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Francis Montgomery Sim

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An Explication of the Logical
Model of Role Systems

presented by

Francis Montgomery Sim

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ABSTRACT

AN EXPLICATION OF THE LOGICAL MODEL OF ROLE SYSTEMS

by Francis Montgomery Sim

Concepts of status, position, role, etc., are
fundamental in social sciences. Increasing emphasis
has been given to defining role concepts in terms of
relationships in a system, such relationships being of
several sorts, e.g., norms, expectations, patterns of
behavior, etc. Any position may have relationships
with several other positions; a role system contains
a set of positions connected in this way.

This work explicates the formal moments of the
concept of a role system. It is asserted that positions
and roles must be specified interdependently; roles are
taken to be sets of relationships between positions, and
each position is specified by its relationships with
other positions. Neither position nor role is substan-
tively prior. The logical structure of this conception
is coordinated to the mathematical structures studied
in the theory of multilinear directed graphs or "nets."
In showing the coordination, it is convenient to take

the concepts of position and relation as primitive;

Francis Montgomery Sim
then an "axiom of relation" is stated, and role and
other related concepts (especially "role sector," and
qualifiers "focal" and "counter") are defined in terms
of the primitives and axiom. A binary matrix repre—
sentation isomorphic to a net is also introduced, since
it is more convenient for some purposes.

In concrete role systems, positions are occupied
by actors, and this requires representation in the
logical structure. The term "actor" is taken as an
additional primitive (though it is intended to include
any social object), and the necessary representation is
entailed by an "axiom of incumbency." This axiom states
that any actor in a position must have the relations of
that position with some actor(s) in each of the other
related positions. It allows any actor to occupy one or
more positions in the system. Further, it guarantees
some "mapping" of actors, and of relationships among
them, into positions and roles. Such mappings depend
on the number and kind of additional restrictions used
to define any particular role system, and some possible
restrictions and substantive interpretations of them
are explored. In general, there are many possible
mappings for any set of actors and relations.

There are several salient features of this
reconstruction of the concepts. It emphasizes the

selection of a set of relations, in terms of which

Francis Montgomery Sim
positions and roles —— and thereby the role system ——
are to be defined, as a crucial analytic requirement
for the investigator. Also, while the model allows
public identification of positions, etc., it does not
require it, and it specifically eliminates identities
as defining characteristics of positions; a role system
may be either latent or manifest (or mixed). Finally,
the incumbency axiom appears to be a unique statement
of a necessary concept. Suggestions are made for con—
tinuations of analysis of the formal characteristics

of role systems.

AN EXPLICATION OF THE LOGICAL MODEL OF ROLE SYSTEMS

BY

Francis Montgomery Sim

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Sociology

1966

ACKNOWLEDGMENTS

This study was done under the direction of
Professors Santo F. Camilleri and William H. Form,
and to each of them I express my appreciation. In
the early work, Professor Form was most tolerant of
my vague explorations. His steady guidance toward
a substantive focus provided a base for whatever
novelty the reader will find. The analytic develop—
ment and present form of the study are due to the
generous and cogent direction of Professor Camilleri.
Of course, I do not wish to implicate him in its
faults —- it is my toothbrush. But I am very glad
of the opportunity to make public acknowledgment of
his wisdom, patience and friendship.

Professors Jay W. Artis and John Gullahorn
were also members of the committee. In the nature of
such arrangements, theirs were less active roles, but
they were performed without stint and sometimes with
little notice. I thank them for their continuing
support and interest.

I have reserved a place of special mention for

Professor Charles F. Wrigley. It is too little to say

ii

‘V

that I should not have completed this work without his
encouragement —- without it I should never have begun.
It is a privilege to be able to dedicate this study to
him. I do so with respect and affection.

This investigation was supported by a Public
Health Service fellowship (number 2-F3—MH—25,Sll—02)

from the National Institute of Mental Health.

iii

CONTENTS

Chapter Page
I. EXPLICATION AND MODELS - . . . . . . 1
Explication . . . . . . . . . 1

MOdelS o o o o o o o o o o o 9
"Explicational Models" . - . . . . 15
Purpose and Overview . . . . . . 28

II. THE ROLE CONCEPTS . . . . . . . . 30

State of Agreement on the Concepts . 31
Selective Emphases and Organiza—

tion of the Review - . . . - . 34

The Elements of the System . . . - 40

The Variety of Concepts . . . . 4O
Benchmarks: Behavior, Structure,

and the Emphasis on Consensus . . 42

Norms . . . .- . . . . . . S4
Expectations . - - - - - - - 56
Minority Views . - . . . . . 62
Summary of Themes . . - - - - 65

Attributive and Relational Con-
cepts of Role . . - - - . - 68
Positions and Actors . . . . . 73
The Role System . . . . . 76
The Graphic Model of Role Systems . 83

III. PRIMITIVES, DEFINITIONS, AND AXIOMS . . 89

Primitive and Defined Terms and
the Axiom of Relation . . . . . . 89
The Incumbency Axiom . . . . . . 96

IV. THE FORMAL MODEL . . . . . . . . .114

Linear Graphs and Matrices . . . .114
Coordination of the Systems . . . .120
The Incumbency of Actors . . . . .126
Comparison With Other Formali—

zations . . . . . . . . . .131

iv

Chapter

V. MAPPING ACTORS INTO ROLE SYSTEMS.

Mapping Procedures
Assumption of Exhaustiveness
Agreement

Some Substantive Coordinations

VI.

Summary of the Model and Some

Evaluation of the Explication
Suggestions for Further Research

BIBLIOGRAPHY

Canonical Form.

Maximality

Logical Restrictions.
Symmetry.

Quantification
Connectedness
Iteration on Gp and
Bookkeeping Matrices
Equivalence

em.)

Problems of Equivalence.
Nesting .
Types of Structures .

ROLE CONCEPTS REVISITED

Interpretations .

Page
.140

.141
.141
.143
.144
.145
.147
.147
.151
.152
.153
.154
.155
.157
.161
.164
.170

.179

.180
.193
.198

.205

CHAPTER I
EXPLICATION AND MODELS

The work reported in this thesis undertakes an
analysis and reconstruction of several related concepts ——
particularly status or position, and role —— widely employed
as elements of social systems, and it purports to show how
we may conceive of the logical form of a role system. The
discussion in this first chapter is presented as a sketch of
some facets of the context in which the writer believes the
work fits and should be seen. We will make a statement on
the general procedure employed before developing the problem
treated with it.

This introduction is directed primarily to eXplaining
our meaning in regard to some terms which appear in the title,
especially "explication" and "model." Received definitions
require emendation in order to be applied here (and, we
believe, elsewhere as well). Our brief critique of them is

a convenient vehicle for the contextual sketch.

Explication

 

In contemporary usage, the term and concept "eXplica—

tion" was given its initial sense in the statements of the

2
philOSOpher Rudolf Carnap,l though it appears to have
acquired further import in use. It refers to character—
istic problems and forms of concept formation and
definition, and generally indicates efforts toward
redefinition which attempt to clarify and perhaps
extend the meanings of concepts. Carnap proposed that
the term be used in situations ". . . where a concept
already in use is to be made more exact or, rather, is
to be replaced by a more exact new concept."2 In this
case, ". . . we call the old concept, used in a more or
less vague way either in every—day language or in an
earlier stage of scientific language, the explicandum;
the new, more exact concept which is proposed to take
the place of the old one the explicatum."3 As these
passages suggest, he lays particular emphasis on the
vagueness of the explicandum and the precision required

of the explicatum.

 

1First, in his "The Two Concepts of Probability,"
in H. Feigl and M. Brodbeck, eds., Readings in the
Philosophy of Science, New York: Appleton-Century—
Crofts, 1953, pp. 438-455; and more fully in Logical
Foundations of Probability, Chicago: The University of
Chicago Press, 1950, Chapter 1.

2Carnap, "The Two Concepts of Probability,"
. . . , p. 455.

3Ibid. The philosophic reasons for this exact
choice of terms are given in Logical Foundations of
Probability, p. 3, but they are not required here.

 

3

Hempel, in his work on concept formation, places
explication in the context of more usual types of defi—
nition. He notes that it is an especially important
kind of "real" (as opposed to "nominal") definition.
He suggests that explication goes beyond the "Meaning
analysis, or analytic definition . . ." of real defi—
nition which aims at ". . . characterizations of
approximately uniform patterns of usage."5 An expli—
cation ". . . combines essential aspects of meaning
analysis and empirical analysis."6 He introduces the
latter element not to indicate that an explication
requires immediate examination of relevant evidence,
but to denote the fact that it is concerned with the
intended range of reference of the concept as well as
with internal consistency of usage. Insofar as this
extended purpose is involved in the analysis, Hempel
suggests that explication transcends definition as
such and becomes concerned with "scientific explanation";
the focus becomes concept formation more than termino—

logical specification.

 

4C. G. Hempel, "Fundamentals of Concept Formation
in Empirical Science," International Encyclopedia of
Unified Science, Vol. II, no. 7, Chicago: The University
of Chicago Press, 1952, especially pp. 2—14. A nominal
definition introduces a (new) term to stand as a shorthand
for an established usage. Hempel notes that explicational
procedure is sometimes ". . . called logical analysis or
rational reconstruction." (p. 11). '

SIbid., p. 10.
61bid., p. 12

4

Carnap notes ". . . one of the puzzling pecu—
liarities of explication,"7 viz., that the explicandum
is vague, thus the ". . . problem itself is not stated
in exact terms; and yet we are asked to give an exact
solution."8 He emphasizes that it is not possible to
decide whether a solution is right or wrong, though it
may be possible to say whether it is satisfactory, or
more or less satisfactory than some alternative.
Similarly, Hempel states that,

An explication sentence does not simply
exhibit the commonly accepted meaning of
the expression under study but rather pro—
poses a specified new and precise meaning
for it.

Explications, having the nature of pro—

posals, cannot be qualified as being either
true or false.

In these circumstances, then, one does not have
recourse to usual logical or material tests of validity;
so that —— in lieu of other cannons —— personal taste,
arbitrary convention, or more likely "whatever works"
would have to be taken as the guide. Both authors stress

criteria for an explication, apparently in order to avoid

a purely pragmatic test of adequacy. Carnap suggests

 

7Carnap, Logical Foundations :1- . , p. 4.
8Ibid.

9Hempel, op. cit., p. 11.

5
that an adequate explicatum should be characterized by
". . . (l) similarity to the explicandum, (2) exact—
ness, (3) fruitfulness, (4) simplicity."lo The last
item is distinctly subsidiary to the others, and amounts
to usual notions of parsimony. The first item is con—
siderably qualified by the assumed vagueness and more
importantly by conflict with the third.ll The second
and third criteria refer respectively to syntactic pre—
cision and to empirical and/or conceptual relevance.
Hempel redevelops them (and the first as well) in

observing that explications,

. . . are by no means a matter of arbitrary
convention for they have to satisfy two

major requirements: First, the explicative
reinterpretation of a term, or —— as is often
the case -— of a set of related terms, must

permit us to reformulate, in sentences of a
syntactically precise form, at least a large
part of what is customarily expressed by
means of the terms under consideration.
Second, it should be possible to develop, in
terms of the reconstructed concepts, a com—
prehensive, rigorous, and sound theoretical
system.12

 

OCarnap, Logical Foundations . . . , p. 5.

lCarnap does not state the matter in just this
way, but his discussion clearly implies this priority;
ibid., p. 5f. In preparation for his discussion on
these requirements he remarks on treatment of the expli-
candum, and we return to this point below.

2Hempel, loc. cit.

 

6

However, these instructions to produce well—
formed and effective constructs do not seem —— at least
to the present writer —— to apply with unique force to
explication as compared to other modes of concept forma—
tion. The flavor is more hortative that directive; the
objectives of the procedure (e.g., precision, clarity,
specificity, extension) are restated, but there are
not additional guides for deciding when they have been
reached, or, more important, when they are being
approached. We stress that the authors have stated a
normative ideal, not a method.

There are several points about this analysis
which need consideration here. It is our opinion that
the emphasis on the vagueness of the explicandum
(especially by Carnap) is overdone. Sometimes the
apparent lack of precision is due to unnoticed instances
of what is technically better called ambiguity ——
Carnap's own analysis of probability is an example. More
than this, there does not seem to be any a_priori reason
for believing that all aspects of any given concepts
would be equally problematic in respect to vagueness,
or even that any would be, or be recognized as such.

Further, it is certain that Carnap does not
believe that the explication proper (i.e., the solution)

begins with the explicandum in its "natural" obscurity.

7

There is a temptation to think that, since the

explicandum cannot be given in exact terms

anyway, it does not matter much how we formulate

the problem. But this would be quite wrong.

On the contrary. . . we must, in order to pre—

vent the discussion of the problem from becoming

entirely futile, do all we can to make at least 13

practically clear what is meant as the explicandum.
This means that reasonably close attention must be given
to the sense and use of the concepts to be explicated.
For example, in our own analysis, it would be improper to
begin by asking, in effect, "What is a role?" and straight-
away to begin looking for a solution, ". . . without first
examining the tacit assumption that the terms of the
question are at least practically clear enough to serve
as a basis for an investigation."14 The fact that we do
not take these preliminaries to be part of an explication
as such also reflects the freedom permitted in choosing
a solution, which both writers stressed in different ways.

In the emphasis on vagueness there seem to be

assumptions that what is vague is also wrong, and that
what is right is also precise. Moreover, there appears
to be a tendency to convert the latter proposition;

i.e., what is precise is also right, or at least more

right than what is not precise. To be sure, exactness

 

l3Carnap, Logical Foundations . . . , p. 4.

l4Ibid.

8
is a condition for assessing the correctness of an
explicative prOposal, but it certainly does not assure
that the proposal is correct. Our own analysis will
yield examples of proposals which are precise, and
precisely wrong. The point we wish to emphasize is that,
of the suggested criteria, "fruitfulness" is both the
salient purpose and test of an explication; the other
"internal" criteria are important in that they contri—
bute to this and to our ability to judge it.

There is one final point of reinterpretation
needed for our work. Although Hempel notes that often
several terms are involved in an explication, both he
and Carnap treat the subject as though the concepts being
analyzed are all of a piece. The explication is to
provide a whole new meaning. Our own view is that it is
possible to undertake explication of certain facets of
a set of concepts without attempting to adjudicate com—
peting formulations of other aspects. In this event we
should not expect improved precision beyond those facets
treated, but we might anticipate that the analysis would
shed some light on other aspects of the concepts. Also,
a partial explication cannot be said to result in replace—
ment of the explicandum in the way suggested above, for it
explicitly recognizes that some of the pre—existing

features have not been qualified directly.

9

To summarize: By explication we understand the
reconstruction and extension of some aspect(s) of a set
of related concepts which are already in established,
though perhaps disparate, usage. The results are in
some respects novel relative to that usage; they bring
to attention what had gone unnoticed and propose appro—
priate incorporations. Thus, the first step is to
assay the state of the concepts, but the proposal is
not bound by them —— it may (perhaps must) transcend
that state, consistent with the objective of improved
understanding. It is this objective which places stress
on precision; exactness is not the test of adequacy,
but it is an important condition for allowing comparisons
with alternatives and empirical referents.

One final note on this subject here. It is the
author's opinion that the appeal of the idea of expli—
cation (under whatever name) is that it breaks out of
the narrow and unreal confines of the traditional
philoSOphic treatments of definition, and comes closer
to formulating the conditions of disciplined Eganalysis
which is characteristic of cumulative concept formation
in science.

Models

The concept of a model has excited considerable

attention in recent scientific and philosophic discussions,

10
and it has been the object of explication as an essential
of the scientific tool kit. Here we attempt to summarize
only some important points and to say what sense is rele-
vant to our analysis.

Particularly on the philosophic side, emphasis
has been placed on the relationship between models and
theories, especially in terms of formal or syntactic
aspects. The key concept in diagnosing this relationship
is isomorphy. The usual development is to say that two
sets of prOpositions -— each of which may have some
conceptual intentions, but not necessarily the same
ones —- are isomorphic if they have the same logical
form, i.e., there is one—to-one correspondence between
their elements and the same kind of relations are
observed among them.15 The relations are usually taken
to be of the form of laws. An alternative way of
stating the relationship is in terms of the underlying
logical calculus or formal deductive apparatus of the
systems, and while this leads to some difference of

emphases the general result is equivalent.16 A model is

 

15This follows Brodbeck's statement most closely;
see M. Brodbeck, "Models, Meaning, and Theories," in
L. Gross, ed., Symposium on Sociological Theory, New York:
Harper and Row, 1959, pp. 373-403.

16E.g., R. B. Braithwaite, Scientific Explanation,
New York: Harper and Brothers, 1960, especially Chapter IV;
or E. Nagel, The Structure of Science, New York: Harcourt,
Brace & World, 1961, Chapter 5 (and also 6).

11
isomorphic to a theory for which it is a model, but they
are not the same and the difference is denoted by saying
that one or the other or both (depending on the author)
is "interpreted"; while the elements of each are formally
coordinate, they have different referents. This inter-
pretation provides a vehicle by which the model becomes
a convenient reasoning device for generating results
relevant to the theory. The manner of emphasis on a
formal deductive system is not uniform among all such
discussions (the emphasis on a strict logical calculus
reaches its nadir in Braithwaite's analysis), but it is
a central theme in the philosophic treatment and the
effect is in part pernicious. It results in a concep—
tual tendency to narrow the range of reference of
"theory" —- and therefore of "model" —— to efficacious
instances. Strictly speaking, deduction is 22: a
defining requirement of theories, though it is generally
supposed to be a requisite of effective ones. Any con—
junction of propositions with the same range of reference
constitutes a theory, even if not a powerful one. At
least, it is only in this sense that we can understand
how scientists use the concept of a model (and still
save the form of the philosophic analysis) when the
object is not a well-formed theory (i.e., an effectively
deductive one). And scientists do this, and do it effec-

tively, as we shall see in the next section.

12

There is another idea of some importance in our
concept of a model which usually receives little, if
any, notice in these discussions, but it is common both
to the model—theory circumstance and to other instances
of modelling as well. We refer to the fact that a
model is always in some sense "less than" what it is
a model of. In deductive systems, we refer to the
interpretation of an axiom set as a model because it
is not as rich as the interpretation. In other cir—
cumstances, a toy which is smaller and less detailed
than its full-scale counterpart is called a model.
Other examples requiring more subtle justification of
the point could be given, but it is essential to our
sense of the term that a model does not account for
all aspects of its object; if it did, it would not be
a model. All of this is quite clear in the discussion
based on deductive systems, but the point is frequently
lost in the tacit assumption that the acquisition of
such systems is, anyway, the really important step in
the development of powerful theory. It seems to us that
the relative poverty of a model needs reemphasis when
the characteristics of the object in view do not consist
in a well—formed deductive apparatus. A model con-
stitutes an attempt to develop by judicious selection

a device for reasoning about the object, but it is not

13
necessarily a complete calculating system.17 We alluded
to this point above in remarking that an explication
may develop only a part of the conceptual equipment
being analyzed.

In view of this discussion, it may seem either
redundant or anomalous that we qualify our analysis as
a "logical model." While we actually develop an axio—
matic treatment of the role concepts, this is not taken
to be a calculus, or a deductive system (or a set of
laws) of importance as such, since the object is not a
theory in the narrow sense above.l8 Rather, this is an
instrument for defining a set of concepts in terms of
certain of their aspects, viz., the logical form of
propositions about them. The term "logical" appears in

the title to denote the character of the contents of the.

 

l7Brodbeck, op. cit., p. 381f., castigates social
scientists for use of the term "model" to refer to
theories which do not conceptualize all aspects of the
phenomena considered. "All theories . . . omit some
variables simply because they are not relevant to the
phenomena to be explained . . . . Selfconsciousness
. . . about such perfectly legitimate omissions seems to
be peculiar to social science." Of course, the key term
here is "relevant," and we feel that Brodbeck misses the
fact that a theory may not be thought to organize all
relevant aspects, i.e., there may be external grounds for
believing that some omitted variables would be included
in a proper theory of the empirical matters in view.

18That is, a theory as a deductively interrelated
set of propositions; this seems to be the common sense of
the unqualified term, but we do not adhere to it in the
sequel.

l4
inquiry, not the form of the results, even though it can
be correctly applied to them. This facet of the role
concepts seems to have gone unnoticed, or at least has
not been properly apprehended before, and thus requires
explication.

Some last remarks on the concept of a model at
this point concern the question of interpretation men—
tioned above. In his discussion, Nagel —— who uses
"model" to signify an interpreted calculus (much in the
way of mathematicians) —— states that there is still a
third component of a theory (besides the calculus and
the model), viz., the "Rules of Correspondence." These
are ". . . a set of rules that in effect assign an
empirical content to the abstract calculus by relating
it to the concrete materials of observation and experi—

ment . . ."19

The significance for present interests
is that conceptual interpretation relevant to a model
does not require "operationalizing the concepts,"
though it does not preclude it. Further, our analysis
does not yield gp_interpretation of the model; it is

developed from several somewhat different interpreta—

tions of the same concepts.

 

l9
Nagel, op. cit., p. 90; and see also pp. 97—105.

15

"Explicational Models"

 

The heading above is entered in quotations because
the phrase is taken directly from the analyses of Berger,
Cohen, Snell, and Zelditch.2O The fact that the expres-
sion combines two of our key orienting concepts would
be warrant for at least considering the relevance of
their conception here. But more than this, their work
is of interest to us both as evidence regarding some of
the directions of the preceding discussion and as a
trenchant source of further considerations on the problems
of scientific concept formation.

In their work this group has emphasized that ——
at least at the present stage of analysis -— the most
useful classification of formal models is by the functions
they are intended to serve with respect to the theories
they are intended to assist.21 They assert that these
goals differ according to the stages of research and

theory development, that characteristics and functions

also vary, and that models are helpful at more than one

 

20Types of Formalization in Small—Group Research,
Boston: Houghton Mifflin, 1962.

21"By 'formalization' we refer to the general
process of making explicit the logical structure of a
set of assertions." Ibid., p. 3, N. 1. In this sense,
our own analysis would—BE'called a formal model.

16
stage. We would add (and they clearly imply) that
models help in getting from one stage to the next;2
models are not passive residues of codification of
earlier thought, they are important agents in cumulative
reconstruction.

In addition to explicational models they also
distinguish two other types which they call "represen—
tational" and "theoretical—construct" models. We will
review all three of them in a brief way first and then
return to a summary consideration of their example of
the explicational case.

An explicational model deals with some key:
concept(s) though not with all of the propositions of a
theory, and its work is to clarify the status and meaning
of the concept in the way we have outlined above. They
suggest that to be fruitful such models must be concerned
with really central concepts, and that the formalization
must be checked closely for its coordination with the
performance of the concept in the theory. Expectably,
explications may yield an additional reward in new con-
cepts, as we shall see in their example.

A representational model is appropriate to
circumstances in which a known social phenomenon (appar—
ently usually a process) is "fitted" by a formalization.
(The example which they analyze is Cohen's formalization

of the Asch effect in conformity experiments.) They

l7
stress as requirements both simplicity of the represen—
tation and that it should give a gggg fit. In fact, a
representational model for a particular process may
produce a better approximation to the empirical material
than a general explanatory theory which intends to
account for a variety of observed processes. The other
side of this accuracy is that this intermediate stage is
in general subject to the criticism of "curve—fitting."
However, it is sometimes the best that can be done, and
the authors seem to imply that it marks a step up in
prediction from the reconstruction of the first stage.

A theoretical—construct model is the full step
up to formalization of a ". . . general explanatory
theory . . ."22 (i.e., one which is well—formed and
efficacious). This puts most constraint on the model
builder, since simplicity and adequacy of empirical
relevance are still required, but one incurs the
additional necessity of embodying a set of propositions.

There are some observations to be made about
this typology of models and about the model of the
research process which we believe to be implied by it.
These can be brought out by considering the terms they
choose to designate the three types. We noted above

that their typology is based on "analyzing" the goals

 

221bid., p. 67.

18

(functions) of any particular formalization, and this is
the source of the type names.23 But it appears to the
writer that there are other constant features of each
situation, and that these could have been used to
designate the types.

This can be seen by the conventions that we
1) explicate concepts, 2) represent phenomena, and 3)
construct theories. Their types could be called 1) con—
cept, 2) phenomena, and 3) theory models respectively,
and better denote the central moment of each. This
amounts to shifting attention from the function of the
formalization to the structure of each type of situation,
and the writer's reading of their work strongly suggests
that it is in fact the character of the situation which
determines the salient kind of formal assistance
required, and not the other way around.

But, we would argue in addition that any par—
ticular kind of assistance (explication, representation,
construction24) is never more than salient; particularly,

in terms of our present interests, it is believed that

 

23"Analyzing" is set off here because in no case
do they attempt to justify their imputation of goals in
a direct way; but in each case the goals are, by any
reasonable standard, self-evident.

24This typology suggests a further line of
analysis, but it would certainly take us even farther
afield into philosophic and methodological questions than
seems necessary to the writer.

l9
explication can be used properly to denote certain
aspects of the process of formalization in all three of
their types. That is, freeing their terms from par—
ticular kinds of situations suggests a useful extension
of the earlier summary of our understanding of explica—
tion. There we said that explication is performed upon
"a set of related concepts," and here it is necessary
to notice only that all representations of a phenomenal
process are made in terms of such a set, and that
"related concepts" similarly are fundamental constituents
of any well—formed theory. In these terms, explication
is relevant to all three of their types of formalization.
Similarly, while it involves some ambiguity in the use
of the term, representation is a function that models
perform for concepts as well as for phenomena, and we
shall frequently use it in this sense later.

We now need to reemphasize that their specifi-
cation of kinds of situations seems to us to be essen—
tially correct, and that insofar as a particular instance
of formalization approximates to being a "pure" type,
the relevant kind of development will be most useful.

In our own work the main aim is to render assistance to

 

25We believe that our commentary also suggests
that Berger and his associates have embodied a dialectic
conception of the interplay between theory and research,
but, again, this observation cannot be pursued here.

25

20

a set of concepts, and in this connection it is helpful
to review the highlights of their analysis of the
example of an explicational model. The example which
they use as a type case is the development by Cartwright
and Harary of the concept of "balance" in Heider's
theory of the same name.26

Heider's theory concerned the organization of
relations among elements of the ". . . life—space of
a fixed person . . . ,"27 which we will loosely call
cognitive structure. This structure is characterized
by two kinds of elements, persons and non—persons, and
two kinds of relations, which Heider denoted by L and U.
L referred to a sentiment and U ". . . covered virtually
every relation . . . which was not a sentiment."28
Because of the "life—space" restriction, a relation
between some other person and another element (of either
kind, including the fixed person) was interpreted as a
thought of the fixed person; his own sentiment or activity
was expressed by designating him as the first element of

the appropriate relation, with another element as second.

The only other restriction on correct formation of relations

 

26Citations to relevant works are in ibid.

27Ibid., p. 11.

281bid., p. 10.

21
between elements was that a non—person could not be the
first element of an L relation, e.g., tables and base-
ball games do not love anything.

In his analysis, Heider considered a variety of
examples in order to define a state of balance in such
a structure, and we will not review them. He was con—
cerned largely with the case of three elements (of which
one was the fixed person), with one relation of either
kind between each of the three pairs, and with defining
balance in terms of the positive or negative character
of each relation.

In general . . . Heider concluded that

balance occurs when either all relations in

a system are positive or any two are negative

and the third positive. In this definition

L and U relations were 'exchangeable' in the

sense that, in classifying systems as balanced,

it made no difference which relations were

negative and which positive, so long as two

were negative and one positive, or else all

were positive.29
He made several propositions about the consequence of
imbalance for reorganization of the structure (or, if
this was not possible, for the tension state of the per-
son), and other matters as well, but these are not

needed directly in our summary of the formalization and

we omit them here. Similarly, Heider's conceptions

 

29
Ibid., p. 14.

22
stimulated a number of interesting experiments which
Berger, et al., review, but we may pass over these with
their summary observations.
It may be said, then, that in general the
body of experimentation stimulated by Heider's
theory provided empirical confirmation for some
of its principal assertions, but that it also
pointed to the need to extend the theory beyond
its original scope and provided evidence that
the original theory was in need of conceptual
clarification.30
Such a clarification was made for the concept of
balance (as defined in a quotation above) by Cartwright
and Harary using the theory of linear graphs. This is of
interest to us both as an explicational model and because
we use the same general kind of formal apparatus in our
own analysis. Since we review concepts of linear graphs
at a later point (see Chapter IV), we will be brief here.
The theory of linear graphs deal with
finite collections of points, and lines
between pairs of these points. A set of

points, all, some, or none of which is
connected by these lines, is called a

graph.31

In our review above we have used the terms "structure"
and "element" and they may be equated to "systems" and
"entity" respectively in the following.

To the formal definitions of graph
theory . . . coordinate the concepts . . .

 

301bid., p. 19.

311bid., p. 20.

23

according to the following rules of inter—
pretation: each point in a graph is an
'entity' in Heider's theory; each line in
a graph is a 'relation' . . .; a graph may
then represent a system of such entities
and relations . . . . For each kind of
graph a definition of a balanced graph can
be made that is consistent with Heider.32

 

If positive and negative indications are attached to each

line of a graph (thus constituting a "signed" graph),

then the concept of balance may be defined in terms of

path and cycle. A path is a connected sequence of lines

which is not redundant —— the same line does not appear
twice -— and a cycle is a closed path. "The sign of a
cycle is the product of the signs on all lines of a
cycle."33 This product is negative if there are an
uneven number of such signs, and it is positive other—
wise. Finally, "A signed graph is balanced if all its

cycles are positive."34

 

Useful features of this explication are quite
directly obvious even in the greatly attentuated form
given here. For example, since Heider considered only

one relation between each pair out of three entities,

 

32Ibid., pp. 20-21.

33Ibid., p. 22.

34
Ibid.

24

there could be only one cycle. In the graphic model,
Cartwright and Harary did not limit themselves to
systems of three entities, and consequently were able
to consider the more general case of having many cycles,
some of which might be balanced and others not.35 This
led to the concept of degree of balance; this development
is well remarked by Berger, et al., in their evaluation
of the explication, and we turn to this now.

The authors summarize the "Characteristics and
Functions of an Explicational Model" in a series of five

major points, which we reproduce with a selected conden—

sation of the exposition.36

l. The model is selective with respect to the
original conceppualization. We have noted this
feature in our earlier development. Here, the
explication deals with the concept of balance, not
with all of Heider's propositions. Another example
is that sentiments of a person toward himself are
not handled since the explication treats irreflexive
graphs.

2. The modelyprovides a means of clarifyipg
the original conceptualization. Heider did not
distinguish between the complement of a relation
and its opposite, but this is easily accomplished
in a graph by the distinction between omitting a
line and attaching a negation to it. Clarifying
nonexistent relations led to the concept of
"vacuous" balance -— a graph with no cycles at
all —- and this improved the fit of experimental
results, the deductive completeness, and the
interpretation of the theory.

 

35The number of possible cycles is increasingly
larger as entities are added and may be represented as a
factorial function.

36Ibid., pp. 24—36.

25

3. The model provides a means of refining
the original conceptualization. We noted above
the extension to more than three points; together
with the ideas of partitioning the set of cycles
according to l) inclusion of a point, 2) the
number of points (or lines) in a cycle, and 3)
the balance of a cycle, this leads to concepts
of "balance at a point," "local pfbalance," and
"degree of balance."

4. The model provides a means of generali-
zing the original conceptualization. Again, the
extension to more than three entities is impor-
tant, but perhaps more important (from the present
writer's View) is the possibility of incorporating
more than one relation between a pair of entities.
This latter development is more programmatic than
others (Cartwright and Harary actually did not
extend the theory to higher order graphs), but it
has served to locate gaps in the original formu—
lation and its development.

5. The model pgovides a means for determining
implications within the original conceptualization.
An especially interesting example occurs in the
Cartwright-Harary development of the "structure
theorem" which shows that in a balanced graph
there must be two mutually exclusive and exhaus—
tive subsets of points such that signs of lines
are always positive within a subset and always
negative between subsets. This appears to be
equivalent to a proposition about balance of
segregated entities made by Heider, and it shows
that it is actually deducible from others.

We have reproduced the example and discussion of
an explicational model in some detail for two reasons.
First, it seems useful to give the reader who is not
familiar with the original work (or some comparable
development) some orientation to the kinds of questions
and tools employed through an example of a particularly
efficacious explication. Second, it is thought that the

explication (!) by Berger, et al., especially the five

26
points summarized immediately above, provides a very
useful extension to our guidelines for evaluation of
the results of our own work. However, there are some
points of difference in the circumstance and we should
take note of them.

There seem to be two salient and related points
of difference between the example reviewed and the corpus
of concepts to be attacked in our work. In the Cartwright—
Harary formalization of balance there was only one source
(or tradition) for the concept and its uses. This has
the beneficial effect (for the explicator) of limiting
the variety of vagueness and ambiguity confronted in the
original conceptions. On the other hand, in the analysis
of role concepts carried out in the following chapters
we are confronted with a much more diffuse array of
meanings and uses, to the point that it is difficult to
make classifications of them to show which ones "go
together."37 The second, and we believe more conse—
quential, point is that the concept of balance was

embedded in a fairly well developed and coherent theory.

In the functions enumerated by Berger, et al., many of

 

37This is relieved somewhat by the focus of our
examination on the logical characteristics of the con—
cepts, as we will indicate in the next chapter.

27
the most important results concern not just the concept
of balance as such but the wider theory and its recon—
struction.38 In dealing with role concepts we must
attend to questions as to what theory exists, and what
the consequences of our explication are in this regard.
In connection with this we must take notice here of a
warning given by Berger and his associates.

It is probable that where the model-builder
attempts to explicate a concept which is not
part of a significant substantive theory,
there will be few . . . constraints on the
way in which he formalizes the concept. It
is probably also true that in such a case
the explication may be a relatively idle
exercise, unless a theory is later developed
which makes use of the model-builder's
particular formulation.39

 

There are some difficulties in interpreting this
injunction on the perils of bootstrap efforts. It is
uncertain whether one should emphasize "significant" or
"theory" or both, though one can make an informed guess
that the last option is intended. It is then necessary
to inquire what they mean by "significant theory," and

one need not guess here that they mean a well-formed one

 

38We take this as confirming evidence of the
points made earlier concerning our views ". . . that
models help in getting from one stage to the next,"
and that one may significantly speak of explication
of a theory as well as of concepts.

391bid., p. 104.

28
which is effective vis—a-vis its intended range of
phenomena. However, it is felt that there is an implied
counsel of excessive caution; in the absence of a well—
formed object for our investigation we had best begin
with what is at hand lest one never be developed. The
writer shares the concern for responsible and responsive
theory construction which is surely behind their notice
of the dangers of free thinking, but he does not share
their pessimism concerning the cumulation of viable
concepts. It is well known that "exercise" sometimes
locates and develops ligature previously unknown.

Of course, all of these remarks are directed to
clearing away objections to our undertaking herein, and
they will be effectively otiose if it proves unsatis—
factory. But they do serve to point up our concern with
the frequently stated fact that "role theory" is more
an aspiration than a reality. We aim to aid the reali-
zation of this ambition in the knowledge that such
attempts should not be made or discarded lightly. And
it seems best to make the statement of our analysis in
as definite form as possible in the realization that

others will accept and treat it as provisional.

Purpose and Overview
It is our intention to review and examine the

concept of role and related concepts as they are defined

29
when being used as elements of a structure or system.
The focus will be on the logical character of certain
features of prOpositions about these concepts so used,
and we try to show the consequences regarding formula—
tion of these concepts and the system. After explicating
our model of a role system, we further intend to explore
the consequences of the model for our conceptions, or in
one of the senses of the now familiar phrase "its feed—
back on the system." Naturally enough, this last task,
is saved for the final chapter. The actual model and a
solution for a problem it brings out are presented in
Chapters IV and V respectively. In order to specify
compactly the appropriate formal character of the
model, in Chapter III we set out a verbal axiomatization
(and some considerations of its justification) which
entails the logical structure developed in the succeeding
chapters. The first burden of Chapter II is to provide
a background upon which the justification of the axioms
can be based. It attempts ". . . to make at least
practically clear what is meant as the explicandum,"4O

and we now proceed to this task.

 

OCarnap, Logical Foundations . . . , p. 4.

 

CHAPTER II
THE ROLE CONCEPTS

The concepts of role1 have a long and useful
history in the literature of sociology, both in theory,
either speculative or analytic, and more recently in
empirical analyses. The history is also one of con-
troversy, of clarification, specification, definition,
and redefinition, of caveats against one or another of
the associated terms or ideas, of great anticipations
and sometimes small realizations, and of appeals for a
common tongue. In spite of their parentage in specu-
lative discourse, the concepts of role have not been
relegated to the intellectual dustbin by modern
empirical social science. The amount of recent effort
toward developing the theoretical refinement and

empirical relevance of the concepts is substantial

 

1In the sequel, we will sometimes, as here, use
the term "role" to stand for a battery of combining and
related terms (especially "status," "position," "role
behavior," and the like) in order to avoid repetitious
locutions. We will also use it in more restricted ways,
and the context should indicate our intention.

30

31
evidence of their felt importance in sociology, and in
the neighboring disciplines of social psychology, social
and cultural anthropology, political science, social
work and education. There is no need in the present
work to justify the importance of the concepts or the

empirical referents they assay.

State of Agreement on the Concepts

The task which we have set for this analysis is
to examine the concepts as they are used by social
scientists with a view to determining their meanings
with somewhat greater precision than has been achieved
to date. What we hope to show is that there are common
features in what sometimes have appeared to be disparate
conceptualizations, and that they form the basis for a
model of a structure or system of roles as a general
conceptual tool. It is clear, however, that other
observers do not share the belief that commonalities are
noticeable in the definitional schemes that have been
offered. Neiman and Hughes in their review of a decade
and a half ago presented a pessimistic appraisal of the
then current condition of the concepts.2 They professed

to find little convergence beyond several rather general

2L. J. Neiman and J. W. Hughes, "The Problem of

the Concept of Role -- a Re-survey of the Literature,"
-30cfial Forces, 30 (1951), pp. 141—149.

r
-..
F.
4-
‘r
..,
r
'.
-.
—
..
.

l.Al:_l

32
assertions about the possibility and general character
of the analysis of human behavior, which did not really
entail the concept role or any of its relatives.3 They
reached the conclusions that the concept was vague,
nebulous, non—definitive, reified, little used in
research, and used for 28.222 explanation of human
behavior. After an intervening decade in which the
concepts had come into wider usage in reports of empirical
research, Biddle similarly inveighed against a babel of
competing formulations and declared that each of the
several summaries which had appeared in the interval
". . . has emphasized the lack of structure in the field
and has called for rectification of . . . [the manifest
dissensusj."4 However, the present view that there is
an important measure of consensus among social scientists
on the role concept finds support in the recurrent
efforts to give expression to the consensus through
reconceptualization, through the observable continuity
given by the adoption by investigators of the formula-
tions offered by others, and in the conclusions of some
of the above—mentioned summary evaluations. Particularly,

Gross, Mason and McEachern, in their review of a rather

‘

3Ibid., p. 147.

4Bruce J. Biddle, The Present Status of Role
Theory, Columbia, Missouri: University of Missouri,
SCNZIal Psychology Laboratory, 1960, p. 3.

 

33

wide range of sources, concluded that certain common
themes did occur in almost all conceptualizations, whether
or not they were explicitly incorporated in stated defini—
tions.5

There is, however, an important difference in
the line of argument to be pursued herein with respect
to conceptual agreement. Almost all of the conceptual
reconstructions alluded to above have taken as their
focus the analysis of the substantial character of
concepts and of the kinds and ranges of empirical phe—
nomena to which they should apply. In the present
analysis we shift the focus to the formal character
of the concepts, or what might be termed their logical
foundations as a conceptual system. It is tempting to
say that what we intend is to "move to a higher level
of abstraction," but his phrase carries some inappro—
priate connotations. As will be seen later, we do not
intend to "abstract" in the sense of eliminating the
substantive content of the concepts. This content must
be used to determine the relevant logical forms, and

these forms must be an adequate reflection of content.

N. Gross, W. S. Mason, and A. W. McEachern,
Egplorations in Role Analysis, New York: John Wiley
and.Sons, 1958, Chapter 2.

 

 

 

.1
ex,-

_n.

. ¢

 

34
This shift of focus does make itypossible we believe to
epecify certain dimensions of agreement which transcend

particular substantive differences.

Selective Emphases and Organization of the Review

It is commonplace, as we observed above, to
notice that different authors and even the same author
have spoken with many tongues in the history of the
develOpment of role theory. This is true with respect
both to the use of different terms or phrases which
apparently have the same referent, and to the use of the
same term with manifestly different meanings. At a
later point, we will settle our own usage on two terms,
position and role, and on some combining forms of them
in a manner to be specified here. To do this we want
to get something of the flavor and the content of these
many uses in order to show that ours is not completely
idiosyncratic. Fortunately, in view of the quite
literally vast interest in the subject, a number of
summaries and reviews have appeared, as was remarked
above, and they will provide some assistance. However,
we will not be able to rely on them entirely, since
some of the facets of the concepts which are necessary
to our purpose have not been adequately established, and
We will want to be able to see.at what point certain

developments of the concepts enter which are important

 

 

35
to our formulation.6 Both the volume of the literature7
and our own interests direct us to employ several cri—
teria of selection for the material to be reviewed.

The first restriction that we impose is that we
will be interested in what may be termed modern or con—
temporary views of role concepts. It is our opinion
that no useful purpose would be served for our explica—
tion by reaching back beyond the last generation of
writers for sources.8 In saying that our object is an
appraisal of current perspective, we are not thereby
denying important antecedents; much of the conceptual
development can be seen as a "working out" of earlier
specifications.

A very salient element of selectivity in our
interest relates to one of the major points of signifi—

cance of the role concepts. It is universally accepted

 

6This last remark should not be taken to mean
that we see a unilinear evolution of the concepts, only
that certain changes to be noted have emerged, even
though they are not uniformly accepted.

7In their forthcoming study of role theory,

B. J. Biddle and E. Thomas include a bibliography of
about fifteen hundred items which they believe to be
incomplete even in the particular areas covered. The
writer expresses his appreciation for their scholarly
aid in making this compilation available.

8This office is well performed elsewhere; see
fFMDtnote 13 on antecedents below, and especially the
Clrtation to W. B. Catton, Jr.

 

36
that roles are the intersection or articulation of the
individual and the larger milieu, however defined, in
which he is situated. It is this linking function which
gives them their wide substantive and empirical rele—
vance; on the one hand they are crucial to analyses of
personality, attitudes, etc., and on the other they
are the building blocks of social structure. In this
study, we are interested in the latter aspect: the
conceptualization of roles as elements of a social
system. This does not mean that we may thereby elimi—
nate social psychological views from our review; many,
if not most, of the authors who have contributed
materially to clarification of the concepts have been
much concerned with the linking function. In recent
years this interest has found major expression in a
proliferation of studies of the sources and effects of
role conflict. Our effort will be to attend primarily
to structural aspects, introducing these parallel con—
siderations insofar as they have immediate bearing on
our concepts of a role system.

In limiting our attention to the structural
relevance of role concepts we thus neglect problems of
socialization, role learning, and development of self—
hood, of role playing and its relationships to
<fliaracteristics of the individual or personality, of

tkhe intra—personal experience and effects of multiple

37
roles, and in general most of the manifold ways in which
role concepts have been employed in analyses of the
internal dynamics of the personality system. In effect,

we adopt the widely accepted assumption that it is

 

perfectly feasible to ignore these questions, specifi—
cally as at present when dealing with roles as parts
of a social system.9

The concept of a system of roles and related
elements provides a general orientation for the present
analysis. The concept of system draws our attention
in the review of existing conceptualizations to a number
of orienting themes and questions: What sort of system
is postulated? What are its elements? And in what way
are they taken to be related? Are sub-systems assumed?
And what other systems (and their characteristics) are
required? Such questions tend to imply a univocality
of response, which we have already agreed does not
exist in detail, and to suggest that simple, explicit

answers can be given. This is not the case, but the

 

9This view has been put forth most insistently by
Talcott Parsons in many writings, particularly in 222.
Social System (Glencoe: The Free Press, 1951). It is a
view which seems "natural" (at least to sociologists!)
and is regarded as self—evidently valid. It is worth
noting that the concept is of fundamental importance in
physical science, where it is referred to as "The indepen—
dence of components," and it is taken as a postulate of
the discipline of physical systems analysis. See H. Koenig,
et al., The Analysis of Discrete Physical Systems, East
I“arising: Michigan State University, Department of Elec—
trfiical Engineering, 1964. The present author's conception
of: a system has been substantially influenced by this work.

38

orientation does have the merit of forcing us to look
for whatever general agreement there is to be found.
For the present it is sufficient to say with Parsons
that what we have in mind is ". . . a system in the
scientific sense . . . ,"10 i.e., a set of interrelated,
or interdependent (or interacting) elements, or parts,
or components. The particular system we have in mind
(or rather what we believe to be implied by the con-
cepts as they have been used) will become clearer as we
proceed, and it will be given explicitly in Chapter IV.

Our review of the concepts will be divided into
two main parts, and these roughly correspond to exami-
nation of the elements of the system and examination of
the composition of the system. We make this do double
duty in dividing time periods of the selected writings
as well. This serves to point up our view that it is
only relatively recently that the idea of a role system
has been worked out sufficiently to provide a basis for
the kind of explication carried through in the sequel.

The selection of particular writings for our
review is somewhat arbitrary. The primary purpose is to
illustrate important themes which have been used in

specifying the concepts rather than a detailed comparison

10 .
Parsons, op. c1t., p. 3.

39
of particular terms. In doing this we have attempted to
use the work of authors who have been thought important
by succeeding commentators. Also, we have avoided using
more than is deemed necessary and consequently we have
not included all of those whom others have commended.
In most cases, the work of a particular author is dis—
cussed at a single point under a topic heading which it
seems to illuminate best, even though it reflects other
concerns as well. This treatment seems more convenient
here than such an alternative as collecting definitions
of a single term together for comparison; this is a
result of the ambiguities of substantive usage, their
essential interdependence which we stress in our recon—
struction, and the relevance of the substantive themes
for determining the logical characteristics which are
the main object of our attention.

While we have not organized the presentation
around individual concepts, it will be an aid to the
reader to point out that our concern in the first section
of the review is with the general sense of role, status
or position, and, in a subsidiary way, actor; and that
we introduce the idea of relation as a fundamental
logical element. The shift of attention to system organi-
zation in the second section introduces several qualifying
terms for position and role which denote different system

afiipects. We will leave their enumeration to that point.

40

The Elements of the System

The Variepy of Concepts
In their review of the literature, Neiman and

Iiughes classified the definitions of the concept role
Jcnown to them in terms of the contexts and contents of
tfliese uses. It is interesting to reproduce the main
kneadings of their classificatory scheme, since they
rweflect the variety of senses of the single term and
eeniphasize the contextual relevance characteristic of the

(jeefinitions. With appended names of writers cited as

ee><amples, the categories are as follows:11

A. Definitions in terms of the dynamics of
personality development

1. Role as a basic factor in the process of
socialization — Park and Burgess, but
preeminently G. H. Mead.

2. Role as a cultural pattern — Linton,
Znaniecki, Parsons.

B. Functional definitions in terms of society
as a whole

1. Role as a social norm, no connection with
status explicit, but implicit — Komarovsky,
Sherif, Benedict, M. Mead, Stouffer.

2. Role as a synonym for behavior, non—
definitive — Neiman, Hughes.

lOp. cit. The appearance of the word "Re—
ESL—‘l.‘1:veyn(see previous footnote) in the title of this
I:‘ticle is somewhat puzzling, since there is no evi—
SEtnce of earlier comparable summarizations.

41

C. Functional definitions in terms of specific

groups
1. Status—role continuity, definitive as
activated status — Linton, Hughes,
K. Young, Hiller, Znaniecki, Merton,

G. Murphy.

2. Role defined as participation in a
specific group - Moreno.

{These category titles are interesting both for features

tidat they emphasize and for ones that they overlook.

EDaIticularly, it is apparent that role has been defined

hxath by what it "does" and by what it "is." For some

vvrriters it "functions" in the socialization process and

:frDr others in the activation of status. Other writers

(53nd sometimes the same ones) see it as composed of

rositterns of behavior, perhaps culturally given, or as

fnéicie up of or oriented by norms. On the other hand,

‘tljee categories seem to emphasize an idea of a role as a
Estrugular element rather than as a factor in relationships

b€3t:ween persons, and these relationships are of consider-

EikDile importance in the literature and in the sequel.

In a later review, Biddle compiled formidable

€3\f:idence that a great variety of different terms have

\
which is generally correlative

12This difference,
points to another

t:C3 the function—structure distinction,
Ei<:tor of selectivity in the formalization to be developed

“quiich was implicit in our earlier remarks. The model to
LhDEE‘ presented is a conception of the structure of a role
535?:stem, and this directs our attention to consider what
tl}1«e concepts "are" as elements in the system.

 

.- o
v' '
a\

our "
.a-‘

42
been used with meanings which are —- at least according
to the definitions of his own array —— the same (as well

as being used frequently with more than one meaning,

again according to his distinctions). A few samples will

suffice to indicate the extent of this proliferation.
I?or his term "position" Biddle finds 16 other different

inords or phrases; for his "expectation" he finds 33

crther expressions; and for "norm" he enumerates 28

crthers. Clearly, there is a lesson to be learned from

tjais depressing display. Both Biddle's analysis and

(3116's own experience suggests that more names than

<:<>ncepts are involved. In the following pages we

vvjrll review some of the important examples from the

Hiarterial, and then attempt to summarize the aspects

Ifealevant for setting out our own definitions.

Eéfiirachmarks: Behavior, StructureL and the Emphasis on

S2§>rdsensus
An attempt to make appropriate selections

FDITeasents a rather difficult task. A simple list of

E1L1‘thors who have written on and contributed to our

C3(31’1cepts would read like an honor roll of the social

Eirlrj behavioral sciences. Similarly, assessment of
EDITLiority or antecedents for concepts leads to an

le"T‘ljpressive array of founders. We have said that we

\

- 13A few of the persons to whom credit for
Tail—gnificant innovation is accorded by recent authors
1‘171c1ude Pareto, Weber, James, Baldwin, Dewey,

43

are mainly concerned with the contemporary condition of

the concepts, but it is sometimes difficult to distin—

guish between history and current events. It seems

fairly representative of the consensus on conceptual
origins to say that two writers whose publications
appeared in the decade of the 1930's (one on the side

(of social psychology and the other on that of social

satructure, though neither was much disposed to observe

rxDundaries) stand as important intermediaries between

‘tlue past and the present, and they are George Herbert

Pfleead and Ralph Linton. Both appear to have served —-

LLII an imperfect analogy —— as "funnels" through which

Licieas have passed into our hands.

y

‘VT- I. Thomas, Burgess, Sumner, Cooley, G. H. Mead,
12‘. E. Park, Moreno, and Linton. See S. F. Nadel, The

£211<eoryof Social Structure, Glencoe: The Free Press,
:LE9ES7, p. 21; E. C. Hughes, Men and Their Work, Glencoe:
inflee Free Press, 1958, p. 56; Neiman and Hughes, op. cit.,

F>- 141; T. R. Sarbin, "Role Theory," in G. Lindzey, ed.,

fijégadbook of Social Psychology, Vol. I., Cambridge, Mass.:

fiKCicjison-Wesley, 1954, p. 223; F. L. Bates, "Position,
F2(bile, and Status: A Reformulation of Concepts," Social
34 (1956), p. 313; Biddle, op. cit., p. 20.

.ELCDJrces,
“M'«- R. Catton, Jr., retrenches further than most in tracing
t:}1<e concepts from Comte, Spencer, Main, Toennies, Durkheim,

ivleiloer, and Simmel. See W. R. Catton, Jr., "The Develop—

jn‘fieant of Sociological Thought," in R. E. L. Faris, ed.,

iiiEggdbook of Modern Sociology, Chicago: Rand McNally,
Pages 936-943 contain a compact summary

S364, Chapter 24.
(335: the historical and recent development of the concepts

CD1? status, role, position, and related terms, primarily
€3~SS components of societal structure. This orientation
eEads to an emphasis on conceptions of ranking which the

1331C7esent account will not share.

44
Our deemphasis of the relevance of role concepts
as used in analysis of the individual might suggest that

Mead's work would have relatively little to offer for

present purposes. It is important to our analysis

fibecause of his profound effect on others coming after.
fIwo main.features of this heritage are the markedly
Iaehavioral quality which he imparted to definitions of

rwole and an emphasis on the consensual character of

srocial structure. Substantively, we are interested in

r1j_s conception of the "generalized other" as an aspect

(Di? the development of the organization of the self,

aarld in his use of "games" and their rules to illustrate

t:r1e formation of the individual's conception of the

ESCDCJial structure. His explicit introduction of the

Wuacin.concept follows.

The organized community or social group
which gives to the individual his unity of
self may be called 'the generalized other.‘
The attitude of the generalized other is the

attitude of the whole community. Thus, for
in the case of such a social group

example,
as a ball team, the team is the generalized
other insofar as it enters —— as an organized

process or social activity -— into the
experience of any one of the individual

members of it.14

IVI‘EEEELd was centrally concerned with analyzing the relation—

$31fii‘Lpof the individual to a total group, and with the way

\

(2 14George H. Mead, Mind, Self and Society,
1I-lfchago: University of Chicago Press, 1934, p. 154.

p
.. .-

 

45

in which this organized the behavior of the individual.

In an abstract, speculative formulation it was not

necessary to raise much question regarding veridicality

or agreement. The use of illustrations in which explicit

rules are common to all heightened the emphasis on the

shared characteristics of outlook and performance in

:functional detail as in the remark,

". . . in a game where a number of individuals
are involved, . . . the child taking one role
must be ready to take the role of everyone

else. If he gets in a ball nine he must have
the responses of each position involved in his

own position.l5

Pie did not depreciate individual differences, but their

.rwelatively low salience for his analysis seemed to lead

13)? conversion to an emphasis on structural integration

earld.a.deemphasis of differentiation in parts of the

S tructure .

Of course, we are not only what is common to
all: each one of the selves is different

from everyone else; but there has to be such

a common structure as I have sketched in order
that we may be members of a community at all.
. . . We cannot have rights unless we have

common attitudes. . . . Selves can exist 16
only in definite relationships to other selves.

In the original, these passages were relevant to

tlit—1e import of the structure of the social group for an

\

lSIbid., p. 151.

lerid., pp. 163-64.

46

understanding of the development and behavior of the

individual. Here, they are of primary interest for the

conception of that structure (and of the ideas of it
The emphasis

held by participants) which they convey.
on the rules and logic of games presents a unitary

Eaicture of group structure. At least some of the credit

ifor the acceptance of interpersonal consensus as a

l>asis and mark of structure should be attributed to the

cusncept of the generalized other. And it is this con-

sseensus which has become a point of some dispute in later

i?c>rmulations.
An emphasis on consensus also characterizes the

i?c>rmulation of the concepts of status and role presented

133’ Ralph Linton in the now classic eighth chapter of
ggklea Study of Man. In contrast to Mead, Linton had a
Ci€EiFinite concern in this work with the nature of societal

53t:r:ucturing of these elements independently of the way
j—rl which they are incorporated into the individual. His
St:eatement is a forerunner explicitly incorporated in

““Eilay later works, or at least a landmark to be reckoned

VVjL‘th. In part, this may stem from the unusually compact

Eirle straightforward statement and exposition of definitions.

._____
l7 ,
The first two pages of the chapter must be
Eirrhong the most frequently quoted passages in the literature

C>j§ sociology.

47
Many, though not all, of the points to be found in later
structural role theory are contained in them, as well as
some which have not survived intact. The following
extract contains the main analytic specifications.

. . . the functioning of societies depends upon
the presence of patterns for reciprocal behavior
between individuals or groups of individuals.
The polar positions in such patterns of recip—
rocal behavior are technically known as statuses.
The term . . . has come to be used with a double
significance. A status in the abstract, is a
position in a particular pattern. It is . . .
cOrrect to speak of each individual as having
many statuses, since each individual partici-
pates in a number of patterns. However, unless
the term is qualified in some way, Epe status

of any individual means the sum total of all

the statuses which he occupies. It represents
his position with relation to the total

society. . . .

A status, as distinct from the individual
who may occupy it, is simply a collection of
rights and duties. . . .

A role, represents the dynamic aspect of
a status. The individual is socially assigned
to a status and occupies it with relation to
other statuses. When he puts the rights and
duties . . . into effect, he is performing a
role. Role and status are quite inseparable,
and the distinction between them is of only
academic interest. There are no roles without
statuses or statuses without roles. Just as
in the case of status, the term role is used
with a double significance. . . .

Although all statuses and roles derive

from social patterns and are integral parts

of patterns, they have an independent function
with relation to the individuals who occupy
particular statuses and exercise their roles.
To such individuals the combined status and
role represent the minimum of attitudes and
behavior which he must assume. . . . [They]

. . . serve to reduce the ideal patterns of
social life to individual terms. They become

48

models for organizing the attitudes and
behavior of the individual so that these
will be congruous with those of the other
individuals participating in the expression
of the pattern.18

The conception of status as a position, i.e.,
location in a pattern, with associated rights and duties,
and of role as what is done in consequence of being in a
status, has survived in substance if not always in name in
rnuch of the later discussion. Linton's own reformulation
eat a later date represents a shift in the content of the
j_ndividual concepts and an extension (or at least develop—

rnent) of the total substance.

The place in a particular system which a
certain individual occupies at a particular
time will be referred to as his status with
respect to that system. . . . The second
term, role, will be used to designate the
sum total of the culture patterns associated
with a particular status. It thus includes
the attitudes, values and behavior ascribed
by the society to any and all persons
occupying this status. It can even be
extended to include the legitimate expec—
tations of such persons with respect to the
behavior toward them of persons in other
statuses within the same system.19

UTTaus, "rights and duties" have been removed from status,

liftzbeing reduced to position, and some extended counterpart

l8R. Linton, The Study of Man, New York:

I) - Appleton-Century, 1936, pp. 113—114.

 

19R. Linton, The Cultural Background of

lEfiersonality, New York: Appleton—Century-Crofts, 1945,
pp. 76-77. .

 

49
with a more explicitly normative ring in the form of
". . . attitudes, Values, . . . and legitimate expec—
tations . . ." has been added to role. It is worth
emphasizing that in neither of the two statements is
a role taken to be the actual behavior of an individual,
since there has been some confusion on the point.20
In so far as it represents overt behavior,
a role is the dynamic aspect of a status:
what the individual has to do [but not what
he does] in order to validate his occupation
of the status.2l
There are several other points of importance about
ISinton's conception to be brought out here. One of these
j_s the root of the concepts in "patterns for reciprocal
.r>ehavior" and "polar positions," which implies that a
£3t:atus and role entail other related statuses and roles.
Igjlnton generally treats statuses as pairs or dyads,
t:Id<3ugh this may be for simplicity of exposition (as some
22

l-Eii:er observations in this chapter tend to show).

55lirnilarly, though he recognizes that individuals have a

\

2OSee M. J. Levy, Jr., The Structure of Society,

EDIT-Z‘aneton, New Jersey: Princeton University Press, 1952,
§:‘- 154; also, M. J. Daniels "Relational Status and the
(3.1.e Concept," Pacific Sociological Review, 2 (1959), p. 44.

lLinton, op. cit., p. 77. Emphasis and brackets

alciCied.

22Some other writers have an explicitly dyadic
§©ncept of statuses. See, e.g., E. T. Hiller, Social
.~J§fi£ations and Structures, New York: Harper, Chapter 22,

 

533i passim.

50

multiplicity of roles (or statuses), there is little
indication as to how these may be conjoined, except by
being played by the same individuals and perhaps by the
concrete situation.23 In his explicit definitions there
is a one—to-one relation between each status and its
role and no more. Another feature, related to the
reciprocity of roles and statuses is notable by its
absence in Linton's concepts.24 This is the factor of
ordering or ranking which long has been (and still is)
associated with the term status. Catton points to this
divorcement of hierarchical evaluation from the concept
of social location as one of the essential steps in the
purification of the concepts.25 Other authors (e.g.,
Gross, et al., and Biddle) have preferred to drop status
in favor of a more neutral term, usually position,
because it is thought still to carry the connotation of
hierarchy even though defined without it.

Perhaps the most enduring elements of the speci—

fication are the idea of position as an abstract model,

 

23See his example of a department store clerk, in
Linton, op. cit., p. 78f.

24This is not precisely true. "Status has long
been used with reference to the position of an individual
in the prestige system of his society. In the present
usage this is extended to apply to his position in each of
the other systems." Ibid., p. 77. ,However, the extension
to other systems removes ranking as essential to the defi—
nition.

25Catton, op. cit., p. 937, 939.

51
and the spatial analogy and the idea of occupancy which
it entails. With one exception to be noted below, these
elements have been retained to the present,26 though not
always uniformly interpreted. Over time, position has
become a more or less defined term in its own right,
rather than the relatively "primitive" (in the logical
sense) concept which Linton used. This is a point to
which we will return repeatedly and in the end quite
extensively below.

The impact of Linton's comparatively brief state—
ments can hardly be overemphasized.27 The basic "static—
dynamic" dichotomy has influenced (one is tempted to say
plagued) much of the later thought,28 though it does not

appear to have been intended quite so literally by its

 

 

 

26E.g., see the definition of "office" in R. L.
Kahn, et al., Organizational Stress: Studies in Role Con—
flict and Ambiguipy, New York: Wiley, 1964, p. 13.

27

Merton is especially emphatic on this point:

"To say that Linton was not the 'first' to introduce these
twin concepts into social science would be as true as it is
irrelevant. For the fact is that it was only after his
famous Chapter . . . that these concepts, and their impli-
cations, became systematically incorporated into a developing

 

 

theory of social structure." R. K. Merton, Social Theopy
and Social Structure, rev. ed., Glencoe: The Free Press,
1957, p. 368, n. 112.

28

E.g., ibid., and R. K. Merton, "The Role-Set:
Problems in Sociological Theory," The British Journal of
Sociology, 8 (1957), p. 110; Parsons, op. cit., p. 25, and
in other writings; C. P. Loomis, Social Systems: Essays
on Their Persistence and Change, Princeton, New Jersey:

D. VanNostrand, 1960, p. 19.

 

52
author. In addition, the unqualified emphasis on the
societally given character of the content of the concepts
has only gradually yielded in the confrontation of
empirical research.

Another formulation of related concepts made
shortly after Linton's original statement appears in the
work of E. C. Hughes. It is of interest here for com-
parative purposes, since it both agrees with the preceding
in some respects and differs in certain emphases.

Status assigns individuals to various accepted
social categories; each category has its own
rights and duties. . . . in its active and
conscious aspect, [it] is an elementary form
of office. An office is a standardized group
of duties and privileges developing upon a
person in certain defined situations.

In current writings on the development of
personality, a great deal is made of social
role. What is generally meant is that the
individual gets some consistent conception of
himself in relation to other peOple. This
conception, although identified with one's
self as a unique being, is a social product;
. . . But role, however individual and
unique, does not remain free of status.
Indeed, Linton says 'a role is the dynamic
aspect of a status.‘ Role fie dynamic, but
it is also something more than status.29

Disregarding the salient personalization of role (in a

manner similar to Mead, which we did not emphasize above),

 

29E. C. Hughes, "Institutional Office and the
Person," The American Journal of Sociology, 43 (1937),
p. 404. I

53
the most important difference for our purposes is the
absence of some idea of location and an associated
spatial analogy as underlying status. This is also
associated with an attributive quality of the concepts;
the idea of social locations as mutually oriented is
not explicit and the relational component appears most
expressly in the function of a social role in the
individual's conception of himself.

In another presentation which, to judge by the
number of secondary citations, impressed others by its
concise and incisive character, Cottrell combined the two
earlier formulations of role in the following way:

. . . I shall . . . [use] . . . the term role
to refer to an internally consistent series

of conditioned responses by one member of a
social situation which represents the stimulus
pattern for a similarly internally consistent
series of conditioned responses of the other(s)
in that situation. Dealing with human behavior
in terms of roles, therefore, requires that any
item of behavior must always be placed in some
specified self—other context.

By way of further clarification it is
necessary to call attention to the distinction
between the use of the term role to refer to
a modal system of responses which constitutes
the culturally expected behavior and the
particular system of responses with which a
specific individual operates. . . . We may
refer then to cultural roles and deviant
roles. The distinction is most obvious when
we have a person equipped with both a cultural
and a deviant pattern.30

 

3OL.S. Cottrell, Jr., "The Adjustment of the
Individual to His Age and Sex Roles," American Socio-
logical Review, 7 (1942), p. 617.

r

11‘

Il'

/

54
While he makes no explicit definition of status or
position, Cottrell recognizes the assignment function
of social categories in determining what roles an
individual has. His emphasis on the non—isolability of
roles and on how the ideal patterns of expectancy are
met in actual behavior weaves together some strands which
were present in the conceptions of Mead and Linton, and
also presages a line of interest which flowered in the

next two decades in empirical research on role conflict.

Norms.

A later contribution to thinking on role conflict
was made by Samuel A. Stouffer, in which he pointed out
that multiple roles may well place an individual under
the restraints of incompatible norms.31 This way of
conceiving the question shifts attention from consideration
of a role as a (usually coherent) bundle to the elements
which determine or compose the bundle; i.e., Stouffer was
not much concerned with roles as units, but rather with
the consequences of their composition. One particularly
influential consequence of this is noted in the statement,

It is the viewpoint of this paper that the

range of approved or permissible behavior as
perceived by a given individual is an important

315. A. Stouffer, "An Analysis of Conflicting
Social Norms," American Sociological Review, 14 (1949),
pp. 707-170

 

55

datum for the analysis of what constitutes a
a social norm in any group, . . . 2

And in his concluding observation,
From the theoretical standpoint, the

most important implication of this paper

may stem from its stress on variability.

. . . it is common and convenient to

think of a social norm as a point, or at

least as a very narrow band on either

side of a point. This probably is quite

unrealistic as to most of our social

behavior. And it may be precisely the

ranges of permissible behavior which most

need examination, . . 33
The question of whether the constituent elements of roles
are in fact unitary is of considerable importance in the
sequel, and there is one observation that we ought to
make on the matter at this point. While Stouffer empha-
sized variation, this does not preclude the possibility
that norms may be taken to be essentially singular for
some purposes. His idea of a permissive range seems
also to indicate that this range has bounds and that
what falls within them is permitted, i.e., the bounds
act as cutting points to establish two areas, that which
is permitted and that which is not. His use of norm is
also terminologically interesting in that it and similar
terms have been incorporated in subsequent uses to

stand for rules in general, whether effected as rights

or duties.

 

321bid., p. 708.

33Ibid., p. 717.

56

Expectations

 

‘We have digressed somewhat from our main line of
structural emphasis and we need now to go further afield
to explore the use of another term which has been used
in a somewhat parallel way to norm, but usually with a
pronounced interest in the behaving individual who is
incumbent in the social location and performs the
associated activities in context of the relevant rules.
The reference is to the concept of "expectations" which
has played an increasing part in the last 15—20 years.
Of course, our attention is directed to the consequences
these usages have for our idea of a role system. We
will consider first three sources -— Parsons, Rommetveit,
and Sarbin. A fourth will be delayed to a summarization
point in later pages.

It may seem amiss to introduce our selections
from Parsons' treatment of role concepts in connection
with otherwise "social psychological" considerations
since his work is intimately identified with the
structural—functional disposition of contemporary socio-
logical theory and the very concept of a social system.
The reason is simply that we believe that he has been
very effective in impressing the concept of expectation
on role analysis with respect to the incorporation of

standards into action via orientations of the actor.

57
We will pass over his early statements, since he
later regarded their context as "completely obsolete."34
A number of statements appearing in the early 1950's

present similar views, and the following extracts from

The Social System are representative. After noting the

 

act (in interaction) as the most elementary unit of a
social system, and status-role as a (most) convenient
higher order unit, he says:

. . . it is the structure of the relations
between actions . . . which is essentially the
structure of the social system. The system

is a network of such relationships.

Each individual actor is involved in a
plurality of such interactive relationships
each with one or more partners in the comple—
mentary role. Hence it is the perticipation
of an actor in a . . . relationship which is
for many purposes the most significant
unit. . . .

This participation in turn has two
principal aspects. . . . where the actor
in question is '1ocated' in the social
system relative to other actors. This is
what we will call his status . . . On the
other hand there is the processual aspect,
that of what the actor does in his relations
with others seen in the context of its
functional significance for the social
system. It is this which we shall call his
role.35

 

4Specifically, a brief taxonomic paper (circa
1941) which was later reproduced in the first edition of
Essays in Sociological Theory (1949). See T. Parsons,
Esseys in Sociolggical Theory, rev. ed., Glencoe: The
Free Press, 1954, p. 12.

35Parsons, The Social System, op. cit., p. 25.
This is the first appearance of the term 'network' in any
of the extracts cited here. Parsons did not introduce the
idea nor did he work out its significance in the relevant

way.

58
As we have seen, the locational and behavioral components
were not novel. A few pages later in discussing "the
integration of the motivation of actors with the norma—
tive cultural standards," Parsons recasts role from the
actor's perspective. After remarking that value stan—
dards have both expressive and instrumental significance
to an actor in his expectation system as "role—expecta—
tions" and "sanctions" respectively, he continues,

The relation between role—expectations and

sanctions then is clearly reciprocal. What
are sanctions to ego are role-expectations

to alter and vice versa.

A role then is a sector of the total
orientation system of an individual actor
which is organized about expectations in
relation to a particular interaction context,
that is integrated with a particular set of
value—standards which govern interaction with
one or more alters in the appropriate comple-
mentary roles. These alters need not be a
defined group of individuals, but can involve
any alter if and when he comes into a particu-
lar complementary interaction relationship
with ego which involves a reciprocity of
expectations with reference to common standards
of value—orientation.36

The concept of expectation is a summary psychological
conception of the characteristics of cognitions under
repeated stimuli.

. . . action . . . does not consist only of
ad hoc 'responses' . . . the actor develops a

 

36
Ibid., pp. 38—39.

59

system of 'expectations' relative to the

various objects of the situation. These may

be structured only relative to his own need—

dispositions. . . . But in the case of inter—

action with social objects a further dimension

is added. Part of ego's expectation, in many

cases the most crucial part, consists in the

probable peaction of alter to ego's possible

action, a reaction which comes to be antici—

pated in advance and thus to effect ego's own

choices.37
It is clear from his discussion of the employment of
expectations that their essential substantive relevance
to the social system is that they become shared. How—
ever, the main point of relevance for present purposes
is that these expectations involve at least two actors
(or social locations), one who expects and another of
whom something is expected. Any explicit proposition
concerning an expectation or expectational state would
always require a predicate of second—order, i.e., the
logical character requires a major operator of relational
form. This point stands out again in the next two
formulations.

Rommetveit's analysis is a highly technical

exposition, and it is even less possible to do justice
to it here than to many other statements. The main

substantive foundation is an explication of the concept

"norm," whose formal specification is then used to

 

37Ibid., p. 5.

60
define role. (He makes residual use of status or
position, accepting it as received.) He finds three
main themes in the concept norm, viz., shared frame of
reference (from Sherif), statistical behavioral uni—
formity, and social pressure or role obligation (from
Festinger and Stouffer), and adopts the third in his
formal definition.

A social norm is a pressure existing between

a norm—sender (A) and a norm—receiver's (E's)
behavior in a category of recurrent situations
(5) manifesting itself as follows:

A expects E to behave in a specific way Xi or
A wishes E to do x. or A is satisfied (dis—
satisfied) when E 1does (not) x. or A applies
overt sanctions when E does (no ) xi

or

E perceives that A expects him to do x. or E
perceives that A wishes him to do x. 03 E
anticipates A's (dis—)satisfaction then x.
will (not) be done or E anticipates overt
sanctions from A upon performance (non—perfor—
mance) of x., x. and x. referring to similar
but not necessaEily identical modes of behavior
wiggin the class of possible behaviors x in

S.

Our reason for quoting this exact form is that the
logically relational form of the constituent elements
is suppressed in the verbal statement of his definition

of social role.39

 

38R. Rommetveit, Social Norms and Roles, Minneapolis:
University of Minnesota Press, 1954, p. 51.

39He also uses extensive symbolic notation to pro-
vide explicit definitions of concepts (reproducing it here
would require extensive explanation) which exhibits the
relational form.

 

 

.I

61

A social role is a system of social norms

directed toward one and the same individual

as member of a group or representative of a

psychologically distinguishable category of

individuals.40

In his psychologically oriented review of role

theory of a decade ago, Sarbin adopted definitions which
represent a meld of several of the themes which had

appeared (which in fact is the case with the greatest

number of the statements in the literature). We cite

 

his formulations here both for the character of their

content and as an avowed stocktaking at a time when

 

developments which are very important for our purpose
were in the offing.

The writer would regard a position in a
social structure as a set of expectations

. . . That is to say, the person learns
(a) to expect or anticipate certain actions
from other persons and (b) that others have
expectations of him. . . . In other words,
a position is a cognitive organization of
expectations . . . These expectations,
organized as they are around roles, may
justifiably be called role expectations.
Thus, a position is a cognitive organization
of role expectations.

A role is a patterned sequence of learned
actions or deeds performed by a person in an
interaction situation.

Two general kinds of expectations are found:
rights and obligations. Rights are role

 

40Ibid., p. 84.

 

62

expectations in which the actor of the role

anticipates certain performances from the

actor of the reciprocal role; . . . Obli-

gations (or duties) are role expectations

in which the actor of a role anticipates

certain performances directed toward the

actor of the reciprocal role; . . .41
It may appear that a new concept of position enters here,
but we regard it as a rearrangement of denotata rather
than any new substance. The locational idea has been
suppressed and the term used to compass contents variously
assigned to status or role by others. The phrase "cog-
nitive organization" might seem to introduce some new
moment, but the idea has been used extensively before and

the emphasis is on the character of expectations. Again

we note the explicit emphasis on the relational form.

Minority Views

We have followed a brief historical route through
a few of the salient contributions to the role concepts
and presently will make a provisional summary of the
material. Before doing so let us examine some suggestions
which have not been incorporated into general use or run
against it. Selections from two authors will suffice.

In the material above, we have seen that most
authors associate the rights, duties, obligations, norms,
expectations, cognitions, behavior patterns or other

non—locational elements of the role concepts with one or

 

4lSarbin, op. cit., pp. 225-26.

63
another of the variety of available terms, but that the
elements of similar content are usually tied to only one
of the terms for any given author. A somewhat different
view which may appear to introduce a distinctive way of
handling the point is represented in the following:
A natural distinction between role and

status . . . suggests itself, i.e., that

role is somehow associated with duties or

obligations, and status with rights, privi—

leges, and sanctions, and that there is no

necessary logical relationship between them.42
That is, in a role the "responsibility" is on ego, and in
a status it is on alter. This way of putting the matter
has a certain persuasiveness since we do frequently
associate "activity" with role.and the converse with
status, and responsibility connotes doing (or giving)
while privilege connotes passive reception. However,
the point which is missed is that while there is no
"necessary logical relationship" between rights and
duties, neither is there any among rights or among
duties. Again, the proposal seems to amount analytically
to little more than a reshuffling of terms, as the author
seems partially to recognize in the statement:

Recognition of the analytic separability
of role and status in no wise denies of course

that the relationship between the two is often
institutionalized in such a manner as to give

«gnu-r

 

 

42
Sociologus, N.S. 6 (1956), p. 30.

 

A. Pierce, "On the Concepts of Role and Status,"

64

an appearance of inseparability. It is not

uncommon however for some of the elements

comprising one to change with no related

change in the other.43
He fails to add that it is also common for such related
changes to occur. The argument seems to proceed from
observed statistical irregularity to logical independence.
The point is however that dropping the archaic use of
status as necessarily associated with privilege is pre—
cisely one of the purifications of contemporary role theory.

Another anomaly is represented by the commentary
of Argyle. His discussion is intended to be a clarifi—
cation of the concepts but leads to a view which is
incompatible with main currents. He does not provide a
concise statement of position but offers several examples,
as social categories (age, sex), membership in formally
constituted groups, and in institutionally established
structures. In terms of these,
There may be said to be a role in the

social structure sense if the behavior of

occupants of a position (as defined above)

[sic] is modally distributed . . . [for

situations] . . . and if the mode differs

significantly in adjacent positions.44

Later, he utilizes the notion of position as follows:

 

43Ibid., pp. 31—32.

4
M. Argyle, "The Concepts of Role Status,"
Sociological Review, 44 (1952), p. 41.

 

 

65

An informal group may be defined as a group

without positions —— though of course
members will occupy positions in the wider
society.45

It appears to us, however, that the notion of a human
aggregate to which the term "social" or any equivalent
locution would be applied and that was devoid of positions
(and thereby of roles) is unusual in sociological thought.
We do not refer to the "real" nature of human groups, but
to how sociologists choose to think about them, and most
do not choose this way. However, we will see later on
that when the choice is less apparent another view is
often accepted which has operational consequences

essentially similar.

Summary of Themes

We come now to the point of intermediate summari—
zation promised earlier. For this purpose we turn to the
excellent review of role and related terms which Gross,
Mason and McEachern provided as a preliminary to their
own reformulation of the concepts. They distinguish
three main categories of definitions of the term role,

". . . which, if not exhaustive, are at least represen—

tative of the major role formulations in the social

 

45Ibid., p. 43.

m 0
K

 

 

 

66

science literature."46 They are as follows:

Definitions of role which either equate it
with or define it to include normative culture
patterns . . . which includes . . . Linton's
often-quoted definition.47

 

 

. . . definitions . . . as an individual's
definition of his situation with reference
to his and others' socialypositions . . .48

 

 

 

. . . definitions which deal with role as

the behavior of actors occupying social
positions . . . this . . . does not refer to
normative patterns for what actors should do,
not to an actor's orientation to his situation,
but to what actors actually do as position
occupants.49

 

 

 

 

For brevity, let us refer to these as (l) normative,

(2) cognitive, and (3) behavioral conceptions. We use

the term "cognitive" for the second set to emphasize the
salience of the actor as the definer; "expectational"

would better suggest the content of such individual defini—
tions, but the term is also associated with ideas of
sharing or consensus, either among a plurality of actors

or as socially or culturally given.

 

46Gross, et al., op. cit., p. 11. A materially

equivalent set of categories appears in D. J. Levinson,
"Role, Personality and Social Structure in the Organi-
zational Setting," Journal of Abnormal and Social
Psychology, 58 (19597, p. 172.

 

 

47Ibid.

48;p;g., p. 13. Expectations are included here.

49Ibid., p. 14.

67

It seems more accurate to regard these ways of
approaching the definition of the concept of role (and
of status as well) as themes or components of the con-
ceptions which recur with frequency in the literature.
Sometimes more than one of these ways of looking at the
question will appear in the same explicit statements,
or the same author will employ each in different places
as he needs to for different purposes.50

There are a number of points to be noticed about
these different conceptual elements. First, the concept
of role is not in itself taken to be logically primitive,
i.e., there is a clear consensus that it is not to be
viewed analytically as an undefined term or element, but
that it should be defined as a collection, or in a more
conventional formal term a "set," of other elements
(which usually are taken as formally undefined, though
some specifications of them may be made). Second, it is
not possible in general to choose from among these kinds
of elements; there is not consensus as to which ones
should be included in the "real" definition of role, each

being required for different purposes. Third, in these

 

50 . . .
E.g., see Parsons, Essays in Soc1oloq1ca1
Theory, rev. ed., op. cit., pp. 230, 337, 393f, The
Social System, op. cit., pp, 25f, 38f, and SLEHCENEE and
Process in Modern Societies, Glencoe: The Free Press,
1960, pp. 171, 251.

68
terms, it might well seem that there are in reality a
series of definitions of role (rolel, role2, . . . ,
rolen) and that there is no point of connection among
them (though there may be substantial connection in
specified theoretical contexts). We intend to argue

that this last point is incorrect.

Attributive and Relational Concepts of Role

The line of the argument was indicated above,
and the particular point of attack was taken up in
connection with the concept of expectations as elements
of role definitions. It has been suggested that the
properties of the logical character of the concepts
taken as elements of role concepts provide a basis for
agreement among the ways in which the otherwise dis—
parate elements are used. What we are particularly
concerned with is what we will call attributive and
relational uses of the concepts.

Attributive concepts are ones which attend
explicitly (or entail operations which require) only a
single object (position, actor) in making specifications
of a role. Consider the way in which we might describe
an occupational role (with any of the "primitive" ele—
ments) such as machine operative in a factory: the
occupant of the position must carry out the actual

performances needed to run the machine, do "a fair day's

 

 

69
work" (though the actual specification would be much
more precise), be prompt, and perhaps we should include
that he display courtesy and friendliness. The accuracy
or completeness of the role description does not con—
cern us, but the form of the statements shows that it
is possible and quite usual to specify characteristics
of a role by stating attributes or prOperties of the
incumbent of the position.

On the other hand, relational concepts are ones
which attend to two or more objects in specifying a
role. The same machine operative would take instructions
from a foreman, receive stock to be processed from fellow
workers and pass completed pieces to still others, and
most particularly courtesy, friendliness, etc., would
specifically imply his appropriate demeanor vis—a—vis
both peers and superiors. (Obviously, attributive and
relational concepts are of a much more general order
than exhibited in this example. Logically, they corres—
pond in syntactic analysis to first- and higher—order
predicates respectively.)

It is our contention that any of the generally
used kinds of elements of role definitions can be
analyzed either attributively or relationally. In
general on the relational side which is our focus of
interest, this follows in almost all formulations from

such assumptions or remarks as "reciprocal patterns of

70

behavior," "selt—other context," "polar positions," etc.
This seems to give little difficulty with respect to
normative or cognitive components. The normative ele—
ments are most usually factored into rights and duties,
and it is sufficient to note the frequent "one's rights
are another's duties." Similarly, cognitive elements
become role—relevant by virtue of their employment in
orientation to alters, as noted above for expectations.
It may be thought that behavioral components present
some difficulty here, since the idea of a "pattern of
behavior" of an actor does not require any notion of
present or even relevant alters. But it is, exactly
on this point that behavioral definitions of role
should be challenged.51

Here we need to take note of a similar formula—
tion by Nadel in order to make clear what is and what
is not intended by our distinction between logically
attributive and relational characteristics of role
definitions. In his very important contribution to
role theory, which we have not referred to explicitly
up to now, Nadel defines two sub-classes of "logically

related" roles in the following way.

 

5le. remarks by M. J. Levy and M. J. Daniels
in the citations in footnote 20, and Cottrell's defi—
nition cited above.

71

One is constituted by roles in the case of
which the logical relation also implies, with
logical compulsion, an actual relationship
between the respective actors. [i.e.] . . .
a given role . . . requires to be enacted
vis—a—vis another counterpart or correlative
role. The second extreme is constituted by
roles in the case of which the logical rela-
tion merely means that actual relationships
between the actors are possible, not in any
logical sense necessary. . . . I will call
the first type of roles dependent, and the
second, independent roles.52

 

 

. . . dependent and independent roles repre-
sent only extremes, between which all con-
crete roles are ranged along a continuous
scale or at least one of many degrees.53
First, we need to notice what he means by "logically re—

lated" roles. He refers by this to sets of roles which

have the same differentiae in the sense of belonging to

 

the same class (e.g., ones that are all occupations).
Nadel has reference to substantive categories, whereas
our use of "logical properties" refers to the analytic
character of role elements. .Second, the distinction
between dependent and independent roles, depends on
whether or not an "actual relationship" is more or less
required, and again this is not what we have in mind by
the attributive-relational distinction. In our View,

roles always entail "actual relationships," albeit

 

525. F. Nadel, The Theory of Social Structure,
Glencoe: The Free Press, 1957, pp. 79—80.

53Ibid., p. 84.

72

possibly not with others which only share the property
of having been named (or thought of) in terms of the
same differentiating concepts by the culture in question.54
And when we speak of a relational definition of a role,
we mean that relationships are to be considered as the
analytic components of a role, and thus it would be
formally incorrect to speak of the relationships as
between roles.

It should be clear that we have not "taken sides"
as to £233 are the contents of roles, but have only
tried to show pgp_the major kinds of such contents can
be handled as to their logical form. Further, we have
not argued that role elements can have only the form
of a relation or only the form of an attribute. It
seems quite clear that both are useful, but for different
purposes. One would use attributes in order to describe
a particular role, or the qualities or performance of an
incumbent of the position, especially with regard to
recruitment into it. That is, it is quite possible when

dealing with a single actor to make many substantively

 

54We should note that "relationships" does not
have the meaning for Nadel which is connoted by the first
passage quoted above. Elsewhere, he defines them as
"determinate ways of acting towards," with the emphasis
on a rule—determined form, and the discussion between
the two passages quoted shows the same intention here.

73
significant statements about characteristics and
behavior without taking explicit notice of any other
actors; and it is possible to extend this analysis to
a collection of actors, describing the properties and
actions of each without reference to any of the others
explicitly, and even to compile the "data" so as to
show summary characteristics of the collectivity.
However, if it is desired to treat such a collectivity
as more than an aggregate, then some explicit notice
must be taken of more than one of the members at a
time, i.e., one must have recourse to statements of a
relational form. What we have tried to show is that
such statements are (or more precisely, may be) about
roles.

Thus, from the point of view of logical form,
we wish to extract a certain common moment from the
concepts of roles, and to say, provisionally, that
they are sets of relations among positions, and that
the substantive content of these relations may be one

or more of several kinds.

Positions and Actors

There are two other terms which have been much
used in our review and discussion, and which we will
require in the development of our model, viz., position

and actor.

74

We have observed above that the concept of a
position enters most discussions as an undefined term.
It is sometimes given some suggestive specification
as a "location in a set of social relationships" or
some similar expressions. However, this is not usually
accompanied by specific instructions on what is to be
understood by "social relationships." Since we have
already taken relations as the formal component elements
of roles, one might begin to wonder whether one or the
other of the terms "role" and "position" is redundant.55
Our answer, which will be developed in the next section,
is that while they can be analyzed in terms of the same
contents and are connected, they are not redundant.

As a preparation for this and in order to com-
plete our review of system elements, let us take notice
at this point of the exceptional definition of position
made by Biddle.

A position is a set of persons who exhibit
similar characteristics, who are treated
similarly by others, or for whom a cluster

of unique cognitions are maintained either
by themselves or others.56

 

55Nadel argues that the static—dynamic distinc—
tion is "not only redundant but misleading," since it
amounts to knowledge-performance of the same rights
and duties, but this is not the same as the question
here. Ibid., p. 29.

56Biddle, op. cit., p. 5.

75
The reference to "a set of persons" is unique, and
replaces explicit mention of "location" or the like.
Biddle remarks that definition by persons or by "loca—
tion in a social system . . . are equivalent in deno—
tation," but does not go on to show why or how this is
true. The definition which we will develop in the
formal model displays the correctness of his assertion
regarding the two modes of the definition. A point of
interest about Biddle's and all other definitions of
position known to the present author is that the positions
in a group, system, or structure are always assumed to
be fixed. Moreover, in most uses they are assumed to
be given, i.e., not open to question by the participants.
As we shall see later, positions differ very markedly
from roles in this respect, and we shall have reason to
inquire why this should be the case.

We turn now to the term "actor." Most conceptuali—
zations of the role concepts are not definite on this
point, but it is usually clear in context that the authors
intend the range of their terms to take in living humans
in social groups. Whether other objects are allowable as
referents is often uncertain, but they are not in general
precluded. Part of the indecision here stems probably
from the fact that most writers seem to have essentially
attributive "pictures" of the concepts in mind and to

attend primarily to the "ego" of the ego—alter pair

76
assumed as the fundamental context. This seems to lead
naturally enough to semantic restrictions of the actor
to single and potentially existent persons. On the
other hand, there are frequent references to the possi—
bility of "empty" (i.e., unoccupied) positions, or
"mythical" ones (i.e., occupied by symbolically signifi-
cant but unreal figures conceived by the participants).
It seems to be necessary to pay attention to the possible
incumbents of the "alter" in order to determine the range
of permissible referents. We do not need to attempt to
decide what kinds of objects can allowably be actors,
since we take the term as formally undefined in our
analysis. However, it is intended that it would include,
operationally, any symbolic referents capable of being

treated as alters.S7

The Role System
We turn now to the problem of how elements of a

role system are related to each other. Definitions of

 

57In a footnote, Mead makes an observation about
the composition of the generalized other which is of
interest here. "It is possible for inanimate objects
. . . to form parts of the generalized . . . other for
any human individual, insofar as he responds to such
objects socially. . . ." G. H. Mead, op. cit., p. 154.
Imaginary persons as incumbents of positions
are used not only by participants, but also by field
workers. See M. J. Herskovits, "The Hypothetical
Situation: A Technique of Field Research," Southwestern
Journal of Anthrepology, 6 (1950), pp. 32—40.

77
role concepts which were generally in use up to about the
middle 1950's postulated a "primitive" positional element
and associated with it in a one-to-one way a collection
of contents which taken with the position constituted
the status and/or role (or office, or other similar
term). Further, it was frequently suggested that each
such status was paired with another status similarly
constituted of position and associated contents in a
dyadic relationship. Usually, no other general specifi—
cation was made and the matter was simply left at the
assertion that each status and position was associated
with one role, and vice versa. Sometimes, following
Linton, it was said that a collection of all of a person's
statuses constituted his concrete status.

A number of different authors have presented
largely congruent reformulations of the relationship
between status and position and role. Merton, who was
one of these writers, remarked in a felicitous phrase
regarding his own reconception that it involved ". . .
initially a small . . . shift in the angle of vision
. . . ," and he gave to the formulation what has come
to be its distinctive title, "the role—set." The full
quotation, which in context was preceded by a restate-
ment of Linton's definitions, follows.

It is at this point that I find it useful

to depart from Linton's conception. The dif—
ference is initially a small one, some might

78

say so small as not to deserve notice, but it
involves a shift in the angle of vision which
leads, I believe, to successively greater
differences of a fundamental kind. Unlike
Linton, I begin with the premise that each
social status involves not a single associated
role, but an array of roles. This basic
feature of social structure can be registered
by the distinctive but not formidable term,
role—set. To repeat, then, by role—set I mean
that complement of role-relationships in which
persons are involved by virtue of occupying a
particular social status.58

Merton's succinct term has persisted, but his own and
others further analysis and use is somewhat clouded by
the fact that the term has been taken to refer to the
set of persons incumbent in the reciprocal positions,
rather than to the set of roles as such which seemed to
be implied by the passage above.

In terms of priorities, it appears that honors

for this "small shift" must be accorded separately 1) to

an article by F. L. Bates59 (which was known to Merton60
and to Gross, et al.,61 whose ideas were also known to

each other), and 2) to a less noticed paper by J. E. Haas.62

 

58Merton, "The Role-set . . . ," op. cit., p. 110.

9Bates, op. cit.

60R. K. Merton, Social Theory and Social Structure,
op. cit., p. 368, n. 115.

61Gross, et al., op. cit., p. 69, n. 26.

62Another development of the same line of argu—
ment, though in quite different terminology, which appears
to be innocent of any knowledge of these is contained in

79
Bates way of developing the underlying idea has certain
defects which we will consider in connection with a
topic of the next chapter. On the other hand, Haas'
presentation is very compatible with the views we wish
to develop, and we will set out the essential points of
it in the following extracts.

. . . in any group where the members interact
repeatedly we can find sets of normative
specifications for behaVior (norms) which apply
to distinct units of social interaction. Such
units . . . refer to the interaction of the
persons who occupy positions within the group.
We may refer to a set of normative specifications
for the behavior involved in a unit of social
interaction as a role. . . . when role is
defined in this manner, the concept 'position'
is usually left undefined. . . . [but] . . .
It is simply a label we give to a whole

system of normative specifications which apply
to a number of units of interaction.63

 

 

 

 

He then introduces the position "doctor" to show how
various roles (as doctor-patient, doctor—nurse, etc.) of
the same position may differ in many respects, but are
related by the fact, at least, that the incumbent "plays"

all of them.

 

Ward H. Goodenough, "Rethinking 'Status' and 'Role';
Toward a General Model of the Cultural Organization of
Social Relationships," Association of Social Anthropolo—
gists Monographs: l, The Relevance of Models for Social
Anthropology, London: Tavistock, 1965, pp. 1-24. "This
is a revised and expanded version of a paper entitled
'Formal Properties of Status Relationships' read at the
Annual Meeting of the American Anthropological Association,
16 November 1961." Ibid., n. l.

63J. E. Haas, "Role, Position, and Social Organi-
zation: A Conceptual Formulation," The Midwest Socio—
1ogist, 19 (1956), p. 34.

 

 

 

80

Thus it seems useful to define position as a

cluster of roles that are conceived of as

belonging together. . . . It refers to no

more or no less than the normative specifi—

cations of which its roles are composed.

. . . A role may be viewed, then, as being a

segment or part of each of two positions. In

the course of describing a position by indi—

cating the roles of which it is composed, there

will be an explicating of at least some seg—

ments of the other positions involved.64

Here, then are the main points of the "break—through"

for defining roles and positions as common elements of a
system. Roles are made up of elements (norms) and posi—
tions are made up of roles, and consequently positions
are also made up of the same elements, though in larger
aggregates. We have tried to explicate the logically
relational character of roles by the requirement of
always designating two positions when specifying the
elements of -— and thereby defining -— a role. It is not
clear that Haas believed that the same requirement applied
as well to a position, i.e., that it could only be con—
cretely specified in terms of statements which also
specified other positions at the same time, though it
necessarily follows from this way of formulation. Also,

he does not take note of the fact that defining a role

in terms of a pair of positions can be done, so to speak,

 

64Ibid., pp. 34—35.

81

from either point of view (as doctor—nurse, and nurse—
doctor). (These points are clear in Gross, et al., to
which we will turn presently.) Thus it develops that
position and role are to be defined interdependently,
and in fact that this is to be done in the context of
a system of them.65 Before leaving our discussion of
Haas' proposals, we must emphasize that we have not
accepted his specification of norms as the elements but
only his formulation of the connections between posi-
tions and roles. Further, contrary to his specifica—
tions,66 we retain the notion that actors "occupy"
positions (and one might say roles as well) since it
is a useful locution that can be given operational
interpretation.

The similar analysis carried out by Gross and
his associates has certain terminologic advantages for
us, though there are conceptual shortcomings in their

treatment (which we shall analyze more fully in the

next chapter) concerning positions. Foreshadowing

 

65Biddle, and Coult after him, has made a point
of insisting that position and role ought to be defined
independently, though for reasons that have nothing to
do with whether they are defined interdependently or not.
In the sequel we will find that it is precisely through
this interdependent definition that we are able to show
that Biddle's definition of position cited before is
equivalent to the locational definition. See Biddle,
op. cit., p. 4, and A. D. Coult, "Role Allocation,
Position Structuring, and Ambilineal Descent," American
Anthropologist, 66 (1964), pp. 30-31.

66Haas, op. cit., p. 35.

 

82
Biddle, they employ a definition of position as "the
location of an actor or class of actors in a system of
social relationships,"67 but their definition is in—
complete in not making clear the nature of the "social
relationships." Their definition of role is "a set of
expectations [which are "evaluative standards"] . . .
applied to the incumbent of a particular position."68
Formally, this is equivalent to Merton's "role—set" in
its original sense and to Haas' "cluster of roles."
They introduce the term "role—sector"69 to correspond
to what Haas and Merton (and Bates) called "role," and
we will adopt the term in connection with specification
of the model to avoid the misleading "role-set" appel—
lation. In their analysis, the author's recognize
that a set of related positions is formally constitui—
tive of a system and that it is necessary to make
"relational specifications of positions."70 In their

definitional development, however, this amounts only to

the enumeration of other positions which are related to

 

67Gross, et al., op. cit., p. 48.

68Ibid., p. 60.

"A role sector is defined as a set of expecta-
tions applied to the relationship of a focalyposition to
a single counter position." Ibid., p. 62. On focal and
counter positions, see below.

 

7OIbid., p. 50f.

83
a given position (again, see the remarks in the next
chapter). Though they define a "system model," their
main analysis is concerned with a single "focal posi-
tion (school superintendent) and its relationships to
any other "counter" position(s). Thus they refer to
their model as "position—centric."71 While their
definitions do not suit our present purposes precisely,
we will adopt the terms "focal" and "counter" to denote
the necessary ordering of positions, etc., noted in the
summary of Haas' concepts. We will use these terms to
modify "position," "role," and "role-sector" as needed,
and when the qualifiers are not used we will intend that

the term(s) be understood as referencing both aspects.

The Graphic Model of Role Systems

In order to complete our review of role concepts,
we need to take notice again of a feature which charac—
terizes most of the conceptualizations, and particularly
those most directly concerned with the structure of the
system, viz., the implied or explicit spatial analogy.
We do this to underline the essential correctness of
the redefinition of position and role as interdependent

and to indicate how the commonly used spatial conception

 

71
Ibid., p. 51f.

84

is inadequate. We should emphasize that, even though
our treatment in this section is informal, the shift in
the focus turns our discussion away from the substantive
concepts and toward major characteristics of their under—
lying calculus.

Many authors, especially recent ones (particu-
1arly Haas, Bates, and Gross, et al.,) use graphic dia—
grams as an aid in their expositions. The use of a l_
geometric conception of some sort seems to be required

by the locational idea. This is well expressed by ,1

 

Linton in his discussion of the concept of a social

system.72

Perhaps the nature of a social system
can best be understood if we compare it to
a geometric figure, . . . Actually, there
is nothing else within the range of common
experience which would be so closely com—
parable. A geometric figure consists of a
series of spatial relationships which are
delimited by points. These points are
established by the relationships and can be
defined only in terms of the relationships.
They have no independent existence. Each
of the patterns which together compose a
social system is made up of hypothetical
attitudes and forms of behavior, the sum
total of these constituting a social rela-
tionship. The polar positions within such
patterns, i.e., the statuses, derive from
this relationship and can only be defined

 

72"Social systems consist of the mutually adjusted
ideal patterns according to which the attitudes and behavior
of a society's members are organized. A society is an
organization of individuals; a social system is an organi—
zation of ideas." Linton, The Study of Man, op. cit.,
p. 253.

85

in terms of it. They have not more inde—
pendent existence than do the points of the
geometric figure. Any status, as distinct
from the individuals whom society may
designate to occupy it, is simply a collec-
tion of rights and duties. Thus the status
of employer derives from the relationship
between employer and employee and can be
defined only in terms of the attitudes and
behavior which the total pattern for this
relationship ascribes to this one of its
two polar positions.73

Linton's statement here corresponds closely to the
"role—set" redefinition of the concepts, and it explicitly
eliminates any idea of delimination of a particular posi—
tion outside the content of the relationships. Such a
definition of a set of points suggests that the spatial
concept is what is technically called a "linear graph."

We will deal with the concept of a linear graph more
fully in Chapter IV in preparation for formally specifying

our model, but a few points about it here should be

 

731bid., pp. 256—57.

74More familiar geometric conceptions have also
been attempted, as in R. E. Park, "The Concept of Posi—
tion in Sociology," Publication of the American Socio-
logical Society, 20 (1925), pp. 1—14, which uses a more
or less Euclidean notion; and in R. Linton, "A Neglected
Aspect of Social Organization," American Journal of
Sociology, 45 (1940), pp. 870—886, which employs a
system of coordinates.

The context of an idealized patterning may also
suggest why Linton did not utilize the graphic concept to
define a total system. If he had been interested in the
concrete "society" (see definition in footnote above),
he might have used the analogy to represent the ramified
pattern of connection between persons. In the abstract
social system it is possible to conceive of isolated
pairs as representing the structure, though there is

 

86
helpful. Roughly, it conceives of a space without
independent referents of distance and direction, and so
differs from intuitive notions of space and from Euclidean
geometry which we usually think of as their formal repre—
sentation. A drawing of such a graph shows a group of
points with lines connecting some, though usually not
all, of the possible pairs of such points. Sociographs
are an example, and are, in fact, one of the stimuli for
the study of the properties of such graphs. We have
said that there are not measures of distance and direc—
tion assumed which are independent of the particular
collection of point and lines, and this is seen in one
of the "peculiarities" of such a geometry: There are
many (in fact infinitely many) different possible drawings
of a given collection of points and lines (i.e., just so
many points and just such connections between them), and
all of them are indifferently equivalent from the point of

view of linear graph theory.75

 

certainly warrant for considering more extensive structures;
e.g., Mead's example of "the ball nine" suggests this.
However, it should be noticed that the source work on

graph theory by D. K6nig was being published in Germany

at about the same time (1936) that Linton's classic
appeared in this country; Linton did not have a properly
worked out geometry available for his purposes.

75Indeed, this point quickly became clear in early
sociography and considerable effort has been expended to
develop ways of using more usual ideas of spatiality to
produce ways of drawing sociographs which "intuitively"
display relevant properties.

87
Another recent statement by Kahn, et al., exhibits
essentially the same spatial analogy, but also brings out
more clearly the point on which the analogy is inadequate.

If the organization is a network of inter—
related roles (or, more precisely, role
behaviors), it follows that the bonds which
connect one role to another may be of many
kinds, not of formal authority alone. Two
roles may be related in terms of authority,
to be sure, but they may also be related
because of the sequence of work flow, or of
information, or because of the liking which
one person has for another, or because of
expertise. . . . The organization is a
complex network of roles connected by such
bonds of expectation and influence, some of
them reciprocal, some asymmetrical. Thus
no role in an organization is intact or fully
separable from others. . . .

We may wish to deal with a single role,
and for that purpose we figuratively pluck
it out of the network of other roles. . . .
When we do so, we find that what we have in
hand is an assortment of duties and obliga—
tions, expectations and rights which state
relationships between this role and various
others. These relational statements dangle
from the role like strands from a knot which
has been cut out of a larger net of which it
was a part. And if we try to eliminate those
bonds and define the role without mentioning
its connections to any other, we make a
startling discovery: there is virtually
nothing left. The role is defined in terms
of its relationships to others, just as the
knot in a net is no more than the intersection
of bonds and disappears if we try to trim too
closely.76

 

 

76
Kahn, et al., op. cit., pp. 388—89. We would
add that "virtually" is unnecessarily cautious —- for

their purposes, and ours, there would be nothing left at
all.

 

88

There is a point of ambiguity in the observation of plural
bonds of connection between a single pair of roles (i.e.,
positions). The graphic and pictorial concepts that are
used are of a sort suggesting a single connection between
each pair of positions. (For example, our picture of a
fish net or a spider web is of a structure such that each
line segment runs between a pair of knots or intersec—
tions, but any given pair is joined directly by at most
one such line.) But the substance declares that there
are many connections of one position to another and that
in general these are conceived as being of different
kinds. In connection with this we shall need to recon—
struct the spatial analogy and more formally the graph
theoretic conception of it in the specification of our
model.

We have remarked on the sociographic image of
the spatial conception, and at this point it is also
possible to say succinctly why we have passed over the
work of sociometric analysis in our review of role
concepts. On the one side, it has not made any unusual
contributions to the substance of role concepts, and on
the other side, while its emphasis on relational
characteristics is essentially correct, the sociographic
model is inadequate for a full definition of the logical

structure of a role system.

CHAPTER III
PRIMITIVES, DEFINITIONS, AND AXIOMS

In the preceding chapter we introduced the gen—
eral ideas involved in the redefinition of the concepts
of position and role that have emerged in the work of
several different authors in the past decade. We tried
to show in an approximate way the points of consensus
in these conceptualizations and some of the underlying
inadequacies. The task of the present chapter is to
restate the consensus in a more rigorous way and to
modify and extend it to permit the statement of a definite
model of the logical structure of a role system in the
following chapter. In order to accomplish this we

develop a somewhat informal and incomplete axiomatization.

Primitive and Defined Terms and
the Axiom of Relation

In the foregoing discussion we listed a collec—
tion of terms which we will require in our analysis, viz.,
position, relation, role, role—sector, and the terms focal

and counter which are to qualify the others as necessary.1

 

lAnother term to be used as a primitive, actor,
will be introduced later.

89

90
The first two terms will be selected as our primitives,
then an axiom will be stated which serves to formulate
the connection between these primitives, and this gives
the basis for defining the remaining terms.

A few words seem to be required about our
selection of the primitive terms (especially since the
choice of one of these, position, may seem to contra—
dict the results of the last chapter where we emphasized
the interdependence of position and role). In general,
there is no formal basis for our specific choice of
primitives here, since others in the main set could be
substituted, or even a single term employed as a primi—
tive, and an equally rigorous and formally satisfactory
development could be made which would provide a logically
identical result. However, it is our judgment that the
two selected provide the best intuitive basis for the
exposition of the system. Moreover, they, they or their
analogs of whatever name, appear most frequently in
functionally equivalent uses in the substantive concep—
tualizations reviewed from the literature. 'In addition,
these terms can be coordinated most easily to coores—
ponding primitives in the standard treatment of the
mathematical equipment which we shall introduce in the
next chapter.

There does not seem to be any difficulty in

accepting the intuitive validity of treating relations

91
as primitive when they are understood in the sense of
their logical use in the discussion of role concepts
which we sketched in the preceding chapter. That is,
rights and duties, norms, obligations, expectations,
etc., and even behavior patterns, when taken as the
elements of status, position, role, office, etc., can
always be interpreted as Operationally requiring a predi—
cate of second degree, i.e., one which needs two arguments
to make a complete expression;2 and such concepts are
always used logically in this way. However, the situation
is a bit different with respect to position.

As we have seen in the theoretical statements,
the positional concept is indeed frequently taken as
undefined, or rather it is effectively assumed that the
reader will assign a meaning to it from his own intui—
tion or experience and that there is a widely shared
meaning (which is generally true since sociography).
However, in our review we applauded the recognition
(specifically by Haas, and by Kahn and associates, but
also by Linton) that a position can be denoted only
definitively by specifying all of its relations with

other positions. Since the same relations also make up

 

2Again, it may well be asked why we should stop
at second degree, when examples requiring higher orders
can easily be given. Provisionally, the answer is that
most conceptualizations do not make this extension.
However, it is still a relevant tOpic and we return to
it in the final chapter.

92
the roles, position and role are interdependent in their
use. Now it would seem that either position is a more
"basic" concept and role is defined in terms of it, or'
they are specified interdependently, but that we cannot
have it both ways at the same time. The conflict is only
apparent, however, and in fact arises from an ambiguity
which shows up in the use of position.

The abstract or ideal concept of position refers
to a location or point of some space, the location being
capable of specification relative to other points. Put
another way, the concept refers to the possibility of
such a point, but not to the existence of any one in
particular. It is in this sense that position is to be
used as a primitive. This concept of position leaves
open the question of where any given position ie located.
In any space, the particular location of a point is not
given by the fact that it is there or even by naming it
(unless the label implicitly or explicitly includes
further information, as in the use of coordinates to
designate the elements of a Cartesian system). "Locating"
requires the use of rule(s), and in the kind of space
which we need here, positions are "located" with reference
to each other. That is, knowing where (or what) a
singular position is requires knowing where it "stands"
relative to the other positions in that system. And this

means knowing what its relations are, or in general what

93
the relations are in the system; this is the same as
knowing what (or where) the roles are in the system.
It is in this sense that we say that positions and roles

are specified interdependently. Empirical knowledge of

 

roles in a system implies positions, and vice versa.
Turning now to some hitherto neglected considera—
tions, we note that in any role system the number of
positions and the number of kinds of relations by which
positions are to be specified are both finite, though
usually of different magnitude. It follows that the
number of relations of whatever kinds which connect any
pair of positions, and the total collection of all such
connections, are also finite. Either this point is
trivially obvious to most theorists, or they do not
accept it, or it is of no consequence in their intended
analyses, and this last alternative certainly seems to
be the likely-one. At any rate, the present author is
aware of only one writer who makes any explicit remark
on the matter. Bates says that, "Within any given
culture there exists a limited number of roles which
were combined in various ways to compose a limited number
of different positions."3 A position ". . . is associated

with a set of social norms," and a role is "A part of a

 

3
Bates, op. cit., p. 315.

94
social position consisting of a . . . sub-set of social
norms . . .," i.e., of the ones for that position.
Thus, positions and roles are taken by him to be finite
(limited) in number in a given system; it is still
possible that he intended to take the number of norms
as unlimited, but this hardly seems likely.

A less obvious point is implied by our use of the
phrase ". . . kinds of relations by which they are to be
specified . . ." in the paragraph above. In general, we
assume that any of the finite set of relations is a
candidate for the connection between any given pair of
positions, and that all of the positions (and roles, etc.)
in a given system can be definitely and completely des—
cribed in terms of some set of relations, properly enum-
erated. On one side, this point is not at all clear in
most statements, since most only deal explicitly to any
extent with a dyadic model;lacking an attempt to define
a total system, there is little interest in considering
whether the set (of whatevers) has a parent, or how it
compares analytically to other such sets. On the other
side, this specification seems to be no more than a
sensible operational requirement. A related assumption
which we shall use is that each relation is binary, i.e.,

that it either applies to a pair of positions, or it does

 

4Ibid., p. 314.

95
not. This is mostly for simplification, and we will see
later that it can be relaxed somewhat.

With this background on our primitives (as a
result of which one might way that they are not quite
undefined),we may state a fundamental postulate of our
model of a role system, which we will designate as the
axiom of relation. Consider any relation, Ri’ and any
two positions, Pj and P (which are not necessarily

k

distinct), each having been selected from its appro—

k

not, and Ri either joins Pk and Pj or it does not.

Taken together these two conditions result in four

priate set. Then Ri either joins Pj and P or it does

possible combinations which we could symbolize as
follows, using "ii" to show that the relation does not
join the pair.

1) PjR.P and p R.P.

i k k 1 j’
or 2) PjRiPk and PkRin,
or 3) PjRng and kain’
or 4) PjR'iPk and PkRin.

This display shows a feature mentioned in the
preceding chapter but not yet recalled here, viz., that
in general relations are taken to be "directed," which
position comes first and which second must be specified.
This feature is required for our formal definitions of

role-sector and role. In the following we will refer to

96

an expression of the form "PjRiP " as a "predicate" and

k
one of the form "Pj" (or "Pk") as an argument for ease
of reading.

1) A role—sector is a collection of predicates
whose first arguments are identical and
whose second arguments are identical.

2) A focal role is a collection of predicates
whose first arguments are identical.

3) A counter role is a collection of predicates
whose second arguments are identical.

4) A role is a focal role and a counter role
whose respective first and second arguments
are identical.

We should notice that each role contains at least one
role—sector, and that it may contain only one, in which
case the denotata of all four terms are the same. Also,
though "Ri" does not appear explicitly it is accounted

for as the remainder when a relevant set of "Ri" has

been collected.

'The IncumbencyeAxiom

 

We now need to formulate explicitly some assump—
tions of considerable importance as a foundation for the
subsequent developments of our analysis. We shall refer
to these assumptions as "the incumbency axiom." It

requires an additional primitive term, "actor," which we

97
introduce without comment at this point. The axiom may
be stated in two parts. Assume that Ai represents an
actor and Pj and P represent two positions.

k

1) Either Ai occupies Pj (or Pk or both) or

he does not. (There is no such thing as

a slight case of incumbency.)
2) If Ai occupies Pj’ then for any role—sector

of P.,
J
constitute the role—sector with some incum—

Ai has all of the relations which

bent of Pk.

Insofar as the writer is aware, this precise
statement of the assumptions does not appear in this
form in the literature of the concepts with which we
are concerned. In fact, some of the drift of theorizing
in this area might well be said to run counter to our
axiom. We must therefore consider its desirability with
some care. Let us begin with some general considerations
and then proceed to examine arguments against it, but
hopefully also some for it, from the literature.

We have said previously that our task in this
work is taken to be one of concept formation (or refor-
mation). In the question at hand, we believe that a
clue to desirable characteristics of our concepts is

available in measurement theory. Even at the lowest

level of measurement, i.e., what is usually called a

98

nominal scale, or otherwise may be thought of as a set
of categories, we require that the categories be
mutually exclusive. That is, that any object for which
the particular scale of measurement is relevant may be
assigned to one and only one of these categories. We
have said that there is a pipe_quite deliberately in
order to avoid giving the impression that measurement
considerations are to take precedence over others, or
that they are a directly adequate model of the empirical
reality particularly as it is, so to speak, experienced.5

What we do want to suggest is that the criterion
of exclusivity carries the kernel of the problem we
confront when we have to decide whether two persons (or
one person in two different situations) are involved in
the same role—sector. The concept of exclusivity implies
that with respect to the variable to be measured (which
may be complex), the object in question has only that
set of properties associated with a particular category
of that measure.6 In our present case, exclusivity would

suggest that the same role—sector exists for actors A and

 

5Nadel does hold a view of this sort in his con—
cept of logical differentiae of role frames. Nadel,

op. cit., p. 74.

 

 

6This way of stating the point may at first
seem strange, since we most frequently think of a measure
as unidimensional, as involving only one property.
However, consider the well-known model of a Guttman scale,

99
B and for actors C and D, only if the same relations hold
between A and B that also hold between C and D.
As an example of a possibly contrary view, let
us consider the case which Bates puts forward in arguing
that the content of a role may not be fixed. The propo—
sition is stated as a "postulate" of his system, viz.,

Postulate 3. . . . When a given role is a part
of two different positions, it tends to contain
slightly different contents in each position.7

He argues that this is justified by two earlier postulates,8
which are stated as follows.

Postulate 1. Within any given culture there
exists a limited number of roles which were
[sic] combined in various ways to compose a
limited number of positions.
Corollary: A. Some roles, though not
all, are found as parts of a number of
different positions.
Corollary: B. Within any given culture,
a given role tends to contain basically
the same norms regardless of the position
of which it is a part. . . .

Postulate 2. Within any given position there
tends to be a strain towards consistency or
adjustment between the various roles composing
a position.
Corollary: A. In the long run, roles
which are part of the same position and

 

which is given as a model for extracting a unidimensional
ordinal scale from a multiplicipy of properties. Further,
notice whathalternatives exist when "the data doesn't
scale." We may either Seek a multidimensional account of
the scale and non-scale (or quasi-scale) types, or we may
treat them as categories of a nominal scale.

7Bates, op. cit., p. 316.

81n which case, of course, it might well be
called a theorem.

100

also inconsistent . . . will either be

changed . . . or at least one of the two

inconsistent roles will be eliminated . . .

Corollary: B. In the short run, various

devices or mechanisms may be develOped . . .

which will allow two inconsistent . . .

roles to be maintained as a part of the

same position.9
There is a logical problem here that we must dispose of
before turning to the empirical example with which Bates
buttresses the postulate (or theorem) cited above. In
his definitions of the concepts of position, role, and
norms, he states that a position "is associated with a
set of norms" and that a role is "A part of a position
consisting of a more or less integrated or related sub-
set of social norms . . . distinguishable from other
sets . . . forming the same position."10 In this con—
text, the "norm" is the element and the content of a
"role" is a particular sub—set of such elements. The
phrase ". . . more or less integrated . . ." refers to
relationships between these norms, not to which ones
are included. However, in the statement of the postu—
lates, he has in effect given primacy as an element to

the concept of role.

If a given role is combined with different
other roles to form two different positions,

 

91bid., p. 315.

lOIbid., p. 314.

101

then, its content, (i.e., the norms of which

it is made) must be slightly different in

each case in order for it to be consistent

with the unique contents of each position.ll

The difficulty in his formulation is that "role"

has been defined in two different ways, once in the
explicit definition, and then again in Postulate l, but
at what we usually call two different levels, or at
least in two different ways, of abstraction. When a
particular role is part of a position, for Bates it
partakes of the "unigue contents" of that position. There
is no question raised here as to whether the same role in
the same position may vary in its contents. But a ques—
tion does arise when this "same" role appears in two
different positions: In what sense are we to take this
as the "same" role? In the context of the passage under
consideration and elsewhere,12 the reason for this
assertion regarding the sameness of roles is that the
term is being considered at two levels at once, culture
and group structure. What is "known" about the culture
is applied to the group structure in order to "see" that

the two roles are the "same." Now we must make very

clear that the present argument is not inveighing against

 

llIbid., p. 316.

12
F. L. Bates, "A Conceptual Analysis of Group

Structure," Social Forces, 36 (1957), pp. 103—111.

 

102
the obvious and considerable benefits for substantive
theory and empirical analysis of such an application of
knowledge. But it must also be recognized that a precise
rendering of concepts does not permit their indiscriminate
use, and in the present case the sameness of the roles as
parts of positions in different group structures does not

arise solely from those structures when defined as sets

of related positions.13

In order to recapitulate these remarks, let us
consider the concrete example used by Bates.

Let us suppose the 'A' represents the role
of 'disciplinarian' and that position 1 is the
position of father in a family while position 2
represents that of foreman in a factory. What
Postulate 3 states is that the father's role as
disciplinarian differs from the foreman's role
as disciplinarian since it is combined with
different other roles in each position. . . .
let it be supposed that the other roles in the
father's position include 'teacher,' 'playmate,'
'spouse of mother,‘ and 'provider'; while those
of the foreman include 'representative of
management,‘ 'technician,' and 'friend.' It is
seen that the father's role of disciplinarian
is conditioned by a totally different set of
other roles than in the case of the foreman.

In order for it to be integrated into each
position and to be consistent with other roles
contained within that position, it must vary
in its contents.14

13At the same time we should note that there is no
implication in Bates' analysis that a role as a culture
element exhibits any intrinsic variability at that level.
We can only assume that it ie_a particular sub—set of norms.

l4Bates, "Position, Role, and Status: . . .,"

op. cit., p. 316.

103
In Bates' conception, each of the named roles shown in
single quotes is one of the "limited number of roles"
available from the culture, and these are put together
in the position. We have raised the question whether
"disciplinarian" is the same role in "father" as in
"foreman," and we have said that at the level15 of group
structure there are two different roles, which are

referred to by the same identifying term and indeed bear

 

considerable resemblance to each other because, so to
speak, they are drawn from the same cultural model.16

At this point, it is somewhat anticlimactic to
observe that Bates' postulation of role variability
actually does not contradict our axiom of incumbency as
it is stated above. The axiom only requires that any
actor who occupies a particular position will have its
roles as given, and we have seen that he does not assert
that "the same role in the same position" will vary in
its contents. So, in effect, we have argued for a bit

more than we need for immediate purposes in suggesting

that the "same" role in two distinguishable positions

 

15"Level" is an unfortunate term here, but it is
hallowed by usage; a more accurate rendering might be
"when considering" or "in reference to."

1
6These remarks should not be taken to mean that

our explicated model does not apply "at the level of
culture." Specifically, there seems little doubt that
the elements of culture that Bates has in mind come in
reciprocally connected pairs.

104

must have the same content. Later it will be seen that
in its narrowest sense our graphic model defines the
logical structure of a role only with respect to the
specific positions with which it is associated. The
argument has been carried through both to help bring
out what the axiom does require and to show how this
particular argument for role variability ought to be
construed.

"Cultural" identification of positions and roles
is also involved in the next set of views which we need
to examine in connection with our axiom. We refer to
the arguments put forward by Gross, Mason, and McEachern.
The questions which their discussion raise flow most
immediately from their denial of "the postulate of role
consensus," but we will need to review other points of
their presentation as well.

They argue (with good cause) against the idea
that the expectations for and patterns of behavior of
the members of a group are social or cultural givens,
and/or that there is complete agreement among members on
these matters. In their third chapter they have amassed
evidence that a number of theoretical discussions are
committed to such assumptions, and they cite relevant
empirical evidence which confounds the position. We are
concerned here with whether their criticisms argue against

the advisability of the incumbency axiom.

105
There are a number of interrelated questions to
be examined, and they stem generally from what Gross,

,17 By

et al., have called the "role definers problem.‘
this they mean that in any empirical research, the
investigator must specify whose definition(s) of the
expectations which are taken to constitute the roles
in question are to be analyzed. Their denial of role
consensus as a postulate is based on the empirical fact
that a plurality of definers may disagree as to how
these expectations apply.

One of the keys here is the term "plurality."
It is quite clear that the authors do not consider the
question of consensus to be relevant if only a single
role definer is being considered as such, whether it be
a person, or an actor more generally (as a collectivity),
or the modalities of a social system in general.18
Ambiguity or vagueness in the cognitions (in Biddle's
sense) of a specified definer are not viewed as theoreti—
cally relevant, and they would probably be treated as

residual errors of measurement, though the authors are

not explicit on this point. In effect, for a specified

 

l7Gross, et al., op. cit., p. 70.

18They do not allow this generality explicitly,
but the intention of their scheme requires it: ". . .
a special point was made of defining concepts without
specifying whether they applied to individual, social,
or cultural phenomena." Ibid.

106
role definer and a specified position incumbent the
expectations for that incumbent vis—a—vis a specified
alter "are what they are," and this is true for all
specified alters.

Epie much agrees with the second part of our
axiom, but what of the first part? Here there seems to
be little difficulty.19 Their views seem clearly and
completely in support of the assertion that an actor
either occupies a position or he does not. However,
there is another element involved in this view of posi—
tions which lends further and substantial credence to
the second part of our axiom, which deals with role(s)
in positions.

We may express the authors' views on positions by
saying that they have adopted a "postulate of position
consensus." Particularly, if a person is a superinten-
dent of schools or a member of a board of education,
they assume that all role definers (or perhaps all
rational ones or the like) will agree to the allocation
of such positions. But what does it mean in their scheme
for a person to occupy a position? It is not simply that

certain "labels or identities"20 are uniformly applied,

 

19The same remark applies to our discussion above
of Bates' scheme, though we did not take note of it here.

2OIbid., p. 40.

107
but that the incumbent is located in a system of rela-
tionships with others; this latter feature is, of course,
their own formal definition of a position. The essential
question here is ppp_positions are located. We are not
referring to the way in which an investigator decides
which positions to study or what descriptions of them
he provides; rather we are asking 1) what it means in
operational terms for an investigator to locate a
position in a system 222.2) what additional consequences
are assumed for the incumbents' behavior?

Gross and his associates provide only a partially
explicit answer to the first part of this query in their
theoretical discussion, but some suggestions about the
answer to the second part may be made on the basis of
their use of the theoretical concepts in their own research.
They propose "two aspects of position specification, the
relational and the situational."21

Relational specification in its fullest use amounts
to the enumeration for a given position of all other posi—
tions to which it is related. We assume they mean this
to be a dependent aspect which requires the situational
specifications; it is difficult to conceive how the
enumeration could be made without some knowledge of the

situation. But the operational meaning of situational

 

21Ibid., p. 50.

108

specification is not made clear in their theoretical
remarks, which consist mainly of examples of "situational
context" but do not provide a hard definition.22 We
believe that this is because, in practice, the concept of
situational specification does double duty in 1) setting
the limits of generalization, which is the salient facet
in their theoretical analysis, and 2) providing constraints
on the system which, in effect, explain and justify the
relational specification. The latter function becomes
apparent in their empirical analysis.

In considering the specifications of the

object population, it is pertinent to make

certain observations about the kind of

system of social relationships in which the

positions of school superintendent and school

board member are involved.23

Following this remark, they list five features of

the board system and discuss their general consequences
for the structure of the situation. That is, at a fairly
general level and without mentioning the term explicitly,
they enumerate features whose operational and behavioral
consequences can only be that they set certain expectations
for incumbents of the positions.

We have gone to some trouble, and perhaps the long

way round, to say that the authors have turned the postulate

 

22Ibid., p. 56f.

23Ibid., p. 99.

109
of role consensus away from their front door, only to
readmit it at the servants entrance. Gross and his
associates have denied the "postulate of role consensus"
and declared that it is a matter for empirical investi-
gation, and one cannot take exception with this. But
when confronted with evidence of consensus, they take
no notice of its consequences for their definitional
system. If we are correct in believing that their
description of a school system as ". . . a form of
organization that has mpg: of the characteristics of

. . . bureaucracy,"24 entails l) uniformity of 2) certain

 

expectations regarding positions, and given their defi—
nition of role as ". . . a set of expectations applied
to an incumbent. . .,"25 then it seems that there is
consensus and it is about roles.

It seems fair to say that the authors recognize
that it is necessary to assume a "fixed" core in the
structure of a system,26 but they choose to conceptualize
it in position and to assignoelements of variation to

role. In some respects this seems a very natural usage.

The concept of position is usually taken as similar to

 

24Ibid., pp. 99—100.

25Ibid., p. 67.

26"That the members of a social system, whether a
dyad or a total society, must agree among themselves to
some extent on values or expectations is a matter of
definition." Ibid., p. 43.

 

110
"point," and the idea of unambiguous fixity is clearly
contained in the common meaning.27 If a role is thought
of as "the dynamic aspect of status," it is easy to
assimilate the idea of change and variability to it.
However, the trend of conceptualization in recent decades
has been toward taking the concept of role up more firmly
into "structure." Even conceived as expectations there
is no inherent reason for requiring it to "soak up the
variance" unless it is tacitly assumed that there can
be no variability about position epg_that it is necessary
to locate variability in one or the other. The present
analysis will try to suggest that there is no need to
locate variation in either. Both position and role can
be taken up interdependently in a model of a "fixed"
structure, and variability can then be conceptualized
in terms of the structure(s) as such.

These remarks have led us ahead of our immediate
interest, and explanation of them must be delayed. There
is yet one aspect of the usages of Gross, et al., to be
noted here. We refer to the use of the "identities" of

positions which were mentioned before. In actual practice

 

27Cf. Bates remarks on the "imperfect spatial
analogy" of having an actor occupy more than one position
in a structure. Bates, "Position, Role, and Status: . . .,"
op. cit., p. 313. However, we do not accept his argument
herein; it is a bad example of allowing representational
techniques to decide form, rather than suiting them to
content.

111
it appears that they believe that the names of positions
are to be used to locate them, even though ". . . posi-
tions have been defined as locations of actors in systems
of social relationships. . ."28 It is of considerable
interest to inquire whether this use of positional labels
is necessary or even possible in all cases. The answers
would seem to be that names are quite handy if they
exist, but that it is quite possible to conceive of an
actor or actors in a system of social relationships
whose location is not manifestly identified by partici-
pants. It seems perfectly sensible to conceive of a
latent structure of positions which are neither "intended
nor recognized" by members of the system.29 The only
possible way then to locate the actors would be in terms
of the "social relationships" themselves. Indeed, this
is what is done in studies of "informal organization."
That is, actors are "located" in the "group structure"

in terms of one or a few relations of whatever content.3O

 

28Gross, et al., op. cit., p. 49. Emphasis added.

29we use "latent" here in the sense defined by
R. K. Merton. The phrase "latent structure" may be infeli—
citous since it is used by P. F. Lazarsfeld in another sense.

3OMany examples could be cited but a few will suf-
fice here: A.Zaleznik, et al., The Motivation, Productivity,
and Satisfaction of Workers, Boston: Harvard University,
Graduate School of Business Administration, 1958; R. S.
Weiss and E. Jacobson, "A Method for the Analysis of the
Structure of Complex Organizations," American Sociological
Review, 20 (1955), pp. 661—68; H. White, "Management Con-
flict and Sociometric Structure," American Journal of

Sociology, 67 (1961), pp. 185-199.

 

 

112
This is a question of much interest to our analysis in
other respects and it is taken up elsewhere, but it is
introduced here for its bearing on the advisability of
our axiom of incumbency. What can it tell us on the
matter?

An unequivocal answer cannot be given, but the
general indications are positive. The term role is
rarely used to refer to patterns of the relations
analyzed in such studies, though the formal definitions
of the concept which we have reviewed would not preclude

this usage. The term position is used frequently, but

 

in a way which parallels the absence of role, viz., as
the location of 32 actor rather than as, so to speak,
the location of a position. When a collection of actors
are observed to have the same pattern of relations with
others, they are usually designated by "group" or some
similar locution, but not as occupying a common position,
although, again, this would be permissible in the usual
definition. However, the main point of interest here is
what happens to disagreements cpncerning relations among
actors, and the answer is that they are taken up as part
of the structure. "Contradictions" may exist, but they
are seen as reflecting different positions (and perhaps
dissensus as well). In terms of our axiom, if an actor
occupies a particular position, he does so in virtue of
the unique pattern of relations for that position vis—a-

vis others which actually defines the position.

113

We have reviewed a number of substantive con—
siderations, including potential objections, regarding
the axiom of incumbency, and we conclude that it sur—
vives these tests. In addition, these considerations
have served to give further direction to our reinter—
pretation of the concepts of position and role and to
suggest features which the conceptualization must
represent adequately.

In an important formal sense, it is unnecessary
to interpret our analysis in this chapter as justifying
the terms and axioms. In the formal structure of a
deductive system, such conceptions are established as
satisfactory by leading to desirable propositional
consequences, in the present case by assisting in the
development of a structural model which properly reflects
important concepts. It will be the burden of later

analysis to show that this is the case.

 

 

CHAPTER IV
THE FORMAL MODEL

In the last chapter, we set out a verbal axiomatiza—
tion of the concepts of a role system which we believe is
adequate to compass the structure we have detected in the
sociological literature. The work of this chapter is to
coordinate these results with an appropriate mathematical
representation which we can study to learn whether such a
representation has properties satisfactory for our needs.

We shall specify two analytically equivalent mathematical
representations, but we will find that they are incomplete
and require some extension to represent our concepts and

that this leads to as yet unanalyzed problems.

Linear Graphs and Matrices

 

Our discussion of graphs and matrices in this
section is mathematically informal, but it is still
technical and quite compressed. There are some impor-
tant points which we wish to emphasize at the beginning
so we may ignore them when it is convenient to do so in
the exposition. We have said that we will specify two

equivalent representations. However, there is really

 

only one system —- the same axioms apply to both

114

115
representations. In Nagel's sense, each of the represene
tations, graphs and matrices, is an interpretation of the
axiom system (i.e., the "calculus"), and each is thus a
model.1 It is useful to describe both since each aids our
understanding in different ways. For example, our primi—
tive terms (position and relation) seem to "come through"
better in the graphic representation, and our defined terms
(role and role—sector) and qualifiers (focal and counter)
do best in the matrix display, though all of these terms
can be discussed in either representation. By convention,
the axiom system is associated with the graphic model. In
view of this one might think of the matrix representation
as only a convenient alternative to the "real" thing.
However, we use the matrix interpretation heavily in the
sequel, and it seems best to the writer to acknowledge that
it ie_equally as legitimate as the graphic.

The theory of linear graphs deals with a kind of
geometric structure which is outside our ordinary intuitive
conceptions of spatiality, though it is somewhat familiar
to sociologists since a sociograph is a specie of the

general kind. Linear graphs are pictorially displayed as

 

lNagel, op. cit., p. 90f.

2Also, the reader familiar with matrix theory
will see that our matrix representation admits only a
small part of the operations of that mathematical disci—
pline. The possible confusion from our use of the term
"matrix" is regretable, but we have not been able to
devise an alternative.

116

collections of points and of lines which join them. They
differ from some other geometric structures in that there
is no assumption of ancillary arbitrary referents by

which distance and direction are determined.3 A point
does not have a location in the sense of the familiar
Cartesian or polar coordinate systems of analytic geometry.
Neither does a line have any length, regardless of the

way in which we might draw it in a diagram, though we
might say that each line has length "1". However, a

point does have a location relative to all the other points
in the system in terms of the lines which join them.

The example of sociography may suggest to the
reader characteristics of such collections of points and
lines which were not strictly intended in the preceding-
paragraph, viz., that each line goes from one point to
another point, so that we might say that it is "directed,"
and also that there is at most one line directed from one
point to another and/or another line in the reverse dir—
ection. In general, we will assume that all lines are
directed,4 but we will pg; assume that only one line may

be directed from one point to another. The term "linear

 

3Concepts of distance and direction are used in
analyzing linear graphs, but they are defined in the
system itself.

4Actually, undirected graphs can be taken as a
special case of directed graphs in which there are always
lines in both directions between a pair of points, if there
is one in either direction; i.e., they are point symmetric.
However, they are of great interest in applications, and the
case of graphs with single undirected lines has been inten-
sively studied because of its importance in physical systems
analysis.

117

(directed) graph" is usually restricted to ones in which
there is only one line in a given direction between a pair
of points. (The term "ppilinear graph" seems to us to be
better.) Also, the terms "edge" or "arc" are usually used
for lines (and we may sometimes call them "arrows"), and

a line is said to be "incident from" or ". . . to" a

point to show its orientation. Finally, a line incident
from and to the same point is sometimes called a "100p,"
and two lines incident from and to the same points are
said to be "parallel."

As we have indicated, another mathematical structure
which can be shown to be completely isomorphic with a uni—
linear directed graph is a matrix, or more specifically a
square binary matrix.5 A matrix is a rectangular array of
elements such that each element can be uniquely denoted by
reference to the row and column in which it appears, that
is, the element lies at their intersection, which is
called a cell. A square matrix is one which has the same
number of rows and columns; a chess board is a pictorial
example of a square matrix with eight rows and columns.

By calling such a matrix binary we mean that there are
only two kinds of elements which can appear in it (one or

the other in each cell), and we usually use the digits

 

5See L. Katz and J. H. Powell, "Probability Dis—
tributions of Random Variables Associated with a Structure
of the Sample Space of Sociometric Investigations," Annals
of Mathematical Statistics, 28 (1957), pp. 442-48.

 

118
"0" and "1" for representation. The rows and columns are
denoted by assigning the integer digits from "1" to "n"
(where n is the number of rows and of columns) to the
members of each set, usually in order and starting from
the upper left, so the first row is the top-most hori—
zontal array, the first column is the left-most vertical
array, and so on.

We can coordinate the elements of a unilinear
directed graph and of a square binary matrix in the fol—
lowing way. In a graph of n points, we assign the digits
1, 2, . . . , n, to the points one—to—one in any arbitrary
order. We associate a given point of the graph with the
row and the column of the matrix which have the same
digital referent, which serves to represent the originating
(rows) and terminating (columns) aspects of the points
with respect to the edges which join them. We then
inspect each edge in the graph, noting the numbers of the
points, which we use to determine the row ("from") and
the column ("to"), i,e., the cell of the matrix in which
we are to place a "1." Having inspected all edges and
entered 1's in the appropriate cells, we enter 0's in all
remaining cells. This is because we earlier had assumed
that there either is or is not an edge of given orienta-
tion between each pair of points. Obviously, a similar
procedure could be followed to "map" in the converse way,
from a matrix to a graph. The following illustration

shows a unilinear directed graph of three points, with

119

arbitrarily chosen index digits and edges, and the

corresponding matrix representation.

 

Graph Matrix
E:::::::3r. 2 1 2 3
,1
l ' 1 o 1 o

 

 

2 l 0 1
<3; 3 3 0 0 1

In order to provide a corresponding matrix repre-

 

 

 

 

 

sentation for a multilinear directed graph (i.e., one
which possibly has more than one edge of given orientation
between any given pair of points, or more precisely, one
which has a plurality of different kinds of possible
edges) we need to think of having a separate matrix for
each of the kinds of edges represented in the graph. We
might think of these as appearing like sheets of paper in
a file drawer, each with the same square two—dimensional
matrix form, but each referring to a different one of the
possible kinds of lines used in the multi—graph. This
suggests extending our matrix into three dimensions, each
plane of the third dimension (besides rows and columns)
representing an edge type. Mappings could then be con—
structed by an extension of the foregoing procedure, from
the multilinear directed graph to the three—dimensional

matrix and conversely.

120

Coordination of the Systems

 

We have described graphs and matrices in an infor—
mal way, and we now need to show that they are indeed
mathematical representations or structures which are
formally equivalent to our conception of the structure of
a role system as stated in the axiomatization of the last
chapter. For this purpose, we examine the primitives and
axioms of multilinear directed graphs and show that they
can be coordinated to our previous results. Our source
is the recent treatment by Harary, Norman, and Cartwright.
We should note that these authors use the term "net" to
refer to what we have called a multilinear directed graph,
"relation" to refer to our unilinear directed graph, and
"digraph" to designate a "relation" with no "loops."6
Their terms are more economical, but we have avoided them
in the preceding description for several reasons. First,
the term "net" conjures an image of only single strands
between knots, which seems to us to be misleading. Second,
we have used the term "relation" in our earlier discussion
and did not want to assign it prematurely to the corres—
ponding mathematical structure. And finally, the restric-
tion of graph theory to "no loops" is not completely
standard (this may be their reason for coining the term

"digraph," which is not used elsewhere). In a sense, all

 

6Such a "digraph" maps into a "hollow" matrix ——

one with 0's on the principal diagonal, the set of cells
each of which has its subscripts identical.

121

of this is by the way, since we will mainly use the matrix
representation in our development. However, matrix theory
includes a good deal more than our rather restricted
representation, and the graphic isomorph is required for
its axiomatization.

The four primitives of nets (and also of rela—
tions and of digraphs) are:

P . A set V of elements called 'points.‘

P2: A set X of elements called 'directed lines,‘
or more briefly, '1ines.‘

P3. A function i whose domain is X and whose
range is contained in V.

P4. A function 5 whose domain is X and whose

range is contained in V.

The first two of these primitives are self-
explanatory. The second two relate the lines to
the points by means of two functions f and s which
serve to identify the 'first' and the_ 'second'
point of each line respectively. . . . In general,
for any line x of X, the image fx of the function
f is called the first point of x _and the image sx
of the function 5 is the second -point of x. Thus
every line of a net is directed from its first
point to its second point.

 

The axioms for a net are:

A
A

1' The set V is finite and not empty.
2. The set X is finite.
The first axiom excludes consideration of an
empty net with no points at all and of a net with
an infinite number of points. Then the second
axiom avoids nets with a finite number of points
but an infinite number of lines. These two axioms
impose no restrictions on the structure of a net

other than the number of its points and lines.7
It will be seen that this is not manifestly identical with
out earlier systematization, so we must examine the

several items to see how the coordination may be made.

 

7F. Harary, et al., Structural Models: An Introduction
to the Theopy of Directed Graphs, N.Y.: John Wiley & Sons,
1964, p. 5.

 

122

for points and P2 for directed

lines may be taken as equivalent to our positions and

First, the primitives Pl
relations respectively. Next, the primitives P3 and P4,
which specify the functions £_and e for first and second
points of edges respectively, serve the same purpose as
our axiom of relation. For our purposes, the earlier

way of stating the assumptions (N.B., primitives are
assumed) is intuitively preferable in that it displays a
catalog of possible decisions which can be taken as
determining entries in the ce11(s) of a matrix. Also,

it explicitly denotes the kinds of relations (edges)
which may appear, though this will later be shown to be

a matter of formal indifference to the theory of "nets."
However, the formulation by Harary, et al., is clearer in
showing how the concept of direction is defined. Finally
their axioms Al and A2 appeared in our commentary, though
not as explicit elements of the axiomatization, with the
exception of the provision that there may be no edges in
a graph. This provision is perfectly acceptable though
only trivally interesting.8 Thus, we see that the formal
concepts of a role system are mathematically represented

in the theory of "nets" (or in our term, multilinear

directed graphs).

 

8It does raise a point which we have not mentioned
before, in that one usually thinks of a position in terms
of its, so to speak, manifest relations with other posi—
tions. But when one shifts attention to the total set of

123

It is also interesting to consider the import of
two additional axioms which we do not use. The first is
required for the theory of relations, and both for the

theory of digraphs, as well as all primitives and axiom

Al above (A2 can be derived from Al and A3).
A : No two distinct lines are parallel.
3. 9
A4. There are no loops.

Either of these would be mischievous in our system, except
in special cases, and we are thus debarred in general
considerations from the aid of the theories of unilinear
graphs. Unfortunately, it is precisely these topics
(particularly systems requiring axiom A3) which have
received most extensive treatment.10 It can fairly be
said that the mathematical theory for role systems has

yet to be written. However, the theories of relations

and digraphs are still of no small comfort since they can

 

relations which may possibly exist between any pair of
positions, it becomes clear that a position is also
defined by the relations it does pg£_have with others.
And the idea of a null position is quite intelligible as
a residual category which is not part of the connected
system. These ideas will become somewhat clearer in

the sequel.

9
Ibid., p. 9.

10We should note that the modern treatment of
these topics as graphs is usually dated from the work
of D. Kdnitg (in 1936) who suggested the name and began
a systematic study of their properties. See C. Berge,
The Theory of Graphs and its Applications, London:
Methuen & Co., 1962, p. ix.fi

124
be applied to single relations (which we shall not do to
any extent in the present work) and they generate concepts
which can be applied by extension to the multirelational
case, and we shall note these as we go along.

We have coordinated our primitive terms and
axioms to the mathematical representations, and we now
turn to coordinating our defined terms which denote
various forms of the concept of role. To do this it
will be convenient to shift our attention to the matrix
representation. We have described this as a three—
dimensional array of cells, each of which contains
either the element "1" or the element'WL" The file
drawer example will again be handy. On each page
(which we will subsequently refer to as a table) in our
file, we have a square matrix whose rows and columns
represent positions in what we may now call their focal
and counter aspects respectively. If we consider a

particular cell (say the ' " one, where i and j are

'i’jth
integers between 1 and n) in a table and its counterparts
in all other tables we may call this a cell vector, and
the pattern of relations reflected by the contents of
these cells represent a role sector (one of two, the
other being the "j, ith" one,1ufless i=j) for the ith and
jth positions. If we consider a collection of all cell

vectors whose firSt position identifications (subscripts)

are the same, then we shall call the collection of

125

patterns of relations a focal role; a counter role is
defined similarly for second subscripts. Such a collec—
tion of cell vectors from the same row (or column) of

our supermatrix is represented by a horizontal (or
vertical) plane, and is itself, of course, a rectangular
matrix. Finally, a role is coordinated to the collection
of contents of a set of two such planes, a row plane and
a column plane with same subscripts (only one subscript
is required to identify a plane).

Thus, as anticipated, we have developed a repre-
sentation of a role system which provides interdependent
specification of positions and roles. That is, we may
think of the concepts, as Linton suggested, as indicating
different aspects of the same content; we might think of
positions as denotative, and roles as definitive. To
rephrase the original aphorism, a position is a locational
aspect of a role, and a role is a substantial aspect of
a position.

In order to have a compact way of referring to
this three—dimensional array (or as we called it above,
supermatrix), we shall adopt some notational conventions.
We will call such a representation of a role system a
position matrix (without prejudice to other concepts, but
in anticipation of some other terms to be adopted later),
or a P—matrix, or just P as the circumstance permits.

On occasion, we shall have need for reference to sub-sets

126
of cells of the sorts used above to define concepts, and
we will indicate them by appending appropriate subscripts,
the first for rows, the second for columns, and the third
for tables, with dots (.) to show a collection over an
entire dimension (as pij' for a cell vector, and P..k

for a table).

The Incumbency of Actors

To complete the main outlines of our formal
model, it remains to consider the representation entailed
by our incumbency axiom developed in the last section of
the preceding chapter. It will be recalled that it was
stated in two parts; the first guaranteed that each
actor could be determinately assigned or denied occupancy
of each position in the system, and the second required
an invariance of relations, that an incumbent of a
position must have each role sector with at least one
incumbent of the relevant other position (N.B., we do
not use the terms focal and counter here, as the
incumbent's position must be considered in both aspects).
There are two features of this axiom, one regarding
each part, that we have not mentioned before, and they
are somewhat easier to conceive when a complete repre-
sentational scheme is in hand.

The first point is that the axiom does not

assume that there are any actors occupying the system.

127

We need to be able to conceive of systems abstractly
(this was the concept Linton had in view in discussing
social systems, and the statuses and roles which were
to compose them), and the P-matrix does this. For
this purpose, the incumbency axiom is supernumerary.

Secondly, while the axiom specifies invariant
relations between the actor and some actors in other
relevant positions, it is silent concerning what rela—
tions he has with the remaining actors in those other
positions. This is quite reasonable since we cannot
require, for example, that a given adult female be mother
to all children of her community, nor that a foreman super—
vise workers in departments other than his own. In addi—
tion, we ought to allow the possibility, say, of friend-
ship in either case. This exhibits the possibility that
these residual alters may be the salient ones for our
hero or heroine in some other position. However, further
examination of these features would take us into a con—
sideration of problems which will be delayed until the
next chapter. We return to the present question of the
representation of incumbency, with these observations as
an aid to see what is implied.

The first part of the axiom, again, says that
every actor is or is not incumbent in every position.
It seems quite natural to display this in a binary rec—

tangular matrix, in which we will take rows to refer to

128

actors and columns to refer to positions, with a "l" in a
cell to show that the actor occupies the position and a
"0" otherwise. We will call this the incumbency matrix,
or the I—matrix, or I, in a way similar to our notation
for the position matrix.

The second part of our axiom of incumbency needs
a bit more attention as it yields an additional matrix
representation which is not quite so obvious from its
statement. The representation has the same form as P, and
in particular the same number of tables. But it need not
have the same number of rows and columns (they will be
equal to the number of rows of I, say m), and its precise
contents are in general not specified from a given P and
I because of the qualifying "egme other actor" in the
statement of the axiom.- An example of this would be a
P representing the skeletal line of an organization
(though most organization charts would not be so complete),
and an I representing the roster of duty assignments of
employees. However, the axiom does guarantee the

existence of this matrix (which we shall call the actor

 

matrix, or A-matrix, or A, as before), which would be
constituted as follows. There will be one table for

each table of P, and there will be one row and column
for each row of I, and each plane will be coordinated

appropriately.

129

We can demonstrate the existence of A with the
following algorithm. For a given row (or columm) of A,
search the relevant row of I; each "1" encountered resides
in a column of I which designates a row (or column) of P;
search this row (or column) of P; each non—empty cell
vector of P designates a pattern of relations and the
column (or row) designates a column of I; search the
column of I and note the row of each "1"; these rows
designate columns (or rows) of A, and at least one of the
cell vectors formed by their intersection with the
originally given row (or column) of A must contain the
pattern of relations secured from P. This completes the
proof of the existence of A.

We should notice particularly that empty cell
vectors of P are idle in this proceeding. This results
from the requirement of "exhaustiveness" of the second
part of the axiom, that each actor of a given position
must be properly connected to some actor of the other
position in question for the role sector to be affirmed
(and this holds for both focal and counter aspects). It
exhibits a fact of considerable importance, that the
axiom of incumbency works a marked effect on the concept
of a role system. This can be conceptualized by observing
that, in general, A cannot be obtained as a set of single—
valued functions of the I and P matrices (as our proof of

the existence of A has shown in the indeterminacy of role

130
sectors between actors), i.e., symbolically A # g (I,P);
but on the other hand, I and P must jointly be determined
by single-valued functions of A, i.e., (I,P) = f(A). We
say "functione_of A," because in general the "f" of our
equation must be thought of as complex, using several such
functions11 (and iteration). (A proof of the existence of
"f" is that one such set of mappings is developed in the
next chapter.)

The more important side of the point is a substan—
tive one. When we accept the incumbency axiom we thereby
accept the consequence that pge_role systems for an
(empirical) group are definitely determined by the relevant
relations specified between the actors in the group. This
emphasizes that role systems are analytic constructs of
the sociologist, even though he will prefer to construct
them in a way which he believes coincident with or repre—
sentative of reality. There is more to be said on the

subject and we return to it in the final chapter.

 

11The mathematical concept of a single—valued
function involves what is called a "mapping" of the
elements of one set into the elements of another set,
under the restriction that each element of the first
must go to only one element of the second, but not
conversely. The first set is called the "domain" and
the second set the "range." The concept is used in
the quotations from Harary, et al., above.

131

Comparison With Other Formalizations

We have specified a formal representation which
we believe to be the correct logical model of the concepts
of role systems. It is now of interest to inquire pro—
visionally about its comparative virtues vis-a—vis other
attempts in this direction. First, let us notice that
every other formulation of which the present author is
aware assumes, sometimes explicitly but usually not, the
theory of graphs or in its most general form "nets."
Since our development, not including the incumbency
axiom, uses the theory of nets as a model, but no more,
all other formalizations must be "special" cases thereof,
i.e., prOperly explicated they are either isomorphic or
utilize additional axioms. If they incorporate additional
axioms, the resulting formal structure will be richer,
but we must question whether it then properly represents
our ideas of role systems in general.12

There are two other formalizations of which the
author is aware that are intended for the same purpose in
general and are sufficiently developed to warrant considera-

tion.13 These are contained in l) a series of articles by

 

'T

12An excellent example of a richer axiom set is
H. White, An Anatomy of Kinship, Englewood Cliffs:
Prentice Hall, 1963. We should be clear that this
author's purpose ie restricted.

l31t is believed that the present analysis is an
acceptable explication of the intentions of Hass, Gross,

 

132
Bates, beginning with the definitional one frequently
cited above, and 2) two papers by Oeser and Harary.
Though they are quite different, they share certain faults.
Chief among these is a functionalist orientation which
leads to a focus on goals (Bates) and tasks (Oeser and
Harary), and in each case this vitiates the possibility
of achieving a proper structural analysis. Equally
unacceptable are the different restrictions made regarding
assignment of actors to positions in a group, but in each
case the restriction is a locution more than a reality
since they subvert it at the first turn, each in their
own way. We will consider each conceptualization as
briefly as possible.

Bates'definitions of the role concepts have been
cited in Chapter III and they do not need to be repeated
here. His extension of these concepts into a definition
of group structure is made in the next two papers in the

series.14 The first of these specifies the concept of a

 

et al., Merton, Goodenough, and Kahn, et al., in the works
cited in preceding chapters, or rather of what their
intentions would have been if fully carried out.

14F. L. Bates, "A Conceptual Analysis of Group
Structure," op. cit., and "Institutions, Organizations,
and Communities, A General Theory of Complex Structures,"
Pacific Sociological Review, 3 (1960), pp. 59-70.
Respectively, these are referred to as "Group Structure"
and "Complex Structures" in the next footnotes.

133

group in terms of position and roles, but the intent and
implications of this only become clear in the second. He
states his definition of a group in terms of two conditions.

Condition 1. A group consists of at least two
individuals who interact with each other as the
occupants of two positions, each of which contains
at least one role reciprocal to a role in the other
position.

 

Condition 2. A group is composed of eil individuals
who occupy positions reciprocal to all other positions
in the group structure and includes no individuals

who gp_pgp meet this condition.15

 

We may interpret this in terms of our own model as follows.
It imposes a restriction on the P—matrix such that it can
be partitioned into sub—sets of positions which are non—
overlapping and dense. By dense we mean that each sub—set
has no empty off—diagonal cell vector,16 and by non—over—
lapping we mean that all cell vectors not contained in such
sub—sets are empty. (It appears that he would also intend
the principal diagonal to be empty, but the point is not
clear.) These requirements may not be immediately clear
from the conditions, but it may help to recognize their

formal equivalence to the definition of a clique as a

maximal sub—set of persons in a sociomatrix who enter into

 

15
"Group Structure," pp. 104-5.

16
."Off-diagonal cell vector" means Pij" such

that 1 ¢ j.

134
reciprocal (friendship) choices.l7 There may be many
such cliques and they may overlap with regard to persons.
However, in Bates' application the positions of each
group are taken to be unique, the linkage between groups
being performed by multiple incumbency of individuals p22
in positions of several different groups. Apparently,
these specifications are motivated by a consideration
introduced in the earliest of his papers, viz., "The
concept of social position depends on an imperfect epatial
analogy since it allows a given individual to occupy two
positions in the same social space at the same time."18
Congruent with this, "the same social space" can compass
only a single group, and other related groups simply are
not in the same or even determinately relatable spaces,
i.e., not in terms of the same operations used to define
each separate space. However, his later usage suggests

that the concept of a group only serves to "save the forms,"

to retain the idea of singular incumbency without serious

 

17L- Festinger, "The Analysis of Sociograms Using
Matrix Algebra," Human Relations, 2 (1949), pp. 153—58;
and R. D. Luce and A. D. Perry, "A Method of Matrix
Analysis of Group Structures," Psychometrika, 14 (1949),
pp. 95—116.

18F. L. Bates, "Position, Role, and Status; . . .,"
op. cit., p. 313, emphasis added.

135
prejudice to the definition of larger enclaves. In an
example used to show group linkage he remarks of the
individuals performing this function, "In other words,
these two actors occupy two positions in the structure of

e_multigrogp system."19 In connectionwifli all of this, we

 

should notice that in our own constructions of the A and P
matrices, each actor and position do occupy unique loca—
tions in the corresponding multigraphs. Multiple incum-
bency is accounted for in the I matrix, and it employs a
quite common spatial concept (in the mathematical sense)
of a mapping of one set of elements into another. In
part, Bates' construction stands on an imperfect under-
standing of spatial analogies and an attempt to fit his
concepts to a predetermined form, rather than allowing
them to generate the form they require.

A similar criticism applies to the formalization
offered by Oeser and Harary.2O In our view, they have
stated a model in search of a theory, but not the one

specified in the titles or context of their articles.21

 

l9
"Complex Structures," p. 60. Emphasis added.

20We do not mean that their spatial concepts are
incorrect; Professor Harary has been one of the foremost
contributors to the development of graph theory, and we
would hardly challenge his mathematical credentials.

21O. A. Oeser and F. Harary, "A Mathematical Model
for Structural Role Theory, I," Human Relations, 15 (1962),
pp. 89—109, and ". . . , II," . . . , 17 (1964), pp. 3—17.

136

Their model does seem to codify some common conceptions
about the structure of formal (complex, large—scale,
purposive, bureaucratic) organizations, but it is not put
forward as such. Fundamentally, the difficulty arises
from their effective acceptance of the proposition that
the structure of a role system is given by the formal
(purposive) organization, which they take to exist in
every social structure, and that this is only subject to
ancillary modification by the informal organization.
Since they take the formal organization to be represen-
table by a collection of related digraphs (which is quite
conventionally accepted), they are able to take the
theory of digraphs as the representational system for
their construction. It is in this sense that we say that
they have fitted the substantive theory to the formal
model in an unacceptable way; the substantive theory is
wrong to begin with. However, it is always possible to
get the right results for the wrong reasons, but a brief
further consideration of their development will show that
they have not represented role concepts prOperly as a
general structural system.

The following quotation is a summary description

of their concepts.

 

elements: persons hi), positions (p ),
tasks (tk). J
relations: R , between persons (the M

 

graph);

137

R1, between positions (the P
graph, or organization chart);

R2, between tasks (the T graph,
or work layout);

R , between persons and positions
(the H—P graph, or personnel
assignment);

R4, between positions and tasks
(the P-T graph, or task allo—
cation);

R , between persons and tasks
(the H—T graph, or induced
personnel assignment).

roles: There are three kinds of role.

(a) The informal role of a
person, R' (h.);

(b) The formal role of a posi—
tion, R' (p.);

(c) The actugl, or operational,
role of a position, which is
the second as augmented by the
the first.22

The general bent of their formulation which we noted above is
rather well indicated by the content of the quotation. All
graphs are unilinear except H, which represents the "infor—
mal social relationships" between persons, and it is not
given extensive consideration. R3 is a function which
assigns each position to one person, but not conversely.

However, R4 allows overlap so that a task may be assigned

to more than one position, and in consequence R allows

5
overlap of person-to—task assignments. R3 is obviously not
a satisfactory formulation of incumbency, but R5 immediately
recovers some of the necessary content.23 Their formal

 

22Ibid., p. 6.

23R is not even a good statement of their own "intui—
tive" observations set out in the first article regarding posi—
tions. For an example of the strain to employ digraphs, see
Ibid., p. 8, n. 4.

138
statement of the definitions of role shows the consequences
of their assumptions.

D3. The informal role pine person, R'(h.), con—
sists of all relationships entailing hi, in

the set R0.

D4. (a) The formal role pine pgsition, R'(p.),
consists of all relationships entailinng..
(b) The actual g£_0perational role pine

position is

 

 

 

AR'(pi) = R'(pj) U R'(hi)

where person hi is assigned to position p..

In other words, the actual or operational

role of a position is the augmentation of

its formal role by the informal relation-

ships of its office—holder.24
This makes difficult reading, as there are several technical
problems as well as substantive ones. The use of R' through-
out is misleading since the relations referenced in D3 and
D4(a) are different, being R0 in the former and R1’ R3, and
R4, in the latter. The appearance of i as a subscript on
the left side of D4(b) is either a misprint or wrong, in

consequence of R and their own verbal statement following

3
the definition. With this correction, it will be seen that
in addition to the uniqueness of every position we have a
correspondingly unique actual role associated with that
position, and only one. Further, while it might be possible
to compare positions (in the P—T graph) for equivalence of
task assignments, and thus arrive at some higher level of

classification of them, there is no formal provision for

comparison of different positions in terms of "informal

 

24Ibid., p. 9.

139
social relationships" for equivalence of roles. And each
of these features must be counted as a defect when com—
pared to actual sociological usage.

There is one final point of comparison with other
models (explicit or otherwise) with which we may close
this chapter and provide a motivation for the next. It
appears that the only other intentional use of epy formal
specification lige the incumbency axiom is the one used
by Oeser and Harary in the material we have just reviewed.
This axiom seems to us to be an absolutely necessary con—
ceptualization for representing the concepts of role
systems in their fullest intention, and it leads to
interesting and potentially rewarding problems which it

is the business of the next chapter to explore.

CHAPTER V
MAPPING ACTORS INTO ROLE SYSTEMS

In the preceding chapter we noted that the incum—
bency axiom generates a relationship between the A matrix
and the I and P matrices that we expressed as (I,P) =
f(A). That is, ii it is assumed that a set of actors
occupy a role system, SEER it must be possible to deter-
mine their incumbency and the positions, roles, etc.,
from the information contained in the A matrix. This is
a consequence of the theoretical assumptions embodied in
the incumbency axiom. As is the case with any formal
postulates of a theory with empirical intentions, the
incumbency axiom is subject to revision and rejection on
evidence of inadequacy. The incumbency axiom, or some
systemic equivalent, must serve as the first step in
specifying the rules to coordinate the purely constructural
concept of a role system to intended ranges of relevant
phenomena. It is the purpose of this chapter to examine
implications of these considerations.

We observed earlier that the displaying of a
candidate mapping procedure in this chapter furnishes

proof that the (multiple) function exists. This display

140

141
will not constitute proof of the uniqueness of the set
of techniques. It is nevertheless possible that they
are a unique solution. In fact, it is the author's
belief that certain of the steps in the procedure to
be outlined do exhaust the possibilities permitted by
the axiom at those points, but they require a further

assumption not stated and not implicit in the axiom.

Mapping Procedures

 

Assumption of Exhaustiveness

 

If we consider a single cell vector of A,
which contains the information about the relevant
relations between a pair of actors taken as focal and
counter, then it seems natural to suppose that the
second part of the incumbency axiom directs us to
take this as a role sector to be mapped into P. How—
ever, the conditional form in which the proposition is
given (which see) only permits this, it does not
reguire it. "Permits" is perhaps an inadequate term
here, since it does not forcefully exclude other
alternatives. That is, what the second statement in
the axiom is intended to mean is that any cell vector
of A can be used only for a determinate assignment of
the relations as a role sector, and of the actors each
to positions in the I matrix. If the cell vector of A
is not used for this purpose, then it has no other

significance, i.e., it is ignored.

142

This possible ambiguity could have been circum-
vented in the statement of the axiom simply by changing
the constituent propositions and their order in the
conditional to read:

Assume Ah’ Ai;

If Ah is focally joined to Ai by a pattern

of relations, then there is a Pj and a Pk

such that Ah and Ai are incumbent in them

respectively and the pattern constitutes

the role sector for Pj focal and Pk counter.
However, this amounts only to burying a necessary assump—
tion of exhaustiveness in the axiom, and there appear to
be good reasons against doing so. It is not altogether
clear that the assumption is acceptable; the sociological
literature is either silent or vague on the matter of
E222.we are to take EEEE as evidence ofla role (sector).1
It seems better to state exhaustiveness as a separate
assumption which can be dealt with independently of
incumbency. The issue is of considerable importance,
and we return to it in the final chapter. Suffice it to
say at present that we accept exhaustiveness essentially
as an additional axiom in the development of our solution

to the mapping problem. Thus, for any mapping of an A

matrix, each cell vector of A must be mapped to P, and

 

1The conceptual situation seems to be similar to

the one summarized in Freud's famous comment on cigars.

143

the associated actors must be mapped to the associated
positions in I.

The statement reveals another facet of the
problem which has not been touched on before. In general
there is always the possibility of more than one P and I
into which A can be mapped, but only one of these mappings
might have a significant interpretation. In the remarks
below we consider the logical decisions which seem to be

required in making a mapping.

Agreement

The cell vectors of A must always be taken as the
elements of the domain (or "values of the variable") which
the specific mapping functions carry into the object range,
I and P. Incumbency requires that all relations of the
set be considered, and a given pattern must be considered
as a unit. The next question is how we shall decide when
two actors are incumbents of the same position, and the
natural answer is that they must have the same patterns
of relations (the same "values" of the units) with incum-
bents of the other positions to which the position is
joined. This does not mean that they must have exactly
the same patterns of relations with all other members of
the set of actors (and with each other), but only that
each will have a pattern of relations with some incumbent

of each other position to which their position is joined,

144
and that for each position taken separately the pattern
of relations will be identical. This does not require
that they be related to each other or even to related
alters. On the other hand, it does not enjoin against
any of these possibilities, and, in fact, we sometimes
wish to use some of them in specifying a position in a
particular role system. For example, for certain pur-
poses we might wish to specify (among other things) that
two assembly operatives occupy the same position only if
they are supervised by the same foreman. But in such a
case is not the statement of the incumbency axiom
inadequate? The answer is yet and no. The statement is
given in the most general form necessary, and it allows
other restrictions as special cases. The burden of our
discussion is to consider the logical alternatives which
can be generated to specify the restrictions.

The proposal above is that we shall take cell
vector identity, or as we shall say "agreement," as the
basis for the first step in our mapping procedure. This
requires that we develop a procedure for determining when
or which cell vectors agree and how they (and their actors)
will be aggregated for consideration as candidates to be

mapped. We will do this in several parts.

Canonical Form

 

First, we define what might be called a "canonical"

form for the A matrix (which applies to P as well). By

145
requiring patterns to agree, we mean that they must have
the same order of 1's and 0’s in cells taken in an arbi—
trary but fixed order, say from "front" to "back." If
r is the number of relations in the set, then there are
2r possible distinct patterns. This generates a geometric
series of weights; so we may assign a unique integer from
the series 0, l, . . . , 2r—1, to each of the possible
patterns by assigning integers 0, l, . . . , r—l, uniquely
to the relations, raising 2 to the same power to be used
as a weight for each relation, multiplying the contents
of each cell by the weight of its relation, and summing
over these products for each cell vector separately.
These sums can be cast into a square matrix (which is £23
binary), and this matrix contains all of the information
which was in the original three—way array. The two forms
are completely equivalent under the weighting function,
but each is an intuitively more convenient representation

for certain further constructions. We call this new

representation G(A) (for "geometric weights over A").

Maximality

We can locate agreeing cell vectors of A by
finding cells of G(A) which are equal. It seems natural
to inquire which of an agreeing set of entries we shall
associate together, and equally as natural to decide to

take all of them, under whatever other restrictions we

146
might want to impose to specify "special cases." That
is, it is prOposed that we adopt a criterion of maximality.
In order to define this it is convenient to employ the con—
cept of partitioning of a matrix. (This is part of the
practical motivation for defining G(A).) Strictly speaking,
the concept of partitioning in matrix theory involves the
separating out of sub-sets of rows and of columns, but
not necessarily the same ones; the rectangular arrays of
cells of these sub—sets then being taken as the elements
of a reduced matrix. However, we only require the idea
of separating out sub-sets of rows and of columns, i.e.,
we wish to construct sub—matrices from G(A). We need to
think of constructing a separate sub—matrix for each of
the 2r possible entries in G(A). We construct each under
the restriction that each row and each column must
contain at least one entry of the value for which the
sub—matrix is being constructed. This satisfies the
requirement that each focal actor must have the pattern
of relations which defines the role sector with some
counter, and vice versa. Our maximality criterion
directs that all such rows (and columns) of G(A) will
be included in the sub-matrix. We should note particu-
larly that each sub-matrix must be thought of as
extracted separately (or at a separate round) from G(A),
since any given row or column may appear in a great many

of them. Now since the defining characteristic of such

147

a sub—matrix is that some entries (properly distributed)
are the defining value, we may conveniently think of
another operation on it such that we replace all entries
of that value by "l" and all other entries by "0." Let
us denote this result as Gp (where p ranges from 0 to
2r-l corresponding to the defining value from G(A) ).
Our following development requires the idea that we

retain G(A) subscripts in Gp.

Logical Restrictions

With the idea of the rectangular binary matrix
Gp in hand, we are now in a position to define the
restrictions which can produce "special cases" as
further substructions on Gp; i.e., since Gp is binary,
the only restrictions on it must deal with further
partitioning to produce sub-matrices of Gp by selection
of appropriate rows and columns. Let us denote any such
sub—matrix of Gp by Gp*, and note that, again, there may
be many, and they may overlap, just as Gp's may overlap
"in" G(A).

Symmetry. We first consider the import of three
logical properties of relations, viz., reflexivity,
symmetry, and transitivity. We are interested in them
not for their reference to relations as such but for
what they suggest regarding selections of actors as

incumbents of focal and counter positions. Transitivity

 

148
is mentioned here only for completeness since logical
usage associates it with the others, but in view of our
limitation to consideration of pairs in the formal
development, it does not appear to have any immediate
application to the question. (It is a question to which
we shall return in the final chapter.) Consequently,
we assume that it is safe to treat all Gp as though the
set of relations represented is essentially non—transitive
(i.e., indifferently transitive or intransitive). Simi—
lar remarks apply to reflexivity, though for different
reasons. It is not difficult to construct examples of
actor-reflexive relationships between positions, e.g.,
a university departmental head who writes a letter to
himself as chairman of a guidance committee, but such
cases do not seem to be of such substantive importance
as to warrant special consideration. We shall assume
that non—reflexivity is the general case, which is
interpreted operationally to mean that the content of
a cell of Gp which has identical row and column sub-
scripts is unimportant.

It appears that symmetry is a most decisive
factor in determining desirable restrictions. Non—
symmetry is already a characteristic of Gp as defined
above, i.e., rows and columns (actors as focal and
counter) with same subscripts may appear or not. How—

ever, it is easy to cite important examples of symmetric

149

and asymmetric relationships, e.g., friendship as a
reciprocal set of expectations (and as operationalized
in the definition of a clique by Festinger, and Luce

and Perry mentioned in the previous chapter), or well—
formed authority as a superior—subordinate relation—
ship. Symmetry would require that any rows or columns
of Gp which do not have matching subscripts in the other
set must be dropped in forming a Gp*, then empty vectors

are dropped, and the procedure iterated. Asymmetry

 

would require that in any case of matching the row or
column must be dropped, but of itself does not indicate
which.

It might seem that maximality could provide a
solution, say that the product of the number of rows
and the number of columns in Gp‘ be as large as possible.
This is the solution used in a technique called Multiple
Agreement Analysis, and it has some application at a
point to be noted below, but it has some defects for
present purposes. First, it is clear that it does not
provide a unique solution, as in the case when there is
only one offending subscript and an equal number of
rows and columns. Second, even if the product solution
were unique for a particular Gp, it might be at the
expense of the total number of 1's in Gp‘. (This
objection disappears under another type of restriction

to be noted below.) Another approach to the solution

150

also developed in Multiple Agreement Analysis, which does
generate unique solutions is to apply maximality to rows
or columns, i.e., to require either that only columns or
only rows be dropped.

Thus, up to this point, non—symmetry leaves Gp* =
Gp, symmetry provides a unique solution, and asymmetry
yields several competitors, two of which are unique. l?
There might seem to be no intrinsic reason why any of
these restrictions must provide unique solutions, i.e.,

neither our model nor the additional criterion of maxi—

 

mality assumed in this chapter require that a given cell
vector of A be mapped to one and only one cell vector of
P. However, our criterion of exhaustiveness might be
interpreted as intending to require this so as to

satisfy in a simple way the assertion (I,P) = f(A), that
"f" is constituted of well—defined functions.' In this
case one would prefer the maximization of rows or columns
as equally acceptable alternatives. However, it should

be made clear that a relaxation of these requirements
would still yield only a finite number of solutions,

and each of these could be treated as a separate function
by denoting different P's, barely saving the form. It
would be equally admissible to choose product maximization
as a first choice which, if it failed to yield uniqueness,
could be replaced by row or column maximization, and this

would yield a unique final result. In general, it seems

151
best to interpret the "f" in the expression as denoting
a complex function and to admit all of the alternatives
as possible "parameters" for the asymmetric case, on
the grounds that the non—unique solutions do yield ones
which are finite egg determinate. That is, each Gp‘ is
equally definite, since though an Aij' may thus map to
several Pij" it will definitely go to each of them.
(Indefiniteness is the reason for the general case of
A # g(I,P), as shown by our proof of the existence of A

in the preceding chapter.)

Quantification. We now turn to a second general

 

set of restrictions on Gp suggested by the logical quan—
tification of propositions. The statement of the incum-
bency axiom says that an actor must have a role sector
with epme_alter of a matching position. A restriction
on this which can be represented as a sub—set of Gp is
the requirement that an actor have a role sector with 311
incumbents of the reciprocal position. Of course, the
restriction automatically guarantees that it applies
equally to incumbents of both positions, so that it yields
a rectangular matrix which we could describe as "off—
diagonally dense," i.e., all cells other than those with
identical subscripts have 1's as entries, remembering
that subscripts come from G(A).

In this case of "each to all" the problem of

multiple solutions applies to symmetric, non-symmetric,

152
and asymmetric choices, though in different ways. The
application to the symmetric case is easily seen from
the definition of a sociometric clique by Festinger
(and others), i.e., such cliques may overlap. However,
there is not any problem of maximization since the Gp*
must always be square. Any non—symmetric Gp* may in
general be taken as made up of a symmetric portion and
a remainder which is asymmetric. The maximization solu—

tion for such a non-symmetric case has the same alterna—

 

tives as for asymmetry outlined above —- which also aply
to asymmetry here —- though they may yield different
results, dependent on the actual contents of matrices.
While we have taken some effort to point out the maxi—
mization problems, we will not need to consider them
extensively in the sequel since we assume that the
existence of determinate solutions satisfies our needs
for (I,P) = f(A). Moreover, they appear to be of
limited substantive relevance except as noted below.

Connectedness. The combinations of "some" and

 

"all" and the various modes of symmetry yield a general
six-fold classification of functions which can be applied
to Gp to generate candidates for incumbencies. There is
yet another logical property of Gp‘ which may be used to
create a further dichotomy of the "some" group. Any Gp*
derived under "all" and symmetry must be totally connected,

each actor reciprocally to every other. Consider an

153
instance in which a Gp yields two Gp* under these restric-
tions, such that they are disjoint and exhaustive of Gp.
The same Gp under symmetry and "some" would yield a single
Gp‘ (=Gp), which contained both of the dense sub—matrices,
but there would be no connection between them. This
could be determined by a partitioning of Gp to reveal the
null matrices which represent the disconnected character.

The concept of connectedness can also be extended to the

_" ."..

 

asymmetric and non-symmetric cases, and it is implicit
in the "all" restriction, so that it does not provide a
way of further dividing those cases.3 The nine—fold
classification which results has certain interesting
substantive applications, but we delay consideration of
these until completion of our mapping procedure and its
ramifications.

Iteration on Gp and G(A). To complete the cri-
teria for application of this set of "parameters" for

extraction of Gp‘, we need to provide that the selected

 

2A connected symmetric matrix has the interesting
property that it must always contain at least one "cycle"
as defined in graph theory. While we have not remarked
.the fact above, perhaps the reader has noted that Gp and
Gp‘ are always digraphic. For the definition of a cycle,
see Harary, et al., op. cit., Chapter 2, and on connected-
ness, see Chapter 3.

3Connectedness goes beyond our restriction to
pairs, but it can be determined in a digraph by the
existence of "semi—paths" between pairs of actors.
Ibid., p. 31.

154
modes are applied iteratively to Gp, with multiple map—
pings of equivalently maximal Gp‘ as necessary, and that
after each "level" of maximality has been exhausted all
entries used for that level are "swept out," i.e., the
1's are set to 0, until Gp is null. This is required to
satisfy our assumption of exhaustiveness. This procedure
must be repeated for all Gp extracted from G(A), with
parameter values as appropriate for the given Gp. It
should be noted that while in general there are 2r possible
Gp, there may not be more than m(m-l) for a given A and set
of relations, where m is the number of actors. Still, the
problem of conceiving how this mass of information is to
be used to construct I and P suggests the utility of some

additional bookkeeping devices.

Bookkeeping Matrices

 

For this purpose, we introduce three new rectangular
matrices which are not essential to the solution but are
convenient as devices for isolating steps in the procedure.
The first stores focal positions from Gp*, the second
stores counter positions, and the third stores role sectors,
all by columns, with rows for actors in the first two and
relations in the third. We shall denote them as F, C, and
R, respectively. All of them have as many columns as there
are Gp* from all Gp. F and C have m rows, and R has r
rows, and we employ binary entries in the now familiar way.

(We could as well use three vectors with geometric weights

 

155
of the appropriate sort, but the binary representation
seems more natural at this point, and is essential for
F and C in connection with "nesting" below.)

We now add an additional row to each of the F
and C matrices in which to record position numbers to
be used in constructing I and P. There are two
apparent candidates for the criterion required to
generate these position numbers, and we shall designate
them as "equivalence" and "nesting." We shall develop
the procedure with respect to the first (which is more
general), and then examine the consequences of the
second. The purpose of the criterion is to determine
which columns of F and C are to be assigned the same
position number. In our discussion, we use subscripts

to denote the columns of F, C, and R.

Equivalence

 

This criterion requires that if two or more
columns of F and C are to be assigned the same position
number, then they must display identical patterns, i.e.,
the actors incumbent must be exactly the same. We begin
by choosing any arbitrary column, say Fl, entering a 1
as its position number, scanning all remaining columns
of F and C for equivalence with the starter, and entering
1's in those which are equivalent. Next, we choose a
column not yet numbered, enter a 2, and scan unnumbered

columns entering 2's in equivalent ones. The procedure

 

156

is repeated until all columns of F and C have been num—
bered, and the final number used is the number of positions
in I and P. I may be generated during the numbering pro—
cedure by recording the appropriate patterns. P is gen-
erated by examining the columns of F, C, and R, in the

sets in which they were given by Gp*, using the new
position numbers of F and C as row and column subscripts
and the column of R as the cell vector of P.

We now examine the consequences of the use of
this criterion to combine the provisional positions
extracted by Gp*. There are two salient points, both
concerned with the parameter values used to define Gp‘.
Since, in general, we do not assume that a unique set of
parameters has been used for all Gp* (from all Gp) it
is possible for a given position to have role sectors
with separate other positions which have been defined by
disparate parameters. However, we do assume that a
given type of role sector (a particular Gp) has been
defined by only one unique set of values of quantifica-
tion, symmetry, and connectivity.

Further, the question arises whether the new
numbering of positions may not require combining role
sectors from R in P, and the answer is that this can
occur only if errors are made in forming the Gp.
Consider the results of two Gp‘, as Fi, Fj, Ci, Cj, and

Ri, Rj. If either Fi ¢ Fj or Ci # Cj, then separate

157

role sectors in P result. If Fi = Fj, and Ci = Cj, then
if R1 = Rj, the Gp* is redundant; or if Ri # Rj, then a
type of element of G(A) has been used in two different
Gp, contrary to assumption. (This is not true for
"complex" structures to be defined below, but as noted
there it causes no problem.)

In the general case, the criterion of equiva— 1
lence is the only rule for equating columns of F and C ,
which insures satisfaction of the second part of the i
incumbency axiom. If any other rule less than complete
identity were employed, an actor might be assigned as
incumbent to a position in virtue of relations to an
incumbent of a reciprocal position, but thus be attri—
buted relations to still a third position in virtue of
another incumbent of his own position. There is a
special case of some interest because of its substantive
applications, and it requires that P be what we shall
call "locally digraphic" (i.e., defined by a single Gp),
and "articulated through maximal positions." A much
simplified example will assist in explaining these points.

Problems of Egpivalence. Consider the following

 

matrix of seven actors and one relation (which we might
take to be something in the nature of authority or
influence) as an A. (A digraphic representation is given

as well for clarity.) Let us assume symmetry and "all"

158

1
1 J,
2 1
3 1 2 3 4
4 1 1 \Lk{/’ u(//
5
6 5 6 7
7

as the criteria for Gp*. Gp consists of the first four
rows and the last six columns of A (which is already

conveniently binary). Gp‘ yields three position pairs

 

which are enumerated in F and C (it is not necessary to
represent R for a digraph, since it is always a vector of
1's). Examination of F and C under the criterion of

equivalence reveals six distinct positions.

F C P I
a b c a b c 1 2 3 4 5 6 l 2 3 4 5 6
1 l 1 1 l 1
2 1 1 2 1 2 1 l
3 l l 3 1 3 l l
4 l l 4 4 1 l
5 1 5 5 l
6 l 6 6 1
7 1 7 l
1 .2. 3 4 5 6

However, a glance at the digraph of A suggests
that this does not reflect facts of considerable substan-
tive interest. There are two "condensations" of the
graph which it would be desirable to be able to extract
in a mapping of A (into different P's). The first would
take actors 2 and 3 as incumbents of a position, and also

6 and 7 as mates, leaving 1, 4, and 5, as occupants of

 

41bid., p. 57f.

159
separate positions. The second would associate 2, 3, and
4, in one position, and 5, 6, and 7, in another, with 1
in a singular position. Examination of the columns of
F and C suggests ways in which this could be accomplished.
A comparison of the second and third columns of F and the
first column of C shows that the logical sum of the first
two yields the third. There are two alternatives for
using this information, and we examine both.

First, the columns of F might be used to deter-

 

mine that the first column of C ought to be broken up,
i.e., a further partitioning on the first Gp‘. This
could be done by adding a fourth Gp*, with l as focal
and 4 as counter, and deleting 4 from the counter of the
first Gp*. This would yield five distinct columns in

F and C, and the resultant P and I as shown which is the

first "condensation." At this point it might be
F C P I
a b c d a b c d l 2 3 4 5 l 2 3 4 5
1 l 1 1 1 1 l l
2 1 1 2 l 2 1
3 1 1 3 1 3 1
4 1 1 4 4 1
5 l 5 5 l
6 l 6 1
7 l 7 1
1 2 3 1 2 4 5 3

observed that entries of P show further association

between the now positions 2 and 3 in their relation to

 

1. Perhaps, this could be utilized by taking P "as an

A" (it should be recalled that A is of the same form as

160
P, and is in fact a special case of P in which each
actor occupies one and only one position) and repeating
the procedure. However, this would not combine positions
2 and 3 (actors 2,3 = position 2, and 4 = 3) since each
appears separately with still another position. This
reveals both a limitation and the character of this
technique of further partitioning of Gp* which we might
call "de-nesting" by comparison with the technique to
be outlined below. If the elements of A (or P if it is
already a result of a mapping) which are associated in
the F and C in a group of columns (which form a logical
sum represented by one or more columns) appear among
these columns as singular incumbents, then repetition
will reproduce the same A. In a sense, it thus
guarantees the lowest possible level of definition,
i.e., the one closest to the original A and consistent
with the several criteria used.

It will be seen above that we have implicitly
used equivalence after the partitioning of Gp* as the
definition of position combination, and this suggests
correctly that partitioning may be used on sets of
columns of F and C matrices resulting from heterogeneous
Gp*. The partitioning would have the advantage of
yielding P's which are as connected as possible since
disconnectedness is always the result of non-equivalent

positions and de-nesting guarantees a maximization of

161

equivalence. However, the restriction to very concrete
representation may outweigh this advantage.

The iteration of the procedure is an important

 

additional technique, but one which does not apply in
general when equivalence is required, i.e., when more
than a single Gp is involved and the criteria for a given
Gp* are held constant. However, if the criteria are
changed at each round from the most stringent ("all" and
symmetry or asymmetry) to less stringent ones ("some,"
disconnected, and nonsymmetry), then iteration is sig—‘
nificant under equivalence and is analogous to a shift

in the "level" of definition from concrete groups to
societal or cultural (or better, "system component")
models. In the present example, such changes would yield
a single position pair, with l, 2, 3, and 4 as focal and
2, 3, 4, 5, 6, and 7 as counter, which represents the

only general type of role in the system.

Nesting

The second alternative for using the information
on the relation between the second and third columns of
F and the first column of C in our example is to take the
logical sum as evidence that only a single position is
involved and thus assign the same number to all and the
sum as a column of I. It is this procedure that we
designate as "nesting" and it is strictly limited to local

digraphs which are articulated to a larger multigraph

162
through positions which are maximal in being logical
sums. In the example, this criterion would first yield

four positions composed of l; 2, 3, 4; 5; and 6, 7. An

F C P I
a b c a b c 1 2 3 4 l 2 3 4
1 1 1 1 l 1
2 l 1 2 1 l 2 1
3 1 1 3 3 l
4 1 1 4 4 l
5 1 5 1
6 1 6 1
7 1 7 1
l 2 2 2 3 4

iteration would then yield three positions by combining

5 and 6,7. It should be noticed that here the iteration

F C P
a b a b 1 2 3 1 3
1 1 1 1 l 1
2 1 1 2 1 2 1
3 1 3 3 1
4 1 4 l
1 2 2 3 5 l
6 1
7 l

is absolutely essential, since the first result yields
a position composed of 2,3,4, which is joined to
separate positions of 5, and 6,7, and 4 is not connected
to the first of these nor are 2,3, to the second, and
this violates the "some" requirement of the second part
of incumbency. The iteration combines 5,6,7, and joins
it to 2,3,4, which satisfies the axiom.

The restrictions stated above are necessary to
avoid violating the second part of the axiom. The

reasoning is as follows: any appearance of a strict

163
sub—set of the logical sum will be assigned the same
position number, and if this sub—set in F and C was
generated by some Gp other than the one being used to
permit the nesting, then the sum will be articulated to
a second position such that some of the sum's incumbents
do not have the second Gp with any incumbents of the
second position.

Nesting is essentially a technique that takes
into account the simplest order existent in a set of
positions, and this entails effectively operating on
more than a pair of positions, even though they are
taken a pair at a time. That is, mapping still proceeds
by pairs, but the effect of iteration is to link pairs.
It is of interest because of the substantive significance
of ordering in producing ranks. In connection with this,
there are some restrictions on the effectiveness of nesting
which are best stated in terms of digraphs. Any set of
positions which are to be nested must be mapped from a
set pf actors for whom the associated digraph of Gp is
no more than "strictly unilateral."5 This means that
for any pair of actors there may be only one "path"
between them: e.g., consider Ai and Aj, then Ai(Gp)Aj
or conversely (or neither) but not both; further, for any

Ak, Ai(Gp)Ak and Ak(Gp)Aj, or conversely (or neither),

 

5Ibid., p. 69.

164
but not both; and so on by extension. In digraphs, this
means that there are no "cycles" (i.e., a closed path) on
Gp in the set of actors. If cycles exist, nesting will
produce ambiguous results.

There is an interesting substantive aside which
is related to the restrictions on nesting. It is well
known that the line and staff structure exhibited in an
organization chart does not represent the "real" structure
of a bureaucracy, because office holders form other
relationships with fellow members of the organization than
those intended by the chart. This results in erstwhile
incumbents of the same position entering in disparate
relations with others, which violates the articulation
restriction of nesting. The chart becomes not merely
incomplete but wrong. Of course, even though position
mates were as much alike as tweedle—dum and tweedle—dee
in their extra-formal relationships, these latter might
still take precedence to the point that the chart while
limitedly correct was trivial. Which underlines the
point that our considerations herein are not thought to
guarantee substantive significance.

Types of Structures
Up to this point, we have taken as assumed that

the procedures described are conceptually applicable to

 

6This example should not be taken to suggest that
ordering is only relevant to formal structure.

165
the mapping of a set of actors into a role system on the
basis of a set of relations considered as a whole. We
now need to consider whether this assumption is required
in all cases, and whether it adequately represents the
operations we wish to conceive. As often, the answer is
both yes and no. On the technical side, it appears best
to assume that every effort should be made to retain the
procedures, as described on the assumption that an entire
set of relations are exhaustively used in a mapping, as
the basis of any elaboration, since any relaxation would
only ramify an already complex conceptual structure.
However, on the substantive side it may seem that the
procedures described are inadequate, and this revolves
around our definition of a position under the incumbeneyg

assumption.

 

When incumbency is assumed, our definition of a
position is that it is specified by its pattern of
connections with other positions through the set of
relations and that it is occupied by a set of actors.
The mapping procedures outlined operate to insure that
this set is unique; though it may overlap with other
sets, it is not identical with any set involved in the
specification of any other position. Further, for any
given position, there may not be more than one role
sector with any other position. And here an apparent

difficulty arises. While most theoretical conceptions

 

166

do not require (and some do not permit) more than one
sector between positions, they do not prohibit (and

some require) that a unique set of actors may occupy
more than one position in different contexts and thus
jointly hold more than one role sector with another
unique set of actors in a larger sense. In any concrete
empirical situation, more than one context may be (and
usually is) relevant to the definition of the situation.

How may we save the form which we have developed
and still incorporate these considerations? There appear
to be two alternatives. The first is to affirm that the
form applies only to a definitive singular context and
that there always may be multiple P's in any empirical
situation, and this seems to be the preference of some
theorists.7 The other alternative is to develop a way of
extending our conception of the formal structure of a
situation, using the ideas of mapping as a basis, and we
will explore this briefly.

The question centers on what we conceive as the
character of the relations in the set used to specify A
and P. We began with the idea that they are contentless,
or more properly that they represent the logical form of

each of several kinds of contents which are frequently

 

 

7
E.g., W. Goodenough, op. cit.

167

used in substantive analyses of the role concepts, and
this is still to be maintained. The problem we want to
consider is how these contents of whatever kinds may be
analytically related to each other. Let us use the
following example.

Suppose that two actors as incumbents of related
positions are related by both authority and friendship,
as is not infrequently the case. As labels, authority
and friendship are distinct, but we might wish to
decompose them into some analytic constituents, and
assume further that one of the constituents of each of
these is communication. (We are not concerned here with
how any of these implied measurements are carried out in
practice, only with the analytic consequence of the
overlap.) If we wished to map a P for authority, we
would naturally take those relations conceived as con-
stituent of it, and similarly for friendship. We could
easily conceive of creating a revised and extended P
(and I) which combined both structures in one representa-
tion, but in order to do this we would have to consider
what to do with the analytic constituent communication.
If it appeared only once in A and P (as we have up to
now assumed), then the authority—friendship structure
would be intermeshed through combined role sectors.

On the other hand, we easily think of a modifica—

tion of our earlier design, such that the authority and

 

168

friendship structures are represented each by unique sets
of role sectors and the two structures are linked through
positions. We do this by supposing that we allow the set
of relations (r in number) to be divided into sub—sets (by
definition, sub—sets of a set may overlap), and that each
of these be used to specify an A. Then each A may be
used to form G(A), and all Gp and Gp* are extracted and
mapped into F, C, and R, and the procedure iterated for
all such A's generated from the parent set of relations.
In order to do this some modification of R (and P) is
required in order to distinguish the results of distinct
A's. This is conveniently accomplished by supposing that
we use geometric weights for this purpose, and that each
successive A requires a single row in R and a single table
in P. (In "complex" structures, role sectors mey_be com—
bined in mapping from R to P, and this avoids redundancies.
We noted above that in simple structures role sectors of R
are never combined in mapping to P.) However, this is only
a convenience, since the weights are completely equivalent
to a binary representation, and in such a representation
each overlapping relation would appear more than once, in
R and P, and in an A expanded in the same way.

As a matter of nomenclature, we may designate our
original development of the mapping procedure as the model
of the simple structure of a role system and the extension

suggested here as the model of the complex structure. Just

169
as there are many possible simple structures under various
combinations of parameters for various Gp*, there are also
many possible complex structures. Since each of the sub—sets
of the r relations might be used in specifying any of the
constituent tables (using the weighting notation) of P
(and A), there are 2r—1 possible tables (excluding the
case in which all relations are irrelevant) which might or
might not appear in P. Each of these in itself generates
a complete simple structure (including G(A), all possible
Gp, selected Gp‘, etc.). Excluding these, there are
22r_l—(2r—l) possible complex P's, each of whose con-
stituent tables may be defined in all of the ways of any
simple structure (1) (Since each relation may or may not
be used to specify the sub—set for the simple structures,
and if used it may or may not appear in a role sector for
a particular structure, there are 3r—l ways —- excluding
the case when all are unused —- of specifying role sectors
for simple structures.) Of course, these potentially
rather large numbers represent an enumeration of logical
possibilities but certainly not all of them could be
given substantively significant interpretations in any
empirical situation. What they do serve to illustrate is
the (literally) exponentially increasing ramifications of
even such a simple "set" definition of an extended
structural concept which seems more nearly adequate to
substantive requirements than our already rather complica—

ted specification of the mapping of a "simple" structure.

 

170

Some Substantive Coordinations

We remarked above that the nine-fold classification
of the main values of parameters for extracting Gp‘ may be
given certain general interpretations in terms of their
use in specifying types of role sectors. It should be
emphasized that they apply only to role sectors and not
to total systems, but in fact some systems are constructed
mainly through the use of one set of values. The following
table reproduces the nine-fold classification with numbered

cells for ease of reference.

 

"some" "all"
disconnected connected
symmetric
non-symmetric 4
asymmetric 7 8 9

 

We noted before that cell 4 represents the most
general definition of a role sector under the incumbency
axiom (with these parameters) and that all others are
special cases of it. As such, it cannot have any specific
application and is of residual interest here. The remaining
cells in the row, 5 and 6, are of interest but are better
examined with and after others.

The difference between the first and last rows of
the table is essentially comparable to that between peer
(or coordinate) relationships and non-peer (or ordinal)

relationships. There can be little question that this

171

reflects a deep chord in sociological concern; the dis-
tinction between associative and differentiating connections
between social elements is repeated again and again in
human life (and in human sociology), and one or the other
or both types of relationships seem to be fundamental
building blocks of the structure of any social group. As
such, it appears that a "parameter" of this sort is
absolutely required in any conception of how such structures q
are built up.8 But it is not particularly surprising
that there should be congruence between ways of conceiving -M
relations abstractly and the ways in which we construct
relationships in role systems.

The implementation of symmetry in empirical groups
and in sociological analysis reveals an interesting facet
of the other main dimension of the table. We have noted
at several places above that cell 3 represents the defini-
tion of "clique" offered by Festinger (and others). It
may seem curious that this definition has been used (to the

best of the author's knowledge) mainly for analytic

 

8The parenthetic reference to the ordinality of
the last row is only residually correct in view of our
general limitation to consideration of pairs (an order
can always be imposed on any pair), but any, usually
hierarchical, order is fundamentally distinguished by
asymmetry. And it is in this connection that we found
it necessary to introduce the idea of "nesting" in
order to represent a fundamental way in which combinations
are formed in a positional structure.

 

172

purposes, and that most empirical sociography has used
definitions that would fall either in cell 2 or in cell 5,
the former for "cliques," and the latter for "crowds." In
existent social groups it has proved difficult to find the
consensus required by the "all" restriction. However, this
is also a function of the kinds of problems, i.e., the
kinds of relationships, which have been subjected to socio— l
metric analysis. Most usually these are what is broadly w
designated "informal" (and usually associative, especially

in the original formulation), and it turns out to be asking

 

a bit much of a collection of persons, however solidary

or well—informed, to agree upon and/or reciprocate all
expressed connections. However, cell 3 actually is used

and with great frequency when the context is "formal" or
effectively so. Collegial relationships of all sorts

fall here, but these are hardly the kinds of phenomena that
one would think of subjecting to a sociometric analysis;

the answers are already known and quite efficiently
represented. The point is that the elementary architectural
decisions are not of a completely different order, but that
it requires broad social givens to establish the complete—
ness of consensus demanded by such a definition. Even

here the reference of such givens is to existent structures,
since the higher "level" specifications slide over to lower
numbered cells. However, the right—most column is used

and with important effects (except as noted below regarding

cell 6).

173

As noted above, the last row of the table is
mainly associated with ordinally differentiated structures,
though not always so, e.g., in some definitions of communi-
cation (when feedback is neglected). Authority (and per-
haps influence) as usually pictured fall in cell 9, which
is the basic component for any "line" structure of a formal
organization. It is also customarily associated with
notions of status9 or prestige, though cell 8 would appear
to be closer to the concept of a pair of strata in a system,
since it does not require that every member be immediately
comparable to every other, only that the comparison can
be made at some determinate remove. Cell 7 serves (as
does cell 1 in the first row) only to indicate the exist—
ence of a mode of relationship in an aggregate.

In all instances, the most general application of
any type of relationship in an existent system tends to
lead to cell 5, since all ways of formulating connections
between locations in groups lead to both symmetries and
asymmetries through multi-relational incumbencies. But
the importance of the requirements on relational recip—
rocality and quantification is rather well exhibited by

the difficulty of discovering position pairs which may

 

9Cf. F. Harary, "Status and Contrastatus,"
Sociometry, 22 (1959), pp. 22-43, in which a canonical
ordinal measure is developed from this assumption.

 

174

be cast into cell 6. It is possible that some examples
may be drawn from deference behavior in kinship systems10
(which are notable instances of formal circumstances in
which both symmetry and asymmetry are combined and
required —— as in husband, wife, spouse —- though usually
with contextual clarity). However, it is of interest that
such deference patterns (in which some of a set defer to
all of another, and others in the first set defer recip—

rocally with some of the second set) are defined "on top"

4. wm... _ I.
.‘o- I
l

or in terms of the kin positions as such, and not conversely.
The purpose of all of these remarks is to suggest

that the parameters of our agreement substruction have

fundamental substantive import, but not to suggest that

they are the only possible or significant ones. Others

might be outlined, but they would not confront a more

basic question which may be raised concerning the concept

of mapping as applied to analyses of role systems. Through-

out the present chapter we have spoken "as if" the mapping

procedure for assigning actors as incumbents in role

systems were a conceptual and empirical reality in the

measurements required by sociological analysis. The

skeptical (or better, realistic) reader may well inquire

whether the "as if" is, or could be, a reality. The

formal model which was exposed in the preceding chapter

 

lOSee W. Goodenough, op. cit.

175
may be all well enough, but does it logically or materially
entail the analysis outlined here? Once more, our answer
is equivocal.

At the outset of the chapter, it was remarked that
the details of the mapping described may not constitute a
unique solution, though some of them seem to be required.

(Particularly, Gp as the fundamental starting point for

agreement substruction, and equivalence as the general M
definition of congruent incumbency, since both are cir—

cumscribed directly by the second part of the incumbency “
axiom.) On the other hand, it is reasserted that the
model, with the additional assumption of exhaustiveness,
requires that egme solutions exist if actors are to be
said to be in role systems. And the assumption of
exhaustiveness may be replaced with any other determinate
directions for deciding when a cell vector of an A is
significant, and the same consequence follows.

So much for the logical necessity, but what of
its real application? And it is at this point that the
"as if" enters. As a means of discourse, it is naturally
preferable to be able to speak as though our statements
designate empirical truths, but it does raise a question
of their validity. It is rather obvious that the full
array of armaments described herein have never been
applied, even separately, to the problems of summary and

descriptive measurement of an existent social system,

176
though certain rather limited cases are in evidence, mainly
in sociography as a degenerate case dealing with a single
relation. Nor is the author led to believe that the
battery of techniques should or could in general be applied
to any good purpose in specific empirical analyses at the
present time. But then what material merit can such an
analysis as the one carried through in this chapter have 1
for sociological thought? 4

Our answer to this question has been given before,

 

and we need to repeat it here for emphasis. The develop—
ment of the details of the procedure for mapping of actors
into role systems is not intended primarily as a contribu—
tion to the methods of measurement in structural analysis,
but as a device to aid in clarifying our concepts of a
role system. And it is asserted that it does serve this
with merit.

First, the very elaborateness in terms of expect-
able detail in any empirical application which has issued
from a few essentially very simple operational specifica-
tions serves to indicate why such a mapping could hardly
be expected to be carried out, except in technically
degenerate cases as noted above. The richness of the
results would make them most difficult to use. However,
we may observe that this embarrassment of riches is in
useless currency just because it is not backed by sub—

stantive coinage. The manifold possible outcomes are all

 

177

about equally plausible in the absence of specifications
which tell us which ones are of interest.ll This circum—
stance underlines an observation that has been noted
repeatedly elsewhere, and which amounts to questioning
whether "structural role theory" ought to be called e
theory or even a theory in an empirically corrigible and
substantively extensive sense.

Second, we have, by our development of the con—
struction, been led quite naturally to notice similarities
between the operational categories devised and extant
techniques of structural description, most notably in
sociography. We have intended these operations as the
working out of the logical form of role systems, but must
this be taken as evidence that all sociography is properly
seen as the analysis of positions and roles? Surely an
individual's choice of luncheon partners is a bit more
ephemeral than the kinds of social order that the role
concepts intend to compass. The logical form is not
decisive on such questions but it leads to their considera—

tion and to some suggestions for answers.

 

11There is a similarity to the rather common
circumstance in 'survey' research in which the investi—
gator (who perhaps began with rather immodest aspirations)
confronted with the wealth of alternative possibilities
in 'multivariate' designs falls back on the use of bi—
variate contingency tables, for lack of a theoretical
basis for making the necessary specifications.

178
Other suggestions as to the conceptual consequences
of our mapping analysis may occur, but all of these go
somewhat beyond the originally rather technical intent of
the present chapter. It is the purpose of the final
section of this work to attempt an appraisal of some of
the questions raised by the formalization of the role

concepts which has been developed.

CHAPTER VI

ROLE CONCEPTS REVISITED

We arrive now at our final stock—taking and
appraisal of the role concepts and the analysis carried
out in the preceding chapters. There are several pur-
poses and parts to this discussion. First, we will L
summarize the logical model of a role system which has
been stated and attempt to specify some interpretations
of this development for other aspects of the conceptual
system which are not directly involved in the statement
of the logical model. Some of these points have been
outlined to a greater or lesser extent in previous
commentary, but we must try to pull them together here
to get an overview of the consequences of our explication.
Briefly, the object is to find out what the model says
about the role concepts. A second objective is to eval-
uate the explication qua explication along the guidelines
set out in the first chapter. This, also, will be pro—
visional for reasons we will note at that point. Finally,
we will try to enumerate additional formal problems of
the conceptualization of role which have not been incor-

porated in our model. The reader may agree with the

179

 

180
present writer that the model at least gives illumination
to some of these topics. Our purpose in the final section
is to indicate what the model does pg: say about the role
concepts, and thus to suggest continuing directions for

further analysis.

Summary of the Model and Some Interpretations

It is possible to think of the model which we have
articulated as having two related representational objectives,
corresponding to the two axioms stated in the third chapter
above. The first axiom (of relation), together with the
primitive and defined terms, sets up a representation of an
"abstract" role system in terms of positions, roles, etc.
While position and relation were taken as primitive terms
for convenience in coordinating the system to its model
in the theory of "nets," we noted that this was not formally
necessary. Both terms were given contextual specification
through the syntactic rules of the axiom itself. A rela-
tion involves two positions, and a position is known by
its relations to other positions. A directed set of rela—
tions between two positions is called a role-sector, and
a role is the set of role-sectors for a position.1 Thus,

both positions and roles may be thought of as specified

 

lFor brevity, we omit focal and counter qualifiers
here.

181
by role sectors. Representation of the system is given
by the three—dimensional array which we call the position
(P) matrix.

The second axiom (of incumbency), together with the
additional primitive term "actor," extends the apparatus to
represent a "concrete" role system, one which is assumed to
have occupants of its positions. Again, the new primitive
term was specified contextually by the additional rules of
syntax -— the axiom states the conditions an actor must
satisfy to be an incumbent. Further, the stipulations of
the axiom are such that the role system is derivable from
relations among actors. The resulting representation is
made in two new matrices, one like the P—matrix with
actors rather than positions, which we call the actor (A)
matrix, the other a two—dimensional array of actors in
positions, which we call the incumbency (I) matrix.

While the A— and I—matrices result from the incumbency
axiom, the I—matrix stands on the otter side of the rela—
tionship between A and P. The axiom entails two associated
mappings, of actors to I and of cell vectors to P, coordi—
nated by positions in each. We denote these mappings by
the expression (I,P) = f(A). Finally, in our development
of a set of mapping procedures, we aCCupted the additional
assumption of exhaustiveness —— each c611 vector of A must

be used in the mapping.

 

182

We will not try here to examine all of the con—
ceptual problems which might be exposed or illuminated
by our considerations of the logical model.

The writer's experience in attempting to formulate
the problem and solution has led to a host of speculative
asides of considerable interest. However, some of these
resulted more from the cognitive exercise than from the
formal considerations, and others remain as yet specula—
tions; both classes are largely deleted from our comments
here. Naturally, we attempt to summarize what seem to us
the most important implications.

Some of the questions we wish to consider here
appear to be so highly interrelated that it is quite dif—
ficult to separate them, if they are, indeed, separate
questions. At various points in our analysis we have
remarked on such problems as 1) whether it is useful to
conceive of social groups without positions and roles,

2) the use of manifestly given names or labels to
identify positions and roles in a system, 3) the possibi-
lity of latent positions and roles, 4) the importance of
specifying role definers and the question of consensus

of expectations which are central to the concepts of
Gross and his associates, 5) whether any and all rela—
tionships between actors must be taken to be components

of role sectors, and so on. Some of our earlier remarks
\x

183

on these questions easily could be seen as talking on
both sides of an issue, even though at different times.2

One line of exploration of the questions enumer-
ated, and a proper reply to them, can be pursued by a
"shift in the angle of vision" on the role definers
problem stated by Gross, et a1. Nadel has observed
". . . that the role concept is not an invention of
anthrOpologists or sociologists but is employed by the
very people they study."3 He goes on to remark that
these scientists, as well as recognizing the public use
of the concept, have made of it ". . . a special analytic
tool." With most other observers, he appears to believe
that this dual usage is a resource of great strength.
But we must also recognize that it is a source of con-
siderable peril.

When Gross and his associates speak of the
problem of role definers, they refer to the use of
concept by the "peOple they study." The authors have,
of course, set down their own definition of the concept,

and we have already asserted that our model adequately

 

2For example, we asserted that it seems unusual to
conceive of role—less groups and that the idea of a latent
structure of positions and roles is natural extension of
sociological usage which is approached though not directly
employed in studies of informal organizations. On the
other hand, we have expressed doubt that sociographic rela-
tionships (or more specifically, the kinds of transitory
choice behavior sometimes employed in such studies) ought
to be seen as evidence of role systems.

3S. F. Nadel, op. cit., p. 20.

 

184

embodies the logical features of their definition. How—
ever, their construal of the actual specification of
role elements as finally and definitively given by the
participants introduces an additional restriction of
their definition and works mischief on the concepts. If
it be allowed it assuredly removes some degrees of free—
dom from the possibility of latent positions and roles.

If the conceptualization of role systems is to be "a

special analytic tool," it must both take account of the
public use of role concepts and transcend them. In brief, ,

the scientist must be admitted as the ultimate "role
definer," though in a fundamentally different sense.

In our explication, our objective has been to
develop the scientific rather than the common sense of
the concepts. The former provides the substantive
definition and the latter is included in the order of
phenomena used for the specification of the relationships
examined in empirical analyses. In this sense, the
public, and particularly the participants', concepts of
relations in the system being examined are part of the
data, and as such they cannot "speak for themselves."
Their employment requires the active conceptual inter—
vention of the investigator.

Simply, it appears that the concept of partici—

pant role definition indicates that every actor has a

 

185

"cognitive mapping" of the relationships among participants.
In general, such a "mapping" could be, but probably is not,
of the same detailed form as our model and its constituents
and mappings; especially, such cognitions are likely to be
incomplete respecting enumeration both of participants and
relationships. The congruence among the "mappings" of the
actors in the system would then be seen as yielding measures
of consensus on participant role definitions.

More important than the simple enumeration of
participants' conceptions of roles is the question of the
perceived content of relations, which is required for
decisively defining the system. In the situation studied
by Gross, et al., are the "expectations" held by superin-
tendents and board members and elicited by the investigators
the only and real content of the roles? In connection with
our statement of the incumbency axiom we observed that they
are not, and there is another source of evidence which is
suggestive on this same point. In their conceptual refor—
mulations, Gross and his associates state definitions of
a battery of concepts. These include role and role—sector,
which are defined in terms of expectations. However, one
may observe that these key terms are used relatively infre-
quently in their subsequent discussion of the actual data
analyzed in their research. There the preference is for
expectations as such, though combining forms, such as

"intrarole conflict" and "role congruency," which restrict

186

the range of empirical reference, are used liberally.
Perhaps this happens only because the detailed analysis
necessarily deals with a restricted range of phenomena.
But the issue does arise as to just what order of
extensiveness and significance of expectations for a
pair of positions must be encompassed before it is appro—
‘priate to refer to the set as a role—sector. No answer
to this question is given.

We have reviewed these questions regarding the

usages of Gross and his associates as a vehicle for

 

reintroducing the general question of the relations taken
as primitive in our model, for this issue is also involved
in our views on the "necessity" of positions and roles,

and on manifest identities and latent systems. We have
said that our aim is to represent the formal character

of the concepts of sociologists rather than of those they
study, and that logical relations are uniformly appropriate
for the several main ways of defining these concepts. But
our model goes beyond this. It introduces the idea that a
igii specification of a-role system as a conceptual tool
can be made only in terms of a definite set of relations

of whatever kinds. In terms of our objective these
relations must be specified by the investigator, on grounds
of substantive import. Participants in a concrete system
mey_have cognitions about the relations and these mey_be

relevant data, but neither is required. The model is indif—

 

ferent to whether a concrete system is latent or manifest.

187

However, since the model can represent a manifest
system, is it not then permissible in such instances to
use the public identities of positions to locate them,
as most investigators have assumed? In our strict con—
strual of the model, the answer is "No" -— the formulation
of the incumbency axiom does not permit this. We noticed
in our summary that A and I are the additional representa—
tion needed for a concrete role system, but that I "stands
on the other side of the relationship between A and P."
That is, the expression (I,P) = f(A) is our notation for
saying that incumbency and positions are functions of
the chosen set of relations as observed in the object
group. If public identities of positions were used, then
this might be expressed in a similar way by P = f(A,I), or
A = f(P,I), dependent on other assumptions. Further, all
of these expressions might yield equivalent A's, P's, and
1's, for a given group and set of relations, as, for
example, in the authority structure of a formal organiza—
tion. But this need not hold in general. The incumbency
axiom is intended to represent what we detect as the
sociologic definition of the concepts rather than their
public specification. It should be clear that we view the
investigator's selection of the set of relations as a
central step in the full specification of a structure.

In terms of this we may see the reason for holding

doubts about the appropriateness of "just any" content for

188
the set of relations used to specify a structure. We
have observed before that investigators sometimes are
reluctant to use role terms to designate detailed
aspects of relational systems, for example in studies of
informal organization. This seems to reflect sensitivity
to the inadequacy of one or a few observed relationships
to delimit "a person's role" or "the total structure."
Moreover, it indicates a diffuse concern with the
matter of substantive significance of the structures
under analysis.4 We also have observed that the
model does not preclude use of any set of relations
whatsoever, but it might better be said that it is
silent on the matter —— the model neither precludes
nor requires treatment of any given relational structure
as a role system. As a conceptual tool it may be used
well or poorly, or it may not be used at all. I£_it is
used, then the role concepts are applied to the chosen
content. Whether it should be used must be decided in
terms of substantive considerations of theoretical
relevance, and these are not resolved by the logical

form.

 

 

4By comparison, role terms and concepts are
used quite freely in discussions of the structure of
small laboratory groups, where there is no implicit
idea of a transcendent social system whose significance
overshadows that of the particular task relationships
formed therein.

‘—

 

189

The final point we wish to make on the questions
of public identities, latency, and relational specifica—
tions, concerns our assumption of exhaustiveness (that
every pattern of the relevant relations between pairs of
actors is a role—sector) which was introduced in connec—
tion with the mapping solutions developed in the last
chapter. We remarked there that sociological literature
is not determinate on the matter of "EEEE.W€ are to take I"
EEEE as evidence" of the existence of positions and roles.

Perhaps we might.say that in the broad context of defini—

 

tional statements the matter is ambiguous, since both
"public identity" and "relationship" criteria are advanced.
Our rejection of the identity criterion and acceptance of
exhaustiveness are related. If positions and roles must
be recognized (and named) as such by participants, then

it is quite conceivable that the recognition might both
occur and be absent in separate instances having the same
relational pattern. This is equivalent to ignoring a
particular cell vector of A in the mapping. On the other
hand, in the use of the relational criterion for positions
and roles, there is no intrinsic reason for discriminating
between patterns in one or another of their instances, and

exhaustiveness is the sensible criterion on grounds of

 

parsimony.
We now turn our attention to another question of

the interpretation of the analysis which has been made.

190

In the beginning of our review of role concepts it was
stated that the concept of a system would provide a
general orientation for our work. Indeed, we have
developed the logical model in terms of elements and
their connections as displayed in several preceding
chapters. We also activated the common assumption that
it is permissible to attend to the social structural
aspects of role concepts without attempting to incor—
porate social psychological considerations in any com—
prehensive way. There is another broad assumption of .
a similar sort which has been implicit in this develop-
ment, but it is on the side of social structure itself.

We have used the term "rOle system" to refer to the
structure whose logical features have been analyzed,

but we have not been explicit as to what relationship

is assumed between role systems as conceived here and
concepts of social system or social structure. Obviously,
it is assumed that there is an intimate and important
connection, but, as often, the received opinions in the
literature are not decisive. There is general agreement
that roles (etc.) are important parts of (social) structure,
which in turn is an essential characteristic of a social

system, but variations in the working out of this formu-

lation are too well known to require more than brief

 

comment. Consider, for example, the differences between

Nadel and Parsons. For Nadel, what we have called the role

191

system ie the social structure (assuming, we suppose,
that the right relationships are chosen). His ". . .
definition of social structure . . ." is given as follows:

We arrive at the structure of a society through

abstracting from the concrete populations and

its behaviour the pattern or network (or 'system')

of relationships obtaining 'between actors in

their capacity of playing roles relative to one

another.‘5
On the other hand, for Parsons what we assume to be equiva-
lent to our role system is only one of two "levels" of
analysis or ". . . points of view, both of which are
essential to completeness. . ." in analyzing the structure
of a social system.

The first is the 'cultural—institutional' point

of view which uses the values of the system and

their institutionalization in different functional

contexts as its point of departure; the second

is the 'group' or 'role' point of view which

takes suborganizations and the roles of individuals

in the functioning of the organization as its point

of departure.6
However, such differences in formulation need not introduce
any fundamental ambiguity concerning the status of the
concept of a role system, since they agree that role systems
can be demarcated as such, whether as the system or as a

sub—system of some larger one, and this is all that is

required here. The converse of this is also important and

 

5S. F. Nadel, op. cit., p. 12.

6T. Parsons, Structure and Process in Modern

Societies, op. cit., p. 20.

192
should not go unnoticed; if we assume that role systems
are capable of being treated in certain respects as inde-
pendent components of larger social structures or systems,
then we also have assumed that the larger system can
accept or account for such a component. It is our belief
that these assumptions are entailed by sociological use,
and this is sufficient for our purpose.

To conclude our remarks on these interpretations of
the model of role systems, we may observe that they are not
the only way of construing the model. We have developed
the model as a means of capturing the logical moments of
the concepts of position and role, but we have been
selective in admitting and reconstructing views of the
concepts. Let us go back once more to the general question
of variability of role contents (or consensus on them).

We have argued that it is not necessary or parsimonious to
assume that such variability should be incorporated in the
concept of a role, that it is better to postulate a deter—
minate structure and to conceptualize variability in
reference to it. Another line of argument could be made
which accepts the model as given in all formal particulars
(A, P, 1, etc.) but places a different interpretation on
the elements. It may be recalled that the concept of a
position is uniformly assumed to be an element of a fixed
structure, i.e., as a location is a determinate matrix of

relationships. Strictly speaking, this is sufficient to

193

entail the formal apparatus, in which case it might be
called a "positional system," and the role terms used for
other concepts. But it is our interpretation that socio—
logical concepts of roles as structural elements imply that
the sets of relations between positions should be designated
by the role terms. The point of this example is to empha—
size that the model is intended to represent important
conceptual facts, but that they are selected aspects whose
fit in the larger conceptual structure may be changed
without requiring modification of the representation. In
this sense, the model may be more enduring and fundamental
than the particular interpretation which we have placed

upon it.

Evaluation of the Explication

In the first chapter of this report we made a brief
review of the concept of an explication and related topics
in order to provide a frame of reference for understanding
and evaluating our analysis of role concepts. Here we will
attempt to apply the guidelines which were suggested to
make an estimate of the results which have been obtained.
In discussing explication we first reviewed some rather
general considerations and later summarized points made by
Berger, et al., which are more specific, and we will repeat
that order of presentation here.

The overarching criterion of explication seems to be

"fruitfulness," though we have questioned whether this is

194
more than an exhortation to do well. It is the writer's
belief that he may reasonably be excused from any attempt
to make a definitive declaration on the question, both on
the grounds that it is an unnecessary jeopardy and that a
litigant ought not be permitted to sit in his own judgment.
However, it does seem reasonable to point out some main
features of the work which ought to be considered in
rendering such an evaluation. Briefly, the writer believes
that the following characteristics are central.

First, the concepts of position and role are
defined interdependently, and this is accomplished in
terms of specific syntax; apparently conflicting or
unrelated conceptions are shown to be compatible, and,
in fact, to entail the same referents under comparable
specifications. Second, models which have been used
previously (and often intuitively) are shown to be
inadequate; in the most general case, role concepts
require multilinear directed graphs, or some isomorph, as
a model. Third, to the best of the writer's knowledge,
the axiom of incumbency is a unique statement of a neces-
sary concept. Finally, the model states a general system
which does not require the restricting use of public
identities to specify the concepts.

Two other features which were stressed in our
initial statement on explication are the likelihood of

novelty relative to established usage and the value of

195

exactness or precision of presentation. It is felt that
the development has exhibited both of these traits in good
measure. Of course, the new features are not in the sub—
stance of the concepts but in correctly apprehending the
common form of usage, and it is this form which makes
possible relatively precise statement. But we need to
emphasize once again that neither of these features is
valuable in itself —— they are of interest because they
are frequently associated with worthwhile results, and in
respect to exactness because it facilitates appraisal.

Let us turn now to the five—point summary of
Berger and his associates, as more concrete measures of
the work. The first of their characteristics is not
important for this purpose, though we have noticed
repeatedly that it is certainly characteristic of the
model, viz., that it is selective with respect to
original formulations. The remaining points are the ones
we require and they are that an explication l) clarifies,
2) defines, 3) generalizes, and 4) determines internal
implications of the original conceptualization.

As evidence of clarification of the concept of
a role system, we may cite two developments in particular.
First, the model makes clear (one might almost say that
the techniques of representation require) that the specifi—
cation of a given role system can only be made with respect

to a definite set of relations; even if these must be

196

further substructed to produce a complex structure, the
model draws attention to the fact that structures cannot
be defined by simply enumerating their "elements," i.e.,
positions and some associated behaviors. Further, it
emphasizes that, whatever its empirical intentions, a given
role system is a construct of the investigator. Second,
the incumbency axiom brings out a necessary assumption,
and it casts the assumption in a form such that its
consequences can be considered independently of other
questions. If subsequent analysis and opinion should find
the assumption untenable, it may be elided without pre-
judice to the abstract conception of a role system.

From the examples which are mentioned in our
first chapter, it appears that the criterion of refinement
refers essentially to developments in the direction of
producing measurement techniques for the concepts under
analysis. It is the writer's opinion that our analysis
of role concepts fares less well on this point that on
any of the others under review. Broadly understood, the
analysis of (I,P) = f(A) carried out in the previous
chapter is a specification of procedures for measurement
of the elements of a role system. But in our conclusion
to that analysis we expressed our doubts regarding the
usability of the mapping techniques which were outlined.
We reiterate the opinion here and further emphasize that

the techniques will neither be used or transcended in the

197

absence of theoretical grounds for specifying structures
in terms of their component relations, of whatever kinds.

With respect to generalization of the concept of
a role system, two key factors should be noticed. The
first of these is the explicit postulation of a model of
a multi—relational system in the theory of "nets," in
place of the more usual and somewhat haphazard use of
unilinear graphs. The second important direction is the
definition of a system which eliminates the competing
assumption of public identity from the specification of
positions and roles; such identities imply relations,
but relations do not imply identities. In consequence,
the concept of a role system becomes a more general tool
for the investigator.

Finally, the model of a role system provides a
means of rigorously showing in what way positions and
roles may be said to mutually imply each other without
also thereby being redundant. This interdependence of
the concepts has been remarked so frequently before in
our discussion that it needs no further comment here.

With these points for consideration, and setting
aside just for the moment our earlier reluctance to judge,
what may we conclude about the explication that has been
carried out? Is it a good, or at least satisfactory, one?
Or if such an absolute appraisal cannot be made, how does

it compare to others? Specifically, say, how does it fare

198
when set beside the Cartwright—Harary formalization of
balance theory which we summarized in the first chapter?
Simply, we have started out with less to work with and
ended up with less striking results. We noted especially
that the concept of balance was part of a relatively
"tight" theory, and the role concepts which we have
examined are not. On the other hand, we would repeat
the assertion made in the concluding section of the
fourth chapter where we argued that the model does the
general concepts more justice with respect to its
emphases than any alternative known to us.

There is further reason for suggesting that no
final evaluation may be made, for the work is not done.
The final pages of this report indicate only a few of
the additional questions which seem to require explora—
tion before the "fruitfulness" of our analysis could be

determined.

Suggestions for Further Research
In these suggestions we attempt to maintain

continuity with the kinds of concerns which have been
evident throughout the report. Thus, the main questions
concern relations, namely, what order and how many of
them must a role system model confront, and what are the
relations among the relations themselves? We will not
ask after what kinds of relations should be included in

terms of appropriate substantive content; we have

 

199
eliminated this order of inquiry up to now, and there is
quite sufficient material yet in the line of thought being
pursued to keep our attention.

In our review of role concepts a distinction was
made between attributive and relational concepts of role,
wherein we recognized that both modes of predication could
be used in describing and analyzing the concepts and their
applications. Henceforth we dropped further consideration
of attributive forms in keeping with our judgment that
relational forms provide a correct basis for the model of
a role system. However, the question remains open whether
it is possible to construct an attributive model, and
whether this would not represent the concepts as well or
better than the model we have devised. A salient candidate
exists in the work of Nadel.7 He develops a notation
system which mainly utilizes first—order predicates (i.e.,
attributes) to symbolize features of relations which we
have detected in the literature. However, though he
correctly notes that the ". . . system approximates to a

calculus,"8 it does not yield a satisfactory representation

 

of a role system. In fact, careful reading of the original

definition and subsequent propositions reveals that it is

 

7S. F. Nadel, op. cit.

81bid., p. 57.

200

based on suppressed relational operators9 (i.e., second-
order predicates), and this development is one of the
sources of the writer's conviction that a role system
model must be fundamentally relational. But persuasion
is not proof, and the problem remains open.

Turning our attention from the reduction of the
number of arguments permitted for the basic predicates,
we may look in the other direction and inquire whether
we need to consider increasing the number. In our develop-
ment of the model we used the term "relation" to refer to
dyadic connections, though its technical meaning does not
require this. The use was dictated by the fact that
almost all of the extant definitional materials assumes
just this much complexity. Is there evidence that our
model ought to be able to deal explicitly with, say,
third-order predicates? The role definers problem which
we have commented on above, as well as Biddle's concep—
tions of "Levels of Cognition,"lO might suggest this
requirement, but in each case it appears that the authors
do not see the actor's conception of the relationship
between others as denoting incumbency and its consequences.

However, Nadel, again, provides an example of the kind of

 

9Ibid., p. 10, et passim.

10B. J. Biddle, OE. cit., p. 14.

201

consideration which might require such an extension in his

concept of ". . . triadization . . . ,"11 which denotes the

 

concern or involvement of a "third party" in a dyadic
relationship. This amounts analytically to a relation
between a position and the role—sectors of another pair of
positions. It should be observed immediately that the usual
graphic representation cannot incorporate this feature with-
out considerable complication, though the matrix represen—
tation could do so by the addition of a fourth dimension.

In either case, the result is awe-inspiring. Perhaps it

is fortunate that the makers of definitions of role con—
cepts have not concerned themselves with third parties to
any significant extent.

There is another numerical question of some interest
concerning relations. In our development we have assumed
that the model requires a definite EEE.Of relations, but
we have not asked how many might be required in any con—
crete system. In general, of course, there is little but
speculation to guide such inquiries, but the work of some
investigators seems to suggest that the number might not

12

be very large. We have emphasized that the actual com—

position of a set of relations used to define a structure

 

11S. F. Nadel, op. cit., p. 86.

12E.g., A. F. C. Wallace, "On Being Just Compli-
cated Enough," Proceedings of the National Academy of
Sciences, 47 (1961), pp. 458—464.

 

202
is a theoretically based decision which has to be made by
the investigator, but there is nothing which need constrain
him to superfluity. If a small number would suffice, the
practical relevance of the kind of considerations outlined
in our fifth chapter might increase sharply.

Practical restriction in the actual number of
relations utilized might also be a heuristic circumstance
for solutions to two connected problems concerning the
relations among the relations themselves. We refer to the
question of ranking which has long been associated with role
systems through the concept of status, and to the property
of transitivity of relations which is fundamental to the
concept of an ordering of a set of elements. Transitivity
is defined with respect to single relations and the repre-
sentation is essentially "digraphic." However, it is some—
times possible to define a single common property of a set
of different relations, e.g., "power" as a component of
both "authority" and "influence," and thus satisfy the
requirement. Or more complex relationships might be
developed which induce transitivity. In the same example,

a three position chain of "influence" followed by "authority"
might be thought to entail "authority" between the first and
third positions, while "authority" followed by "influence"
might not do so. A still broader approach might be made

by the use of multidimensional scaling techniques, and this

would have much to recommend it. The problems of multiple

203

dimensions of ordering in social structures have excited
much interest, forxexample in such relatively recent con-
ceptualizations as status crystillization, congruence, or
equilibration, though such interests have been directed to.
investigating ways in which multiple orders collapse into
single dimensions.

Another kind of interest in relations among
relations concerns their compatibility, and while this
requires substantive considerations of the content of
relations, there are interesting formal features which a
model may isolate. The outstanding source of contemporary
interest in this topic appears in the proliferation of
studies of role conflict. The focus of these interests is
often in the dynamic consequences for the participant,
and on this such a model as ours has little to say. But
the antecedent stress is assumed to be structural, and it
would seem that a useful model ought to provide some
representation for this. For example, in the P-matrix
representation of the role system it is possible to form
pairs of cells for comparisons regarding compatibility
in eight different ways by enumerating ones with like
and unlike subscripts in all three dimensions. In fact
this might be taken to represent a basis for a more
general typology of general structural conflict within a
system since it allows cases which concern elements of

role sectors between completely different position pairs.

204
But such observations are little more than suggestive,
and their import goes beyond our purpose here of
noting another possible extension of analysis.

To repeat, the intent of this final section
is to underline the writer's belief that there are
manifold possibilities for fruitful extension and
reconstruction of the formal characteristics of our
conceptualizations of role systems. We conclude in
agreement with another recent observer of the field.

The final way of dissecting this conceptual
complex has doubtless not yet occurred.l3

 

13W. R. Catton, Jr., op. cit., p. 942.

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