strictly increasing function and P(x) = fly=, xip’, Pj f 0, for some j.

4. Further questions We close by noting some open questions. (1) AczCl, Roberts and Rosenbaum (1986), AczCl and Roberts (1989), Roberts (1990), and AczCl (1988) have proven many results analogous to ours without the assumption of continuity of u. However, continuity seems essential in the proof of Theorem 1. In Theorem 1, the continuity of u is used to prove the existence of two distinct points which have the same images under u. Is Theorem 1 true under a weaker regularity condition than the continuity of u? The same question applies to the other theorems. (2) We consider only the cases of total independence of different transformations on ratio or interval scales. It would be desirable to prove similar results on the possible mappings under partial independence of different transformations on ratio or interval scales.

36

S.-R. Kim / Scientific laws

References J. Aczel, Lectures on Functional Equations and their Applications (Academic Press, New York/London, 1966). J. Aczil, Determining merged relative scores, Centre for Information Theory and Quantitative Economics, University of Waterloo, Waterloo, Ontario, 1988, mimeo. J. Aczel and F.S. Roberts, On the possible merging functions, Math. Sot. Sci. 17 (1989) 205-243. J. Aczel, F.S. Roberts and 2. Rosenbaum, On scientific laws without dimensional constants, J. Math. Anal. Appl. 119 (1986) 389-416. R.C. Buck, Advanced Calculus (McGraw-Hill, Kogakusha, 1978). R.D. Lute, On the possible psychophysical laws, Psychol. Rev. 66 (1959) 81-95. R.D. Lute. Comments on Rozeboom’s criticisms of ‘On the possible psychophysical laws’, Psychol. Rev. 69 (1962) 548-551. R.D. Lute. A generalization of a theorem of dimensional analysis, J. Math. Psychol. 1 (1964) 278-284. D.K. Osborne, Further extensions of a theorem of dimensional analysis, J. Math. Psychol. 7 (1970) 236-242. F.S. Roberts, Measurement Theory, with Applications to Decisionmaking Utility, and the Social Sciences (Addison-Wesley, Reading, MA, 1979). F.S. Roberts, Merging relative scores, J. Math. Anal. Appl., in press, 1990. F.S. Roberts and Z. Rosenbaum, Scale type, meaningfulness, and the possible psychophysical laws, Math. Sot. Sci. 12 (1986) 77-95.