STRUCTURAL SiMPLECW’Y AND THE
SELECTEON OR REJECTEON OF
FORMULATEONS EN 'E'HE SOCEOLOGY
0F EDUCATBON

THESIS FOR THE DEGREE OF P-H.D.

M SCH EGAN STATE U N iVERSITY

FRAN KLEN D. CORDELL

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Michigan State

University

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This is to certify that the

thesis entitled

STRUCTURAL
SIMPLICITY AND THE SELECTION OR
REJECTION OF FORMULATIONS IN
THE SOCIOLOGY OF EDUCATION

presented by

Franklin D. Cordell

has been accepted towards fulfillment
of the requirements for

Ph, D. degree in Education

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ABSTRACT

STRUCTURAL
SIMPLICITY AND THE SELECTION OR
REJECTION OF FORMULATIONS IN
THE SOCIOLOGY OF EDUCATION

By

Franklin D. Cordell

This dissertation is an inquiry into the
possible role of structural simplicity as a criterion
to be used by the sociologist of education and the ed-
ucational practitioner in the selection of formulations
in the sociology of education. Starting with Nelson
Goodman's explication of simplicity and Richard Rudner's
suggestions that structural simplicity might be an effec-
tive and objective criterion in the selection of
nontheoretic formulations in the social sciences, the
inquiry moved to focus upon the decision making process
in the contexts within which the sociologist and educator
operate.

It was concluded that the characteristic pur-
poses, attitudes and the conceptual frame of reference
of each decision context has a direct bearing upon the

decision making process and that there are some

“i ‘- -

 

 

EFEZHEi 1n '1)": Cor—dc 11""

significant differences between the contexts of the
philosopher of science, the sociologist of education
and the educator. The philospher is primarily con-
cerned with formal or logical matters in the context

of validation, while the sociologist is primarily con-
cerned with the development of concepts and their
application to social phenomena in a context of contin-
uing inquiry, and the educator is primarily concerned
with the power to be derived from employment of socio-
logical formulations in a context of use.

The problem of the evaluation of simplicity in
the sociologist's and educator's context is considered.
The specification of the role of simplicity in these
two contexts is described in terms of six criteria de-
veloped and specified for evaluation of criteria to be
directly used in the selection of formulations. Those
criteria are: appropriateness, objectivity, effective-
ness, compatibility, sequential significance and
overall significance. It was found that Goodman's
calculus of simplicity is inappropriate for use in such
contexts because it can be applied only to formalized
theoretic formulations and at present none exist in so-
ciology of education. It was argued that simplicity
might be made appr0priate by formalizing formulations,
but if this were done, it would demand premature

closure of formulations for sociologists, and this

Franklin D. Cordell

would be incompatible with the criterion of systematic
vagueness. If such a strategy were used in the educa-
tor's context, simplicity would not necessarily be
incompatible with other criteria, but would be sequen-
tially insignificant in the light of the weight placed
upon functional adequacy of formulations by the educa-
tor. It was argued that simplicity might be weakened
or altered to be made appropriate to nonformalized
formulations. This strategy is rejected because it led
to a trivialization of simplicity.

The overall conclusion drawn is that simplicity
as conceived by Nelson Goodman at this point in time
has an insignificant role to play in the selection of

formulations in sociology of education.

STRUCTURAL
SIMPLICITY AND THE SELECTION OR
REJECTION OF FORMULATIONS IN
THE SOCIOLOGY OF EDUCATION

By

Franklin D. Cordell

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Social and Philosophical Foundations in
Education

19 68

)Copyright by
FRANKLIN D. CORDELL

1969

To my son Franklin and my wife Louise who
pretended that I was human through the whole episode.
Without them I could not have completed the task, and

without them I would not have desired to do so.

TABLE OF CONTENTS

Page

Chapter
I. INTRODUcrIONO O O O 0 O O O O O O O O O O l

1.1 The need for research of this kind
1.2 Some definitions and distinctions

II. RUDNER'S SUGGESTIONS CONCERNING STRUCTURAL
SIMPLICITY AND SOCIAL SCIENCE
FOWULAT I CNS 0 O O O O O O O O O O O O O 28

2.1 The classificational scheme for
simplicity

2.2 The importance of simplicity in
science

2.3 Some problems in using simplicity

2.4 Some observations concerning the
usefulness of Rudner's suggestions
to the sociology of education

2.5 Summary of Chapter II

III. A REVIEW OF GOODMAN'S EXPLICATION OF
SIMPLICITY. O O C O C C O O O O O O O O 62

3.1 Goodman on the use of simplicity

3.2 The calculus of simplicity

3.3 Some observations concerning
Goodman's explication of simplicity
and the sociology of education

3.4 Summary of Chapter III

IV. SIMPLICITY AND THE SELECTION OF
FORMULATIONS IN SOCIOLOGY OF EDUCATION. 95

4.1 The problem of different contexts

4.2 The problem of evaluating criteria

4.3 An evaluation of Goodman's
arguments in other contexts

4.4 The role of simplicity in each
context

V. SUMMARY AND CONCLUSIONS . . . . . . . . . 140
BIBLIWRAPHY . O O O O O O O O O O O O O O O O O O 153

iii

LIST OF FIGURES

Figure Page

I. A General Model for the Selection
of Scientific Formulations . . . . . . . 123

II. An Example of the Sociologist's
Selection of Formulations. . . . . . . . 124

III. The Educator's Selection of
Formulations . . . . . . . . . . . . . . 125

IV. Questionnaire Given to Doctoral

Candidates Concerning the Selection
of TOpics and Criteria . . . . . . . . . 162

iv

CHAPTER I
INTRODUCTION

The first chapter of this dissertation
includes a rationale for research of this kind, a
discussion of the purposes of the inquiry, an elabo-
ration of the general plan of the inquiry, and some

preliminary definitions and distinctions.

1.1 The Need For Reseagch Of This Kind

This dissertation is based upon the assumption
that education finds its raison d'etre in what it can
contribute to human welfare and the preservation of
man. It is also assumed that the size and quality of
that contribution is determined in part by the knowl-
edge of the educative process attained by educators.

Sociology of education has promised to make
intelligible certain aspects of the social context
within which education is carried on, and it has accom-
plished this task to some extent, but has failed at
least in part, because of the very enthusiasm with which
new and different theories and formulations are pro-
duced. A brief look at the literature of the field

discloses a shocking proliferation of overlapping and

competing formulations. This proliferation of
formulations seems to have blocked the very intelligi-
bility that is so diligently sought.

The reports of people that have made more than
passing studies of the state of sociology of education
tend to confirm this somewhat pessimistic assessment.1
In fact, the present state of formulations in sociology
of education is such that Sidney Morgenbesser recently

2 Talcot Parsons, along

asked whether it is a science.
with several other self-conscious theoreticians, has
stated that the most disappointing single thing about
the work done in sociology is that theory has failed
to be cumulative.3
The question of the present state of theory in
the sociology of education will be analyzed later. For
now it will suffice to say that there is a destructive
proliferation of formulations in sociology. A result
of this proliferation is that a given term is used in
more than one formulation and is assigned similar, but

destructively different meanings in each of those for-

mulations. Brookover, for instance, comments that

 

1Richard Hoyme, "The Current Status of Educa-
tional Sociology," The ournal of Educational
Soc'olo , XXXVI (19635, 130.

2Sidney Morgenbesser, "Is It A Science?,"
Sociological Research, XXXIII (1966), 255-271.

3Harry Alpert, "Some Observations On The State
of Sociology," Pacific Sociology gevieg, VI (1963), 45.

such central terms as "social class" are given several
diverse meanings in the literature.“ This state of
affairs adds up to confusion on the part of the ed-
ucator seeking to use rather than develOp formulations.
For the theoretician it compounds the problem of
providing a cumulative body of knowledge or set of
concepts characterized by logical integration,
clarity, power and theoretic fruitfulness.

It would seem that a definite contribution
could be made to sociology and to educational theory
and practice if some of the problems surrounding the
proliferation of formulations and the consequent misuse
of terms could be clarified. The problems involved in
dealing with this proliferation of formulations are
complex, but undoubtedly their solutions will involve
the rating and subsequent selection of some formulations
for use by the educational "consumer" and further devel-
opment by the sociologist. The develOpment of an
adequate decision procedure requires among other things,
the clarification of certain logical issues and problems
not the least of which is the development of adequate
criteria by which formulations can be rated. This
dissertation is the report of an inquiry aimed at the
clarification of the possible role of structural

“Wilbur Brookover and David Gottlieb, g
Sociolo of Education (2d ed.; New York: American
Book Co., 19 4 , 154.

simplicity as a criterion for the selection of
alternate possibilities in the sociologist's process
of formulation selection and development.

It should be noted that there is no clear con-
sensus of opinion as to the relationship of the process
of proliferation of formulations in sociology of
education to the eventual achievement of an integrated
body of theory, nor concerning the relationship be-
tween the existence of large numbers of diverse yet
similar formulations to the usefulness of sociology of
education to educational practitioners. It would seem
that some sociologists think that "variety is the
spice of life" and that the proliferation of formu-
lations is a necessary, if not sufficient, condition
for progress in sociology. Perhaps these sociologists
have misconceived the task of science and confuse em-
pirical research with creative theorizing.5 On the
other hand some sociologists focus directly upon the

6

need for integrating theory. The educator's rela-

tionship to this proliferation is unclear since it is

 

5This is based upon the relative rarity of
sociological literature concerned with theoretic
integration.

6Sociologists such as Zona and Charles Loomis
have made major contributions to comparative studies
of sociological theories in their book Modern Social

Theories: Selected Acericac Wzitecs (2d ed. rev.;
New York: D. Van Nostrand Co., Inc., 1965)

hard to say exactly what the educator's relationship
is to any body of theory. It must first be assumed
that educational practitioners are users of sociolog-
ical formulations, and second it must be assumed that
the proliferation of formulations diminishes the
clarity and hence the usefulness of certain terms and
formulations for educators. It follows that the pro-
liferation diminishes the power of the educator to
control the social context of education by depriving
him of useful (clear, precise) concepts. A corollary
to this latter assumption is that a reduction of the
number of formulations in use in sociology of educa-
tion will make it more useful to educators. This

inquiry is motivated by that assumption.

1. 2 The Problem

The term "formulation" will be used inter-
changeably with what Professor Rudner calls
"scientific formulations," which he defines as follows.

"The written statements and parts of statements
that are the product of social-science inquiry
may be referred to conveniently as "formula-
tions." Some, but not all, of them are
formulations of theogy. We shall refer to
formulations of theory as theoretic formula-
tions; formulations that, like typologies,
definitional schemata, classificational sche-
mata, etc., are not theories we shall refer to
as nontheoretic formulations."

7Richard Rudner, Philoso h of Social Science
(Englewood c11rrs: Prentice-Hall, 19865, p. 10:

 

Rudner uses the term "empirical theory" to
refer to a set of empirically testable systematically
related statements, including some law-like generali-
zations. Rudner suggests that one of the important
differences between theoretic and nontheoretic formu-
1ations in the social sciences is that nontheoretic
formulations contain no empirical laws and hence are
not confirmable or disconfirmable in the ordinary sense
of those terms.

It must be stated that if the term "law" is
construed as Braithwaite does to include the specifi-
cation of a sufficient condition for the application
of a classificatory term in the statement:

"The laws may be merely the generalizations

involved in classifying things into various

classes. But to classify a whale as a mammal
is to assert the generalization that all
infant whales are provided with milk by their
mothegs, and this proposition is a general
law."
then the term "law" is not being used as Rudner does
in the sense of a universal or statistical empirical
generalization, but rather as an analytic statement.
Such statements do not yield to the test of empirical

confirmation--unless empirical confirmation is trivi-

ally reduced to mean simply finding an "entity" that

 

8Richard B. Braithwaite, "Scientific Explana-

tion: A Stud of the Function of Theor Probabilit
and Law in Science (Cambridge: The University Press,
1964) p. l.

"fits" the specification. It seems that even if the
term "law" is thus construed, the problem of selecting
one formulation, that is, one specification and subse-
quent division and organization of a universe of
discourse over another remains. It is precisely this
problem of selecting a nontheoretic formulation that
does not yield to confirmation or disconfirmation,
and it is this problem, among others, that is faced by
Sociologists of Education. Rudner states the problem
nicely:
"Now, empirical theories are assessed for their
acceptability on the basis of being confirmed
or disconfirmed by experimental and other
tests. But this sort of assessment is pre-
cisely what is logically unavailable for
nontheoretic formulations. By their very na-
ture, nontheoretic formulations (since they are
neither single hypotheses nor sets of systemat-
ically related hypotheses) are not susceptible
to confirmation or disconfirmation. But if
they cannot be confirmed or disconfirmed, by
what criteria can nontheoretic formulations be
assessed or validated or justified?"9
According to Rudner, any nontheoretic formula-
tion might be included in a theoretic formulation if
and when it generates at least one empirical law and
to that extent becomes empirically testable. On the
other hand, a theoretic formulation might be viewed as
a nontheoretic formulation that has generated at least
one empirical law. This notion gives rise to a dis-

tinction used in this dissertation. The term

 

9Rudner, Philosophy of Social Science, 40

8
"nontheoretic component of a theoretic formulation"
will be used to designate that part of any theoretic
formulation considered apart from any empirical laws
it might generate. For instance, the nontheoretic
component of a social stratification formulation is
the part of the formulation that specifies a universe
of discourse and partitions it by forwarding classifi-
catory terms such as the predicate ". . . belongs to
middle-class" or ". . . belongs to upper-class."

Now Since confirmation cannot be used as a
test in these instances, other criteria must be used.
Rudner and Ackermann have suggested that structural
simplicity be used as a criterion to select nontheoretic
formulations. Their suggestion follows from two con-
siderations: First, other criteria such as confirm-
ability, clarity and power are found unsatisfactory
for one reason or another, and second, simplicity has
traditionally been used in the rationales for the
selection and rejection of theoretic formulations.
Even though Rudner suggests that simplicity might be
used in selecting and rejecting nontheoretic formula-
tions his suggestion is untested and the applicability
and hence the fruitfulness of simplicity in such cases
is as yet untested.

The problem of calling simplicity into service

is further complicated by the fact that an investigation

9

of the literature of simplicity reveals a number of
explications of that term, and that two approaches to
the explication of the term presently enjoy favor
among philosophers of science. One approach taken by
Carl Popper and John Kemeny is associated with the
"curve-fitting problem." This approach attempts to
isolate a criterion for the selection of one of a
number of alternative hypotheses advanced to explain a
set of data. The simplest hypothesis is represented
by the "smoothest curve" that will "satisfy" or "fit"
the graphed points. A

The other approach is that taken by Nelson
Goodman. It is an attempt to define a criterion that
can be applied to the nonlogical primitive predicates
of formulations, but only to those formulations that
are formalized. Briefly stated, the simple formulation
is the one that has few nonlogical primitive predicates.
Just what is meant by this might be made plain by con-
sidering an easy example. Consider two text books on
the topic of plane geometry. Suppose both of the texts
contain a number of undefined terms, that is, terms
that are normally considered part of geometry such as
"point," "line," "plane" and so forth, from which other
terms are defined. Both the defined and undefined
terms are used in the postulates and theorems developed,

and both texts have equal numbers of terms, postulates

10

and theorems but one text presents only Ehgcc_of ten
terms as undefined terms and the remaining terms are
defined "in terms of" the three undefined terms. The
undefined terms would be considered primitives.lo
Say the second text presents fi1c_undefined or prim-
itive terms and five terms defined "in terms of" them.
Goodman would say that the first text presented the
simpler formulation of plane geometry.

In sum then the problem is this: Educators
who seek to bring intelligibility to the social context
of education are faced with a large number of similar
yet destructively different formulations in sociology
of education. They are forced to make choices, yet
they cannot rely upon "confirmedness." Can they rely
upon "simplicity?" Sociologists are also forced to
make decisions concerning the selection of formula-

tions, and even though their purposes might be different

from those of the educator, they also need a criterion

 

10The terms "primitive" and "predicate" are
used throughout this dissertation. The term
"primitive" will be used to designate terms that are
not defined explicitly within a system. The meaning
of a primitive is presupposed in the construction of
a logical system. The term predicate is used to des-
ignate a prOpositional function of one or more
variables. Said in another way a predicate is a rule
that is applied to linguistic entities. In common
parlance a predicate is an attribution of something,
a quality or a characteristic, to something else. For
the purposes of this dissertation a predicate might be
applied to either a linguistic entity or to the inter-
pretation of that entity. Some examples of predicates
might be: ". . . is a man," ". . . belongs to the set
of infinite.sequences," ". . . is a plane rectalinear
figure," or ". . . is middle-class."

11

or set of criteria to make those decisions. Will
simplicity satisfy their needs? This dissertation is

a report of an inquiry aimed at the eventual improve-
ment of educational theory and practice and sociological
theory by determining the feasibility of using the cri-
terion of simplicity as explicated by Goodman in the
selection of formulations in the sociology of educa-
tion. To accomplish this purpose, Goodman's

explication and Rudner's suggestions concerning the
application of simplicity, are taken as a point of
departure in search of an explication to be applied to
decision problems involving choice between formulations
and between alternate paths of development of formula-
tions in sociology of education. As the inquiry
develops, questions concerning the relation of simplicity
to other criteria, the applicability of simplicity to
sociology of education formulations, and the evaluation

of such a criteria are raised and answered.

1.13 Importance of the Problem
The development of a method of analyzing and

evaluating partially-formalized nontheoretic formula-

tions in the sociology of education is an important

task for at least two reasons. First, partially-for-
malized nontheoretic formulations not only carry on an
important function in the logical structure of sociology

of education themselves by specifying and "dividing up"

12

and hence making intelligible a universe of discourse
for educators, but sometimes they also become the basis
upon which theoretic formulations are logically con-
structed and hence are very important to sociological
theoreticians. That is to say the scientist's dis-
covery of regularities and his consequent statement of
them as empirical laws, is dependent upon the way he
defines and partitions the universe of discourse. The
definition and organization of a universe of discourse
is often accomplished through the construction of a
nontheoretic formulation of some sort. This is neces-
sarily logically prior to the statement of any

11 For example, the dis-

empirical generalizations.
covery of any laws descriptive of relationships
between a child's social class and his school achieve-
ment necessarily presupposes some delineation of
society, an organization of the elements of that
universe of discourse (society) into subgroups (social
classes) through some set of categories, and finally

some formulation delineating a second universe of

discourse comprehending school behavior and organizing

 

11This is a logical, rather than a temporal
priority. In "science as a process" it might be the
case that the theory is developed over a period of
time and the nontheoretic component is chosen on the
basis of theoretic fruitfulness. In this case the
"laws" might be formulated before the final refinement
of the nontheoretic base. The "laws" would require
reformulation after the nontheoretic component was
reformulated or refined.

13
that universe of discourse in such a way that "school
achievement" can be clearly designated and differenti-
ated from other patterns of behavior in the same
universe.

Second, even though confirmation is seen by
many researchers as the most important criterion in
evaluating formulations, it has already been pointed
out that it has some severe limitations in dealing with
c;;,nontheoretic formulations and ccpc_theoretic for-
mulations for reasons mentioned above, and by itself,
it is not an adequate criterion to select theoretic
formulations, since an unsystematic formulation (if
such a thing can be conceived), or one that generates
vacuous or analytically true statements might, in one
sense, be confirmed.

Other criteria might have been investigated
but, as Rudner points out, simplicity shows promise of
being both objective and effective where other criteria
fail.12

It is widely held that the theoretic results
that is, the empirical laws generated by a formulation
can be no better than the primitive base (a nontheoretic
component) of the formulation itself. If the nonlogical
primitive predicates of a formulation are empirically

empty or unclear, the generalizations generated from

12Rudner, Philosophy of Social Scicnce, 47

14

them will reflect this emptyness and lack of clarity.
It is suspected, and an attempt will be made to
establish as fact, that if a scientist can reduce the
set of nonlogical primitive predicates to a set con-
taining only those predicates that are characterized

by certain desiderata, e.g., clarity and definite em-
pirical content, then the explicitly defined terms and
consequently the statements generated using those terms

will somehow reflect those desiderata.

1.2 Some Definitions apd Distinctions

Even though the problems discussed in this
dissertation are important to the educational practi-
tioner, the problems themselves prOperly "belong to"
the philosophy of social science. That is to say, the
problem has been defined and treated in the philosoPhy
of social science. This statement does not completely
clarify the frame of reference however, since there are
several areas of disagreement among philosophers of
social science. For that reason, several definitions
and distinctions are made to make the basic assumptions
of this dissertation clear.

It is to be noted that the definitions and
distinctions offered here are not the only definitions
that might reasonably be put forth for the terms de-
fined, and that some of the distinctions made might be

attacked as useless, or as positively damaging to

15
progress in the philosophy of social science,
sociology of education, and education. In fact, the
usefulness and adequacy of some of them is questioned
later in this dissertation. The definitions and dis-
tinctions offered here are not the "natural"
distinctions, or the "correct" definitions, but rather
a set of definitions and distinctions currently used
by part of the community of philos0phers of social
science. Only those definitions and distinctions
needed to set the general frame of reference are offered
here. Others will be offered later.

Richard Rudner makes several relevant distinc-
tions early in his book Philosophy of Social Science.
The first of those is the distinction between method
and technique in science. For Rudner, a scientific
technique is a specific mode of observation, experi-
mentation, or experimental control. The techniques
employed by the biologist vary from those used by the
physicist or the psychologist. Method however, does
not vary from field to field, but is the rationale on
which all science bases the acceptance and rejection
of hypotheses and theories. The following is his
statement concerning this distinction:

"To claim that there is a difference in method-

ology between two disciplines or two types of

disciplines is, by contrast, to make a very
radical claim. For the methodology of a scien-

tific discipline is not a matter of its
transient techniques but of its logic of

16

justification. The method of a science is,
indeed, the rationale on which it bases its
acceptance or rejection of hypotheses or
theories. Accordingly, to hold that the social
sciences are methodologically distinct from the
nonsocial sciences is to hold not merely (or
perhaps not at all) the banal view that the
social sciences require a different logic of
inquiry. To hold such a view, moreover, is to
deny that all of science is characterized by a
common logic of justification in its acciptance
or rejection of hypotheses or theories."

As Rudner points out, this distinction goes a long way
toward solving the problems raised by the "verstehen"
discussion in the social sciences. For Rudner the dis—
pute is resolved by allowing "verstehen" as a technique
appropriate to the social sciences in the "context of
discovery." This leads to the second important dis-
tinction made by Rudner. Concerning the controversey
he says:

"That social science is methodologically dis-
tinguished from the other sciences is a claim
that falls within the area of philosophical
interest called the context of validation, or
the context of justification, in contrast to
that area of empirical inquiry which is called
the context of discovery. . .

Now, in general, the context of validation
is the context of our concern when, regardless
of how we have come to discover or entertain a
scientific hypotheses or theory, we raise
questions about accepting or rejecting it. To
the context of discovery, on the other hand,
belong such questions as how, in fact, one
comes to latch on to good hypotheses, or what
social, psychological, political, or economic
conditions will condufe to thinking up
fruitful hypotheses." 4

 

13Rudner, PhiIOSOphy of Social Science, 5
l“Rudner, PhilOSOphy of Social Science, 6

 

17

A third distinction to be made is that between
science as a product and science as a process. Rudner
says:

"On the one hand (as process term), it is used
to refer to the activities or workings of sci-
entists or scientific institutions, i.e., to
experimenting, observing, reasoning, reading,
organizing research projects, etc. But on the
other hand, the same term is employed to refer
to a result of these activities or processes,
to the product of scientific activity, i.e.,
to a corpus of statements purporting to
describe one or another aspect of the universe
and embodying what counts as our scientific
knowledge.

To understand the following chapters, it
is important to distinguish between science-
as-product and science-as-process. In partic-
ular, it must be noted that 'science' (as
product) refers to linguistic entities ggc,
only, 'science' (as process) refers to extra-
linguistic phenomena."l

Since this dissertation will be discussing
criteria applicable to linguistic formulations it is
necessary to make some major distinctions between
kinds of formulations. This schema is based upon one
offered by Richard Rudner in Epilosophy of_Soci§1_
Science.

Rudner defines a formulation as the "written

statements and parts of statements that are the

 

15Rudner, Philos0phy of Social Science, 8.
Note here that Rudner first seems to include all of

what might be considered the product of scientific
activity. This might be a linguistic formulation apg
its interpretation. He then goes on to limit science-
as-product to linguistic entities. He is either
inconsistant or destructively vague here.

18

16 All

product of social-science inquiry."
formulations are divided into two categories: theoretic
formulations and nontheoretic formulations. A theoretic
formulation is a set of systematically related state-
ments, including some law like generalizations, that

is empirically testable. Theoretic formulations enjoy

a range of formalization. A fully formalized theoretic
formulation is a completely articulated deductive
system. Said another way, a fully-formalized theoretic
formulation is a theory that has a set of nonlogical
predicates that are taken explicitly as primitive
(undefined in the formulation) predicates from which

all other nonlogical predicates are defined. The fully-
formalized theoretic formulation will also be character-
ized by a set of statements explicitly designated as
axioms, and a set of transformation rules. All other
well-formed statements are derived from the set of
axioms by application of the transformation rules. The
fully-formalized theoretic formulation is interpreted

in some model, that is, each of the terms and state-
ments of the formulation are assigned a meaning in some
other linguistic or empirical system by rules of inter-
pretation. These rules are commonly called rules of
correspondence or semantical rules. Rudner says:

"A model for a theory consists of an alternative

 

l6

Rudner, Philosophy of Social Science, 10

19

interpretation of the same calculus of which the
theory itself is an interpretation."17 He gives an
example of a Boolean algebra being interpreted in
terms of elementary algebra. Hence the axiom:
a0(bOc)=(aOb)oc, in Boolean algebra is interpreted as:
x+(y+z)=(x+y)+z, in elementary algebra. Here a, b, c
are interpreted as variables x, y, z, and the O is
interpreted as the operation of addition. Parentheses
are used in the same way and identity is used in both
systems.

Theory T_would then be made up of calculus S
as it is interpreted in model a, while theory T}, might
be calculus §,interpreted in model E). If the model
in which a calculus is interpreted is a linguistic
system, as in the case of interpreting a Boolean al-
gebra in elementary algebra, then the theory is an
analytic theory. If on the other hand the Boolean
algebra were interpreted in such a way that the a, b,
c were the weights of piles of sand and Q,was the cp-
eration of heaping the piles together, then one would
say that the Boolean algebra and its model were an
empirical theory.

A partially-formalized formulation is a for-
mulation that has at least one explicit definition, or

at least one theorem. A non-formalized formulation

 

l7Rudner, Philos0phy of Social Science, 24

20

will be one that has no explicit definitions and no
theorems, that is, all nonlogical predicates are prim-
itive, and all statements are axiomatic or empirical.

It is easy to see that the fully-formalized
theoretic formulation will of necessity be clear in the
logical sense. That is not to say that it will be
clear in a psychological sense.

The second division of formulations is the
nontheoretic formulations. Rudner names four types of
nontheoretic formulations: the definitional schemata,
the analytical conceptual schemata, the classificational
schemata and typologies. Rudner defines the defini-
tional schemata as follows:

"This type of system of definitions consists

of two sets of linguistic entities: a set E_

of predicates, and a set Blof definitions which

refer to those predicates. fl.will be exhaust-
ively divisible into primitive predicates and

defined predicates. 3 will be related to 11

(and indeed will systematize fl_by determining

which members of fl_are primitive and which

are not) in the following ways:

1. Every predicate in fl_that is referred
to by an expression comprising a de-
finiendum in any element (i.e., in any
definition) of E, is a defined predicate
of the system.

2. Every predicate in fl_that is referred
to only in definientia of B, or does not
occur in any element of R gt all, is a
primitive of the system.“-1

This might be made clear by an example. Suppose

 

l8Rudner, Philosophy of Social Science, 50

21
a researcher is interested in developing a definitional
schema to be used later to describe family relation-
ships in a primitive society. He might develop two
lists, one of which will be composed of such terms as
male, parent, sibling and so on--all the terms he
thinks he will need. He will then define some of the
terms, and take some of them as primitives. A very
limited schema might look like this:
Undefined or Primitive terms:
I. Male
II. Biological parent
III. Sibling

Defined terms:
1. Father =df male parent
2. Mother =df non male parent
3. Brother =df male sibling
4. Sister =df non male sibling
5. Uncle =df father's brother
Terms I, II, and III are primitive terms and 1,
2, 3, 4, and 5 are defined terms. The E,that Rudenr
speaks of would be I, II, III, and 1, 2, 3, 4, 5. .1;
is the set of definitions to the right of 1, 2, 3, 4,
and 5.
Rudner goes on to point out that a particular
definitional schema is accepted or rejected as a

component of a theory on the grounds of its general

22
adequacy for a field of study, that is, does it

include enough definitions for fruitful theory. Other
criteria mentioned are structural simplicity and power
of the system.

The analytical conceptual schemata is a type
of schema that contains a definitional system as a
subpart or a component, but it also contains a set of
logically true or "truistic" sentences. These sentences
are adjudged true on the basis of logical evidence
rather than by reference to empirical evidence. If one
looks at the example given of the definitional schema
it is easy to see that if one stipulates that a person
cannot be both male and non male at the same time, other
analytic statements can be made such as "an uncle cannot
be a mother." This statement is said to be true by
analysis rather than by empirical reference.

Both the definitional schemata and the analytical
conceptual schemata are pre-theoretic formulations inso-
far as they are destined for inclusion in some social
science theory.

Another important kind of nontheoretic formu-
lation mentioned by Rudner is the classificational
schema. In general a classificational schema is a kind
of nontheoretic formulation that lays down conditions
for the applicability of its categorical or classifi-
catory terms. It is to be noted that any classificational

schema might be reformulated as either a definitional

23
schema or an analytical schema, while the converse is
not true. Rudner makes a point of this:
"While some classificational schemata are thus
appropriately construed as definitional, the
converse is not true: not all definitional
schemata are classificational. . . Such
systems are usually quite loosely presented,
either wholly discursively or in only partially
formalized manner; and where they are explic-
itly dubbed classificational schemata, they 19
are also sometimes presented diagrammatically."
Every classificational schema will presuppose 1
its universe of discourse which it will divide in
either mutually exclusive and jointly exhaustive sets
or some set of categories that either fail to partition
some of their respective universes of discourse exclu-
sively or exhaustively, or, if they do partition all
of them exclusively and exhaustively, it will not be
by virtue of logical means, but due to some feature of
the empirical world to which they relate.

A relevant example of a classificational schema

in the educational literature is that proposed in

Eguality of Educational Opportunity to classify

students.20

In that report school populations were
partitioned into six groups by the following categories:

Mexican-American, Puerto Rican, Indian American,

 

l9Rudner, Philosophy of Social Science, 32
20U.S. Department of Health, Education, and

Welfare E ualit of Educational 0 ortunit
(Washington: U.S. Government Printing Office, 1966)

24

Oriental American, Negro, and Majority. No attempt
was made to "order" the categories to say that the
Mexican-American was more or had more of something than
the Majority group.

The final nontheoretic formulation that will
be used in this dissertation is that of typology.
Typology will be the kind of nontheoretic formulation
that is characterized by the following:

"1. a concept determining the typology's
. universe of discourse.

2. some relation (3) that determine (5) an
ordering (e.g., as serial or quasi-serial
ordering) among the members of the
universe of discourse.

3. statements implying that certain features
(e.g., transitivity) characterize the
relation (s) (if these statements are
analytic the typology is theoretic)

4. a set of concepts (frequentl a "polar-
pair", or "extreme-opposites usually
designating some specific members of the
universe of discourse that are "distant"
from each other or at opposite ends of
the array into which the ordering
relation (5) order the members of the
universe of discourse."21

The typology then has all the characteristics
of a classificational schema and beyond that, it has
a concept for the explicit determination of its
universe of discourse, an ordering concept or set of

concepts, and often a set of "polar-pair" concepts.

 

21Rudner, Philoso h of Social Science, 37-38

25
Stratification formulations usually carry the import

of a typology even when they do not satisfy the
conditions that Rudner prescribes. The concept "social
class" might well be construed as a concept determining
a universe of discourse. The basis for allocating
individuals to social classes varies considerably from
formulation to formulation, but classes are "ordered"
according to some such notion as prestige, political
power, income or property owned. It is common to read
in sociological literature that the mean income of
"lower-class" individuals is less than that of "middle-
class" individuals, and that the mean income of.
"middle-class" individuals is less than that of "upper-

22 In such an instance it is clear

class" individuals.
that the formulation is to be construed to designate
a set of "polar-pair" concepts that is, "lower-class"
and "upper-class." It is also clear that social
classes are to be ordered; in this instance on the basis
of income. It is on these two points that the strati-
fication formulation differs from a simple classifica-
tional schema.

The sociology of education is defined in many

different ways, and since the purpose of this disser-

tation is to investigate particular formulations in

 

22This of course would be an analytic
statement if the "ordering concept" were income, and
not be analytically true, but might be empirically
true if the "ordering concept" were personal prestige.

26
relation to structural simplicity, the selection of a
definition of sociology of education is important only
insofar as it must include the formulations chosen for
discussion here. Wilbur Brookover forwards a definition
in his pOpular textbook: "The sociology of education
is the scientific analysis of the social processes and
social patterns involved in the educational system."23
No distinction is made between sociology of education
and educational sociology.

This dissertation will later "test" the concept
of structural simplicity in decision problems concerned
with the selection of a classificatory schema, a
typology and a theoretic formulation. The classifi-
catory schemata to be studied will relate to the
classification of students in various ways, while the
typologies and the theoretic formulations will be drawn
from the stratification literature that is associated
with schools and education. Given the vagueness of
certain central terms in Brookover's definition of
sociology of education, it is easy to construe the
definition to include the formulations to be discussed.
It might also be pointed out that the formulations to
be considered are in fact drawn from what is called

the literature of sociology of education.

 

23Brookover, A Sociology of Education, 11

27

Chapter two is a review of Richard Rudner's
suggestions concerning the use of simplicity in the

social sciences in general.

CHAPTER II

RUDNER'S SUGGESTIONS CONCERNING STRUCTURAL
SIMPLICITY AND SOCIAL SCIENCE
FORMULATIONS

In the past several years Richard Rudner has
shown considerable interest in the problems associated
with explicating and using the term simplicity.l In
his writings on that tOpic he has repeated the theme
that formal simplicity, as explicated by Nelson Goodman,
is important in a logic of induction, and that it shows
some promise of being an effective and objective crite-
rion in some situations where other criteria are
unsatisfactory. Rudner's discussion will serve as a

point of departure for this inquiry.2

 

lRudner's interest is first apparent in Philoc-

Ophy of Science Vol. 28, No. 2, April, 1961 where his
article "An Introduction to Simplicity" appears. In
his book Philoso h of Soci S ience (Englewood
Cliffs: Prentice-Hall, 19 , p. 43. Rudner relates
Nelson Goodman's explication of simplicity to the
problem of selecting nontheoretic formulations in the
social sciences. Rudner also states his belief that
Goodman's explication is important in his article

"Nelson Goodman," EncyclOpedig of Ehilosophy, ed. Paul

ZRudner's discussion includes a classifica-
tional schema for explications of simplicity, arguments
for the weight of simplicity and a discussion of some
problems in the use of Goodman's explication of
simplicity.

_28

29

In the last part of the chapter certain
questions are raised concerning the apprOpriateness of
Rudner's frame of reference to problems in sociology
of education. For instance, Rudner talks only about
simplicity in relation to "science as a product." It
is indeed possible that, because of its present stage
of develOpment, sociologists of education are more
interested in "science as a process." Rudner attrib-
utes importance to simplicity on the basis of its
possible function in a "context of justification."
Some sociologists, after Merton, would argue that so-
ciology is best served by exerting effort toward
develOping "middle range" formulations in the "context
of discovery." Theories of the middle range would be
theories of limited scope, such as a theory of small
group interaction or of teacher socialization, as
Opposed to theories on a grand scale, formulated to
explain the workings of whole societies. This disser-
tation is written assuming that this approach may have
been misleading, and that constant attention must be
given to the logical aspects of formulations as they
are developed. This is necessary to eliminate the
wasting of resources through attempts to confirm poorly
conceived and overlapping formulations, and to elim-
inate the confusion that grows with the proliferation

of formulations.

30

2.1 The Classificational Scheme for Simplicity

Rudner forwards a classificational schema
based upon one devised by Robert Ackermann.3 The
basic classifications are the same as those used by
Ackermann, but the terms used are different. The
classificational schema as altered by Rudner first
appeared in PhilosOphy of Science in "An Introduction
of Simplicity" in April of 1961, and is repeated
virtually without change in his book Ehilosophy of
Social Science, in 1966. The following rather lengthy
quote contains the gist of his schema:

"Now, allusions to simplicity in
present-day science and in the historical
literature of the sciences and of philosophy
are immensely varied in intent and
nuance. . . . all of the immensely varied
references to simplicity (can be placed) under
a relatively uncomplicated classificational
schema.

Uses of 'simplicity,' then, may be

classified either as ontological (i.e., extra-
linguistic)

 

3Robert Ackermann, Simplicity gpd the

ACceptability of Scientific Theories, Doctoral disser-
tation, Michigan State University, 1960.

31

or descriptional (i.e., linguistic). Sub-

classifications under both these main rubrics

are subjective (i.e., psychological) and ob-
jective (i.e., nonpsychological). Moreover,
under descriptional it is useful as well to
distinguish notational and logical (or struc-
tural) as subclassifications of the kinds of
simplicity."

Professor Rudner points out that in the first
category, that of ontological simplicity, statements
of simplicity supposedly attribute some quality to the
universe or the way we perceive the universe. If the
statement is meant to attribute a characteristic of
the universe independently of how we perceive it, then
it is Objective-Ontological Simplicity. If on the
other hand, it is taken as a predicate of our responses
to the universe, whether we see the universe as simple
or not, then it is Subjective-Ontological Simplicity.
Rudner rejects this set of concepts as "fundamentally
obscure to me."5 This obscurity might be a function of
Rudner's distinction between "science-as-process" and
"science-as-product." Insofar as Rudner hesitates to
speak of simplicity in relation to "science-as-process"
and ontological simplicity would lead to assertions
about "science-as-process" Rudner may be rejecting an

approaCh to philosophy of social science as well as one

notion of simplicity.

 

“Rudner, Philosophy of Social Science, 42
5Rudner, Philosophy of Social Science, 42

32

The second main category is that of

descriptional or linguistic simplicity. Subcategories

under descriptional simplicity are notational and

logical.

Rudner describes this distinction as follows:

"An attribution of simplicity to a de-
scription on the basis of such a notational
prOperty of it as the number of (e.g., alpha-
betical) characters it contains, independently
of anyone's psychological responses to such a
property, will be classed as an instance of
Objective-Notational simplicity. On the other
hand, an attribution of simplicity to a de-
scription on the basis of the familiarity of
the notation, or its elegance, or its conven-
ience, or its efficiency for manipulation, or
any aesthetic quality it has, etc., will count
as an instance of Subjective-Notational sim-
plicity. . . .

In the sense just indicated, then, neither
objective nor subjective notational simplicity
is our topic of concern in what follows. What
is being attended to is the logical (or, alter-
nately, the formal,or structural) simplicity
of descriptions--and especially those descrip-
tions which constitute scientific theories.
Moreover, since our interest is not in how
people psychologically respond to logical
properties of theories, we may characterize our
field of attgntion as ObjectiveeLogical
Simplicity."

Here Rudner charges to the heart of the problem

of simplicity as it is interesting to the philosopher

of social science. He is interested in Objective-

Logical Simplicity.

Professor Rudner goes on to note that philoso-

phers and scientists, (with the notable exception of

 

ity,"

6

Richard Rudner, "An Introduction to Simplic-

Philos0phy of Science XXVIII, No. 2 (1961)

111,.112.

33

Mario Bunge) have taken two major approaches to the
explication of ObjectiveéLogical Simplicity. The two
approaches are fundamentally distinct and Rudner has
some misgivings regarding the application of this clas-
sification to one of them. The approach taken by
Nelson Goodman has shown considerable positive results.
The second line of inquiry, an older one, is associated
with the "problem of curve-fitting." This approach
assumes the notion that in confrontation with data
represented as a set points on a graph, the scientist
should "induce to" the simplest (smoothest) curve or
function descriptive of the data.

The second approach has produced various expli-
cations of simplicity by Harold Jeffreys, K. R. Popper,
and John Kemeny. Professor Rudner rejects the approach:

"The concept of inductive simplicity as
elaborated by Jeffreys, Popper, and Kemeny
comes to depend on some such notion as the
number of "freely? adjustable parameters" which
occur in alternative hypotheses. . . . Even if
such treatments were otherwise wholly success-
ful, it is doubtful that they would furnish

any tenable criterion or test of simplicity-

especially of formal simplicity. Thus, for

example, there seems to be no good reason for
believing that a hypothesis with n+1 parametric
expressions is less simple in the sense of

Goodman's calculus than a hypothesis with n

parametric expressions."

Professor Rudner goes on to point out that it
will not in general be the case that the set of predi-

cates from the hypothesis of n+1 parameters will have

 

7Rudner, An Introduction to Sim licit , 115

34

a higher complexity value than sets of predicates from
alternative hypotheses of less than n+1 parameters.
This will depend on, among other things, whether or how
some of the predicates in such sets are definable by
others in the same set. According to Rudner, this will
lead to conflict of judgments with the assessment of
simplicity by Goodman's calculus.

Rudner points out that all this proves is that
8

the two concepts are not identical. But since this is

so, he goes on to ask why they are not identical. If

 

8W. V. O. Quine points out that the "curve
fitting" problem, as a problem of selecting the
"smoothest curve" may involve something as simple as
the fact that the probability that subsequent data will
confirm the hypothesis is raised when "round figures"
are used in constructing a curve. He says: "True, we
do not customarily say "simple hypothesis" in the
round-number case. We invoke here no maxim of the sim-
plicity of nature, but only a canon of eschewing
insignificant digits. Yet the same underlying prin-
ciple that operates here can be detected also in cases
where one does talk of simplicity of hypotheses. If we
encompass a set of data with a hypothesis involving
the fewest possible parameters, and then are con-
strained by further experiment to add another parameter,
we are likely to view the emendation not as a refutation
of the first result but as a confirmation plus refine-
ment, but if we have an extra parameter in the first
hypothesis and are constrained by further experiment to
alter it, we view the emendation as a refutation and
revision. Here again the simpler hypothesis, the one
with fewer parameters, is initially the more probable
simply because a wider range of possible subsequent
findings is classified as favorable to it. The case of
the simplest curve through plotted points is similar:
an emendation prompted by subsequent findings is the
liklier to be viewed as confirmation cum-refinement,
rather than as refutation and revision, the simpler the
curve." This occured in "On Simple Theories of a
Complex World," 5 nthese,.XV, (1963), 106.

35

the "curve-fitting approach" measures something other
than what Goodman's calculus measures, he would like to
know just what it is. He concludes that whatever the
cogent criteria may be for ordering a selection of
hypotheses in the "curve-fitting" situation, it would
be misleading to classify them as considerations of
ObjectiveéLogical simplicity.9 Rudner's rejection of
this approach is taken as sufficient grounds for elim-
inating it from further consideration here.

Professor Rudner points out that Goodman's
approach is the only one that can reasonably fall under
the rubric of ObjectiveéLogical simplicity. He points
out that it is not implausible to assume that the for-
mulations of science all share a common apparatus of
logical expressions. This turns his attention to the
extralogical expressions in any formulation. It is
not unlikely that one would first attempt to assign
simplicity on the basis of the set of axioms or pos-
tulates of a system. He points out that this approach
immediately runs into a problem since any formulation
can have its number of axioms trivially reduced to one
by the logical operation of conjunction. This leads
to the sorry conclusion that every formulation would be

trivially equivalent by reduction to maximal simplicity.

 

9Rudner, An Introduction to Sim li it , 117.

36

This leads to the assessment of simplicity on
the basis of the characteristics of the extralogical
predicate bases of formulations. Rudner points out
that this is exactly the approach that Nelson Goodman
has taken, and points out that Goodman has been able
to construct a calculus of simplicity which provides
a means of assessing the simplicity of the predicate
bases of fully formalized systems. "In general (and
here, vaguely), Goodman's assignments of simplicity-
value may be thought of as depending on the manner in
which the extralogical primitive predicates of a
system organize (by virtue of such of their logical
properties as reflexivity or of symmetry) the entities
comprised in the total extension of the system."10
This is a limited but precise description of Goodman's

explication of simplicity.11

2.g The Importance of Simplicity In Science

Professor Rudner argues the importance of
simplicity on two points: (1) simplicity is important
to the development of an adequate logic of induction,
and (2) due to the lack of other adequate criteria,
simplicity may be a very important criterion in the

 

10Rudner, PhilOSOphy of Social Science, 44

llGoodman's explication is discussed more
fully in the following chapter.

37

selection of nontheoretic formulations in the social
sciences. Concerning the importance of simplicity to
a logic of induction he says, "the problem of acquiring
an adequate measure of simplicity is crucially important

12 A "logic of in-

for the logic of induction itself."
duction" is an adequate body of methodological princi-
ples upon which decisions concerning the selection and
rejection of formulations can be based. This need for
a body of principles is necessary for the simple reason
that no one criterion is applicable to all decision
situations.

Simplicity is seen as an important criterion
because it is so closely related to systematicity, an
essential characteristic of scientific descriptions.

To make the point, Rudner quotes Goodman:

"We are inevitably concerned with simplic-
ity as soon as we are concerned with system at
all: for system is achieved just to the extent
that the basic vocabulary and set of first
principles used in dealing with the given sub-
ject matter are simplified. When simplicity
of basis vanishes to zero--that is, when no
term or principle is derived from any 23 the
others-~system also vanishes to zero."

Insofar as simplicity is related to systemati-
city, it is also related to "eXplanatory power." Even
though the notion of "explanatory power" is probably

reducable to power of primitive basis and fruitfulness

 

12Rudner, Philosophy of Social Science, 45-46

13Nelson Goodman, "The Test of Simplicity,"
Science, CXXVIII (1958), 1064 .

38
of a formulation, these characteristics are often
sought under the rubric of "eXplanatory power." Sci-
entists feel that they increase the explanatory power
of a formulation when they systematize or bring together
a number of formerly discrete or logically disconnected
laws or generalizations. The Newtonian physics for
example was thought to have greatly increased the ex-
planatory power of extant physical laws by tying them
together logically. What the Newtonian formulation
accomplished probably increased the heuristic value--
or said another way, it surely made the physical world
more intelligible, and it probably made physics more
"learnable." Suffice to say, systematicity is a
primary desiderata of science.

It is upon this basis that Professor Rudner sees
simplicity as important to a general logic of induction.
Simplicity is also important by default, since other
criteria considered are inadequate. Clarity is com-
monly thought to be important by scientists, but Rudner
rejects this criterion as ultimately important on the
grounds that it is subjective, since it is relative to
person, place, time and even mood. Obviously a set of
predicates or hypotheses might well be clear to their
inventor, and yet be quite unclear to others.

It would certainly seem that there is a legit-

imate sense in which the term clarity might be

39
attributed to a formulation. Surely we might say that
a classificational schema is logically clear if it is
made up of a set of mutually exclusive and jointly
exhaustive categories, rather than a set of categories
that are indeterminant in extension, or worse, of inde-
terminable extension. In empirical formulations there
is a second sense in which one formulation might be
said to be more clear than another. An empirical
formulation that employs predicates that have direct
empirical reference rather than reference to "the
world" through theoretical constructs might be said
to be more clear. For instance a clasSificational
schema for members of a community based upon skin
color might be said to be more clear than one based
upon national origin of the parents of the individual
members, or the marital status of the individual
members.

Another criterion, theoretic fruitfulness, de-
fined as the number and scope of empirical laws generated
by a formulation, is also a most important criterion
but obviously since it cannot be applied to formulations
until they have been fully elaborated, its usefulness
is restricted in two ways: (1) it cannot be applied
to nontheoretic formulations at all, (2) nor can it be
applied to theoretic formulations until they are well

developed.

40

A third criterion, power of primitive base,
that is, the scope of the extension of the primitive
terms, is also important, but again, by itself power
is not adequate since it cannot be used to choose be-
tween egually powerful bases. For example, one might
look at attempts to axiomatize arithematic. Of the
two major attempts, both formulations have equally
powerful bases, since they both are adeguate bases
for the system.

It should be noted that no criterion is suf-
ficient on its own, but becomes adequate when used at
a given time, at a given stage of development. For
instance, in the early stages of the development of a
particular formulation a scientist is often more inter-
ested in the intuitive clarity of the formulation than
he is in the precision of the statements generated.
Later, the criterion of confirmability will ascend
the throne of the most crucial criterion. At another
stage the scientist will be interested in the coherence
with the rest of science manifested by his formulation.
An example that is often used in science is the rejec-
tion of certain approaches to "psychic phenomena" on
the grounds that those approaches do not "fit" modern
science. The approach might well be clear, and
possibly even confirmable, but it is rejected on the

grounds of lack of coherence with psychology and

41
other basic concepts of modern science.

For example, scientists who would study the
phenomenon of clairvoyance are in general shunned in
the scientific community because the possibility of
anyone knowing future events is inconsistent with
present psychological conceptualizations of learning
and beyond that it violates the notion of time that
is pretty well accepted by the scientific community.
To say that Jean Dixon sometimes makes remarkably ac-
curate predictions about the future is to say one
thing. Social scientists predict election results
and riots too. But to say that Miss Dixon in some
way "sees" the future violates a notion of time and
relations basic to modern science. To "see" the
future requires that a future state exiSt in the pre-
sent and this somehow destroys the distinction between
present and future. In other words to admit certain
formulations or even certain approaches to studying
phenomena demands a rejection of the present structure
of modern science; and most scientists are unwilling
to do this.

It ought also to be noted that there is no
guarantee that all criteria that play an important

role in scientific decision making are compatible.

Mario Bunge eloquently points out that a scientist

 

42
might seek simplicity to the loss of "transcendent"
terms.14

The second point is that simplicity may be
important in selecting nontheoretic formulations. If
it is, then simplicity is especially important for the
social sciences, since they are poor in theory and
rich in nontheoretic formulations. Professor Rudner
makes a definite point of this in the following
statement:

"Now, empirical theories are assessed
for their acceptability on the basis of being
confirmed or disconfirmed by experimental and
other tests. But this sort of assessment is
precisely what is logically unavailable for
nontheoretic formulations."15
Due to the special weakness of other criteria, con-
firmability being the singularly most important, in
the area of assessment of nontheoretic formulations,
simplicity takes on added importance.

An important question, one that Rudner did not
adequately treat, is now beginning to emerge. It
might go something like this: What is expected of a
criterion that is to be used in selecting nontheoretic

formulations. Said another way, what is to be ad-

mitted as evidence that simplicity is a "valid" or

L A_

l“Mario Bunge, "The Weight of Simplicity in
the Construction and Assaying of Scientific Theories,"
Philos0phy of Science, XXVIII, No. 2 (April,_l96l),
122.

15Rudner, Philoso h of Social Science, 23

43
"good" criterion in such decision situations. Since
this dissertation sets out to define the role of sim-
plicity in the development of formulations in
sociology of education, this question is of central'
importance to its logical develoPment.

Rudner suggests certain criteria in his dis-
cussion of clarity, power and fruitfulness. It has
already been noted that clarity is not "objective,"
power is not "effective" and fruitfulness is not
"appropriate." This gives a hint of some of the cri-
teria that might be seen as relevant, and will be
used as the core of a list of criteria by which a

16

criterion might be selected or rejected.

2.3 Some Problems in Using Simplicity
Even though the calculus of simplicity holds

some promise for evaluation of nontheoretic formulations
in the social sciences, the calculus is applicable only
to theories that have been at least sufficiently for-
malized to enable discrimination of their primitive
predicates. Its utility as a factor in assessing the
acceptability of formulations in the social sciences

is severly limited by the fact that there are few

 

16The criteria to be used in selecting
criteria is discussed more fully in Chapter IV.

44

formulations that make the primitives explicit.l7
Rudner suggests that this situation would be allevi-
ated if some means could be found either to bring
those formulations to a degree of formalization that
allowed the calculus to be applied, or if some modifi-
cation of the calculus could be made to make it
applicable to less than fully-formalized systems.

Professor Rudner sketches an outline of what
a relevant partial formalization of formulations would
involve. He also suggests some problems that might be
presented when such a partial formalization was
attempted. For Rudner, a partial formalization is one
of "those negligibly formalized systems exhibiting
even one supposedly deductive connection among its
statements, or which determine explicitly the usage of

18 Anything less than

even one constituent concept."
full formalization would be partial formalization,

(with the exception of formulations excluded by the
above statement). Since there is such a wide range

of partial formalization, the question, what aspects

of formalization are relevant to an assessment of

 

l7Sanford Dornbusch is almost apologetic for
even considering formalization as a technique for im-
proving theory in sociology in his introduction to
the recent book T es of Formalization in Small-Grou
Research (Boston: Houghton Mifflin Co., 1962). This
seems to reflect_a general distrust of logical
techniques in sociological research.

18Rudner, Philoso h of Social Science, 47

45
simplicity arises.

A hint at just what aspects of partial formal-
ization might be important can be taken from Goodman's
system of measuring simplicity. What is lacking in
most partial formalizations is knowledge of what con-
cepts are to be counted as primitive in a future
formalization. Thus what is needed is a method of
making sound projections about the logical character
of such future formalizations on the basis of infor-
mation available. This suggests to Rudner the study
of systematic presuppositions in theories. He defines
systematic presupposition as the ”implicit presupposi-
tion of large segments of a field or discipline other
than that to which the theory being constructed is
indigenous."19

Since the technical terms of any formulation
are either defined or undefined in the system, it
follows that any term not defined in the system must
either be a primitive of the system or be nonindigenous.
Rudner points out that concepts that are indigenous
to a theory, may be given the status of tentative
primitive. ”Concepts not indigenous to the theory .

. . may have a distinctive status still to be deter-
mined.”20

19Rudner, Philosoppy of Social Science, 48
20Rudner, Shilosoppy of Social Science, 48

 

I./_-

46
An example of the use of nonindigenous terms
in a formulation from sociology of education is seen
in a brief look at Wilbur Brookover's work on self-

21 In a footnote

concept of academic ability.
Brookover notes.

"We use the term significant others, as pro-

posed by George H. Mead, to refer to other

persons whose role a given individual takes

in appraising himself in a particular role

performance. See G. H. Mead, Mind, Self and

Society (Chicago: University of Chicago

Press, 1934).2
Here Brookover has taken a concept from a psycholog-
ical formulation, or a social-psychological formula-
tion and included in his own. He might later define
the term explicitly and hence make it a defined term
or he might designate the term a primitive and assign
it a meaning in some other way than alluding to its
meaning in Mead's conceptualization.

The obvious import of focusing upon nonindige-
nous terms in a partially formalized formulation rests
in the fact that some portion of the results of some
other discipline or area is being presupposed in the
theory. This is important in that a wholly explicit
deduction of testable hypotheses mentioning any of

the nonindigenous terms would involve the explicit

use of laws from the donor discipline. The important

 

21Brookover, A Sociology of Education,
chapter 2 and 16.

22Brookover, A Sociology of Education, 455

47

aspect of presupposition is that if a scientist is
ignorant of those presuppositions, they will be unex-
amined and unevaluated; and to the extent that they
are unevaluated, confidence in the theory's adequacy
must be qualified. When the presuppositions are
taken from a relatively well confirmed and clear for-
mulation, the risk tends to be minimal, however, when
the presuppositions are taken from so-called "common
sense" language, the risk may be too great to be
tolerated.

The inclusion of nonindigenous materials in a
formulation is often done through what Rudner calls
quasi-deduction. He states: "We shall designate as
quasi-deduction any putative inference in a theory
that purports to be deductively elaborating such
theories . . . but that fails to meet the requirements
for being a deduction. . . . Quasi-deduction in a
theory is not confined solely to the implicit assump-
tion of nonindigenous material as premises. It is
frequently the case that the statements omitted in
quasi-deductive inferences are ones indigenous to
the theory."23

Under the rubric "relative primitives"
Professor Rudner discusses briefly the rationale or

justification of the concepts that occur in a

 

23Rudner, Philosophy of Social Science, 50

48
partially formalized theory. In this discussion he
presents three modes of concept presentation or con-
cept introduction:

"1) Concepts may be introduced by explicit

definition or some equivalent of explicit

definition. ‘

2) Concepts may be introduced through some

specification in the theory, that yields a

sufficient condition for their application.

3) Concepts may be presented, not explicated

in the modes of l and 2, as relative ‘

primitives.n24

Explicit definition will differ in partial
formalizations from fully formalized formulations
only in that some or all of the primitives are not
explicit in the partial formalization.

Statements of a theory that specify a suffi-
cient condition for something's being a thing of
given kind, also yield a sufficient condition for
the application of the term or concept that indicates
the kind. Thus, the statement 'If an animal suckles
lits young then it is a mammal' not only specifies a
sufficient condition for something being a mammal,
but also furnishes a criterion sufficient for correct
application of the term 'mammal'.

Professor Rudner notes that specification of
sufficient conditions for the application of a term
does not give the full meaning of that term. Any

theory might give a number of sufficient conditions

for a term, thus a theory might contain the statement,

 

2“Rudner, Philos0phy of Social Science, 50

49

'If an organism has a backbone it is an animal,' and
also, 'If an organism is warmblooded, it is an animal.'
Each of the statements partially determine the use of
meaning of the term, and taken together they more
fully determine the usage of the term, but neither
separately nor together do they fully specify the
meaning of the term. The temptation is of course to
designate such terms ultimate primitives. This is
risky though, for there is promise that the term will
get explicit definition in the theory eventually.

Rudner notes that "finally, there will usually
occur in partially formalized theories, terms that are
not introduced in either of the two ways just
mentioned."25 These will be the terms that will be
labeled relative primitives. These terms do not have
unknown meanings, on the contrary, some sufficient
condition for the application of the term might be
available, or the term might have sufficient ante-
cedent clarity from "ordinary usage."

It is important to note that in a theory that
has not been fully formalized, the status of terms
not introduced by explicit definition, or some spec-
ification of conditions for application, are not
automatically that of primitives. Such terms may
only be counted as presumptively, or tenatively

primitive.

 

25Rudner, Philosonhypof Social Science, 51

50

Formulations are being changed constantly,
hence a term or set of terms that are today thought
to be primitive, might with the acquisition of new
knowledge, be judged derivitive of other terms.
Hence the tentative primitive might lose its status
as primitive. This means that any assessment of
simplicity made upon the basis of tentative primitives,
will also be tentative.

of Rudner s Su
Sociology of Education

Even though Rudner’s suggestions are discussed

2,4 Some Observations Concerning the Usefulness
ggestions to the

and tested for their plausibility in Chapter IV, some
preliminary comments concerning them seem appropriate
here. That being the case, several observations of a
general sort are made here. There are four types of
observations: (1) those concerned with the appropri-
ateness of Rudner's purpose and general frame of
reference, (2) those dealing with his more specific
suggestions concerning the use of simplicity as a
criterion to evaluate nontheoretic formulations,

(3) those concerning a program for partial formali-
zation, and finally (4) those observations dealing
with Rudner's suggestions that Goodman's explication
of formal simplicity be reformulated to be applied

to nonformalized formulations.

51

Before any appraisal of Professor Rudner's
suggestions can be made, a few observations concerning
his purposes and his general frame of reference should
be made. It is to be remembered that Professor Rudner
is a philosOpher of social science rather than a
social scientist, and for that reason his professional
and intellectual orientation is somewhat different
from that of the social scientist. For Rudner the
philOSOphy of social science is to be construed as a
branch of epistemology and is interested, "not with
the substance of any social theory nor with a view of
what makes a good society, but with the logic of any
theory construction in social science and with the
logic of justification of (any) social-science
theory."26 It is to be noticed that his interest is
in methodology rather than content, and this allows him
to make a number of distinctions that may bring clarity
to the methodological issues discussed, but which are
difficult to apply to the process of theory building.
The distinctions are quite useful in the area of philos-
ophy, where clarification is at stake, but they may be
less useful when applied to the task of building a
theory. His interests are also quite different from
those of an educator who relies upon theoretic knowledge

in his day to day decisions. The educator is in effect

 

26Rudner, Philos0phy of Social Science, 3

52
"processing" with a "product."

Professor Rudner distinguishes between the
"context of discovery" and the "context of valida-
tion," and between "science as a process" and
"science as a product." He then proceeds to discuss
"science as a product" in the "context of valida-
tion." The very fact that Rudner makes these
distinctions indicates that the kind of thing done
by the philosOpher is quite different from that done
by the scientist. The question that first appears
is in what ways can knowledge or insight gained in
viewing science as Rudner does as static and complete,
be applied to the process of building and using
scientific theory. It seem that the scientist is
engaged in what Kaplan calls the "logic-in-use"27
which is essentially a process that involves contin-
ually moving from the "context of discovery" to the
"context of validation" and back again. The scientist
does not simply "latch on to a good hypothesis" in the
context of discovery and then confirm or disconfirm
that hypothesis, but he constantly revises and refines
a hypothesis on the basis of information gained when

attempting to confirm the hypothesis.

 

27Abraham Kaplan, he Conduct of n uir
(San Francisco: Chandler Publishing Company, 1964),

53
Kaplan strikes a blow for the scientist in the

following:
"A great deal hinges on whether science

is viewed as a body of pr0positions or as the

enterprise in which they are generated, as

product or as process. An account of the

norms bearing on the finished report of an

investigation might well be expected to differ

from one concerned with the conduct of the
investigation itself."28

Had Kaplan been writing on the question of
selecting a formulation to be used the way an educator
might use a study of social stratification in his
community,he might well have added that while the
philos0pher of science is interested in system and
the social scientist is interested in explanation and
prediction the educator is interested in control.

Each of these interests overlaps to some degree, but
the fact remains the interests are different.

Rudner might well respond that even though
these interests are different, the educator's concern
with control pre-supposes the sociologist's concern for
explanation and prediction which, in turn, pre-supposes
the philosOpher of social science's system. This anal-
ysis holds only if one defines scientific exPlanation
as subsumption of particular phenomena under general
laws, and defines system as a set of deductively related

set of statements. It might be the case, however, that

the sociologist views explanation in a different light

 

28Kaplan, The Conduct of In uir , 7

54
than does the philoSOpher of social science. And it
is surely the case that the sociologist would view the
notion of systematicity in a different light insofar as
he would include statements of empirical correlation as
one non-deductive method of systematizing concepts and
statements.

It would seem that the sociologist in his adher-
ence to "functional analysis" is assuming a notion of
explanation and of system that is more in line with
Scriven's explication of the key feature of explanation
as, "The production of understanding; this often amounts
to the reduction of the incomprehensible to the compre-
hensible,"29 than it is in line with the Hempelian
notion of explanation as subsumption under general laws.
Scriven goes on to elaborate his notion of explanation:

"Explaining consists of exhibiting the not

understood as a combination, of an understood

kind, of understood components.

Now this notion of the reduction of a
phenomenon into acceptable components is very
closely connected, where the concept of a model
is relevant, to perception of the phenomenon
in terms of a familiar, i.e., understood,
model. The crucual point about a model is that
it has familiar component parts, and the
relationships between these parts is under-
stood. (I am using the term "model" in the

sense in which it is overextended to include
mere mathematical descriptions.)" 0

 

29Michael Scriven, "The Contribution of Philos-
Ophy of the Social Sciences to Educational DevelOpment,"

Philoso h d Educ tional Develo ment, ed. George

Barnett (New York: Houghton Mifflin Co., 1966), p. 58.
30Scriven, "The Contribution of PhilosOphy of

the Social Sciences to Educational Development," 59

55

It also seems that the sociologist uses
nontheoretic formulations as a model in Scriven's sense,
and that Rudner's emphasis upon nontheoretic formula-
tions as mere pre-theoretic formulations is misplaced.
It is clear that the sociologist looks upon a stratifi-
cation formulation as a model through which certain
social phenomena manifest by a community might be
"explained," while the philosopher of social science
would not admit that explanation results from the
application of such a formulation.

In short then, it would seem that the logical
system demanded or at least prescribed by the philoso-
pher is little more than a sufficient condition for one
kind of explanation, but not sufficient for all kinds
of explanation appropriate to sociology.

Without belaboring the point, it is enough to
say that consideration of simplicity only in the
"context of validation" gives no direct knowledge of
simplicity in relation to "science as process." On
the other hand, "science as process" is not entirely
unrelated to "science as a product" since the process
can be conceived as being made up of steps; a very
important one of which is validation.

Rudner's use of the "context of validation" is

56
misleading insofar as it is defined as the context

within which the question of selecting or rejecting a
formulation is raised. It seems that a further dis-
tinction must be made to deal with the question of
selection and rejection of formulations by different
peOple for different reasons. The context within
which a sociologist selects a formulation is different
from the context in which the educator selects one.
Rudner suggests that aside from being impor-
tant to a logic of induction, simplicity might be very
important to social science theorizing on the grounds
that it might be useful in the task of assessing and
selecting nontheoretic formulations. One comment
must be made concerning the use of nontheoretic for-
mulations in sociology of education. Nontheoretic
formulations play a very important role in sociology
of education. It seems that much of what educators
and sociologists are doing today boils down to com-
paring school districts, communities, cities and even
countries on a multitude of points ranging from
racial composition to methods of instruction and
reporting the facts issueing out of those comparisons.
Classificational schema and typologies are the ve-
hicles of those comparisons, and it will be through
those formulations that significant differences will

be discovered and documented. Whether simplicity is

57
a significant aspect of the desirability of those
formulations is another question.

Professor Rudner suggests that perhaps a
tentative assessment of simplicity for nonformalized
theories can be achieved by partially formalizing
the theory in such a way that relative primitives can
be designated. In order to accomplish the task
of designating some terms as relative primitives, he
suggests that the formulations in question be ana-
lyzed to determine what presuppositions are being
made, and when the formulations are elaborated on the
basis of ”quasi-deduction.” It is well to note that
the kind of partial-formalization recommended by
Rudner will by its very nature have implications that
go beyond the mere assessment of simplicity. It is
also to be noted that several questions are raised,
such as at what point in the develOpment of a for-
mulation should the theoretician formalize, and
when can the scientist afford to formalize a formulation.3

Finally Rudner suggests that Goodman’s eXpli-
cation of simplicity might be reformulated in

 

3'Constantine Politis noted some of the limi-
tations of formalizations in a paper contributed to
the 1964 International Congress of Logic, Method-
ology and Philosophy of Science. This was later
published as "Limitations of Formalization,”
Philosophy of Science, XXXII, No. 4 (1965),.356-360.

58
such a way as to be applicable to nonformalized
formulations, but he does not suggest what form such
a reformulation might take. If Goodman's explica-
tion can be so reformulated it will be far more
valuable. It seems that there are a number of forms
that a reformulation might take. First, the cal-
culus might be construed as a general "rule of thumb"
for the theoretician and go something like this:
Nonlogical predicates should be explicitly defined
whenever possible. This "rule of thumb" or a number
of such rules might be devised with varying degrees
of strength and generality.

A second approach would be to draw directly
upon the calculus for specific rules calculated to
direct the theoretician to construct predicates of
certain kinds deemed preferable in choice situations.
If this approach were taken, the rules would tend to
be more precise and certainly would be more restric-

tive. The two approaches are taken up in chapter IV.

2.5 Summary of Chapter II
Richard Rudner's writings on simplicity are
important to this dissertation because Rudner has
made an attempt to classify and evaluate various
attempts to explicate the term simplicity. Rudner

first divides all attempts at explication into two

59

categories, Ontological Simplicity and Descriptive or
Linguistic Simplicity. Ontological Simplicity would
be applied to those explications of simplicity that
presume to predicate something of the universe or of
the way we perceive the universe, while Descriptive
or Linguistic Simplicity would be applied to those
explications conceived to be predicated of man's
descriptions of the universe. Rudner rejects the
notion of Ontological Simplicity as fundamentally
obscure and irrelevant to the tasks of science. He
selects as relevant to the tasks of science the sub-
division of Descriptional Simplicity called Objec-
tive-Logical Simplicity, and notes that of two
approaches to the explication of that notion, Nelson
Goodman's approach is most promising. Goodman has
devised a calculus of simplicity, the application of
which to a formalized formulation will yield a sim-
plicity value based upon the number and relevant
kind of the extralogical primitive predicates.
Professor Rudner notes that the empirical
test of confirmation does not apply to all scientific
formulations and points out that among other criteria
(clarity, power, theoretic fruitfulness) simplicity
is the most promising for evaluations of nontheoretic
formulations, and since the social sciences use a

large number of nontheoretic formulations, simplicity

60

shows promise of being an important criterion there.

The immediate promise of simplicity as a
criterion in the social sciences is dimmed by the
fact that Goodman's calculus of simplicity can be
applied only to formalized formulations, and there
are few such formulations in the social sciences.

To remedy this situation, Rudner suggests that for-
mulations might be partially formalized and an as-
sessment of simplicity be made on the basis of
relative primitives, and suggests some problems that
are met in attempts to partially-formalize formula-
tions.

Some comments were offered concerning the
usefulness of Rudner's suggestions concerning sim-
plicity and social science formulations in relation
to sociology of education. It was specifically noted
that present weaknesses in theory in sociology of
education are related to problems of assessing for-
mulations. Several areas of sociology of education
are quite concerned with the use of nontheoretic
formulations. It was noted that the concerns and
perspective of a philOSOpher of Social Science might
well differ significantly from those of either a
social scientist or of an educator. It was also
noted that the program for partial-formalization

outlined by Rudner shows some promise of being

61

valuable in each of these tasks. Rudner's suggestions
in somewhat modified form were seen as promising in
general then, and will be tested further by applica-
tion to specific formulations in Chapter IV.

Chapter III is a review of Goodman's eXpli-
cation of simplicity with particular attention to
Goodman's arguments for the importance of simplicity
and to his determination of the context within which

formal simplicity is useful.

CHAPTER III

A REVIEW OF GOODMAN'S EXPLICATION
OF SIMPLICITY

Since the stated purpose of this dissertation
is to investigate the relation of Nelson Goodman's
notion of simplicity to the field of sociology of
education, any review of literature must be a review
of Goodman's writings on simplicity. This chapter
includes a review of those writings and some general
comments concerning the apprOpriateness of Goodman's
frame of reference to some problems in the sociology
of education.

The first section of Chapter III is little
more than an integrated paraphrase of Goodman's
exposition of the calculus of simplicity. Since the
purpose of the dissertation is to determine whether
Goodman's calculus of simplicity can profitably be
applied to certain decision problems in the sociology
of education, the paraphrase is focused primarily
upon Goodman's introduction and definition of the
situation in which the calculus can be applied. When

the calculus proper is quoted however, it will be

62

63

quoted directly rather than paraphrased.l

3.1 Goodman on the Use of Simplicity
The development of Goodman's thoughts on

 

simplicity can be traced through a series of articles
first published in the Journal of S bolic Lo ic, but
such a tracing would be a wasted effort, since his
later articles contain a clearer and more fully de-
veloped elaboration of his calculus. For that reason,
the calculus will be presented as it presently stands
and is elaborated in the second edition of his book,
The Structure of Appearance. In that book Goodman
states: "In Chapter III, old Sections 3 through 7
have been replaced by the new Sections 3 through 10,
incorporating results of the continuing investigation

of structural simplicity."2

Even though the calculus
is complete in Goodman's latest work, a complete
understanding of the deyelopment of the problem ad-

dressed by the calculus may depend upon material

 

1This is not the first paraphrase of Goodman's
work. Others with their own emphasis have been
offered by the following: Rudner, Phi10§0phy of
Social Science, 44-47; Donald Hillman, "The Measure-
ment of Simplicit ," Philosophy of Science, XXIX,
No. 3 (July, 1962 , 225-252; Robert Ackermann,
Simplicity and the Acceptability of Scientific
Theories, dissertation, Michigan State University; and
Lars Svenonius, "Definability and Simplicity," 2pc,

Journal of Sypbolic Logic, XX (1955), 235-250.

2Nelson Goodman, The Structure of Appearance,
(2nd ed. rev.; New York: The Bobbs-Merrill Company,

Inc., 1966), p. vii.

64
taken from the earlier articles. In that case
materials from them is integrated into the paraphrase
where it seems apprOpriate.

The calculus of simplicity itself will not be
presented as a formal system, but will be treated in-
formally. This seems in keeping with the general
purpose of the dissertation.

The first three chapters of The Structure of
Appearance discuss Goodman's notion of System. For
him system plays a central role in science and in
fact is tantamount to both science and philosophy.
This follows from the fact that the purpose of science
is to render precise the relationships it eXplains in
order to be able to make precise analyses and predic-
tions, and from the fact that a system, being a
deductively related body of statements, does this
best. Formalization or systematization is the best
way to make explicity the logical structure of a
system and hence make precise the relations between
terms. And beyond that simplicity is not only tanta-
mount to systematicity, but is also "inextricably
involved in the selection of what is regarded as a

true system."3

 

3Nelson Goodman, "Axiomstic Measurement of
Simplicity," he ou n of hi 080 h LII (1955),
709-722.

65

Formalization is usually thought of as a
method of making the logical structure of a theory
explicit and deals with the common logic of theories.
Goodman explicates simplicity in terms of formalized
theories. The result is that his explication of sim-
plicity is directly applicable to formalized theories.

Structural or formal simplicity is a charac-
teristic of, or is predicated of the extralogical
base of a formalized theory. The extralogical basis
of a system is discerned in its postulates, and con-
sists of all the terms that are not included in the
list of basic "logical terms". These are the terms
peculiar to a given system. .The extralogical terms
are not unique to one system and some of the terms
might overlap or "belong" to more than one system.
For example the term "electron" might "belong" to
both physics and chemistry. The extralogical terms
are the "substance" of a system and are the terms
that areassigned meanings in an interpreted system.

Goodman focuses upon certain formal aspects
of the extralogical basis, and for him simplicity is
a measure those formal characteristics of extra-
logical predicate bases.

The adOption of a term as a primitive in the
extralogical base of a system is to introduce it

into a system without defining it explicitly within

66

the system.“ And insofar as the primitive is not
clear, an explanation of it might be provided in a
metalanguage. For instance, in a system of geometry
the term "line" might be taken as primitive and de-
fined or explained in common language. It would not
be defined exclusively in terms of the extralogical
base of the system.

For Goodman all that is demanded for the
interpretation of a primitive predicate is a deter-
mination of its range of application; and in
explaining a primitive, we may use an enumeration
or a listing of individuals belonging to the exten-
sion of that primitive wherever that is convenient.
In most cases however such a list is not available
and the meaning is conveyed by explanation.

Goodman points out that when considering
alternate formalizations of a theory there may be
alternate sets of primitives within the given extra-
logical base and asks what decision procedure is
needed for the selection of one set of terms over
another. It is within this frame of reference that

he explicates simplicity.

 

“Goodman has conceived of a theory with only
primitive and nonprimitive terms in its extralogical
basis where the nonprimitive terms are eXplicitly
defined in terms of the primitives. This is con-
sistent with his assumption of formalization. It
will be remembered that Rudner made a further dis-
tinction between indigenous and nonindigenous terms.

67
In a given system, a term is indefinable

because it is taken as primitive, not because it is
indefinable in some absolute sense. Goodman would hold
that no term is absolutely indefinable even within a
system. As in the case of geometry given above, the
term ”line" might be taken as primitive and ”point, "
”triangle”.and "angle," be defined in terms of ”point, "
and so forth. Whatever is taken as primitive is then .
defined or eXplained outside of the system in a
metalanguage.

The indefinability of a term in a system due
to its primitive status is not to be taken as incom-
prehensibility. Incomprehensibility has no place in
a system. On the contrary it is usually the case that
an indefinable term in one system is quite definable
in another system, and the terms taken as primitives
(indefinables in a system) come with a high antecedent
clarity. In Goodman's view then, primitiveness is
attributed to a term on the basis of its definitional
relation to tother terms in the extralogical base
of a given system. There is no absolute set of primitives
for a given system.

At this point it might be well to ask just

why it is important to designate primitive terms at
all, since all terms have to be defined or explained

in some language somewhere. What is the difference

68

whether a term is defined within a system or outside
it? This question is probably best answered by con-
sidering the purpose of building a system. A system
is built to clarify terms, to make explicit the
relations between terms or to facilitate the manipu-
lation of concepts by develOping rules of replacement.
System itself is achieved when explicit (definitional)
relations between terms and statements are determined,
and any term that is explicitly defined is guaranteed
precision and clarity, at least insofar as the primi-
tive terms are precise and clear. Beyond that, the
manipulability of any system is increased by an in-
crease in the degree of replaceability among terms.
This replaceability is present only when explicit def-
initions are available. On the other hand a "system"
made up of only primitive terms, and hence no explicit
definitions, is no system at all, and insofar as the
relations between terms are not eXplicit, clarity and
precision are unknown. Systematization increases the
number of explicit relations while at the same time it
reduces the number of unexplicated or "uncontrolled"
terms (the primitives).

The fact that primitives are not absolute to
a given system and hence alternative arrangements are
possible raises the question of what criteria will be

used to select the primitives. It must be noted here

69

that the criteria for the selection of a criterion
needed to make decisions concerning alternate sets of
primitives are not specified by Goodman. From what
Goodman says it can be inferred that the criteria to

be applied to primitives should at least be "objective"
and "effective." "Objectivity" here meaning free from
the influence of such variables as person, time and
mood, while "effectiveness" means being able to make a
definite choice in each decision situation. Goodman
does not specify any criteria however.

Goodman points out that some of the criteria
by which a scientist or philosOpher chooses primitives
are obvious. For instance, contradictory and incom-
prehensible predicates are never satisfactory
primitives within a system. Obviously a predicate
like ". . . is circular and not circular" cannot be
included in a system without logical disaster. The
criterion of noncontradictoriness is both objective
and effective. .

Goodman goes on to point out that fundamen-
tally obscure or incomprehensible predicates are
unsatisfactory choices as primitives. An example of
such unsatisfactory predicates would be "predisposed
to telepathic communication with supernatural aid."5

Goodman does net make an attempt to elaborate the

 

5Goodman, Structure of Appearance, 65

7O
meaning of "obscurity" and "incomprehensibility" hence
some question arises concerning whether one speaks of
obscurity and incomprehensibility in relation to a
linguistic entity or of the interpretation of a lin-
guistic entity. It seems that Goodman would reject
comprehensibility as somewhat less objective than non-
contradictoriness. It would seem that it falls further
into the realm of psychological criteria.

Goodman goes on to say that clarity is commonly
thought to be the most important criteria for the se-
lection of primitives, but in fact a scientist often
takes an unfamiliar term, or even coins a new term and
includes it as a primitive. Goodman points out that
any term that is selected must be made clear, but this
is quite different from selecting a term on the basis
of antecedent clarity. Take an example of the "theo-
retical" term in science. The theoretical term is
part of a logical construct and is in principle an unob-
servable. Hence theoretical terms are never clear in
the sense there is clearly something in the world that
corresponds to them. They must however be precise, in
the sense that an explanation of their meaning can be
given. Some terms are unsuitable for primitives be-
cause of their presystematic use. A term such as
"ectoplasmic" defies clarification even in common

parlance and would be unsatisfactory for that reason.

71

Clarity then for Goodman might mean ease of
explanation of the term rather than the ratio of de-
cided to undecided cases belonging to the extension
of the term.6 This ease of explanation is important
since any misuse of a primitive that might result
from the lack of clarity would infect the whole sys-
tem and any clarity might be reflected in the system.
Clarity is a rather elusive criterion, since it is
relative to person, time, situation and even moods.
For that reason, as important as clarity may seem to
be, it does not provide a criterion that can be re-
liably applied. Clarity is neither an objective,
nor an effective criterion.

The purpose for which a system is constructed
has much to do with the selection of primitives.
Surely when a system is constructed to say something
about geometry the primitives must be relevant to ge-
ometric questions. Mass and force would not be
appropriate to such a system. Limitations are imposed
on the choice of primitives by the very problem or set
of problems addressed. On some occasions there

will be no choice of primitives, but in general the

 

6Goodman's notion of "ease of explanation"
needs some work. Perhaps Rudner's distinction
between indigenous and nonindigenous terms may be
helpful. It would seem that a term taken from an-
other formulation might well be easily explained
while a coined term or one taken from common par-
lance would not be easily explained.

72
set will not be uniquely determined by the problem

addressed.7 The problem will usually only circumscribe
a sphere of possible primitives. This is exemplified
in various attempts to axiomatize geometry and arith-
metic. Relevance therefore seems to be an objective
but not an effective criterion for the selection of
primitives.

An added limitation on the choice of primi-
tives is imposed by the fact that primitives chosen
must form an adequate basis for all of the definitions
required; but adequacy, insofar as it is attainable,
could be achieved by adOpting all predicates, other
than those excluded by the problem, as primitives.
Hence adequacy is objective and effective but in
science, since systematicity is sought, what the sci-
entist seeks is not only the adequate basis, but the
simplest adequate basis. It is in this respect that
Goodman sees simplicity important to science. For
Goodman then simplicity promises to be a criterion
that can objectively, effectively and uniquely deter-

mine the most satisfactory primitive basis among

 

7It will be argued in chapter IV that the
educator's choice of formulations is most often
dictated by the purposes he holds. If he is trying
to comply with civil rights legislation he will
choose one formulation, if he is trying to please a
particular community he will choose another, and
when he is trying to maximize learning or do all
three things he will choose another.

73

alternate bases available.

Goodman asserts that since the primary purpose
of constructing a system is to interrelate its predi-
cates, that purpose is also directly served by
reducing the number of primitives in the extralogical
basis. This is done by eXplicitly defining some of
the terms in the extralogical basis in terms of the
others. But again, there are alternate ways of re-
ducing the number of primitives. Every definition
within the basis both increases the coherence of the
system, and diminishes the number of primitives.

Goodman points out that the replacement of one
basis for a system by another basis effects no genuine
economy if the replacement is purely routine. He then
asks why no real economy is achieved if a basis Bl
consisting of two one-place predicates, is replaced by
a basis B2, consisting of one two-place predicate.
Intuitively the answer is that there is no real savings
because it can always be done without effecting the
number of "entities" in a theory. 0n the other hand,
economy might be achieved by subtracting a primitive
from a system, but economy can never be attained by
adding a primitive. Again this is true because a
primitive can always be added, but not always be sub-
tracted without destroying the adequacy of a basis.
This also means that replacing a primitive by one that

74
has more places effects no saving, since a predicate
can always be traded for one having more places.
Intuitively this reduces the economy (simplicity) of
the basis. With this in mind it is becoming clear
that for Goodman a measure of simplicity must be based
upon the assignment of values to some predicates that
are known to be simple, such as one-place predicates,
and then defining a clear set of equivalence or sub-
stitution rules for manipulations that are other than

routine manipulations.

3.2 The Calculus of Simplicity

Goodman starts by making a general rule that
if every basis of a given kind can be replaced by
some basis of a second kind, then no basis of the
first kind is more complex than every basis of the
second kind. Professor Goodman holds that only struc-
tural characteristics are admissible in determining
appropriate kinds, but not all structural character-
istics can be admitted as relevant. It has already
been pointed out that structural characteristics which
enter into only routine replacement are not relevant.
He finds that the relevant specifications are those
pertaining to the number of predicates in a basis, the
number of places in each predicate in a basis, and re-

flexivity, symmetry, and self-completeness. Such

75

statements as "consists of one two-place predicate"
is a relevant specification; and so also is "consists
of one two-place symmetric predicate". Note that
"consists of one two-place predicate" is a relevant
specification because it limits the kinds of bases
that can be substituted which will still be an ade-
quate basis for the system in question. 0n the other
hand the number of words in the predicates of alternate
bases is not a relevant specification since the number
of words does not limit the set of possible replacing
bases. Also any conjunction or disjunction of rele-
vant specifications is itself a relevant specification.
On that basis if §_and L.are relevant specifications,
then §L_is a relevant subspecification of each, as
each is of Ip-or-Ig.
Professor Goodman then summarizes as follows:
"Now replacement is routine when, given only
elementary logic including identity together
with the information that a basis b answers
relevant specification K, we can define a
basis b', answering relevant specification
L, and can redefine b from b'. Where finding
a b' or recovering b requires anything more,
replacement is not routine. In what follows,
the replacement under discussion is always
routine repéacement unless otherwise
indicated."
The rest of the discussion of Goodman's
calculus is simply an elaboration of the calculus of

simplicity. It is not here set up as a fully-formal-

ized deductive system. It is presented only in such

 

8

Goodman, Structure of A earance, 7l

76

detail as the purposes of the dissertation warrant.
Since the theory is not presented as a formal deductive
system, it would seem inappropriate to ask metatheoret-
ical questions about consistency and completeness.
That task is left to the Logician.

The symbols employed here are not formal marks,
but are signs with their customary interpretations
from logic and mathematics. Proofs for the theorems
are available, but will not be offered here and any
question of the specification of proof procedure is
waived.

The logical and mathematical framework sup-
porting Goodman's theory is the first order logic
with identity. The primitive symbols in the theory
have their meaning from logic. Predicates are defined
extensionally in terms of individuals or names. Hence
a one-place predicate is defined in terms of the indi-
viduals to which it can be applied. The two-place
predicate is defined in terms of pairs of individuals,
and the three-place predicate is defined in terms of
the triples to which it applies. The sequences alluded
to in the theory are those ordered groups of individ-
uals, pairs, triples etc. to which the predicate in
question is applicable. The structural properties of

predicates discussed, reflexivity, self-completeness,

77
and symmetry and their refinements, are defined by
the sequences of individuals belonging to the exten-
sions of the predicates. For example, a two-place
predicate is symmetric if the reverse of each of its
pairs is also among its pairs. Further, the pairs of
the predicate S, . . . is the sibling of . . ., are
all the pairs of individuals that might belong to the
extensions of the predicate. The predicate is sym-
metric because for all x and y that belong to the
extension of S, if Sxy, then Syx.

The systems to which the theory is applicable
are first-order predicate calculi, which for the
purposes of this dissertation may be regarded as for-
malized theories, or as interpreted calculi.

The understanding of Goodman's theory is
dependent upon the understanding of certain refine-
ments of logical structure. Those refinements are
introduced by Goodman through definitions in the
calculus.

Professor Goodman also uses certain abbrevi-
ations that must be understood in order to understand
his elaboration of the theory:

"I shall use "vb" as short for the "complexity

value of the basis b". Again, "VK" likewise

reads "the complexity-value of the relevant
specification K," but since complexity-values
are initially assigned to bases rather than
specifications, a further ellipsis is in-

volved here: vK, the number associated with
the relevant specification of K, is the

78

lowest number such that under the postulates

of our calculus, K guarantees that no basis

answering K has a higher value."9

Goodman's overall strategy is to develOp a
way of comparing all extralogical primitive terms in
a way that will produce a simplicity ordering. He does
not establish a measure for all systems, but only for
systems that have first-order predicate calculus for
their logical calculus. With respect to scientific
theories that are formalized, the predicates refer to
individuals which might refer to any concepts that can
be made clear in observational terms. Predicates are
then compared by noting the way in which they divide
up the range of individual names corresponding to the
individuals which comprise their extension. With
these preliminary remarks in mind, the theorems, pos-
tulates and definitions of Goodman's system will

follow. The fundamental postulate of Goodman's system
is:

 

P3.31 if every basis answering a relevant
specification K can be replaced by some basic

answering a relevant specification L, then
vK §vL. .

 

 

 

 

A few elementary theorems that follow
obviously from the first postulate or from the
definitions of "vb" and "vK" are:

3.311 If every basis answering relevant spec-
ification K can be replaced by some basis
answering relevant specification L, and every
basis answering L can be replaced by some
basis answering K, then vK = vL.

 

9Goodman, Structure of appearance, 71

79

3.312 If a basis b answers relevant specifi-
cation K, then vb é vK.

3.313 vK §'v(K-or+L)

3.314 v(KL) é vK

3.315 vK 4 VI. 9 vL -.: v(L-not-K) (called
subtraction

3.316 vK = vL 9 (K-or-L) = vL (called
summation)

3.317 vK <. v(K-or-L) ‘3 v(K-or-L) == vL.

Also since every specification of 1-
place extralogical predicates that is to be
admitted as relevant is satisfied by the
same predicates as any other, 3.311 will
also yield:

3.318 All extralogical l-place redicates
have the same complexity-value." 0

A second postulate is necessary for the deri-
vation of further theorems:

"P3.32 The complexity-value of every extra-
logical predicate is a positive integer; and
the complexity value of a basis is the app of
the values of the extralogica1_predicatespgp
it.

A predicate is extralogical unless de-
finable soley in terms of elementary logic
including identity. Only extralogical pre-
dicates are counted as contributing to the
complexity-value of a basis; for the logical
apparatus may be regarded as constant for all
the systems in question. Thus a basis con-
sisting soley of logical predicates has the
value 0. Justification for the limitation to
integral values is evident enough. The number
of predicates in a basis, the number of places
in a predicate, the number of different sequen-
tial patterns exhibited by the arguments of a
predicate, and the number of places or place-
sequences with respect to which a predicate
is reflexive, symmetric, etc. are all inte-
gral. No intermediate numbers are needed;
and our eventual assignments of values can be
such as to leave no gaps-that is, where v§_= p,
there will be for every ;_less than p.some

 

lOGoodman, Stppcture of Appearance, 72-73

80
subspecification L_of E_such that vL_= ;, . . .
A useful immediate consequence of P3.32 is:

3.321 If the extralogical predicates in a
basis p_are some but not all of the extra-
logical predicates in a basis p], then

vb < vb'.ll

Goodman introduces some more abbreviations:

"In what follows, use of square brackets
will result in a name for the relevant speci-
fication abbreviat d within them. Thus the
full reading of " EZ-pl. irref.; two l-pl.] "
is "the relevant specification 'basis con-
sisting of one 2-place irreflexive predicate
and two l-place predicates'"; and the full
reading of "v [2-pl. sym.] " is "the com-
plexity-value associated with the relevant
specification 'basis consisting of one
2-place symmetric predicate'". But I shall
normally give much less combersom elliptical
readings such as, for the first case, "basis
consisting of one 2-place irreflexive and two
l-place predicates"; and, for the second case,
"the complexity-value of a basis consisting
of one 2-place symmetric predicate"--or even
just "the complexity-value of 2-place sym-
metric predicates". Where "p?, "p", etc.
occur within the square brackets, we have in
effect a specification-schema that becomes a
specification when these letters are sup-
planted by integers. By way of illustration,
consider such elementary theorem schemata as:

3.322 If pé p, then v [3:le ‘.‘=‘ v [m-pl.]

3.323 If £9 p, then vtp preds.] f? v [m preds.)

The consequent of the former reads: "the
complexity-value associated with the relevant
specification-schema 'basis consisting of an
n-place predicate' is no higher than the
complexity-value associated with the relevant
schema 'basis consisting of an m-place pre-
dicate'"; or elliptically, "the value of
n-place.predicates is no higher than the

 

11Goodman, The Structure of Appearance, 74

81

value of m-place predicates".12

Since all extralogical l-place predicates
have equal value as seen in 3.318, l-place logical
predicates have the value 0, the lowest value is
assigned to l-place predicates in the third postulate:

"P3.33 v [l-pl.] = 1.".13

This puts the l-place predicate at the low
end of the complexity scale.

The consequence of 3.33 are:

"3.331 If a basis p,contains any extralogical
predicate, then v[l-pl.] é vp.

3.332 Unless K guarantees that all predicates

in every basis answering it are logical, then

v [l-pl.] é vK."1‘+

With the general theorems and postulates
stated, Goodman goes on to analyze the properties of
predicates which explain purely mechanical replacement
of one predicate by another. The first property to be
analyzed is that of reflexivity. "Reflexivity has to
do with the occurrence of repetitions within the se-
quences (belonging to the extensions) of a predicate."15

Goodman makes a number of distinctions to refine the

notion of reflexivity. He says:

 

12Goodman, TheiSpructure of Appearance, 75

13Goodman, The Structure of Appearance, 75
lLlGoodman, Structure of Appearance, 75

15Goodman, Structure of Appearance. 76

 

82

"Every pair of elements is either an
identity-pair or a diversity-pair. The impor-
tant pr0perties of reflexivity depend upon
the relationship between the given predicate
and the logical predicates "z" and "=", and
between the identity-pairs and diversity-
pairs of the predicate itself. If all pairs
of a predicate are diversity-pairs, the
predicate is irreflexive; if all are identity-
pairs, it may be called redundant. A pred-
icate having at least one identity-pair and
one diversity-pair is composite. If all
identity-pairs are pairs of a predicate, it
is totallypreflexive; if all diversity-are
pairs of a predicate, it may be called
totally'diversive."

Professor Goodman goes on to point out that:

"A 2-place predicate that is both redun-
dant and totally reflexive is co-extensive

with ":"; and a 2-place predicate that is

both irreflexive and totally diversive is co-
extensive with "=". Thus the complexity-
value of such a predicate is 0 and we have the
theorems:

3.411 v [2-pl.t.r.,red.] :: o.
3.412 v [2-pl.t.d.,irref.] = 0.

Again, only imapplicable 2-place pred-
icates--those with null extensions--are both
redundant and irreflexive; and only universal
2-place predicates are both totally reflexive
and totally diversive. Specification of
either combination of prOperties thus results
in logical difinability, and hence:

3.413 v [2-p1.red.,irref.] = O.
3.414 v [2-pl.t.r.,t.d.] = o

A 2-place predicate having no or all
diversity-pairs in its extension can always
be replaced by a l-place predicate; and
conversely, a l-place predicate can always
be replaced by such a 2-place predicate.
Hence:

83
3.415 v [2-pl.red.] 1
3.416 v [2-pl.t.d.]

Goodman then goes on to deduce a number of

16

1.11

theorems:
"3.417 v [2-pl.irref.] = vLZ-pl.t.r.]
3.418 v [2-pl.red., t.r.] = 1
3.419 v [2-p1.-red.-or-2-pl.-t.d.]:= 1
3.421 v [2-pl.non-comp. I = v [2-p1.irref.]
3.422 v [2-pl.comp. ] = v [2-pl.irref.] + 1.
3.422 v [2-pl.] =-. vIZ-pl.irref.;l-pl.] =
v [2-P1.irref.] + l"17 4
Professor Goodman adds some further refinements
and goes on to make other theorems:

"For example, suppose that the identity
pairs contain all and only those elements
that occur in the diversity-pairs. We may
call such a predicate join-reflexive; and,
admitting this as a relevant specification,
we have the theorem:

3.424 v_[2-pl.j.r.] = v [2-p1.irref.] .

Likewise, we may admit as relevantly
specifiable; meet-reflexivit , where an
element occurs in an identity-sequence of
the predicate if and only if that element
occurs as left component of at least one
of the diversity-sequences and as right
component of another; left-reflexivit ,
where an element occurs in one of the iden-
tity-sequences if and only if it occurs as
left component of at least one of the
diversity-sequences; and ri ht-reflexivit ,
analogously defined. Then:

 

l6

Goodman, Structure of Appearance. 76

l7Goodman, Structure of Appearance, 77

general

84

3.425 v [2-pl.m.r.] = V'[2-p1.irref.]
3.426 v [2-pl.1.r.] v [a-pl.irref.]
3.427 v [2-pl.rr.] = v [2-pl.irref}]

Where any of the above relevant speci-

fications for a basis consisting of a
2- lace predicate has a value less than

[£-pl.irref.;l-pl.] , we call such a basis
regular. Regularity thus enters as relevant
by definition as the disjunction of total
reflexivity, irreflexivity, redundancy, total
diversification, meet-reflexivity, join-
reflexivity, left-reflexivity, and right-
reflexivity. And by summation:

3.428 v [2-pl.reg.] = v [2-pl.irref.]

Professor Goodman expands the theorems to'a

,18

scheme including n-place predicates:
"The general schema is:

3. 437 v [n-p12]2= v[n-p1. .:i.rref.;hl n - l-pl.irref.;
-pl. irref.]+ 1.,

where for ea8h2 j, the number h. of n - j-pl. irre-

flexive predicates is j

n - j (n - j - r)n
E (-l)r
r = 0 (n - j - r)! r!

For any n-place predicate that is regular in
one way or another, some predicates can be
dropped from the equivalent set of irreflexive
predicates. But the value of any relevant
specification can be equated with the value of
some specification of bases consisting entirely
of irreflexive predicates. Hence if values
are determined for irreflexive predicates of
all kinds, the value of any basis may be
computed. Let us therefore confine our atten-
tion for a time to irreflexive predicates."19

 

l8

Goodman, Structure of Appearance. 78

19Goodman, Structure of Appearance, 83

85

The second structural characteristic considered
by Goodman is that of transitivity. His discussion of
transitivity includes a number of refinements upon
which are based a number of theorems. Transitivity
itself is found to effect complexity but it is similar
to the structural characteristic called self-complete-
ness. Goodman has this to say:

"A 2-place predicate R'is transitive in the
usual sense if and only if it satisfies the
condition:

(x) (y) (z) (Rx,y . Ry,z.3 Rx,z).

This property does not affect complexity, and
so is not relevantly specifiable. But a
kindred property is highly relevant. We re-
move the requirement that the first two pairs
be linked, and akk the requirement that all
three be diversity-pairs. Predicates
answering the resulting specification:

(x) (y) (z) (w) (Rx,y.Rw,z. x£y.w#z.
X22. 3 RX, Z) ,

may be called self-complete. Where the "z"
clauses omitted, only meet-reflexive pred-
icates would be self-complete. With them
inserted, we can ignore identity-pairs in
determining self-completeness, which is thus
rendered largely independent of reflexivity-
properties. In particular, an irreflexive
predicate may be self-complete even though
the first components of some of its pairs
are the second components of others." 0

Professor Goodman has derived the following
theorems concerning self-completeness:
I "3.511 v [2-pl.irref.,s.c.] = 2
3.512 2 = v [2-pl.reg.,s.cJ

 

20Goodman, Structppe of Appearance, 83

86

3.513 v[2-pl.irref.,s.c.] = v [2-pl.reg.,s.c.]
3.514 2‘? v[ 2-pl.irref.,s.c.]

. . . from 3.511 and 3.514 we arrive at the
wanted equation:

2

3.515 v [2-pl.irref.,s.c.]
3.516 v [n-pl.irref.,s.c.] = n

To generalize further upon 3.516, an
n-place irreflexive predicate has the same
value as a set of irreflexive predicates
corresponding to the minimal partitions of
the predicate--asset, that is, containing
for each minimal partition an irreflexive

predicate of the same number of places as
that partition:

3.517 V'Ln-pl.irref. with minimal partitions

of k,k', e c. places] = v [k-pl. irref.;

k'-pl.irref,; etc.] ."21

Professor Goodman defines the degree of self-
completeness, s.c, of a predicate as "simply the number
0f divisions between minimal partitions--or the number
of minimal partitions minus 1. All one-partition
predicates, hence all l-place predicates, have

22 He goes on to point out that a 3-place

so = 0."
predicate has so = 1 if it has a l-place and a 2-place
minimal partitions and so = 2 if it has three one-
place partitions. The maximum degree of self-complete-
ness for an n-place predicate is of course n - 1.

The effect of the theorems on reflexivity and

self-completeness is that the theoretician is able

 

alGoodman, Structure of A earance, 84-87

22Goodman, Structure of Appearapce. 88

87
"in effect to take apart and put together again
certain predicates and so equate them in complexity
value with sets of other predicates."23 Any re-
flexive predicate can be replaced by'a set of
irreflexive predicates. For any irreflexive self-
complete predicate an equivalent set of shorter
irreflexive predicates can be found.

The third characteristic of importance in
calculating simplicity is that of symmetry. A
2-place predicate is symmetric if the reverse of each
of its pairs is also among its pairs. In general, an
n-place predicate is symmetric if every permutation
of each of its sequences is also among its sequences.

Goodman points out that an n-place predicate
may, however, be symmetric with respect to some
rather than all its places. "For example, a 3-place
predicate "Q" such that

(x) (z) (z) (Qx,y,zDQx,z,y)
is symmetric with respect to its second and third
p1aces--or briefly, (2) (3) symmetric--but not neces-
sarily with respect to all."24

In order to deal with this contingency,

Professor Goodman introduces the notion of degree of

symmetry:

 

23Goodman, Structure of Appearance, 88

a“Goodman, Structure of A earance, 89

88

"The de ree of s mmetr , SI; of an
n-place predicate symmetric with respect to
k of its places is k-l; thus the maximum
degree of symmetry is n-l. For 2-place
symmetric predicated, syzl; for 3-p1ace
symmetric predicates, sy=2; and for partially
symmetric 3-place predicates (e.g., "Q" just
above), sy=1. A 3-place predicate symmetric
with respect to each of two pairs of places
is symmetric with respect to all three places,
and has sy=2. A l-place predicate is degen-
erately symmetric with respect to its only
place, thus having sy=0.

A predicate of more than three places
may be symmetric with respect to each of two
or more disjoint sets of its places. The
degree of symmetry for each such symmetric
subset of k places is k-l; and the degree of
symmetry for the whole predicate is the sum
of its maximal symmetric subsets-i.e., those
contained in no others. For example, a
4-place predicate may be (1) (2) and (3) (4)
symmetric without being symmetric with respect
to all or to any three or to any other two of
its places, here sy: l+1=2. An asymmetric
n-place predicate is in effect symmetric only
with respect to each of its places taken
singly; and has sy=n(1-l)=0."25

Goodman concludes that the degree of symmetry
of an n-place predicate with k segments is then n-k.
He goes on to point out that a predicate might embrace
both lower-level and higher level symmetries at the

same time and has this to say:

"Some predicate embrace lower-level
symmetries within higher-level ones. The
degree of symmetry of a 4éplace predicate that
is (l) (2) and (5) (4) and (1.2) (3,4)
symmetric is not the sum (l+l+2) of these
three symmetries. Rather, since the speci-
fication of this layered symmetry guarantees
all four places coextensive, the prggicate
has but one segment, and sy=4-1=3."

 

25Goodman, Structure of Appearance, 90
26Goodman, Structure of Appearance, 93

89

With these definitions, and the formula that
have already been develOped Goodman derives the

following theorems:

1
1.:

3.611 v [2-pl.irref.,s.c.,sym.] —

ll
1...:

3.612 v [n-pl.irref.,s.c.,sym.]

3.613 v [n-pl.irref.,s.c.,] - v
n-pl.irref. ,s.c. ,syzk] = k

3.614. v [n-pl.irref.,s.c.,sy=h] -v
[n-pl.irref.,s.c.,sy=k] = k :- h.

3.615 V'[n-pl.irref.,sym. with respect to k
disjoint h-valued partitions] = h."

With these postulates and theorems, some of
which make complexity assignments and others of which
act as rules of replacement, Goodman devises a final
formula for the assessment of the complexity value of

any set of primitive predicates.

LA

27Goodman, Structure of Appearance, 88-93

3.3 Some Observatipns Concerning Goodman's
Explication of Simplicity and the

Sociology of Education

After a look at Goodman's explication of

 

simplicity it seems apparent that Rudner's commenda-
tions and cautions concerning it are well placed. As
Goodman eXplicates the term, it is an objective crite-
rion when applied to possible alternate formalizations
of a given theory, and since it is applied to the
predicate base it can be applied regardless of the
number of empirical generalizations a formulation gen-
erates. 0n the face of it, then it might be applied
to nontheoretic formulations. Finally, it has become
apparent that the calculus cannot be applied to non-
formalized formulations, at least as it presently
stands.

It also appears that the criticisms that were
leveled at Rudner might also be leveled at Goodman.
One of the major criticisms of Rudner's work was that
the particular frame of reference that he selected, in
treating only "science as product" in the "context of
validation," was inapprOpriate to the needs of the
scientist and the applied scientist. The second crit-
icism of Rudner concerned his rejection of certain
other criteria. It was pointed out, for instance, that
clarity might in some sense be objective rather than

simply psychological. A similar criticism is to be

90

91
made of Goodman's work. Goodman argues the
inadequacy of other criteria in his context, but
insofar as his context is seen as inadequate to the
needs of the scientist proper and the applied scien-
tist his rejection of other criteria must be
reevaluated. It will be argued that the contexts
within which the scientist and the applied scientist
work is different from that of the philosopher of
science and for that reason the relative weight
attached to each criterion varies. Finally, it was
pointed out that Rudner's treatment of just how one
is to evaluate a criteria such as clarity or simplic-
ity was inadequate. This criticism stands against
Goodman also.

It will be the task of chapter IV to inquire
into four questions that arise from these considera-
tions. First, what are the relevant differences in
the contexts faced by the philosopher of science, the
sociologist of education, and the educator. Second,
how does one rationalize the selection or rejection
of criteria such'as clarity and comprehensibility in
the selection of primitives (and by that, formulations)
in the different contexts. And closely associated
with this is question three, the question of whether
Goodman's arguments for the centrality of simplicity

hold in the other contexts. The final question asks

92
what the role of Goodman's explication of simplicity
will be in the sociologist's context of open inquiry

and the educator's context of use.

3.4 Summapy of Chapter III
In chapter three Nelson Goodman's explication

of simplicity was reviewed. It was found that
Professor Goodman's explication of simplicity turns on
a particular notion of system, one that Goodman
eXplicates earlier. For Goodman a system is a spe-
cialized language that is constructed within a larger
language. A system is cnnstructed within a larger
language. A system is contrived by scientists for
some special purpose. Within a system the terms are
precise in extension, the relations between terms are,
where possible, explicit through definition.and the
logical mechanisms for relating pr0positions are
explicit. The system has two kinds of nonlogical
terms: primitives which are explained by metalin-
guistic devices in the metalanguage, and nonprimitives
which are defined explicitly within the system. When
a theory is formalized, that is, systematized in the
prior sense, a decision problem concerning the selection
of primitives arises. Goodman's central argument for
simplicity was that the most powerful formulation
with the sleast possible primitives is always desirable.
Given formulations of equal power, as would be the

case when selecting alternate formalizations for a

93
given theory, the most economical formulation is the
simplest formulation, that is, the alternative with
the fewest primitive terms.

Professor Goodman goes on to point out that
the number of primitives in a given system can be
reduced by various means, some of which are counted
as spurious, and others of which are counted as
legitimate simplifications. He argues that the rel-
evant characteristics of predicates in the assessment
of simplicity are the number of predicates in a base,
the number of predicate places in a base, and the
characteristics of reflexivity, self-completeness,
and symmetry.

Other arguments for simplicity turned upon
the rejection of other criteria for the selection of
primitives. It was noted however that the context
named as apprOpriate for the application of simplicity
is far more restricted than the context within which
the theory builder or the user of theory works. This
raised the question of the weight of Goodman's argu-
ments for the rejection of other criteria in those
other contexts. It was finally argued that four
questions remain to be taken up in Chapter IV:

(1) What are the relevant differences in the contexts
faced by the philosopher of science, the sociologist
of education and the educator in relation to the use

of simplicity? (2) How does one rationalize the

94
selection of rejection of criteria such as clarity,
comprehensibility and simplicity in the three contexts?
(3) Do Goodman's arguments for centrality of simplicity
hold in all three contexts? (4) What is the role of

simplicity in each of the contexts?

CHAPTER IV

SIMPLICITY AND THE SELECTION OF FORMULATIONS
IN THE SOCIOLOGY OF EDUCATION

Chapter IV is addressed to the tasks of:
(1) making distinctions between the contexts within
which the educator, the sociologist of education and
the philosopher of social science select or reject
formulations, (2) drawing up some systematic means of
evaluating a criterion such as simplicity, (3) testing
Goodman's arguments for the centrality of simplicity
in the context of the educator and the sociologist,
and (4) evaluating the role of simplicity in the selec-
tionand rejection of formulation in the sociology of

education.

4,] The problem of different contexts

The question addressed in this section is:
what are the relevant differences in the contexts
faced by the philosopher of science, the sociologist
of education and the educator in relation to the use

of simplicity as a criterion in the selection of

95

96

formulations.1 It was pointed out earlier that formal
or structural simplicity might relate differently to
decisions by sociologists and educators than it does to
the decisions of the philosopher of science. An ade-
quate differentiation of the contexts within which the
sociologist, the educator and the philOSOpher of sci-
ence make choices is necessary prior to the determina-
tion of the role of simplicity in each of those contexts

The task of this sub-section is to differentiate
between the contexts. The inquiry will be pursued by
seeking answers to the following questions: (1) What
questions are central to each context, and (2) do the
central questions bear directly upon the selection of

criteria for the selection of formulations.

 

1The notion of ”context” is not absolutely clear
at this point. Several factors might be included under
this rubric, ranging from the goals and interests of the
members of the community, through the traditions and
practices of a field of inquiry, to the conceptual
frame of reference common to the field of inquiry. Here
the term will be used as a rubric to cover the central
question adked by a particular community of interests.
It will be assumed that educators, sociologists of
education, and philosophers make up such communities
on interests. It will, to that extent be an empirical
question. To say that the educator's decision context
is different from that of the sociologist or the
philosopher is to say that the educator asks different
questions about social phenomena than does the sociologist
or philos0pher. It is this that makes the sociologist
interested in education different from the educator
interested in social phenomena.

97

Problems are immediately evident when an
attempt is made to pin down the contexts. A brief look
at the present state of sociology of education shows
the difficulty of coming to any authoritative defini-
tion of sociology of education or any authoritative
stipulation of the legitimate goals and subject-matter
of sociology of education. This vagueness makes the
task of defining or delimiting the sociologist's
context a hazardous undertaking, but some knowledge of
the central question asked by the sociologist can be
gathered by looking at the way sociologists define
sociology of education.

Emile Durkheim gave sociology of education its
initial impetus. He considered education "to be some-
thing essentially social in character, in its origins
and its functions, and that as a result the theory of
education relates more closely to sociology than any
other science."2 Durkheim went on to specify the
appropriate areas of research for the sociologist of
education:

'"l. Identification of the current social facts

of education and their sociological function.
2. Identification of the relationship between
education and social and cultural change.

3. Cross cultural and.comparative research in
various types of educational systems.

 

2

Emile Durkhaim, Education and Sociolo
(New York: Free Press, 19563.

98

h. Investigation of the classroom and school
as an ongoing social system."

Since Durkheim's time sociology of education
has been defined in many ways, ranging from "a study of
the structural elements of the educational system,"h
through a "focus of sociological perspective on educa-
tors, schools and educational institutions in social
and cultural contexts,"5 to "the scientific analysis of
social processes and social patterns involved in the
educational system."6

After citing some 21 statements by various so-
ciologists and sociologists of education Brookover
concluded: "It becomes apparent that there has been
little consensus among sociologists and educators as to
what the proper dimensions of this field should be."7
It seems, then, that the vagueness of the field of
sociology of education is substantiated in the litera-

ture, but even with this vagueness, it seems that there

is a common attempt to make intelligible the range of

 

3Brookover, A Sociolo of Education, a

“B. v. Shah, "Sociology of Education--An Attempt

at Definition and Sc0pe," Sociological Eulletin, XIV
(September, 1965), 6h-69.

5Donald Hansen, "The Responsibility of the
Sociolo ist to Education," Harvard Educational_Review.
XXXIII Summer, 1963), 312-325.

6Brookover, A Sociology of Education, 11

7Brookover, A Sociolo of Education, 10

99
social phenomena that surrounds the educational enter-
prise. The sociologist of education focuses upon the
central question of how one might best describe and,
in a broad sense, explain a set of changing social
phenomena.

There is evidence that educators ask different
questions than sociologists even when treating the
same discipline. This is manifest in the fact that
sociologists choose different topics for treatment in
sociology of education than educators. Richard Hoyme
reports that a sample of he sociologists and 128 pro-
fessors of education, all of whom were teaching a
beginning graduate course in educational sociology, were
asked to list, in order of importance, the ten topics
most appropriate to the field of educational sociology.
Two quite different lists were proposed by the two
groups. The sociologists listed the following ten
topics as appropriate to a beginning graduate course in
sociology of education:

"1. Social status

2. Social mobility
3. Social change and social conflict
A. Social class structure
. The school and its social implications
. Social roles, ascribed and achieved
The community

I The American Society
. Classical sociological studies

\omqmm

100

10. School-community relations."8
On the other hand the professors of education named the
following topics most appropriate to such a course:

"1. Social class structure

2. Democracy in education

3. Race relations

A. Social status

5. Social roles, ascribed and achieved
6. Social objectives of education

7. The teacher as a social being

8. Teacher-student relations

9. Teacher-adminis rator relations
10. The community."

Since only four topics are common to both lists, and
since the order of importance was different, there seems
to be a significant difference between the educator's
central question and that of the sociologist.

The educator probably assumes that the goals of
the philosopher and sociologist are good and legitimate,
but where the philosoPher focuses upon the quest for
clarity and the sociologist seeks insight, the educator
seeks both clarity and insight for realizing some pur-
poses. The educator seeks to understand social
phenomena so that he might have power over a situation,
and in order to gain that power the educator must under-
stand not only social phenomena, but also cultural and

individual psychological phenomena. The educator's

 

8Hoyme, "The Current Status of Educational
Sociology," 130

9Hoyme, "The Current Status of Educational
Sociology," 130

101
central question is then: How can I most effectively
achieve the goals of American Education.

This leads to the statement of a criterion that
is to be used in the educator's evaluation and selec-
tion of formulations from the sociology of education.
That criterion will be called "functional adequacy".
Very briefly, the functionally adequate formulation will
define and make explicit the relations of the elements
of a domain in such a way that it will conceive and
comprehend the significant elements, variables, and
relations necessary for the realization of an end. For
example, a doctor seeks to realize health, and to
accomplish this he needs to know not only physiology but
also psychology and the relationships between physical
and emotional phenomena. He must know how, why and when
physical malfunction is related to emotional problems
and vice versa. To be totally effective he must go'
beyond physiology and psychology because either or both
are functionally inadequate for his purposes.

The educator seeks to change individuals, and in
order to do so he needs functionally adequate knowledge.
For him that means knowledge of what man ;§,in his
present socio-cultural environment, what man can become,
and some value assumptions about what man ought to be.
For the educator, functional adequacy cannot be predi-

cated of knowledge from one discipline.

102

complexity of his task, the educator needs information
that can come only from a number of disciplines, as
they are presently conceived.

Functional adequacy is differentiated from
power by its link with the realization of some end. The
term power is usually attributed to formalizations that
have large extensions, that is, to formulations which
eXplain a broad range of phenomena without reference to
a specific concrete end. One might speak of a power-
ful psychological formulation, but for the educator's
purposes such a formulation willznot be functionally
adequate. The term functional adequacy is used here
to overcome the notion that science, conceived as a
set of discrete disciplines each of which seeks to
explain a rather well defined set of phenomena, is
adequate for the applied scientist's purposes. The
applied scientist such as the doctor or educator is
eclectic insofar as he draws upon a number of disci-
plines. Ideally he would synthesize knowledge from each
discipline. Since he cannot, he seeks formulations from
each discipline that can be related to formulations
from others.

Some formulations descriptive of man can be
be related to the socio-cultural reality only with
great pain, while others are conceived with these relatins
in mind. The latter kind of formulation is more

103

desirable on the grounds that it can become part of a
functionally adequate formulation eventually. A con-
ceptualization that describes man, his potential, and
the "good man" through a list of virtues such as
"truthful," "trustworthy," "patriotic" and so forth
carries with it implicit assumptions concerning the
relation of man to the socio-cultural reality, and is
not easily related to formulations that comprehend
social structure, social activity, and social proc-
esses. Because of the difficulty in relating the
traditional concepts of virtue to a set of manipulable
social circumstances, those concepts are functionally
inadequate for the educator's purposes. Take as an
example a third grade teacher trying to "educate" a
seven year old student who habitually lies. If she
views him simply as a liar, that is, one who is inher-
ently bad or evil, she can do little in a reliable way
to solve the problem. Since she conceives the act of
lying as an act of a bad will her common sense tells
her that a boy who lies should be punished. But pun-
ishment might merely aggravate an emotional problem of
fear or insecurity which moved the boy to lie in the
first place. In general it is difficult to build a
teaching strategy, or a set of social relations and
processes that will minimize such problems, if human
behavior is analyzed in terms of such concepts as the

traditional virtues.

104

If on the other hand an educator selects a
conceptualization that defines man, his potential and
the "good man," in social-psychological terms which
relate what man is to socio-cultural phenomena rather
than some notion of free will, he then can build an
effective strategy aimed at the creation of the "good
man." In that case the educator can conceive of a
seven year old boy's lies as a mode of accomodation
to a socio-cultural context and see that certain sit-
uations tend to encourage lying and certain others
tend to be accomodated to in other, happier ways.

A psychological or a sociological formulation
can never in itself be functionally adequate for the
purposes of the educator, but some formulations are
cast in terms which allow them to be related logically
to the other formulations used by the educator. Take
as an example the area of social stratification. It
is conceivable that the sociologist will be happy with
an adequate description of a stratified society. The
educator is not so much interested in the "facts" of
social stratification as he is in the psychological
differences of classes, the institutional response to
different classes and how the social situation in the
school can be altered to encourage apprOpriate modes

of accommodation.

105

In brief then the sociologist asks how he can
describe and explain social phenomena, while the edu-
cator asks how he can manipulate the social context of
education in order to encourage certain changes in
students. Upon examination it appears that the differ-
ences between the contexts of the sociologist and the
educator is no greater than those between either the
sociologist and the philosopher or the educator and the
philos0pher. It might be said that the community of
philOSOphers of social science to which Goodman and
Rudner belong have the central purpose of clarifying
certain logical and methodological issues surrounding

10 Rudner's work, as reviewed, can

the social sciences.
well be viewed as an attempt to clarify and solve
certain issues that surround the "logic of inductiOn"
of the social sciences. Since clarification of logical
issues is both Rudner's and Goodman's avowed purpose,
their central question might be: What is the best form
to use in expressing a set of concepts. The sociolo-
gist shares these purposes with the philos0pher, but he
also ascribes to a much larger set of goals. The soci-
ologist is interested in develOping concepts, formula-

tions and techniques through which certain social

phenomena can be made intelligible by providing an

 

loSee Rudner, Philos0phy of Social Science,
VII and Goodman, The Structure of Appeargpce, XVII.

106
eXplanation and prediction of further phenomena. The
sociologist is interested in making social phenomena
comprehensible, that is, in making eXplicit the rela-
tions between diverse phenomena. This means that he is
indeed interested inlogic, but also much more, he is
interested in the content of his formulation.

One of the main differences between philosophers
of science, such as Rudner and Goodman, and sociologists
is that, due to their interest in form, the philos0phers
focus upon an instantaneous or terminal stage of scien-
tific inquiry, and must necessarily do so when talking
about the formalization of formulations. Philosophers
of this bent assume that the meanings and relations of
terms to be used in a formulation are relatively precise
and stable, while the sociolOgist, in his interest with
content, questions whether the phenomenon is stable.
Constantine Politis draws the reader's attention to
problems of using axiomatization or formalization:

"It should be noted that the axiomatic
form constitutes a representation of a logical
or mathematical field in its final expository
form and should be clearly distinguished from
the exploratory form of a field in the process
of being develOped, so that axiomatization
does not reflect the process of mathematical
discovery but the conventional form of com-
municating or stating mathematical results."11

Abraham Kaplan indirectly argues that the con-

texts are different on three grounds. First, he argues

 

11Politis, "Limitations of Formalization,” 356

107

that there are in principle no terminal contexts in
inquiry:

"New problems always arise, confronting
us with unforeseen issues of both meaning and
truth. The penumbra of vagueness is not a
"surplus meaning," scientifically expendable.
Because meanings are Open, our conceptual-
izations can reach out for a firmer grip on
reality."12

Kurt Mayer seems to speak to the same point when dis-
cussing stratification concepts:

"The need for continuous conceptual
clarification and theoretical reassessment
arises in a large part from the increasing
complexity of modern social structures and
their rapid changes. By the time we have
fashioned some conceptual tools to analyze
our stratification system and tested them
in empirical research, the underlying
social reality has already become dis-
torted."l3

Kaplan's second attack centers on the question
of whether or not the formalistic approach is contra-
dicted by scientific practices:

"Now the logic of science is not prOperly
restricted to an analysis of instantaneous
states of knowledge. . . We are interested not
only in the timeless relations of entailment
among abstract prOpositions but also in the
processes by which in time one concrete scien-
tific assertion gives way to another, and
usually an assertion making use of new, or at
least modified, concepts.

The point is that we cannot accommodate
these facts about the norms of scientific
usage without providing for Openness of meaning.

 

12Kaplan, The Conduct of In uir , 7O

13KurtB. Mayer, "The Changing Shape of the
American Class Structure," Social Research, 1963, XXX,

“'58 -68 o

108

If meanings are taken to be always wholly
fixed, logically speaking, relations between
an old concept and new ones can be described
only as overlap, inclusion, or complete
replacement. The details of the process of
change, its intermediate steps, its locus and
justification-~all remain unspoken."14
Finally Kaplan argues that the specification
of meaning of terms is a complex task. A term like
"bald" can be eXplained by itself while terms like
"the" have sentential meanings, that is, they can be
explained only in the context of a sentence. Kaplan
goes on to add a third point.

"That point is that there are terms which
require for a specification of their meaning
not one sentential context but the context of
the whole set of sentences in which they ap-
pear. Such meanings I have previously called
"systemic meanings". Each sentential occur-
rence is a partial determination of the
meaning, but only as we encounter the term in
more and more contexts of varying sorts do we
come to understand it more fully."15

One might conclude that as new sentences are
added to a formulation, new or further meanings emerge.
This also would tend to argue against the relevance of
terminal stages of inquiry for the sociologists
purposes.

Berger gp_§l.note that the accretion of a theo-
retical tradition "does not consist of the addition of

endless numbers of gd hoc findings to the mounting

 

14Kaplan, The Conduct of Inguipy, 69-70
l5Kaplan, The Conduct of Inquipy, 64

109
stock of knowledge, but of the continual modification
of the systematic structure of definitions and assump-

l6

tions that is called a theory." What is of importance
here is that he notes the need for continual
modification of formulations.

The question of a theoretic tradition and
problems associated with the choice between develOpment
of new formulations and the reformulation of tradi-
tional formulations does not arise in the philosophic
context. It was pointed out earlier that sociologists
teaching graduate courses in sociology of education
chose t0pics related to the traditional areas of socio-
logical investigation, while professors of education
teaching similar courses chose tOpics that were more
directly relevant to the problems faced by teachers and
educational administrators. This seems to indicate
that the sociologist is interested in developing a body
of theory from traditions already existing, and would
place more emphasis upon coherence of formulations with
the tradition in their selection of formulations than
would the philosopher.

Professor George McKee notes that American
Sociologists tend to select topics or content areas on

the basis of their relevance to social problems than do

 

16Berger, et al, Sociological Theories in
Progpess, x

110

European Sociologists. He points out however that even
the Americans view the selection of apprOpriate subject
matter as being tied to previously developed
conceptualizations.17

The philosopher of social science places a good
deal of emphasis upon the development of precise local
theories. The sociologist is more interested in the
adjustment of various formulations so that they might be
integrated into new and more powerful conceptualizations.
The possibility of integration of theories is to a great
extent linked to the Openness of the conceptualization,
or better said, to the systematic vagueness of the terms.
For example the sociologist is not only interested in
develOping an accurate and precise stratification theory,
and a theory of motivation, but he is interested in in-
tegrating both such formulations into one conceptualiza-
tion so that he can talk about social class and level of
aspiration. This goes a step beyond the work Of philos-
ophy of science and makes other demands upon the
sociologist. As was pointed out earlier the educator
has broader interests yet.

In sum then the sociologist asks how he can best
describe and explain a set of changing social phenomena,

and he evaluates formulations on the basis of their

 

17From an unpublished lecture in History of
Sociological Theory, given at Michigan State University,
January, 1967.

111

systematic Openness, their coherence with a tradition,
power, clarity, and fruitfulness. The educator asks
how he can change individuals by manipulating the
social context, and judges formulations first in the
light Of their functional adequacy and then asks ques-
tions about simplicity, precision and so forth. The
philosopher of social science asks what is the best
form for the elaboration of a set of concepts, and in
asking this question selects formulations on the basis
of their logical characteristics. Hence he is more
interested in logical precision, simplicity, and com-

pleteness than other criteria.

g.2 The problem of evaluating criteria

The question addressed in this section is:
how does one rationalize the selection or rejection of
a criterion such as clarity, functiOnal adequacy, or
simplicity. 'It has already been pointed out that there
are a number of criteria that might be used in the
selection or rejection of formulations. But since it
takes a set of criteria to select, order, and attach
weights to other criteria, the question remains as to
what criteria might be used for that purpose. Both
Goodman and Rudner reject certain criteria for the
selection of primitive terms and formulations, but
neither explicitly treats the criteria used in that

selection. For that reason the tOpic of criteria for

112

the selection of criteria must be discussed briefly.

In the interest of clarity a distinction will
be made between criteria used to select formulations,
and criteria used to select and order those criteria.
Criteria used directly to select primitives and formu-
lations such as confirmedness, simplicity and clarity
will be called "first order criteria." Criteria used
to select first order criteria will be called "second
order criteria." The question at hand then is: How
are decisions concerning the selection of formulations
made in each of the contexts? It is Obvious at the
outset that scientists and educators use a number of
criteria in the selection of alternate formulations.
There is however no precisely defined decision proce-
dure, that is, given a decision problem in the two
contexts, there is no precise set of criteria available
nor is there a definite sequence of application or a
calculus of weights for each of the possible criteria.
The task of this subsection is the specification of a
model of such a procedure and set Of criteria.

In the decision procedures being questioned
here, there is no adequate reconstructed logic.18 And

even though this dissertation is not an attempt to

 

18Kaplan, The Conduct of In uir , Chapter 1.

113

construct such a logic, it must be concerned with
clarification of some of the variables that must be
considered in order to understand the place or legité
imate role of simplicity in the decision procedure.

Perhaps a discussion of some of the factors
that influence the sequence of application of crite-
ria, and the weight attached to each Of them will help
clarify the preceeding point. If one were to ask a
sociologist or educator what first order criterion is
of primary importance in selecting primitives and for-
mulations, he might well get an answer like: "That
depends on a lot of things." The chances are that the
sociologist or educator would add that the selection
of criteria, the weight to be attached to each of them,
and the sequence in which they are to be applied is

determined by some Of the following factors:

1. The stage of development of the formula-
tions in question is relevgpt. For instance, in

selecting formulations in the early stages of develOp-
ment a sociologist might well select a formulation on
the basis of clarity or intuitive plausibility, while
in the selection of well-develOped formulations he
might be looking for a large backlog of confirming
evidence. At the same time the educator might well

choose between formulations in the early stages of

114
develOpment on the basis of its relevance to the
educational task at hand, or its close relation to
other relevant formulations, such as psychology. In
choosing between more fully developed formulations he

too might look for much confirming evidence.

2. The state of development of related disci-
plines is one relevant aspect. For instance, if the

sociologist is dealing with formulations concerning
socialization, he might reject off hand a formulation
that contradicted certain well-confirmed learning prin-
ciples from psychology. The educator might well accept

or reject a formulation on similar grounds.

3. The logical status of a fopmulatipn ig
relevant to the choice being made. If an educator is

choosing between empirical theories, his first move

 

will be to some pragmatic criterion of truth. He would
ask which theory is most well confirmed. 0n the other
hand, if an educator is choosing between two classifi-
cational schemata, he will likely not raise a question
of confirmation, but will look to the structure of the
formulations and ask whether the classificational terms

are clear, or relevant to his task.

4. The purposes of the pepsons or community
selecting a formulation will have a dipect influeppe

upon the selection of criteria. The educator will
likely have a different set of purposes in the

115

selection of topics and formulations in sociology of

education as has already been pointed out.

5. The philOsOphic, theological or ideological
assumptions that the selector makes might often hgve ap

influence on the criteria that he invokes. For in-
stance, a realist might reject a formulation cast in
phenomenalistic terms. It is to be noted that Russian
sociologists tend toward conflict models in sociology
while American sociologists more Often choose equilib-
rium models.19 It is difficult indeed to definitely
locate the source of these differences, but it seems
reasonable to assume that ideological difference play

some role in this instance.

6. The effectiveness of other criteria pre-
viously employed influences the selection gpd

appropriateness Of other criteria. For instance, if
a sociologist were making a choice between theoretic
formulations, and had already found that theories in
question were about equally well confirmed he might
make the decision of the basis of clarity, or some
other criteria. But if he were choosing between for-
mulations not equally well confirmed he would probably

place greatest trust in the more well confirmed

 

19This idea was presented in an unpublished
lecture by Charles Loomis in February, 1967 at
Michigan State University.

116

formulation even though it was not as clear as the
unconfirmed formulation.

Take the following case as an example of the
contextual nature of first order criterion selection:
If an educator were attempting to build an effective
community relations program and were trying to de-
cide what course of develOpment he should follow in
the early stages of development of a classificational
schemata in sociology, one which he intends to use in
the comparison of structural characteristics of Amer-
ican Society relevant to Negro poverty, and had made
the assumption that poverty is not desirable or nec-
essary due to some inexorable laws of theology,
psychology or economics, and that man does not have
"free will" in the traditional sense, he would prob-
ably select a broad range Of categories and try them
in an attempt to identify significant variables. The
Coleman Report Equality of Educational Opportunity
seems to uphold this thesis. This report is essen-
tially an attempt to identify the significant factors
influencing equality of educational Opportunity and
is a study of a broad range of factors, rather than
an in-depth study Of a few factors.

The educator will also be interested in the
functional adequacy of the formulation, but in the

early stages Of investigation will not know just what

117
factors or variables need to be considered to bring
about functional adequacy. After a preliminary inves-
tigation, he is in a better position to determine just
which factors or variables are most directly related
to his purposes and he will be able to define func-
tional adequacy. Hence, after the educator has done
some initial investigation and has acquired a backlog
of evidence, the order of selection of first order
criteria will change. In the first instance he is
interested in identifying relevant variables and
chooses the most powerful formulation for that pur-
pose, but after some of the "results are in" he will
turn to other criteria. In this case the educator
will look to functional adequacy, and he will no doubt
attach more weight to precision and eventually perhaps
simplicity.

With these things in mind the role Of simpli-
city in a particular context will be defined and
evaluated in terms of the following second order cri-
teria: (l) Objectivity, (2) effectiveness, (3) appro-
priateness, (4) compatibility, (5) sequential
significance, and (6) overall significance. The first
three criteria are somewhat independent from the par-
ticular contexts discussed. Appropriateness of a
first order criterion is a necessary condition for its

inclusion in any list of criteria. If a criterion is

118

not apprOpriate to a set of decision problems it
cannot be objective, effective and so forth in rela-
tion to that set of problems. Both objectivity and
effectiveness are desired of all first order criteria
and are to that degree independent of any context.
There is a second kind Of second order cri-
teria however. The weight to be attached to each Of
them is related to the particular context in which
first order criteria are being selected. But before
this contextual relationship is discussed it would be
well to specify the meaning of the second order cri-
teria and give some explanation Of each of them. It
should be noted at the outset that second order cri-
teria may be applied to first order criteria in
relation to either particular decision problems, or
sequences Of decision problems.
(1) Objectivity: The first order criterion g,is said
to be Objective if it can be applied mechanically or
routinely to appropriate cases, or if an algorithm
can be constructed for its application. For instance,
a criterion is said to be objective if it can be ap-
plied by a machine, or by any person with access to it.
Measures of length and duration are Objective when
Standards and appropriate instruments for their appli-
cation are available. A criterion such as beauty, as

it is commonly used, is not an Objective criteria

119

because it is dependent in principle upon private
knowledge, moods and attitudes.

(2) Effectiveness: The first order criterion g’is said
to be effective if it can discriminate in all cases to
which it is appropriate. In the case of a man sorting
potatoes the criterion "big potato" and "little potato"
are not effective criteria unless "big" and "little"
are operationally defined since there will be a number
of cases which cannot precisely be judged either "big"
or "little." "Equal to or less than three ounces" and
"greater than three ounces" would be more effective
criteria because they could discriminate in more case.
"Greater than ten" is absolutely effective when applied
to the domain of integers.

(3) Appropriateness: The first order criterion §,is
said to be appropriate to a decision problem if there
is something in the subject matter that "answers to"
the criteria. The criterion "confirmedness" is not
appropriate in the selection of nontheoretic formula-
tions because confirmedness (as has been used here) is
a characteristic of empirical laws, and nontheoretic
formulations qu,nontheoretic do not generate or contain
empirical laws. In the same sense "ripe" is an inappro-
priate criterion in the selection of swimming pools,
and "invalid" is inappropriate in the selection of ap-

ples. It is easy to see that if a first order criterion

120

is inappropriate for a decision problem or set of
problems, then that criterion falls out of further
consideration. Intuitively apprOpriateness is the
first criterion of second order criteria used in the
selection of first order criteria.

(A) Compatibility: A first order criterion y,is said
to be compatible with a first order criterion y_in
appropriate cases when the attainment Of condition y.
does not require loss of condition y, For instance,
when dealing with black raspberries, raspberry p_may
be either "red" or "ripe" but it cannot be "red" and
"ripe" at the same time. Certain of the first order
criteria used in the selection of formulations may be
incompatible, for instance it may be the case that a
simple formulation may not be clear. In cases where
two first order criteria are not compatible, the se-
lection of an assignment of weight to one of them will
be determined on the basis of the purposes Of the
person or community making the decision. For instance,
if the educational community were faced with the task
of selecting a stratification formulation, it might
well be the case that the choice is between a rela-
tively clear and precise formulation, and one that
promises tO be functionally adequate but is not as
clear and precise, it would probably choose the func-

tionally adequate formulation.

121
(5) Seguential significance: A first order criterion
x,is said to be sequentially significant in appropriate
cases when it is the first in a sequence Of ineffective
criteria to be applied. This again is a contextual
matter, and might be conceived as a matter of degree.
First order criterion y.might have a higher sequential
significance in the educator's context than it does in
the sociologist's context. For instance, functional
adequacy will have a high degree of sequential signif-
icance in many decisions faced by the educator, while
it will have less sequential significance in the soci-
ologist's context. Sequential significance will be
directly related to purposes and as purposes change the
sequential significance of first order criteria will
change.
(6) Overall significance or weight: The first order
criterion y,is said to have a high degree of overall
significance in a sequence of problems if it is appro-
priate to a high percentage Of cases and has a high
degree of sequential significance in those appropriate
cases. For instance, a football coach might assign
high overall significance to the criterion "speed" by
always first asking how fast a prospective player is.
Overall significance is contextual in nature, since it
is dependent upon the sequential significance. High

overall significance will characterize first order

122
criteria that are tied to the central purposes of a
context. For instance, perhaps the most important
goal Of science is adequately explain "the world."
Since confirmedness of theory is tied to that goal it
displays a high degree of overall significance.

In sum then, the scientist and applied scien-
tist are Often faced with the task of deciding between
formulations, and in order to carry out the decision
in a systematic way, or in order to rationalize the
decision after the fact he must use certain criteria.
The problem is complicated by the fact that there are
a number Of criteria that might be used, and the sci-
entist is faced with the problems Of making a choice
between criteria, Of attaching weight to one or more
of them, and ordering the sequence Of their application.
In order to do this he must look to other criteria,
here called second order criteria. Of the six second
order criteria defined, three of them might be univer-
sally applied in the selection of first order criteria,
and the other three take on content or meaning within
a particular context because they are related to the
purposes of individuals, and communities and other
contextual matters.

Figure I is an attempt to represent the rela-

tions of first and second order criteria.

A selection of
formulations
will be made on
the basis of
some ordering
of first order
criteria

A selection of
and ordering
of first order
criteria will
be made on the
basis of second
order criteria

The selection
and ordering of
first order
criteria will
be made on the
basis of two
sets of second
order criteria.
One set will be
independent,
the other tied
to the question
asked by the
individual or

community making

the selection

123

 

Formulation

 

I

 

[Formulation] FO

III

 

possible choices

 

 

 

 

actual
choice of
formulation
I III
First First First
Order Order Order
Criterion Criterion Criterion
X Y z

 

 

 

 

 

 

 

— possible choices-

{*1

 

A

 

 

 

Actual
Choice
of first
order
criteria

mulation

I

 

 

Independent
Criteria

 

 

ApprOpriateness
Objectivity

Effectiveness

 

 

FIGURE I

 

Contextual
Criteria

 

Compatibility

Sequential
significance

Overall
significance

A GENERAL MODEL FOR THE SELECTION
OF SCIENTIFIC FORMULETIONS

 

A selection of
formulations
will be made on
the basis Of
some ordering
of first order
criteria

A selection of
and ordering

of first order
criteria will
be made on the
basis of second
order criteria

The selection
and ordering of
first order
criteria will
be made on the
basis Of two
sets of second
order criteria.
One set will be
independent,
the other tied
to the question
asked by the
individual or

community making

the selection

AN EXAMPLE OF THE SOCIOLOGIST'S

124

 

 

Formulation Formulation Formulatio
[ I I II I III
[-possible choices
actual
choice Of
formulation

 

Clarity

 

 

 

 

 

 

 

 

 

Independent
Criteria

 

Appropriateness
Objectivity

Effectiveness

 

 

FIGURE II

Simplicity [ Power
_ /f :\

 

-—possible choices -—J*

 

FL]

SELECTION OF FORMULATIONS

 

 

 

Actual
Choice
Of first
order
criteria

 

 

Contextual
Criteria

 

Compatibility

Sequential
significance

Overall
significance

 

 

125

A selection of
formulations I
F

 

will be made on ormulationI IFormulationI IFormulationI

 

 

 

 

 

 

 

 

the basis of I III
some ordering - -*
of first order I. -J er\
criteria possiblel choices
actual
choice of
formulation
A selection of ' ‘ I ‘
and ordering Power Simplicity Functional
Of first order I I Adequacy
criteria will ’ ' ' ‘
be made on the ’f}‘
basis of second

order criteria

 

 

 

 

_ possible choices _J

 

 

 

 

 

 

 

 

 

 

 

Actual
Choice
of first
order
criteria
The selection ’ ' I
and ordering Of Independent Contextual
first order .Criteria Criteria
criteria will
be made on the ApprOpriateness Compatibility
basis of two
sets Of second Objectivity Sequential
order criteria. significance
One set will be Effectiveness
independent, Overall
the other tied significance

 

 

 

to the question
asked by the
individual or
community making
the selection

FIGURE III

AN EXAMPLE OF THE EDUCATOR'S SELECTION
OF FORMULATIONS

126
Now that a specification of criteria has been
made, simplicity can be evaluated in each of the con-
texts. Before that is done however, Goodman's argu-
ments for the centrality Of simplicity will be tested

in the sociologist's and educator's decision contexts.

3.3 An evaluation of Goodman's

arguments in other contexts

 

In Goodman's arguments for the centrality of
simplicity in the selection of primitives for a system
he suggests that often predicates are considered def-
initely inappropriate for selection as primitives if
they are inapplicable or Obscure, but that selection
of primitives from the remaining predicates cannot be
so definitely made. He also argues that clarity is
not an effective criterion since scientists Often coin
terms to be used as primitives. He also argues that
the problem area to be treated seldom determines
uniquely the choice of primitives and hence in the ab-
sence of other effective criteria, simplicity takes on

20 The question to be raised here

greater importance.
is whether Goodman's arguments carry weight in the
sociologist's and educator's respective contexts.

There is no doubt that inapplicable predicates

 

20Goodman, The_§tructure of Appearance, 64-66

127
are inappropriate choices in any formulation, but some
question is raised by what is meant by "Obscure" pred-
icates. Goodman gives as examples of "obscure"
predicates, "Predisposed to telepathic communications

21 These pred-

with supernatural aid," and "denotes."
icates are questionable on several grounds, but a
question remains as to just what constitutes "Obscure"
and whether or not the demand for rejection of "obscure"
primitives is appropriate to the purposes of the soci-
ologist. Are such terms as "society," "social order"
and "culture" Obscure? Surely they must be considered
so until their systemic meaning is elaborated.22 It
would seem that the sociologist faces a paradox if he
takes seriously the rejection of "Obscure" predicates,
while he admits that some predicates have systemic
meaning. Such predicates are necessarily Obscure until
they are assigned meaning through elaboration of the
formulation, so the rejection of "Obscure" predicates
as primitives has the affect of eliminating all pred-
icates that have a high degree of systemic meaning.

If this is what Goodman is arguing, it is probably a
valid argument. It does however point up the fact that

the sociologist places himself at a disadvantage when

he takes the constructionist's stance.

 

21Goodman, The Structure of Appeargpce, 64-65
22Kaplan, The Conduct Of Inquiry, 6h

128

The educator will probably lack interest in
the role of Obscurity in the selection of primitives
predicates, and be more concerned with the Obscurity
in the total set Of extralogical predicates. 1

Finally, Goodman argues that the problem to be
dealt with seldom determine uniquely the primitives to
be used.23 If this is true in the constructionist's
context it is probably less likely to be so in the
sociologist and educator's contexts. After all, the
sociologist and the educator are treating different
problems. The constructionist's problem is to bring
logical system to a set of concepts, and interpretation
of those concepts is secondary. The sociologist in his
interest in content is constantly frustrated by the
demands of the pragmaticsah of the situation. The
educator's interest in functional adequacy complicates
the problem further and restricts his range of appro-
priate choice to that degree.

In sum then, Goodman's minimization of certain
concerns and his subsequent maximization of the impor-

tance of simplicity is probably less valid in the

 

23Goodman, The Structure of Appearance, 66

ZIThe sociologist seeks not only a logically
neat system, but a true one in the pragmatic sense.
He is therefore less free to systematize a formulation
than the constructionist. The sociologist must give
the world a chance to "Object" while he is developing
an Objective formulation.

129
sociologist's and educator's contexts than it is in his
own. The following section is an attempt to point out
that while simplicity as Goodman eXplicates it is an
objective and effective criterion, it has little sequen-
tial significance in the sociologist's and educator's

decision contexts.

. The role of sim licit in each context

This section Of the dissertation is addressed
to the question of the role of simplicity in the contexts
within which the educator and sociologist makes deci-
sions. In section h.2 the problem of evaluating criteria
such as clarity and simplicity was discussed and a number
of criteria for that purpose were specified. In this
section the question of the Objectivity, effectiveness,
appropriateness, compatibility, sequential significance,
and overall significance of simplicity in the context
Of the sociologist and the educator is addressed.

In the sociologist's context Of inquiry, what
is the role of simplicity as explicated by Goodman in
the selection and rejection of formulations? In the
light of the criteria elaborated in n.2, and in the
light of Rudner's comments elaborated in chapter two,
the answer at present is that simplicity is an inappro-
priate criterion for use in the selection of formulations.
It is inapprOpriate on the grounds that there are at

present no formulations formalized to the extent necessary

130

for the application Of the calculus. There is nothing
in the formulations of sociology of education that
"answer to" the criteria.25
Any role that simplicity might play in the
future will be dependent upon something similar to
Rudner's suggestions for formalizing formulations or
altering the explication in such a way that it might
be applied to partial-formalizations. The first alter-
native has already been discussed and it has been
pointed out that formalization makes demands and
assumptions that conflict with other desiderata of
sociology of education. Hence, if simplicity is to be
made appropriate by formalization of sociological for-
mulations, it will become incompatible with other
desiderata that fall higher on the hierarchy of values
of the sociological community, and hence simplicity will
become sequentially insignificant. Take as an example
26

the area of stratification theory. In stratification

 

25The term "formalization" is used by social
scientists in a number of ways. As far as could be
determined it is always used in a weaker sense than
necessary for the purposes here described. In Types
Of Formalization in Small-Group Research, Berger says,
"By 'formalization' we refer to the general process of
making explicit the logical structure of a set Of
assertions."

26Stratification theory is chosen because it is
of interest to both sociologists and educators, and it
contains formulations that fall under both the "theo-
retic" and "nontheoretic" rubrics.

131

theory the present thrust of the sociological
community is to keep the conceptualization Open enough
so that progress can be made in the development of the
concept of social class while allowing enough systematic
vagueness in the terms employed to make legitimate in-
quiry. Two things have become apparent in recent years
concerning social class: (1) Sociological research can
be useless if it is infected with a pripri categories,
and (2) all communities do not have similar social
characteristics, hence the notion of social class must
be systematically vague.

Kaplan once said that if you give a boy a
hammer he will find that everything he encounters needs
pounding.27 Sociologists and scientists in general
have agreed that a similar phenomenon exists among sci-
entists-~give them a set of categories and they will
see to it that all phenomena are ordered according to
them. The general response of the scientific community
to this has been to distrust a prippi judgments and to
include an openness or systematic vagueness in their
technical terms. It seems that the general stance of
the Constructionists in their demand for precise concepts
conflicts with the sociologists desire and need for
systematic vagueness.

Melvin Tumin reports on the systematic vagueness

 

27K'aplan, The Cpnduct of nguipy, 28

132
of stratification formulations in pointing out that
stratification refers to five attributes Of the
phenomena of stratification:

"1. It is social, i.e., patterned in character.

2. It is ancient, i.e., it has been found in
all past societies.

3. It is ubiquitous.

4. It is diverse in its forms.

5. It is consequential, i.e., the most impor-
tant, most desired, and Often scarcest
things in human life constitute the basic
materials wh'ch are distributed
unequally."2

Brookover notes: "The concept of social class
has many diverse meanings, all of which have to do with
differentiation in the population of a society."29
Hodgkinson discusses the problems in measuring social
class:

"In general, there have been four views Of
social class structure that have dominated re-
search done in this field. (There are about one
thousand articles and books published since 1925
which deal with social classes in America, so
that we are talking about a large segment of the
field of sociology. )"3
Just what causes this proliferation of formula-

tions is still problematic. Several people have tried
to explain why it exists, and in doing so have attested

to the community goal of Openness Of concepts.

 

28Melvin Tumin, Social Stratification
(Englewood Cliffs: Prentice-Hall, 1967), p. 13.
29Brookover, A Sociolo of Education, 154

30Harold Hodgkinson, Education, Interaction,

and Social Chan e (Englewood Cliffs: Prentice-Hall,
1967), p. 88.

133

Zetterberg laments the hiatus between theoretical and
applied sociology, and points out that established
sociological knowledge is not being used by sociological
practitioners because they reveal a "common inability to
draw upon codified knowledge."31 Implicit is the as-
sumption that sociologists ppghp_to draw upon codified
knowledge, and this in turn seems to indicate an openness
of "codified knowledge." Alpert notes four limitations
upon general sociology in the United States: (1) inade-
quate historical and comparative perspectives; (2) lack
of critical scholarship with respect to past sociological
accomplishments; (3) excessive concern with methodology;
and (4) failure to establish close relationships with
biology.32 Number two seems to make a similar appeal to
the Openness of past accomplishments.

In an earlier article in Synthese, Leyton attri-
butes the failure of sociologists to draw upon codified
knowledge and their lack Of critical scholarship with
respect to the past sociological accomplishments to so-
ciologists lack of "verbal responsibility" in their

33

communications.

31Zetterberg, Hans, Social Theo and Social
Practice. (New York: Bedminster Press, 1962), 190.
32Alpert, Harry, "Some Observations on the

State of Sociology," Pacific Sociolo ical Review
1963, 6.2, Fall, 45- .

33Leyton, Alvin C., "Sociology and Obligations
Of Interpretation," S these, Vol. XI (1959), p. 177.

 

134

In a recent book Berger, Zelditch, and Anderson
lament the fact that students are not taught to work
with theory, and set out to write a book, "Its purpose
is to expose students, by example, to the pitfalls,
the problems, and the difficulties of construction
systematic theory."3[+

These men were moved to write such a book by
Parson's statement: "But the most disappointing single
thing about it has been the degree to which the results
of this work (empirical research done in our field) have
failed to be cumulative."35 Each of these criticisms
assumes that research should be cumulative and it seems
that a desirable if not necessary condition for such a
cumulative effect is the systematic Openness of
formulations.36

Rudner argues that the first order criterion
theoretic fruitfulness is inappropriate to the selection

of formulations which have not been fully elaborated.

3“Berger, et al. Sociolo ical Theories in
Progress. (Boston: Houghton Mifflin Co., 1966), xi.
35Berger, Sociological Theories in Progress, xii.

36NO distinction has been made between systematic
Openness of formulations and systematic Openness of a
conceptualization. Systematic Openness Of conceptual-
izations may or may not make logical demands upon the
forms of formulations. It has been stated that the
axiomatic form of formalized theories assumes that an
incumbent formulation is in a terminal stage of devel-
Opment, and this seems to demand a closure of the
underlying conceptualization.

 

135

It is possible that if simplicity is made appropriate

by formalizing formulations, and that formalization is
possible only in terminal stages of inquiry, that the-
oretic fruitfulness would be made apprOpriate by the
same process of formalization. It is also quite prob-
able that theoretic fruitfulness would have considerably
more sequential significance than simplicity in the so-
ciologists context.

It is doubtful whether formalization of particular
formulations would serve the general purposes Of the so-
ciological community. A second argument might be entered
at this point claiming that formalization of particular
formulations might make them easy to communicate and
work with. This argument is so weak however that it will
not be considered here.

The second_approach to the problem of making
simplicity apprOpriate to decisions in the sociologists
context would be to weaken it in such a way that it
could be applied to more cases. It has already been
pointed out by Goodman himself in his rationale for the
calculus that if simplicity is measured by counting prim-
itive predicates or by counting predicate places in
extralogical primitives is ineffective because the number
of primitives in any formulation can be reduced to one
by conjunction.

Goodman's explication might also be modified by

136
reducing it to a "rule Of thumb" like "extralogical
predicates should be explicitly defined whenever pos-
sible." The vagueness of such a rule argues against
it however, since it would be difficult to determine
when such definition is possible. In certain cases
such a rule might be used as an analytic tool however
and help clarify certain terms. It would be question-
able whether explicit definition Of all possible terms
would allow for the systematic vagueness spoken of
earlier. To the degree that explicit definition frus-
trated this purpose it would be unwarranted. It should
also be pointed out that while explicit definition
would clarify the syntactical relations of a formula-
tion, other means such as metalinguistic explanation
might work better to specify the meaning of such a
term.

The number of formulations that the educator
has to draw upon is identical with that available to
the sociologist and the present inapprOpriateness Of
Goodman's explication of simplicity is just as defi-
nite in the educator's context as it is in the sociol-
ogist's context. Formalization as a method of making
simplicity appropriate will not generally be as anti-
thetical to the educator's purposes as it was to the
sociOlogist's purposes. Whereas the sociologist

sought to develop formulations that reflected a

137

systematic vagueness of concepts, the educator is not
as interested in systematic vagueness. The educator
employs sociological formulations for one Of several
reasons. He may want to compare classrooms, school
districts, or alternate teaching strategies through

the use of a classificational schemata, he may want

to identify social factors influencing academic
achievement, or he might seek a law-like generalization
to be used in an explanation or prediction Of a future
state of affairs. There is one common element, the
educator will use sOciological formulations or general-
izations as tools in the educational enterprise. He
does not seek the kind of Openness that the sociologist
must seek, but he seeks precision and functional ade-
quacy in particular formulations. Formalization does
not conflict with the general purposes of the educator
then, at least insofar as formalization does not Ob-
scure and make the content Of a formulation unintelli-
gible on practical grounds.

The question of the sequential and overall
significance of simplicity as a criterion in the edu-
cator's context remains after it is conceded that
simplicity might be made appropriate to such choices
by formalization of formulations. It was pointed out
earlier that the concerns of the educator as an applied

scientist are directed toward the solution of particular

138

problems, and for that reason the educator as other
applied scientists must seek formulations that are
functionally adequate for his purposes. Out Of the
large number of stratification formulations available
to the educator, it is unlikely that he will be faced
with a decision problem unless he is faced with a
problem that is to be analyzed in terms of stratifi-
cation theory. When the educator is faced with the
problem Of providing equality of educational Oppor-
tunity, or with the problem of turning out equally
well educated individuals who bring different back-
grounds tO the educational context, he then becomes
interested in social class differences, but even then
his interest in social class differences will be limited
to variables that have a bearing upon the educational
task. He can, as an applied scientist, abstract from
much of stratification theory, and focus upon certain
particular elements of the formulation. In the same
way, a doctor setting a broken bone does not have to
know everything about bones at that point in time.

It is then doubtful whether simplicity would
be sequentially significant even if it were to be made
appropriate by adequate formalization of formulations.
It would seem that the first order criterion of func-
tional adequacy takes on such sequential and overall

significance that simplicity would be insignificant by

 

 

 

8881'! It

139
comparison.

The second tack could be attempted, that is of
forming a nonformal analogue Of Goodman's calculus, but
it has already been pointed out that a weakening Of the
criteria serves to trivialize it.

In sum then, simplicity as explicated by Goodman
is inappropriate to the selection Of formulations in
sociology of education because there are no formali-
zations to which it can be applied in that area.
Simplicity could be made apprOpriate in one of two ways,
first by formalizing formulations in sociology of edu-
cation, and second by weakening simplicity itself.

When simplicity is made appropriate by the first method
it becomes incompatible with the systematic vagueness
sought by sociologists, and hence becomes sequentially
insignificant in the sociologist's context, and it is
similarly insignificant in the educator's context be-
cause of its relationship to functional adequacy. A
weakening of simplicity results in a trivialization

of it on the very grounds that every formulation can

be made simple by conjunCtion or substitution of

predicates.

CHAPTER V
SUMMARY AND CONCLUSIONS

This inquiry was based upon the assumption that
education finds its reason for existence in what it can
contribute to human welfare and the preservation of man,
and that the size of that contribution is in direct
proportion to what educators know about the educative
process. Sociology of education has promised to make
intelligible certain aspects of the social context
within which education is carried on, and it has accom-
plished this task to some extent, but has failed at
least in part because of the very enthusiasm with which
new and different theories and formulations were produced.
A brief look at the literature of the field disclosed a
shocking proliferation of overlapping and competing for-
mulations. This proliferation of formulations seems to
have blocked for educators the very intelligibility that
was so diligently sought.

It was assumed that the solution to the problem
Of proliferation would involve the rating and subsequent
selection of formulations for use by the educational

"consumer" and further development by the sociologist.

140

141

It was also assumed that the development Of an adequate
decision procedure would require among other things the
development of adequate criteria by which formulations
could be rated. This dissertation was to be the report
of the inquiry into the possible role of structural
simplicity as a criterion in the selection Of alternate
formulations in the sociology of education.

Richard Rudner's writings on simplicity were
important to this inquiry because Rudner made the initial
suggestion that simplicity might play an important role
in the selection formulations in the social sciences,
and because in them he made an attempt to classify and
evaluate various attempts to explicate the term simplic-
ity. Rudner first divided all attempts at explication
of simplicity into two categories: Ontological Sim-
plicity and Descriptive or Linguistic Simplicity. On-
tological Simplicity was to have been applied to those
explications of simplicity that presumed to predicate
something of the universe or of the way men perceived
the universe, while Descriptive or Linguistic Simplicity
was to have been applied to those explications conceived
to be predicated of man's descriptions of the universe.
Rudner rejected the notion of Ontological Simplicity as
fundamentally Obscure and irrelevant to the tasks of
science. Instead he selected as relevant to those tasks

the subdivision of Descriptional Simplicity called

142

Objective-Logical Simplicity, and he noted that of two
‘approaches to the explication of that notion, Nelson
Goodman's approach was most promising. He also noted
that Goodman had devised a calculus of simplicity to be
applied to formalized formulations to yield simplicity
values based upon the number and relevant kind of extra-
logical primitive predicates Of each formulation.

Professor Rudner argued that the empirical test
of confirmation does not apply tO many formulations in
the social sciences since it can only be applied to for-
mulations that contain empirical laws, and many Of the
formulations used in the social sciences are nontheoretic,
that is, they are classificational schemata or typologies
that do not generate empirical laws. He also argued that
among other criteria (clarity, power, theoretic fruitful-
ness) simplicity was the most promising for evaluation of
nontheoretic formulations. This made simplicity the most
promising for the social sciences, since they contain
large numbers of nontheoretic formulations.

The immediate promise of simplicity as a crite-
rion in the social sciences was dimmed by the fact that
Goodman's calculus of simplicity could be applied only
to formalized formulations, and since there were few
such formulations in the social sciences, simplicity was
only potentially useful. Rudner suggested that simplic-

ity might be made appropriate to the social sciences in

143

one Of two ways: first, the formulations could be
formalized or partially formalized, or second, the
calculus could be modified to be applied to nonformal-
ized formulations.

Some comments were Offered concerning the

usefulness of Rudner's suggestions concerning simplicity

p
and social science formulations in relation to sociology E
of education. It was specifically noted that present I
weaknesses in theory in sociology of education are re- 3
lated to the problem of assessing formulations, and that 3

several areas of sociology Of education are quite con-
cerned with the use of nontheoretic formulations.

As the inquiry progressed it was further noted
that the concerns and perspective of a philOSOpher Of
social science might well differ significantly from
those of either a social scientist or an educator. It
was specifically noted that Rudner's view of "science-
as-product" within which simplicity showed so much
promise, might be inadequate for the sociologist's and
educator's purposes. A

In chapter three Nelson Goodman's eXplication
of simplicity was reviewed. It was found that Professor
Goodman's explication of simplicity turned on a partic-
ular notion of system; one which Goodman had explicated
earlier. For Goodman a system was a specialized lan-

guage constructed within a larger language, for some

144
special purpose. Within a system the terms are precise
in extension, the relations between terms are, where
possible, explicit through definition and the logical
mechanisms for relating propositions are explicit., The
system has two kinds of nonlogical terms: primitives
which are explained by metalinguistic devices in the
metalanguage, and nonprimitives which are defined ex-
plicitly within the system. When a theory is formalized,
that is, systematized in a constructional system, a de-
cision problem concerning the selection of primitives
arises. Goodman's central argument for simplicity was
that the most powerful formulation with the least pos-
sible primitives is always desired in science. That is,
given formulations of equal power, as would be the case
when selecting alternate formalizations for a given
theory, the most economical formulation is the simplest
formulation; that is, the alternative with the fewest
primitive terms is the most economical.

Professor Goodman went on to point out that the
number Of primitives in a given system can be reduced
by various means, some Of which are counted as spurious,
and others of which are counted as legitimate simplifi-
cations. He argued that the relevant characteristics
Of predicates in the assessment of simplicity are the
number of predicates in a base, the number of predicate

places in a base, and the characteristics of reflexivity,

 

I I! II

145

self-completeness, and symmetry.

Other arguments for the importance of simplicity
turned upon the rejection of other criteria for the
selection of primitives. It was noted however that the
context named as appropriate for the application Of
simplicity is far more restricted than the context
within which the theory builder or the user of theory 3
works. This raised the question of the weight of E
Goodman's arguments for the weight of simplicity in 3
those other contexts. It was finally argued after ac- ii
cepting the calculus as effective in the context defined
by Goodman that four questions remained to be taken up
concerning the use of simplicity in the selection of
formulations in sociology Of education: (1) Are there
differences in the contexts faced by the philosopher of
science, the sociologist of education and the educator
that affect the relevance of simplicity to each context?
(2) How is selection of rejection of criteria such as
clarity, comprehensibility and simplicity rationalized
in the three contexts? (3) Do Goodman's arguments for
the centrality Of simplicity hold in all three contexts?
(4) What is the role of simplicity in each of the
contexts?

An inquiry into the three contexts within which
the philosopher the scientist and the applied scientist

works showed that there is a significant relation

146

between the purposes held by a community and the
criteria that are used in selecting formulations. It
was argued that the context within which the philoso-
pher of science, at least those of the breed of Rudner
and Goodman, move assumes that the most important as-
pects of theory building and evaluation are logical.
This assumption allows them to put considerable stock
in the use of formal or constructed systems. The un-
fortunate part of this is that the axiomatic form
assumes that the content of the system is the result

of a terminal stage of inquiry. This is to be con-
trasted with the context within which the sociologist
moves. In this context, the sociologist is more
interested in the content of a conceptualization than
in the form that it takes, and along with this interest
in content comes the interest in continuing inquiry.
Where the philos0pher can well afford to look at in-
stantaneous states of inquiry, the sociologist is
interested in the development of a set of concepts that
underly particular formulations. The educator is also
primarily interested in the content of formulations,
and is consequently less interested in the formal as-
pects of formulations. The educator however has a
different interest in sociology than does either the
sociologist or philOSOpher. The educator is interested

in the power that can be gained through sociological

147

knowledge, and in the selection Of formulations he is
interested in the functional adequacy of the formulations.

It was concluded that there are some significant
differences in the contexts within which the philosopher
the scientist and applied scientist move, and that these
differences play an important role in the selection of
formulations and criteria for the selection of formula-
tions.

It was further argued that neither Goodman nor
Rudner had adequately treated the problem of selecting
criteria for the selection of formulations. In the
second part of Chapter Four the question of the evalua-
tion of criteria was taken up. It was concluded that
the selection of criteria for the selection of formula-
tions requires the specification of a second level of
criteria, and that some of the second order criteria are
directly related to the purposes Of the community or
scientist making the decision. It was argued that there
are at least six second order criteria that might be
used in the selection of first level criteria. Three of
those six are independent and might be used in the eval-
uation Of any criteria. Appropriateness, objectivity,
and effectiveness might well be required Of all first
order criteria. Compatibility, sequential significance,
and overall significance are directly related to the

context within which a decision is being made. The

148

question of the compatibility of first order criteria
is raised when more than one criteria is required for
a particular decision. Sequential significance and
overall significance are attached to criteria within
a particular context on the basis of the purposes Of
decision makers within that context. A

Since the calculus of simplicity can be applied
only to fully formalized formulations, and at present
there are no fully formalized formulations in sociology
Of education, Rudner's conclusions that simplicity is
inappropriate under present conditions is seen as accu-
rate. This is true in both the sociologist's and
educator's decision making context. And beyond that,
since appropriateness is a necessary condition for fur-
ther consideration of a first order criterion,
simplicity cannot be either Objective or effective in
those contests as they presently stand.

There are a number Of strategies that might be
employed to make simplicity appropriate to the selection
of formulations in sociology of education. Those strat-
egies fall into one of three categories: those that
would formalize formulations and take the calculus of
simplicity as it stands, those that would modify the
calculus to make it appropriate to nonformalized formu-
lations, and those that would modify both the formulations

of sociology of education and the calculus of simplicity.

 

149

Each of these approaches might be attempted in each of
the contexts.

In the sociologist's decision making context,
simplicity can be made appropriate by formalizing for-
mulations, but since formalization is apprOpriate to
the terminal state of inquiry or preterminal stages Of
inquiry viewed instantaneously as terminal states of
inquiry, and since one Of the primary goals of the so-
ciological community is to continue inquiry and
conceptual development, the very process by which the
calculus is to be made appropriate frustrates an impor-
tant goal of the sociological community. To that degree
simplicity is incompatible with systematic Openness Of
formulations. In cases of incompatibility of first order
criteria, a judgment must be made to place a value on the
conflicting first order criteria. Such a decision,
insofar as it is systematically made, will be made by
reference to the primary purposes of the decision maker.
It is likely that in such a case the philosoPher of
science would refer to the goal of systematicity and
choose the criteria with logical import, while the so-
ciologist would refer to a different goal or purpose,
probably that of content develOpment and tend to choose
other criteria such as systemic openness. Just what the
individual sociologist's or sociological communities'

choice might be in this case is problematic, and can be

150

resolved by empirical investigation, but it is likely:
that the sociologist would choose the criterion of
systematic openness. If this were the choice in a par-
ticular instance, simplicity would be sequentially
insignificant in that case. If it were often the case
that the sociologist chose systematic Openness over
simplicity, this would mean that simplicity is of little
overall significance.

In the sociologist's decision making context
simplicity could be made apprOpriate by modifying it
in one of several ways to be applicable to nonformalized
formulations. But if this strategy is employed, sim-
plicity is trivialized, that is, it becomes ineffective.
If the assessment of simplicity is reduced to the
counting Of predicates or predicate places, Goodman has
already established that all formulations can be made
maximally simple by conjunction or redefinition and
simplicity cannot be used to choose between equally
simple formulations. Hence simplicity is ineffective
as a result of this kind of weakening. If simplicity
is reduced to a "rule of thumb" or a canon of formula-
tion building such as "make maximal use Of explicit
definitions," or "always reduce the number of primitives
when possible" then simplicity becomes either destruc-
tively vague, or it requires a closure that violates

systematic Openness. This is not to say that such a

151

canon would not be useful and instructive in some cases,
but it seems that insofar as it has any import for the
formulation builder, that import can be reduced to a
call for logical clarity.

The sociologist might finally take the tack of
developing partial formalizations in which tenative or
provisional primitives are named and a tenative assess-
ment of simplicity be made upon that basis. This is the ,
suggestion made by Rudner. Surely some benefit is to be
realized from such an exercise, but the benefit will
more likely be from the kind of analysis that is neces-
sary prior to the assignment of provisional primitives
than from the tenative assessment of simplicity that
would result. In the light Of the overall insignifi-
cance of final assessments of simplicity, it is hard to
imagine just how a tenative assessment of simplicity
could be any more significant.

In the educator's decision making context, it
seems that attempts to make simplicity appropriate by
weakening the criterion would have the same affect as
those suffered in the sociologist's decision making
context. A weakening of the calculus would result in
making the criterion ineffective. This is probably also
true ofthe approach of using partial formalization and
a tenative assessment of simplicity.

In the educator's decision context, formaliza-

tion does not immediately seem to be anthithetical to

152
the educator's purposes it was to those of the
sociologist. It seems to be the case however that if
simplicity is made appropriate to decisions by formal-
izing formulations, simplicity will still be
insignificant in the face of the overall significance
placed upon the criterion of functional adequacy by the
educator. It seems that the educator's or the educa-
tional communities' purposes are centered upon the use
of formulations to gain power to achieve a number Of

educational objectives, and insofar as this is true they

ermW..xialu-Lhm w
. ‘. ' 4.

must first look to the functional adequacy of a formu-
lation. It seems to be the case that simplicity is
not a necessary condition for functional adequacy,
since that is true, simplicity seems insignificant in
the educator's context.

The general conclusion of this dissertation is
then that simplicity, as explicated by Nelson Goodman,
does not offer promise as being an important criterion
in the selection of formulations in sociology of educa-
tion. This is attributed to the fact that it has been
conceived as a criterion applicable only to formula-
tions, which are and can only be the terminal stages of
inquiry, and the scientist is primarily interested in a
continuing process of inquiry, and the educator is in-
terested in the power that is to be derived from
sociological inquiry as a set of methods or techniques

and the product of inquiry as a set of concepts.

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'7V1’Iflunh‘mu.'\.mm
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.4 . ‘

   

APPENDIX A

159

 

,

 

 

160

The purpose of this study was to determine how
educators and sociologists differ in their selection of
tOpics for study in the sociology of education, and to
determine what criteria that they would use in the se-
lection Of alternate formulations.

The population tested was seventy doctoral
candidates at Colorado State College, sixty one of whom
were candidates for degrees in some area of education
and nine of whom were studying for degrees in the social
sciences.

Each of the candidates was given the questionaire
attached. (See Figure II)

Twelve Of the candidates from education answered
the question about the criteria that they might use in
the selection of alternate formulations. (question 3.)
Of those twelve, four pointed out that their selection of
criteria would depend upon the particular choice situa-
tion, five said that they did not know what criteria
they would use and three said that they would choose
formulations cast in terms through which a strategy for
educating the "self actualizing man" could be realized.
The candidates for degrees in education all chose some
form of educational or social problem as apprOpriate
topics for study in sociology of education, and they
all chose them peppppp.they were educational problems.

The candidates for degrees in social science

 

161

all answered the question concerning the criteria by
which alternate formulations would be selected. Six
of them answered that they would choose formulations
that were most "well developed" or most "well con-
firmed" while three Of them pointed out that they would
choose formulations which were part of a particular
"school of thought." The candidates for degrees in
social science chose topics that were problem oriented,
but three Of them mentioned classical studies in soci-

ology relevant to the schools.

Figure IV. Questionnaire Given to Doctoral Candidates
Concerning the Selection of TOpics
and Criteria

Degree program Major Minor

Resident Student

 

Years of teaching experience Subject taught
Level

 

Years of Administrative experience Position

Other educational eXperience

 

 

Do you have any special interest in sociology of educa-

tion?

 

 

(1) What topics do you see as apprOpriate for study in
sociology of education?

 

 

(2) What criteria have you used in the selection of

tOpics?

 

 

(3) In several areas in sociology of education there
are a number of alternate formulations or conceptual-
izations available which may be used to describe and
analyze those areas. Given this situation what criteria
might you use to select alternate formulations?

 

 

 

 

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