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AN EVENT-RELATION APPROACH TO
A METATHEORY 0F ACCOUNTING

presented by

Carl Torben Thomsen

has been accepted towards fulfillment
of the requirements for

Ph.D _ Business — Accounting
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ABSTRACT

AN EVENT-RELATION APPROACH TO
A METATHEORY OF ACCOUNTING

By

Carl Torben Thomsen

PROBLEM

The purpose of This sTudy is To develop a meTaTheory (or a
Theory of Theories) of accounTing which will provide a common
framework for explaining and forming accounTing Theories.

The need for such a sTudy arises from a marked lack of
generaliTy in presenT accounTing Theories. Emphasis upon specific
problems of accounTing pracTice and The lack of generaliTy make iT
difficulT To discern unifying or pervasive elemenTs wiThin accounTing.
CurrenT accounTing liTeraTure provides ample illusTraTion of The
dissaTisfacTion wiTh exisTing accounTing Theory.

MeThod

AccounTing is firsT defined in The broadesT Terms possible.
The elemenTs of ThaT definiTion Then form The basis for The
deveIOpmenT of The resT of The sTudy. In The developmenT of The
meTaTheory, a sharp disTincTion is drawn beTween The pure Theory
(synTacTics) and The connecTion of The pure Theory To The real world
(semanTics). In The pure Theory, a bare minimum of maThemaTical
concest are selecTed To form The basis for a simple sTrucTure of
The core of accounTing. The essenTial elemenTs necessary To connecT
ThaT sTrucTure To The real world in a meaningful way are Then

considered.

Carl Torben Thomsen

DirecT confirmaTion of The meTaTheory is made difficulT by
iTs greaT generaliTy. BuT a limiTed verificaTion of iTs validiTy is
carried ouT by examining how Two of The more formalized approaches
To accounTing Theory fiT inTo The framework of The meTaTheory.
Findings

The evenT-relaTion meTaTheory looks upon accounTing as The
sTudy and communicaTion of a seT of evenTs and a seT of relaTions
on Those evenTs. In ThaT definiTion is summarized The core of The
Theory, namely evenTs and relaTions. The exacT naTure of an evenT
is IefT Open To definiTion, subjecT only To The resTricTion ThaT,
however defined, iT musT fiT inTo The formal framework. The concepT
of relaTion is borrowed from maThemaTics, buT cerTain kinds of
relaTions are idenTified as having special relevance To accounTing.

In The pure Theory, an evenT is defined To be a member of
a sequenTiaI memory. The sequenTial memory Thus consisTs of a
number of evenTs, each of which carries a unique descripTion. The
evenTs are ordered by Time of occurrence and also classified inTo
groUps called evenT series wiTh each group conTaining similar evenTs.
The concepT of The enTiTy is obTained by considering TogeTher a
sequenTial memory and a seT of relaTions Thereon.

The relaTions selecTed fall inTo four caTegories. Equivalence
relaTions form The basis for cIassificaTion of evenTs and relaTions.
Order relaTions make possible The concepT of Time. DefiniTionaI
relaTions combine evenTs in various ways To form new quanTiTies. And
Transfer relaTions seek To find cause-effecT relaTionships beTween
cerTain groups of evenTs. In oTher words, one seT of evenTs is

sTaTed as a funcTion of anoTher seT of evenTs.

Carl Torben Thomsen

The meTaTheory does noT sTaTe exachy how The connecTion wiTh
The real world is To be made——ThaT is The Task of a specific Theory--
buT raTher iT creaTes a general framework wherein hypoTheses making
These semanTicaI connecTions may be formed. The seIecTion hypoTheses
IimiT The field of aTTenTion by seIecTing relevanT evenTs and relaTions.
The sTrucTural hypoTheses are Those of descripTion, ordering, and
cIassificaTion. They are necessary To make each of The Three
dimensions of The sequenTial memory meaningful. The conTroI hypoTheses
are Those of verificaTion (idenTifying valid relaTions), error (dealing
wiTh errors in The descripTion of evenTs), and evaluaTion (dealing
wiTh criTeria for The evaluaTion of all oTher hypoTheses).

In The consTrucTion of The meTaTheory few concest are
included and very broad classes of hypoTheses are considered. BuT
The generaliTy of The meTaTheory discloses a basic framework ThaT is

subjecT To fuTure developmenT.

AN EVENT-RELATION APPROACH TO
A METATH EORY OF ACCOUNTING

BY

Carl Torben Thomsen

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Accounting and Financial Administration

1973

Q CopyrighT by
CARL TORBEN THOMSEN

I973

TABLE OF CONTENTS

LIST OF EXHIBITS. . . . . . . . . . . . . . . . . . .

LIST OF FIGURES O O I O O O O O O O O O O 0 O 0 O O O O O O O O O

ChapTer

INTRODUCTION. . . . . . .

STaTemenT of Purpose. . .
STaTemenT of The Problem.
MeThod of DevelopmenT . . . . .
LimiTaTions . .

THE SETTING FOR THE METATHEORY. . . . . . . . . . . . .

InTroducTion. . . . . . . . . . .

The EnvironmenT of AccounTing . . . . . . . . .
A DefiniTion of AccounTing. .

RooTs of an EvenT- RelaTion Approach .

Summary . . . . . .

THE THEORY: SYNTACTICS .

InTroducTion. . . . . . . . . . . . . . . . .
MaThemaTical PresupposiTions.
EvenTs and The SequenTiaI Memory.
Equivalence RelaTions . . . . . . . . . . . . . .
Order RelaTions . . . .
DefiniTionaI RelaTions. . . . .
The FundamenTal RelaTion. . . . . . . . . . . . . .
AriThmeTicaI RelaTions.
STaTisTical RelaTions . .
Transfer RelaTions. . . . . . . . . . . . . . . . . .
Special Cases of The Theory .
CounTing and AccumuIaTion .
Special Time lndices. . . . . . . . . . . .
Closed SysTems. . . . . . . . .
A Special Case of The FundamenTal RelaTion.
Summary . . . . . . . . . . . . .
NonmaThemaTical CommenTary. . . . . . . . . . . . . . .

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18
23

24

24
25
26
27
28
29
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31
31
31
32
32
32
33
34
35
36

ChapTer Page

IV. THE THEORY: SEMANTICS. . . . . . . . . . . . . . . . . . 4O

InTroducTion. . . . . . . . . . . . . . . . . . . . . . 4O
SelecTion . . . . . . . . . . . . . . . . . . . . . . . 42
DescripTion . . . . . . . . . . . . . . . . . . . . . . 43
CIassificaTion. . . . . . . . . . . . . . . . . . . . . 44
Ordering. . . . . . . . . . . . . . . . . . . . . . . . 44
Error . . . . . . . . . . . . . . . . . . . . . . . . 46
VerificaTion. . . . . . . . . . . . . . . . . . . . . . 47
EvaluaTion. . . . . . . . . . . . . . . . . . . . . . . 48
Summary . . . . . . . . . . . . . . . . . . . . . . . . 49

V. THE THEORY APPLIED. . . . . . . . . . . . . . . . . . . . 50

InTroducTion. . . . . . . . . . . . . 50
Synopsis of The ljiri and MaTTessich Theories . . . . . 51
Comparison wiTh Ijiri' 5 Theory. . . . . . . . . . . . . 52
Comparison wiTh MaTTessich's Theory . . . . . . . . . . 57
Summary . .. . . . . . . . . . . . . . . . . . . . . . . 61

VI. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . 64

Overview of The Theory. . . . . . . . . . . . . . . . 64
EvaluaTion of The MeTaTheory. . . . . . . . . . . . . . 66
ImplicaTions for AccounTing . . . . . . . . . . . . . . 68
FurTher InvesTigaTions. . . . . . . . . . . . . . . . . 69

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . 71

LIST OF EXHIBITS

ExhibiT Page
1. Four DefiniTions of AccounTing. . . . . . . . . . . . . 17
2. LisT of Symbols in The SynTacTics . . . . . . . . . . . 39

3. Comparison of Three AccounTing Theories . . . . . . . . 62

LIST OF FIGURES

Figure Page
I. The EnvironmenT of AccounTing. . . . . . . . . . . . . . . Ii
2. Framework of Johnson's EvenTs Theory of AccounTing . . . . 22
3. STrucTure of ljiri's AccounTing Theory . . . . . . . . . . 54
4. STrucTure of MaTTessich's AccounTing Theory. . . . . . . . 59

5. STrucTure of The EvenT—RelaTion MeTaTheory . . . . . . . . 65

CHAPTER I

INTRODUCTION

STATEMENT OF PURPOSE

 

1 of

The purpose of This sTudy is To develop a meTaTheory
accounTing which will provide a common framework for explaining2
oTher accounTing Theories and for guiding in The developmenT of
new Theories. The Theory will be consTrucTed wiTh These characTer-
isTics in mind:

I) GeneraIiTy and wide applicabiliTy. The Terms of The
meTaTheory should be capable of describing presenT accounTing
Theory wheTher financial or managerial. And iT should also be
capable of describing macro-accounTing (naTionaI income accounTing
and reIaTed areas). A specific economic or insTiTuTionaI seTTing
is imporTanT for individual Theories, buT a meTaTheory does noT

1A meTaTheory is a Theory of Theories. Thus, raTher Than
having as iTs objecT The explanaTion of aspecTs of The real world,
iT has as iTs objecT The explanaTion of a seT of Theories abouT
The real world. This usage is esTablished boTh in science and in
accounTing. See, for example, Mario Bunge, ScienTific Research I:
The Search for SysTem (New York: Springer—Verlag, 1967), p. 32;
Richard MaTTessich, AccounTing_and AnalyTical MeThods (Homewood,
Illinois: Richard D. Irwin, Inc., 1964), p. 31; and ReporT of The

CommiTTee on AccounTing Theory ConsTrucTion and VerificaTion,
The AccounTing_Review, SupplemenT To VoI. XLVI, 1971, pp. 58-63.

 

2ExplanaTion is considered To be The sysTemaTic and general
descripTion of The sTrucTure of a sysTem and of The relaTions among
The elemenTs of The sysTem. For a fuTTher eIucidaTion of The
naTure of explanaTion, see ErnesT Nagel, The STrucTure of Science
(New York: HarcourT, Brace & World, Inc., 1961), chaps. 2 and 3.

1

2
need To be Tied To a parTicular seTTing. Double-enTry should be
a special case raTher Than a general feaTure of The Theory.

2) SimpliciTy. The aim is To use as few concest as
possible and To use The simplesT of maThemaTical Tools. Cohen and
Nagel noTe: "Science aims aT The simplesT accoUnT which will
sysTemaTize The whole body of available knowledge."3 They
characTerize simpliciTy as follows:

One Theory will Therefore be said To be more simple or general

Than anoTher if The firsT can, while The second cannoT, ex-

hibiT The connecTions iT is invesTigaTing as special insTances

of The relaTions iT Takes as fundamenTal.
SimpliciTy in This sense is very similar To generaliTy.

3) Uniqueness. Because of The many exisTing approaches
To accounTing Theory,5 This sTudy seeks To be unique by means of
(a) The Terms chosen as fundamenTaI, (b) iTs generaliTy which
makes The Theory funcTion under any objecTives or environmenTal
assumpTions and makes iTems, previously considered fundamenTaI,
special insTances of The more general Theory, and (c) iTs develop—

menT using a sharp disTincTion beTween The pure Theory (synTacTics)

and iTs connecTion wiTh The real world (semanTics).

STATEMENT OF THE PROBLEM

 

RecenT years have seen numerous aTTemst To provide a

general sTrucTure for accounTing. There has been widespread

 

3Morris R. Cohen and ErnesT Nagel, An lnTroducTion To
Logic and ScienTific MeThod (New York: HarcourT, Brace & World,
Inc., 1934), p. 384.

4|bid., p. 214.
5For one analysis of These, see Daniel L. McDonald,

ComparaTive AccounTing_Thegry (Reading, Mass.: Addison-Wesley
Publishing Company, 1972).

 

3

disconTenT wiTh mosT of These approaches. This disconTenT can
be seen in The commiTTee reporTs of The American AccounTing
AssociaTion,6 in analyses of broad Trends in accounTing Theory,7
as well as in numerous criTicisms of individual works. Two acTions
of The American InsTiTuTe of CerTified Public AccounTanTs may also
be viewed as expressing reservaTions abouT currenT accounTing
Theory. These reservaTions are reflecTed in The publicaTion of
STaTemenT of The AccounTing Principles Board No. 4 and in The
esTablishmenT of a Financial AccounTing STandards Board To replace
The presenT AccounTing Principles Board.

AnoTher example of The apparenT inadequacy of accounTing
Theory is The almosT compIeTe lack of aTTenTion given To The area

of accounTing in ForresTer's IndusTriaI Dynamics. ForresTer

 

aTTemst To creaTe a general meThod of modeling firms. AccounTing,
as a financial model of The firm, would seem a naTuraI sTarTing

poinT for such a developmenT. BuT ForresTer devoTes slighTIy

8

more Than a page To a discussion of accounTing, apparenle because

he saw in iT noThing ThaT would conTribuTe significanle To a

I

6ReporT of The CommiTTee on Foundations of AccounTing
MeasuremenTs and ReporT of The CommiTTee on AccounTing Theory
ConsTrucTion and VerificaTion, The AccounTing Review, SupplemenT
To Vol. XLVI, 1971, pp. 1-79; and ReporT of The CommiTTee on
Research MeThodoIogy in AccounTing, The AccounTing Review, Sup-
plemenT To Vol. XLVII, 1972, pp. 399-520.

7Louis Goldberg, "The PresenT STaTe of AccounTing Theory,"
The AccounTing Review, XXXVIlI (July, 1963), pp. 457-69; John W.
Buckley, Paul Kircher, and Russell L. MaThews, "MeThodoIogy in
AccounTing Theory," The AccounTing Review, XLIII (April, 1968),
pp. 274-83; and Richard MaTTessich, "MeThodoIogical PrecondiTions
and Problems of A General Theory of AccounTing," The AccounTing
Review, XLVII (July, 1972), pp. 469-87.

8Jay W. ForresTer, lndusTrial Dynamics (Cambridge, Mass.:
The M.l.T. Press, 1961), pp. 335-36.

 

4
general understanding of The firm and because The scope of account-
ing was too limiTed as a description of The structure of The firm.
One of The other problems That accounting Theory suffers

from is its lack of generality. There has been a fear by some
accountants That an extension of accounting Theory will encompass
other fields and that accounting will lose its unique orientation.
However, several recent studies have successfully explored account-
ing from The standpoint of related disciplines.9 The need for
generality is noted by Nagel:

Many of The outstanding Theories in The sciences are capable

of explaining a much wider variety of experimental laws and

can Thus deal wiTh an extensive range of materials ThaT are

qualitatively strikingly dissimilar. This feature of Theories

is related both to The facT that Theoretical notions are not

Tied down To definite observational facts and that because of

The complex symbolic structure of Theories more degrees of

freedom are available in extending a Theory To many diverse

areas.

As The above remarks indicate, There is still a need for

further probes into The foundations of accounting.

Thus, Today, perhaps more Than at any Time in The history of

accounting, we are in need of Theoretical concepts which can

provide the basis for a logical, consistent, and articulated
set (or sets) of accounting practices.11

 

9Yuji ljiri, The Foundations of Accounting Measurement
(Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1967); Raymond
J. Chambers, Accounting, Evaluation and Economic Behavior
(Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1966); Richard
MaTTessich, Accounting_and Analytical Methods (Homewood, Illinois:
Richard D. Irwin, Inc., 1964); Thomas R. Prince, Extension of The
Boundaries of Accounting_Theory (Cincinnati: South -Western
Publishing Co., 1963); and Baruch Lev, "Accounting and Information
Theory" (unpublished Ph.D. dissertation, University of Chicago, 1968).

 

10ErnesT Nagel, The Structure of Science (New York:
Harcourt, Brace & World, Inc., 1961), p. 89.

11Report of The CommiTTee on Research Methodology in
Accounting, The AccounTing,Review, Supplement To Vol. XLVII,
1972, p. 438.

 

5
This study is presented in the conviction That proper theoretical
concepts can best be provided when accounting is viewed in its
mosT general aspects Through a meTaTheory.

On The other hand, the amount of activity in the area of
accounting Theory, besides expressing dissatisfaction, also indicates
a renewed interest in both broadening and strengthening The foun-
dations of accounting. AT the present time there is no single study
That has achieved The kind of generality that will serve to integrate
the discipline of accounting.

The present state of accounting research resembles a jigsaw
puzzle where some areas slowly grow into meaningful con-
figurations but without yielding the entire picture. Indeed,
The individual fragments seem to spread outwards and not
Towards a common center.

The need for a deliberate effort of integration in our

discipline is by no means of recent Vintage, buT it becomes ever
much more urgent in a Time of opulent growth and specialization.

12
This study attempts to provide an alternative approach to such
integration.

In summary, the problem can be stated in logical form as "which
is the x, such that x is a meTaTheory of accounting?" Different

researchers may propose different x's. This study is a preliminary

attempt To find one such x.

METHOD OF DEVELOPMENT

 

Three general guidelines are considered in the construction of
the meTaTheory.
1) This study proposes a definition of accounting, The

fundamental terms of which are subsequently expanded in The theory.

 

12Richard MaTTessich, "Methodological Preconditions and
Problems of A General Theory of Accounting," "The AccountinggReview,
XLVII (July, 1972), pp. 482-83.

 

6
This is similar To the work of MaTTessich in which he proposed
a definition of accounting and Then presented his Theory as an
extension and development of That definition.13
2) The meTaTheory is built around The concepts of sets

and relations. These concepts are borrowed from mathematics. In
algebra generalized number systems called rings, domains, and fields
are built using the concepts of sets and binary operations (a

special kind of relation).14

In this study a similar approach is
Taken, except that The sets considered are given an elementary
structure and The relations go beyond only binary Operations.

3) A sharp distinction is made between syntactics and
semantics. Sterling has suggested that theory construction in
accounting proceed on a basis similar to That of the study of
languages, since theories are expressed in language. The three
areas in the study of language are syntactics, semantics, and
pragmatics. They are defined as follows:

SynTacTics—-The study of the relation of signs to signs.

Semantics--the study of the relation of signs to objects or
events in the real world.

Pragmatics--The sTudy of tge relation of signs to the users
of the signs.

 

13Richard MaTTessich, Accounting and AnaIytical Methods
(Homewood, Illinois: Richard D. Irwin, Inc., 1964), pp. 19-20.

14W. E. Deskins, Abstract Algebra (New York: The MacMilIan
Company, 1964), chapter 7.

15Robert R. Sterling, "On Theory Construction and
Verification," The Accounting Review, XLV (July, 1970), pp. 445—56.

 

A somewhat similar distinction is made in Nagel's view on the content
of theories. In his thinking, theories have three components:
(I) An abstract calculus that is the logical skeleton of the
explanatory system, and that "implicity defines" The basic
notions of the system; (2) a set of rules that in effect
assign an empirical content to The abstract calculus by
relating it to The concrete materials of observation and
and experiment; and (3) an interpretation or model for The
abstract calculus, which supplies some flesh for the skeletal
structure in terms of more or less familiar conceptual or
visualizable materials.16
The validity of the two-fold distinction between the pure or formal
theory (syntactics) and the connections of the theory to The real
world or its interpretation (semantics) is well recognized within
science. And support is also given to make a similar distinction in

17 In This study a chapter will be devoted To each of

a meTaTheory.
The two, syntactics and semantics.

Chapter II provides the setting for the rest of the study.
By means of a diagram, accounting is placed within a decision frame-
work. A general definition of accounting, under which can be
subsumed several other definitions, follows. The central concepts
of events and relations between events are included in That definition.
Since these concepts form The basis for The development of The
meTaTheory, their roots in accounting writings are briefly examined.

Also some of The works directly anticipating the development of an

evenT-relation meTaTheory are discussed.

 

16Ernest Nagel, The Structure of Science (New York: Harcourt,
Brace & World, Inc., (1961), p. 90.

17J. H. Woodger, "The Technique of Theory Construction,"
Foundations of The Unity of Science, Vol. II, ed. Otto Neurath,
Rudolf Carnap, and Charles Morris (Chicago; The University of
Chicago Press, 1970), p. 456.

8

Chapter III presents the syntactics, the pure theory without any
connection to The real world. Using events in a sequential memory
and a few simple mathematical Tools from set theory, four classes
of relations are developed and defined. The world is seen as con-
sisting of a mass of facts or events. The concepts selected and
related in this chapter are the minimum tools needed to organize and
interpret the mass of events. The chapter concludes by showing some
special cases of The theory. The abstract nature of double-entry is
discussed in connection with closed systems as one of these special
cases.

Chapter IV, the semantic part of the Theory, discusses the
assumptions necessary To connect The theoretical model of Chapter
III To the real world. The role of specific theories is to spell
out these assumptions in detail. Since this study takes The approach
of a meTaTheory, it will only discuss the broad classes within which
those assumptions fall. One set of classes follows directly from The
structure of the syntactics. Another set of classes is necessary to
assure that the results (the events and relations selected) are
meaningful.
Chapter V Takes the place of the empirical confirmation of The

Theory. Being a meTaTheory, it is not subject to immediate verifi-
cation. Instead, a limited validation of the meTaTheory is carried
out by showing how The theories of Ijiri and Mattessich fit into the
framework of The meTaTheory. The Theories of Ijiri and Mattessich are
chosen because they represent the most formalized approaches to
accounting Theory and therefore provide the most precise bases for

comparison.

9
Chapter VI summarizes The theory in diagram form and shows

some implications for future research into accounting.

LIMITATIONS

 

This study is not directed Toward the solution of immediate
practical problems faced by accountants. And it does not seek to
explain Today's conflicting and inconsistent accounting principles.
Neither is the study a normative or prescriptive theory. Rather,
the meTaTheory provides The framework within which a great number
of different accounting constructs can be deveIOped to suit either
a particular decision maker or a decision model in practically any
economic environment. Because the study is exploratory in nature,
it cannot aim for completeness in every detail. During the study
substantial changes have been made in the formulation of the theory,
and there is no reason To believe that any number of improvements
cannot continue to be made. For These reasons and because the
meTaTheory is subject only to limited verification, it is presented
as tentative.

An ordinary theory would have to specify what should be
measured and how it should be measured. In a meTaTheory these
questions assume a subsidiary role and the framework within which
these questions are asked becomes of primary importance. Therefore,
the question of measurement is not addressed directly.

Behavioral considerations (pragmatics) are also excluded from
the discussion, since their inclusion would only obscure the main
structure of the study. The meTaTheory does not preclude pragmatics,
but iT will be The role of lower level Theories to fit behavioral

aspects into The framework of the meTaTheory.

CHAPTER II

THE SETTING FOR THE METATHEORY

INTRODUCTION

 

The purpose of This chapter is to discuss the nature of
accounting and to trace some previous works Taking an approach to
accounting similar to this study. First, the general nature of
accounting is discussed by means of a diagram showing accounting
as part of a feedback system. Then the definition of accounting
That will be used in this study is formulated and its general
nature is illustrated by subsuming under it Three other definitions
of accounting. Finally, The roots of the event-relation meta-
theory are examined both in general and in terms of specific works

that have attempted an approach similar to This study.

THE ENVIRONMENT OF ACCOUNTING

 

Figure I shows a simplified model of The context of ac-
counting. lts basic features are similar to diagrams drawn by
other authors relating to The nature of theory and to information

feedback systems.18 On the left of the diagram is the universe

 

18Robert R. Sterling, "On Theory Construction and Veri-
ficaTion," The Accounting_Review, XLV (July, 1970), p. 448; John
G. Kemeny, A Philosopher Looks at Science (Princeton, N. J.: D Van
Nostrand Company, Inc., 1959), p. 86; and Report of the Committee
on Accounting and Information Systems, The Accounting Review,
Supplement to Vol. XLVI, 1971, pp. 287-350.

10

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(The real world), and on The right is theory, which is a set of
statements about certain aspects of the real world. In other
words, Theory can be considered as a model of The real world.
Actually, all The elements of the diagram are part of the real world
(universe), but the items shown have been segregated To illustrate the
key aspects of interest in accounting. The diagram does not profess
to be complete but merely shows, in simplified form, the relationships
between The key elements. The diagram follows the convention of
showing input or output as circles and processes as boxes.

The processes of selection and description go together.
Selection is relevant because of the limited capacity for sensation

19 and limitations of available

and observation of individuals
measuring instruments. In the following, both individuals and
instruments will be referred To as instruments. Given an instru-
ment, there are only certain aspects of The universe with which it
can deal. Those aspects of the universe it translates into a
different form That hopefully is more understandable. The process
of Translation is called description, which is a general term that
stands for both measurement (quantitative) and observation (non-
quantitative). The output of any description is an organized set
of data, whether it is quantifiable or not. That which is selected
is described, but only That which can be described is selected.
Selection involves not only the limitation of the field of inquiry

to that which is describable, but it also implies the exclusion of

undesirable elements of The universe which are often described as

 

19Raymond J. Chambers, Accounting, Evaluation and Economic
Behavior (Englewood Cliffs, New Jersey: Prentice-Hall, Inc.,
1966), p. 35.

 

I3

"noise." Selection seeks To provide for the instrument a limited
aspect of the world as noise free as possible. These elements of
accounting will be given more attention in the discussion of seman-
tics in Chapter IV.

The next area of the diagram deals with Theory or syntactics.
Theory can be considered a model of The real world. As noted
before, the aim of this study is not the construction of a theory
as such but rather a meTaTheory, forming the framework within which
other accounting theories can be formed. Theory formation and
theory verification are only imperfectly understood and involve
more elements Than shown in the diagram.20 But as shown in the
diagram, a Theory, or set of statements about the environment, is
verified by a correspondence between data provided by measurement
and data provided by performing Theoretical operations on other
measurement data. The Theoretical operations will in many cases
result in predictions of data that can be measured independently.

Decision making Is oriented Toward the future and therefore
requires predictions about the future supplied by theoretical
operations. Here decision making may either refer to a decision
model and its required data inputs or simply to a decision maker
as an individual, who may not necessarily request specific data
but is rather provided with some data and then makes decisions.

Decisions lead to directives, which are plans of action in Terms

 

20For an analysis of These processes in general see Mario
Bunge, Scientific Research I: The Search for System (New York:
Springer-Verlag, I967), ch. 8. For an analysis of how They apply
to accounting see Report of the Committee on Accounting Theory
Construction and Verification, The Accounting Review, supplement
to Vol. XLVI, 1971, pp. 51-79.

 

 

I4
Of a set Of commands. These directives are then executed or
carried out in a manner that may or may not agree with The intentions
Of the decision maker.

Each Of the processes in the diagram is Of concern in
accounting; but because of the level Of generality Of the meta—
Theory, The processes Of decision making and execution are not
considered. However, specific Theories must Take these processes

into account in order to be relevant.21

A DEFINITION OF ACCOUNTING

 

In the context Of Figure I, accounting can now be defined

in general Terms. The definition will not specifically limit The
subject matter Of accounting, nor will it necessarily distinguish
accounting from other related disciplines. Instead it will be

indicative Of a broad area within which accounting operates.
Accounting is
the study and communication of

a set of events and

a set of relations on those events.
The definition has four elements: study, communication, events,
and relations. The subject matter of accounting consists of
events and relations. The exact meaning Of the terms "event" and
"relation" will be further explained in Chapters Ill and IV. The
kind of events and relations that form the subject matter is left

Open. In traditional accounting, economic events are usually spe-

cified. What is to be done with the subject matter is also divided

 

2lFor an example Of an accounting theory in which decision
making plays a crucial role see Robert R. Sterling, Theocy Of the
Measurement of Entegprise Income (Lawrence, Kansas: The University
Press of Kansas, 1970).

 

15
into two parts. First events and relations are studied and Then
They are communicated. The word "study" covers the semantic and
syntactic area of accounting. It implies a clear understanding Of
The connection between the output Of an accounting system and the
real world. Only when events and relations have been placed in
a sound semantic and syntactic setting can they be communicated
meaningfully. The following discussion will therefore not deal
directly with communication, but rather it will emphasize The
creation of a sound theoretical background against which meaningful
communication can take place.

This new definition of accounting will now be compared to
three accepted definitions. The American Institute of Certified
Public Accountants has defined accounting as follows:‘

Accounting is the art Of recording, classifying, and sum—
marizing in a significant manner and in terms Of money,
Transactions and events which are, in part at least, of a
financial character, and interpreting The results thereof.2
The definition fits into the framework created by the new defini-
Tion. The activities involved in "study" are here specified as
"recording, classifying, and summarizing." "Events" are further
specified as being "Transactions and events . . . Of a financial
character." "Interpreting The results thereof" might broadly
be viewed as the communication process. This definition lacks a
specific statement about The interest of accounting in the relation

between events, except maybe as it might be included in "inter-

preting the results Thereof."

 

22American Institute Of Certified Public Accountants,
Accounting Research and Terminology Bulletins (final edition;
New York: AICPA, 1961), p. 9.

16
The American Accounting Association defined accounting as
the process of identifying, measuring, and communicating
economic information to Bgrmit informed judgments and decisions
by users Of InformatIon.
This again is similar to The new definition if "identifying" and
"measuring" are Taken as examples Of the processes involved in
"study". The last part of the definition (dealing with the pur-
pose Of the economic information) could be considered either as
an independent statement or as a part Of the "communication,"
since meaningful communication will usually have some purpose.
"Economic information" does not imply any specific theoretical
structure, while "events and relations" imply a definite theo-
retical structure to be explained in Chapter III.
The Accounting Principles Board defined accounting as
a service activity whose function is to provide quantitative
information, primarily financial in nature, about economic
entities that is intended to be useful in making economic
decisions--in making reasoned choices among alternative
courses Of actIon.2
This definition is basically similar TO that of The American
Accounting Association definition, with the exception that the
concept of the entity is introduced.
The elements Of These four definitions are also compared
in Exhibit 1. The proposed definition does not delimit accounting

or differentiate accounting from other disciplines as much as the

other definitions. But, rather, it shows a general framework inTo

 

23American Accounting Association, A Statement Of Basic
Accounting Theory (Evanston, III., 1966), p. I.

 

 

24Accounting Principles Board, "Basic Concepts and Ac-
counting Principles Underlying Financial Statements of Business
Enterprises," Statement Of the Accounting_Principles Board NO. 4
(New York: AICPAL 1970) par. 40.

 

17

 

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18
which accounting must fit with appropriate specification. Note
also that In the prOposed definition, the purpose is left Open To
specification. The other Three definitions all include a statement
Of the purpose Of accounting. The distinguishing feature of the
new definition is its generality and commitment to a simple
Theoretical structure based on events and relations. The defi—
nition should thus be considered in the light of the following

two chapters.

ROOTS OF AN EVENT-RELATION APPROACH

Accounting literature contains numerous references T0
events or Terms used synonymously, such as occurrence, phenomenon,
transaction, flow, activity, or happening. In none of Those places
has the term been used as a fundamental term upon which a whole
Theory was built. But the extensive and widespread use of the term
gives support To The attempt To use it as a basic term.

The beginning of a strong emphasis on events starts with

VaTter's The Fund Theony Of Accounting, Some quotations from that

 

work follow:

Accounting, like other quantitative methodologies, is based
upon a set of Eremises as T0 relations between past and
future events. 5

The notion of a unit of business is but a means Of specifying
the area of attention--a delimited and prescribed set of activi-
Ties which give rise to the kinds of data with which accounting
is to deal. 6

 

25William VaTter, The Fund Theory of Accounting and Its
Igmlications for Financial Reports (Chicago: The University of
Chicago Press, 1947), p. 5.

26Ibid., p. 10.

19

The fund was a means of limiting the area of attention by
defining The group of activities or operations with which
any set of accounting records is concerned.27

There is good reason to argue That the accountant should
confine himself to the reporting of business events, without
attempting to report in a single figure (or a set of calcula-
tions directed at that single figure) the final result of
Operations.28

In his book Vatter also gives strong emphasis to The service
concept of assets. Assets are viewed as "embodiments of future
want satisfaction in the form of service potentials."29 Without
doing violence To the Text, one could replace "service" by "event"
in most places. But the term "event" has a more neutral connotation
Than The Term "service", which usually indicates something favor-
able.

More Than twenty years later, Sorter attempts To provide
an approach to accounting Theory by using some of Vatter's con-
cepts. He suggests

that The purpose of accounting is To provide information
about relevant economic events That might be useful in a
variety of possible decision models. . . . Instead of pro—
ducing input values for unknown and perhaps unknowable
decision models directly, accounting provides information
about relevant economic events that allows individual users
To generate their own input values for their own decision
models. In other words, given the state of The arts, less
rather than more aggregation is appropriate and the user,
rather Than the accountant, must aggregate, assign weights
and values To the data consistent with his forecasts and

utility functions.30

Sorter's article explores the events viewpoint from the standpoint

 

27lbid., p. 22.
281bid., p. 38.
29IpId., p. I7.

30George H. Sorter, "An 'EvenTs' Approach to Basic Account-
ing Theory," "The Accounting Review, XLIV (January, 1969), p. 13.

 

20

of having less aggregation in The financial statements. The paper
indicates how the events vieWpoinT affects the interpretation of
the balance sheet, the income statement (statement of operating
events), and The funds statement (statement of financial and
investment events). For example, the balance sheet is viewed

not as a value statement nor as a statement of financial

position but rather as an indirect communication of all

accounting events that gave occurred since the inception

of the accounTIng unIT.
Sorter's article "represents only a rough and underdeveloped
first approach toward a new orientation for accounting Theory."32
This study will attempt to put some of the ideas of The events
approach on a firmer, more organized, and more general basis.

In response to Sorter's article, Johnson tries to provide

a basic framework for developing The events approach. After
discussing four Terms used in theory (observational, inferential,
constructual, and Theoretical), he notes That The events Theory
prefers observational Terms. Accordingly he defines event to mean

"feasible observation of specified characteristics of an action

with regard to which a geporter could say,_]l foresaw that and saw

 

it happen myself.'"33 Of key importance here is predictability

 

and verifiability. In fact, Johnson poses the criterion of The best
forecast as a means of evaluating alternative accounting procedures.
He also discusses some of the problems encountered in setting up

such a criterion. He summarizes his events theory as follows:

 

 

33Orace Johnson, "Toward an 'EvenTs' Theory of Accounting,"
The Accounting Review, XLV (October, 1970), pp. 643-44.

21

In order for interested persons (stockholders, employees,

managers, suppliers, customers, government agencies, and

charitable institutions) to better forecast the future of

social organizations (households, businesses, governments,

and philanthropies) The most relevant attributes (charac-

teristics) of the crucial events (internal, environmental,

and transactional) which affect The organization are aggre-

gated (Temporally and sectionally) for periodic publication

free of inferential bias.
The framework of his theory is also stated as a set of equations
that here will be presented in diagram form as Figure 2. In
summary, Johnson presents an overview of The task to be accomplished
by the events theory, but he does not try to develop an integrated
Theory. And his emphasis upon the importance of forecasts leads To
the neglect of other important relations That will be discussed

in This study.

Statement of The AccountinggPrinciples Board No. 4 empha-

 

sizes events throughout. Events change resources and obligations

of the entity and are classified into external events (exchanges,
nonreciprocal transfers, and other external events) and internal
events (production and casualties).35 This classification scheme

of events is the basis for presentation of generally accepted ac-
counting principles of selection and measurement. Other than this
classification scheme and the extensive use of the Term, There is

no evidence That "event" is used as an underlying theme To which the
other concepts are related. But its prominent use throughout gives

support To The search for a theory in which "event" is a basic term.

 

34Ipid., p. 650.

35Accounting Principles Board, "The Basic Concepts and
Accounting Principles Underlying Financial Statements of Business
Enterprises," Statement of the AccountinggPrincjples Board No. 4
(New York: AICPA, 1970), par. 62.

22

 

 
    

primitive

  
  
 

 

 

 

  

i
0 events
accounting
- chan es
. f t' prIor .. . 9
S In orma Ion u m

utilities

  

system utilities

 

 

 

 

   
 

  
  

  

other
variables
affecting
utility
change

  

events to
be reported
or processed
further

   
       
   
     

 

 

 

 

 

 

 

  
 

transmitted
n mmunicatio

  

subsequent
reality

 

 

 
  
 

   
 

  

real world
interventions
preventing
perfect imple-
mentation

  

barriers
to perfect
perception

   
 

   
          
   

 

 

 

 

 

   
 

  
 

  

non-

  
 

accountin

      
 

       

D data as other accounting action
inputs to f a inputs to decisions
forecasts decision

 
     

makino

 

 

 

 

 

 

 

forecasts

 

 

 

 

 

 

 

¢ 7U 0.

FIGURE 2

 

FRAMEWORK OF JOHNSON'S EVENTS THEORY OF ACCOUNTING

23

In An lnqpiry Into the Nature of Accounting, Goldberg

 

considers events (occurrences) as the atoms of accounting and

36

ventures (a series of events) as its molecules. Like Sorter

Goldberg also views the financial statements as summaries of

events.37

SUMMARY

An examination of The most general nature of accounting
provides the background for a broad definition of accounting
based on a simple theoretical structure involving events and re-
lations. Three other definitions are shown To be special cases
of the more general one. The Term "event" has been used casually
by a number of different accounting writers but does not appear
to have been used as The basis for a general and formal frame-
work by any of them. The widespread use of The term supports the
attempt To provide a general definition of "event" and then use it
in a formal way as a fundamental term in a meTaTheory. This is

The task of Chapter III.

 

 

36Louis Goldberg, An Inquiry Into The Nature of Accounting

 

(American Accounting Association, 1965), p. 98

37Ipid., p. 87.

 

CHAPTER III

THE THEORY: SYNTACTICS

INTRODUCTION

 

This chapter discusses The pure theory, the syntactics.
No real world references are given and the Terms in the syntactics
have no other meaning than that given by definition. A Three-
dimensional array consisting of event descriptions is defined To
be a sequential memory. Different kinds of relations on This
sequential memory are then considered. And an entity is defined
to consist of a set of events (sequential memory) and a set of
relations on those events. Because of the generality searched
for, examples are not given for most of The relations. It wil
be noted that the mathematical concepts used are few in number and
low in complexity. The concepts were kept few and simple to
insure That only the essential skeleton of the syntactics be
exposed. But the concepts used can form the basis for a fuller
and more complete future development. A nonmathematical com-
mentary on some of the concepts of the syntactics follows at the
end of the chapter. Those readers who wish an intuitive picture of

The syntactics should read That commentary first.

24

25

MATHEMATICAL PRESUPPOSITION838

 

 

Definition 1: Given n sets, X1, X2, . . . , Xn’ the Cartesian product

of the n sets is the set

{(x ,x ,x ) | x e X x e X , . . . , x s X } .
1 n n

2’ 1 I’ 2 2 n

Definition 2: A relation is a subset of a Cartesian product.

Definition 3: A binary relation on a set X is a subset of the

 

Cartesian product of X with itself.
Definition 4: A binary relation R, defined on The set X, is an

equivalence relation on X if

 

i) (a,a) e R for all a in X. (Reflexive)
ii) (a,b) c R ="’(b,a) s R for all a and b in X. (Symmetric)

iii) (a,b) c R and (b,c) s R ==(a,c) e R for all a, b, and c in X.
(Transitive)

Definition 5: Given an equivalence relation R' on a set X and an

element b of X, The equivalence class of b is the set

 

as = {a | (a,b) s R'} .

Definition 6: An order relation is a Transitive binary relation

 

which is not an equivalence relation.
Definition 7: Given an order relation R" on a set X and an element

b of X, the order class of b is the set

 

R3 = {a | (a,b)c R"} .

Definition 8: A partition of a set A is a set of subsets of A,

A ,A , . . . ,A , such that
1 2 n

i) Ail? Aj = O for any i # j and

ii) AIIJ AZIJ . . .IJ An = A.

 

38Similar definitions in a purely mathematical context can
be found in W. E. Deskins, Abstract Algebra (New York: The MacMilIan
Company, 1964), pp. 10-13 and in H. L. Royden, Real Analysis
(2d ed.; New York: The MacMilIan Company, 1968), pp. 7, 22.

 

 

26
EVENTS AND THE SEQUENTIAL MEMORY

3
Definition 9: The Cartesian product of p given sets,

D1,D

 

2, . . . ,Dp, is called the description space. And each
Di is called a description characteristic.
Definition 10: Let a sequential memory, E, be a finite three-

dimensional array with elements evtd’ where

 

v = 1, 2, . . . , m (event series index)
t = I, 2, . . . , n (sequence index)
d = 1, 2, . . . , p (description index)
and
evtd s Dd.4O
Definition 11: An element of E, evtd’ is called an event description.
Definition 12: The seT
evt = (evtl’evt2’evt3' ' ' ' ’evip)

is called an event if it is non-empty.
Definition 13: The set

e = (e e e . . . e
V V1, V2, V3, : vn)

is called an event series.

 

Definition 14: The set

e.t = ( )

e]+,e2+,e3+, . . . 'emt

is called a Time instance.

 

Definition 15: A specific event series in E, ec, is said To be a

minimal Time index on E if for all t, eC+ # O.

 

39For a special application, The p sets will be given by a
set of description hypotheses. (See Chapter IV.) Usually The sets

will be numbers That are assigned to different characteristics of
an event, but they need not be limited to numbers.
40

For simplicity, the reader may consider the case of p = 1,

which reduces E to an m x n matrix with elements e , where v is The
, , vt

row Index and t The column Index.

27
Definition 16: The set consisting of E, a sequential memory,
and R, a set of relations on E, is called an entity and is

denoted by (E,R).

EQUIVALENCE RELATIONS

 

 

Definition 17: Two events evt and e are said to be contemporaneous
wu

with respect to e (any event series) if

k

i) t = u, or

II) t < u and ek,++1 = ek,t+2 = . . . = eku = O, or
III) t > u and ek,u+l = ek,u+2 = . . . = ek+ = O.
THEOREM 1: The relation C "__js contemporaneous with ___with

k)

respect to ek" is an equivalence relation on the sequential memory,

E, on which Ck is defined.

Proof: The first Two properties of equivalence relations, reflexivity
and symmetry, are seen To hold by direct application of the definition,

since (ev+,eV+) s C then (ewu,e ) c C .

. .f
by (I), and I (eV vt k

k T’ewu) E Ck

To prove transitivity it must be demonstrated that if

(evt’ewu) e Ck and (ewu’exy) e Ck’ then (ev+,exy) c Ck.

Consider e . Let
wu

I = '
T mIn(T), such that (ekt'ewu) E Ck
and let
t" = max(t), such that (ekt’ewu) 6 Ck'
Then it follows That t"S u, since (e ,e ) 8 C and That t" 2 u
ku wu k

for the same reason. Therefore T' S u S t“ and

ek,t'+1 - eh,“+2 = . . . = ek,T" = 0.

Hence any Two events with sequence indices in The interval [t',tfl

are contemporaneous with respect to e , and any event contemporaneous

k

28

with ew with respect to e must have a sequence index in the
u

k
interval [t',t"]. Then

(ev+,ewu) g Ck implies that T' g t g T"
and

(e ) 6 0k implies That +' S y S t".

wu’exy
Both t and y lie in The same interval and Therefore (evt’exy) s Ck’
which demonstrates transitivity.

With every equivalence relation on a sequential memory, E,
is associated a partition of E, and with every partition of E is

associated an equivalence relation.41

The concept of classification
which results in a partition of a set is Therefore intimately as-
sociated with equivalence relations. The concept of event series
partitions The sequential memory by row index. With this partition
is associated The equivalence relation

"__js a member of The same event series as _;F

The two equivalence relations, contemporaneity and event series,

thus each uniquely partition E.

ORDER RELATIONS

 

Definition 18: For any two events, e and e ,

 

 

vt wu

if T < u, 9 occurred before e ; , (order relation)
vt -—--- Wu

. _ . . .42

If t — u, ev+ occurred SImultaneousiy_WIth ewu’

if t > u, e occurred after e . (order relation)
vt --- wu

41

For a proof of these statements in general, see Deskins,
op. ciT., pp. 12, 13.

2This is an equivalence relation. Note That if two events

are simultaneous, They are also contemporaneous.

29
With each of the above order relations, There is associated a set
of order classes. Other order classes on events may be defined
as follows. Let the function 0 select a single numerical charac-

teristic for any event in E, 0(e ) = evtd*’ where d* is The

vt

characteristic selected. For any given event ewu’ E is partitioned
into two order classes and one equivalence class:

i) the set of all e such That 0(evt) < 0(e ),

vt

II) the set of all evt such that 0(ev+) > 0(ewu). and

WU

iii) the set of all e such that 0(e ) = 0(e ).
vt vT wu
The order relations thus play a role in the development of a time

concept and also in the comparison of events with one another.

DEFINITIONAL RELATIONS

 

The relations in this section define one or more new
characteristics of a single event or group of events given another
single event or group of events. This kind of relation is usually
called an operation, buT for greater generality the term "relation"
will continue To be used.

Definition 19: A relation R is called a definitional relation

 

if for every r e R, r = (r1,r2, . . . ,rg), r is partitioned into
Two non-intersecting sets, r = rI LJrD and rI Fer = 0, such That
given rl a rule exists that uniquely determines rD. The events in

rI are called independent events and the events in rD are called

dependent events.

 

Three kinds of definitional relations are discussed. The
most important is the fundamental relation which defines rD by means
of a special kind of summation. Arithmetical relations define

rD by the common arithmetical operations. And finally, statistical

30
relations Treat the elements of rl as random variables in developing

familiar statistical results.

THE FUNDAMENTAL RELATION

The fundamental relation uses the operation of summation
on a subset of E to define a state. The method of selecting The
subset must be given by the semantics (see Chapter IV) and will
therefore appear as arbitrary here. The selection proceeds by
choosing certain of The indices for each of the Three dimensions
of E:

the set of all v, such that v 5 Ni or v 5 Pi;

the set of all t, such that a S T s w ; and

the single numerical characteristic d d*.

As before C(e ) = e In other words, The function 0 selects

vt vtd*°
the characteristic d*. The specification of the values for Ni’ Pi’
a,w, and d* must be given by The semantics.

Definition 20: The fundamental relation defines a state, Sw i’ as
a,

(.0 (L)
8:,i = VEPi fga 0(evt) - VENi +Za0(eV+).
Theorem 2: For a s y s w , 3:,i = 5:,i - s;::.
Proof: Since addition is associative
Y‘1 Y-1
83" = ngi +Ea0(eV+) - VENi +Za0(eVT)
0.) (.0
+ VEP. +£yo<eVT) - VZNI +5Y0(evT)
I
or Sm . = ST"1 + Sm .
0,. 0,] Y,l
Rearranging S” = S” _ SY-i.

y;1 0,1 a:'

31
ARITHMETICAL RELATIONS
These relations have three elements. Two independent event
descriptions are operated on to yield a result, the Third element.
The arithmetical relations are introduced to allow the four ordinary
arithmetical operations on the sequential memory. The set of inde-
pendent variables, rl, consists of two evenT descriptions and rD

consists of the sum, difference, product, or quotient of the two

event descriptions.

STATISTICAL RELATIONS

Statistical relations define new quantities by means of
The commonly accepted tools of means, standard deviations, cor-
relation coefficients, and many others. For these purposes, some
event descriptions from the sequential memory are selected and
treated as random variables. Thus r consists of a set of event

descriptions and r is the result of some statistical operation

D

on those event descriptions.

TRANSFER RELATIONS

 

Transfer relations seek to identify groups of independent
and dependent variables and to find the relationship that exists

between them. In definitional relations rD was determined strictly

by the r by means of a given rule. In Transfer relations rD and

rI are boTh given, and the rule that explains r by means of rl

D
is sought.

32

SPECIAL CASES OF THE THEORY

 

COUNTING AND ACCUMULATION

Definition 21: For any event series, e , define a new event series,
v

eJ, such That

ejtl 0 If evt = O

and

eth = 1 If evt ¢ O.

The count of eV for The interval from a To w is Then

w
tgaejflo
Note that the count is a special case of the fundamental relation,
where Pi = j, Ni = O, and d* = 1.

Definition 22: An accumulation of event series eV for the interval

 

from a to w on characteristic d* is

U)
Tzaevtd*'

Note that This is again a special case of the fundamental relation.

SPECIAL TIME INDICES

In The fundamental relation the summation on the t-index
is usually from a specified cxto a specifiedcp. These might be
defined in Terms of two specified values, Ca and Cw , of a count
or accumulation, ST,k’ as

a = min(t), such that 3+ - Ca; and w = max(T), such that S1L = C .43

bk— 1,k w

 

43For an interesting case where Time was defined in terms of
an accumulation, see Arthur C. Nichols and Dennis E. Grawoig,
"Accounting Reports with Time as a Variable," The Accounting Review,
XLIII (October, 1968), pp. 631-39.

33
CLOSED SYSTEMS
Definition 23: Given an entity, (E,R), a subentity, (EC,RC),
consisting of Ec’ a subset of The sequential memory E, and Ro’

a subset of the relations R, is called a simple closed system if

 

The following hold:

i) RC contains only a finite number, k, of fundamental

relations, 8“ ., i = 1,2, . . . ,k.

a,I
ii) EC contains no more elements Than necessary To define
all 8” ..
(1,1
iii) For any eV e Ec’ v 6 Ni and v e Pj for exactly one i

and exactly one j. (Duality)

iv) For fixed a and any w > a

k
Z S” i = C, some constant. (Conservation law)
i=1 “’

Parts (iii) and (iv) of The conditions are actually interdependent
as shown by The following theorem.

Theorem 3: In a simple closed system a conservation law implies
duality and duality implies a conservation law.

Proof: By the conservation law
k k
2 5”“! = f s“ . = 0.
i=1 8" i=1 0"

But this can be written
k (0'1

Hi I e..- I ”Elem

1:1 VePi T=a VENi T=a

(I “few I ieVn-‘M

1 vePi T=a veNi t=a

II
HM}—

 

44For simplicity of notation °(evt) is denoted by evt here
and in the following pages.

34

The right side contains exactly the left side plus events with

index t = w and can therefore be written

k w k w_1 k
S . - S . + e — e .
1:1 “" iZI 0" I;1[ng. Vw véNi V“)
I
But since
k k _1
2 5w = SS i’ it follows That

i=1 a" i=1 ’

Assume now That at w There is only one event eV . By (i) and (ii)

(A)
v is a member of some PJ 0r Ni' But for the above equality to hold,
v must be a member of both some Pj and Ni' Hence, duality has been

shown. The above argument could be reversed to show The second part

of the Theorem, That duality implies a conservation law.

A SPECIAL CASE OF THE FUNDAMENTAL RELATION

In closed systems a special case of the fundamental relation
occurs when The events with index in Ni are defined by the events
that have index in Pi' Consider state i

w w

w

S,=ZXe-Ze,

1" vePi t=1 Vi t=l “1

where for simplicity N. has only one member, h, which in turn is
I

defined by the event series with index in Pi

where 0 s fvst s I and a]; ffvst s I.

35

Then
A I i i 2 I
S , = e - (e ° f )
1" vePi T=I V+ t=1 vePi 5:1 VS VST
w w 'I‘
:2. 2.21% - 2 2% ° w
i
(L) (L) (L)
= VEP (tEIeVJr - SET t-Z—s(evs fvs+)).
i

In the last double summation interchange s and T to get

(L) (A) 0.) (U
81" : VEP (+EIeVT - tgl S§+(ev+ . foS))
i
(.0 (L)
= VZP tEI[ev+ — s§T(eVT fVTS))
i
2 i i
= e (I - f ,
VePi t—l VT s=t VTS

the fraction of eV+
that has flowed out

the fraction of evt
That remains

This is a form that occurs frequently in historical cost accounting.

It is a more general form of ljiri's value allocation rule.45

SUMMARY

The entity, consisting of a sequential memory and a set of
relations, forms the basis for The Theory. And this chapter outlined
a possible structure for the entity. The sequential memory was
described as a Three-dimensional array consisting of event descriptions.

Relations were categorized as equivalence relations, order relations,

 

5Yuji Ijiri, The Foundations of Accounting Measurement
(Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1967), pp. 90-98.

36
definitional relations, and transfer relations. The fundamental
relation, a definitional relation which defines states, played a key
role in the development. The meanings of The symbols used in this
chapter are given in Exhibit 2. And a nonmathematical commentary
attempts to give the reader an intuitive feeling for the concepts in
this chapter.

The theory, as presented in this chapter, is distinguished by
its brevity and its sterility by itself. Mathematically, the Theory
does not have much significance, because iT extracts from mathematics
only those definitions and relations that are absolutely necessary in
The meTaTheory. But The significance of The concepts chosen will be

seen in Chapter IV.

NONMATHEMATICAL COMMENTARY

 

This chapter has made extensive use of some simple mathe-
matical Tools. Any attempt To give a nonmathematical representa-
tion of a mathematical system will meet with limitations. The
following discussion is therefore not intended as a complete
exposition of the entire chapter. But, rather, it is an attempt to
give the reader an intuitive feeling for the most important concepts
in the chapter. This discussion, therefore, does not add any new
material or interpretations but rather amplifies The mathematical
material.

The chapter is built around The concept of a sequential

memory. A sequential memory may be visualized as a cabinet

 

containing m x n drawers in m rows and n columns. Each drawer
has p compartments. Each compartment may carry in it a particular

kind of event description characteristic. For example, the first

 

37
compartment might always indicate weight in pounds. Altogether
there are p classes of description characteristics, and each
characteristic has many possible values. For example, weight in
pounds may range from zero to a very high number. The descrip-
tion characteristics may be combined in many different ways. All
the different possible combinations of descriptions are called The

description space. If some of the compartments in a drawer contain

 

a description, that drawer is then called an event. A whole row

of drawers (events) is called an event series. A whole column of

 

drawers (events) is called a Time instance. The columns are

 

numbered from left to right, so if one drawer is to the left of
another, it occurred before The second drawer. Two drawers in

The same column are said To be simultaneous. Another time related

 

concept is that of contemporaneity. Any two drawers are said to

be contemporaneous with respect to a given row of drawers, if all

 

the drawers in the given row in The columns between The Two other
drawers are empty. In other words, two events are contemporaneous
with respect To a given row, if no event occurred in that row between
the two events.

The second key concept is that of a relation. A relation
is basically a string of elements related in a certain way. The
elements in The string (2,3,5) are related by the rule: add the
first two elements To get the third element. A relation has always
at least two elements, but it may also have many more elements.
In The chapter, the most important relation is the fundamental

relation. The fundamental relation adds and subtracts a single

 

characteristic of a group of events to obtain a state (e.g. cash

38
balance). IT is basically a generalization of The common accounting
concept of an account. Three Things must be specified To use the
fundamental relation. First, the event series That are to be
added (e.g., cash receipts) and those that are to be subtracted
(e.g., cash disbursements) must be specified. Then The time span
for which The combination is to be carried out must be indicated
by giving the times (column numbers) of the first and the last
events in The group. Finally, for each event series The part of
the description To be used for the addition and subtraction must be
specified (e.g., monetary value).

The Two key concepts are then combined To form an entity. A
sequential memory and a set of relaTions on That sequential memory
are called an entity. Using the previous analogy, The entity is
thus the cabinet of drawers and a way in which some of The contents

of some of the drawers are related To one another.

vtd

'J'

UOO<+110337¢‘“
7?

R"

39
EXHIBIT 2

LIST OF SYMBOLS IN THE SYNTACTICS

index of description characteristic of E.
an event description.

an event.
an event series.
a time instance.

a specific member of Ni'

index of a state.

The number of states in a closed system.

the number of event series in E.

the number of time instances in E.

number of classes of description characteristics in E.
a set of independent variables in a relation,

a set of dependent variables in a relation.

sequence index or Time instance index of E.
index of event series of E.
The relation "__js contemporaneous with ___with respect To ek."

a specified value of a count or an accumulation,

the description space, the Cartesian product of The description

characteristics, Di'

a sequential memory.

an entity.

a set of event series indices, The events of which are subtracted
in The fundamental relation.

a set of event series indices, The events of which are added

in the fundamental relation.

any relation or a set of relations.

an equivalence relation.

the equivalence class of b.

an order relation.
the order class of b.

state i defined by a fundamental relation over the interval
from a to w.

The sequence index at which summation begins in a fundamental
relation.

the sequence index beTween a and w.

the sequence index at which summation ends in a fundamental
relation.

The function that for a particular event evt selects a
specific event description ev+d*.

CHAPTER IV

THE THEORY: SEMANTICS

INTRODUCTION

 

In mathematics, theory by itself is of interest. But as has
been recognized by mathematicians, even the most profound results are
mere Tautologies. Since accounting is an applied discipline, having
its justification in the real world and in particular within the
business world, where an activity is presumably undertaken only if the
benefits provided are greater than the costs, accounting Theory must go
beyond syntactics and also deal with semantics. AT the level discussed
in Chapter III, the syntactics itself does not have much content apart
from the great number of relations it could borrow from mathematics
and statistics. The real significance of accounting Theory lies in
semantics, namely the connection of the theory with the real world.

The content of semantics will largely be determined by the
syntactics. Some of the Terms in the pure Theory must be given a
meaning in the real world. It must be possible To Translate The real
world into theoretical Terms and then to interpret The results of the
theory back inTo real world terms again. When one considers syntactics
and semantics together, a constraint is placed on the syntactics by
the semantics. For example, certain theoretical relations may be re-
jected either as irrelevant or as not corresponding to reality.

40

41

As noted before, the meTaTheory will only provide The frame-
work within which individual theories may be formed. This chapter
continues the very general development of the meTaTheory. Within
the framework created by the meTaTheory, individual theories will
give accounting its specific character.

The word "hypothesis" will be used frequently in This
chapter. In the search for generality in accounting, many different
Terms have been used for statements of various degrees of generality.
Some of these are: principles, standards, rules, axioms, assumptions,
and hypotheses. To avoid entering The argument as to what kinds of
statements are appropriate under each heading, This sTudy will use
the word "hypothesis" To indicate any general statement. Some
hypotheses may have only empirical content, others only Theoretical
content, and still others a mixture of boTh empirical and theoretical
content. The hypotheses assumed to exist in this chapter are state-
ments about the connection of the real world and The syntactics of
Chapter III.46

The choice of hypotheses in this chapter is not arbitrary
but rather follows as a logical consequence of the structure of
The entity as given in Chapter III. The hypotheses can be divided
into Two parts, The structural hypotheses and the control hypo-

Theses. The first four sets of hypotheses (structural) follow

 

46This use of hypotheses is similar to that of Mattessich.
See Richard Mattessich, Accounting and Analytical Methods (Home-
wood, lII.: Richard D. Irwin, Inc., 1964), chaps. 2 and 7.
Mattessich distinguishes between scientific hypotheses and empirical
hypotheses on The basis of their degree of validity. Scientific
hypotheses have the highest level of validity, while empirical
hypotheses may not be capable of proof but are accepted because
they are somehow superior to competing hypotheses.

 

42
directly from The definition of the sequential memory. And of
these, three correspond directly to The Three dimensions of the
sequential memory. Since the hypotheses are interdependent,
the order in which They are presented is not to be interpreted
as having significance. The last Three sets of hypotheses (control)
are necessary to assure that The results of The system are sig-

nificant.

SELECTION

There exists a set of hypotheses for The selection of
the events and relations That make up The entity.

The purpose of these hypotheses is to limit the field of attention.
They actually consist of two sets, one dealing with selection of
events and The other with the selection of relations. There are
so many events and relations as to make The selection process
extremely difficult. But some factors aid in The selection
process. The limitations possessed by man and by any measuring
instruments to a large degree determine the events that are
selected. Furthermore, the larger The entity in Terms of events,
the larger is The cost of gathering and storing This information.
The selection process may be governed by either the wishes of
decision makers, the specifications of decision theories, or
else by requirements of law, contract, or desire for control.

In selecting relations iT is desirable To have all the
events that are necessary and sufficient factors (both dependent
and independent variables) in the relations included in the entity.
In all but the simplest relations, this will be impossible. Even

scientists in their laboratories have extreme difficulty in

43
removing unwanted influences from phenomena to be studied. The
selection process can at best hope to include those factors That
are most significant to a specific relation. With respect To
relations, the selection hypotheses also serve the role of spe-
cification. In The fundamental relation, for example, Ni’ Pi’
a an , and 0 must be specified by some selection hypothesis.
The selection process is not carried out arbitrarily, but rather

it is guided by the evaluation hypotheses to be explained later.

DESCRIPTION

 

There exists a set of hypotheses for the definition and ‘
description of the events of the entity.

The term "event" does not necessarily carry the conventional
meaning in this study. Rather it is subject To definition. The
description hypotheses will among other things define the nature
of events. Conventionally, event carries with it the idea of a
happening, occurrence, phenomena, change, or maybe even a flow.
All These conventional meanings are possible for The term as used
here. But "event" is not restricted to these. An event is said
to have occurred if a certain set of conditions are met. The
conditions are specified by the description hypotheses.

Once an event has been defined, its description is usually
intimately connected with some kind of description or measurement
apparatus. The event description is generally The output of some
such apparatus. The apparatus may be a procedure for description
or it may involve some measurement instruments.

Event description also involves the specification of the

method and the media of recording events. The term "event description"

44
rather Than "measurement" is used because of its greater generality.
Recall from Chapter III That each event is described by a member of
a Cartesian product. The description hypotheses specify the sets
That make up the Cartesian product (description space) and how a
particular event is To be associated with a subset of That product.
The descriptions may include both measurements and non-quantitative

descriptions.

 

CLASSIFICATION

There exists a set of hypotheses for the initial classi-
fication of events into event series.

These hypotheses lean heavily upon the description hypotheses.

All The events That fit a certain set of descriptions are said to
belong to the same event series. These hypotheses must also assume
That some events are similar or repeatable. Otherwise it would

not be possible to identify other Than unique and individual events.
Many times The classification will involve The description ap-
paratus. And an event series will be the set of all outputs from
a particular description apparatus. In other cases, the outputs
from a particular apparatus will be grouped into different classes

based on the descriptions of The individual events.

ORDERING

There exists a set of hypotheses for the initial sequential
ordering of events.

The hypotheses involve the element of Time. The syntactics assumes
that events occur in some order as indicated by The sequence index
(the second subscript of evf). The ordering hypotheses will indi-

cate how the actual events are To be assigned a sequence index.

45
Once an initial order has been imposed on the events, it is pos-
sible to more clearly define time. Any of the event series may
be picked as a time index, and all other events may be related
to that series by means of the equivalence relation "is contem-
poraneous with". The new Time index will be given by the count
of the event series used as a Time index.

Since presumably events will keep occurring, the sequence
index of the sequential memory is continually increasing. All
events in the sequential memory are historical or past events.
But decision making is concerned with The future. Therefore,
prediction seeks by means of known relations on the sequential
memory to say something about events with a sequence index greater
Than the present one.

Some fundamental problems arise in sequentially ordering
events. The question of when to recognize an event must be
addressed by the ordering hypotheses. Many events can be sub-
divided into smaller subevents. Should an event be recorded only
when all the subevents have occurred or when a substantial number
of the subevents have occurred? The problem might be solved by
recording the individual subevents. But each of These could
probably again be subdivided.

Many Times there is also a considerable delay between the
actual occurrence of an event and the Time at which it reaches the
sequential memory. This delay may lead to events appearing in

different order than the order in which they occurred. The
ordering hypotheses must deal with these problems. One solution

to The problem of delay in recording lies in having the event

46
series which is to serve as a Time index available for immediate
association with The event when it occurs. In this case the
description of the event contains a Time index. When the record
of the event arrives in The sequential memory, it is stored and
after a Time delay, L, the events can be sorted on the time
index and correctly represent The order of occurrence. Assume
that the lags (the time between occurrence and recording) have
a probability distribution f(t) with the probability of a lag
greater than L equal to P(lag>L) =cEf(t). As L increases,
the probability of an event being eIZTEded from the sequential
memory decreases. But The memory is only reliabile for P(lag>L)
being less than some specified number. For some systems, the
problems created by lags may be minimal, in others they may be

critical. The question of reliability is also dealt with in The

next set of hypotheses.

ERROR

There exists a set of hypotheses regarding The reliability
and error content of the enTiTy.

A sequential memory is said to be reliable if all the events that
occur in a given event series are recorded and no other events are
recorded as part of that series. If not all the events in a series
are recorded, a type I error occurs. If some extraneous event is
recorded as part of the series, a type II error occurs. The same

is true as in statistics, where an attempt To reduce the probability
of one kind of error will usually increase the probability of the
other kind of error.

Some error usually accompanies every description. The

47
exact nature of The error may not be known, but it should be
possible to have hypotheses that state the range and distribution
of errors in the descriptions. When the original events are ag-
gregated and related To other events, there should also be a set
of hypotheses specifying how The original description errors
affect The subsequent aggregation or other relations. There may
also be errors in classifying and ordering of events. The error
hypotheses must deal with each of the three dimensions of the

sequential memory.

VERIFICATION

 

There exists a set of hypotheses for identifying the
relations That are valid.

The error hypotheses deal primarily with events, while the veri-

fication hypotheses deal primarily with relations. The process

of ascertaining The validity of a relation is called verification.

What is meant by validity and how the verification is To be

carried out must be specified in The verification hypotheses.

In general, verification may take place when at least Two sets

of events are consistent with The relation being verified.
Verification is especially important for transfer relations.

A transfer relation may be written as rD = f(r ), where f is the

l

rule applied To the independent variables to obtain the dependent
variables. The rule, T, was obtained on the basis of a set of

observed values of rI and r0. When applied to another set of

values for The independent variables, The rule predicts values

for rD. These predicted values will usually be different from

the observed values of rD. If these differences are

48
"sufficiently small" over a "sufficiently large" Set of values
of rD, the rule, f, has been verified. The role of The verifi-
cation hypotheses is to specify The meaning of "sufficiently
small" and "sufficiently large".
In other cases verification is attained by arriving at The
same rD using two different relations. This may sometimes be

spoken of as verification of rD instead of as verification of f.

EVALUATION

 

There exists a set of hypotheses for the evaluation of

how well particular events, relations, and other hypo-

theses fulfill a given set of goals.
These hypotheses serve as the crucial element both in the original
formulation of all the other hypotheses and in their reformulation
to adapt to a changing environment. To avoid circularity, the
goals against which the evaluation Takes place must not be derived
from the rest of the Theory but must rather be given exogenously.
The goals may be specified by other theories or may be imposed
arbitrarily.

The uniqueness of accounting will be found in the evalua-
Tion hypotheses. For it is the evaluation hypotheses that guide
in The development of the other hypotheses. And The other hypo-
Theses will be consequences of The particular evaluation hypotheses
if these are well formulated. Many other disciplines could have

a similar theoretical structure, but The goals To be attained make

accounting unique.

49

SUMMARY

This chapter has discussed semantics as two sets of hypo-
theses. The first set, The structural hypotheses, are necessary
To connect the real world To the sequential memory. The second
set, The control hypotheses, are necessary to assure that the
connections and The relations are meaningful. In accounting
theory these hypotheses need to be spelled out in detail. This
study being a meTaTheory, the hypotheses were only assumed to exist.
The next chapter will show how these hypotheses Together with The
syntactics will be able to explain two previous, generalized

approaches To accounting.

CHAPTER V

THE THEORY APPLIED

INTRODUCTION

 

There have been notable attempts To present a general
foundation for accounting in the past. Of these, two are chosen
for comparison with the evenT-relation meTaTheory because of their
conciseness and formality.

Only when Two theories are formalized or axiomatized can we
properly compare them, because only then are the essentials

upon which a comparison rests jaid bare--only Then do they
possess a definite structure.4

The Two Theories chosen are those of Mattessich48 and Ijiri.49
An examination of the broad outline of Their theories will show
how their terms and assumptions fit into the framework created

by The meTaTheory. By this comparison the meTaTheory is partially

validated. It is hoped that comparison with other theories as they

become more formalized will further validate The meTaTheory. Throughout

 

47Joseph H. Woodger, "The Technique of Theory Construction,"
Foundations of the Unity of Science, Vol. II, ed. Otto Neurath,
Rudolf Carnap, and Charles Morris (Chicago: The University of Chicago
Press, 1970), p. 521.

 

48Richard Mattessich, Accounting and Analytical Methods
(Homewood, 111.: Richard D. Irwin, Inc., 1964).

 

49Yuji Ijiri, "Axioms and Structures of Conventional
Accounting Measurement," The Accounting Review, XL (January, 1965),
pp. 36-53.

 

50

51
this chapter the event—relation meTaTheory will be represented by
the abbreviation ERT.

First, The theories of both Mattessich and Ijiri will be
summarized briefly to indicate their general scope and how They
differ in approach. Next, each of the theories will be presented
in diagram form with comments on how each item relates to The ERT.
Finally, a tabular presentation will show how the two theories fiT

into the ERT framework.

SYNOPSIS OF THE IJlRl AND MATTESSICH THEORIES

 

Ijiri sets out to describe as simply as possible conventional
accounting by an axiomatic system and a seT of measurement rules. By
examining current accounting practice and abstracting therefrom, he
derives a system containing three axioms and three measurement rules.
The axioms are needed first to identify the property seT of a subject,
Then To assign a measure to The various classes of the property set,
and finally to recognize changes in the property set by means of
exchanges. The numbers assigned To the exchanges are specified by
means of The measurement rules, which are developed using the
concept of a basic class. Because of the simplicity of the system
there are a number of notions that it cannot handle: market values,
proprietary investments, capital surplus divisions and reverse
exchanges.50 Subsequent work has modified the theory slightly, but
it remains one of The simplest and most concise explanations of

historical cost accounting.

 

50lbid., pp. 49-50.

52

Mattessich, on The other hand, sets out To provide a more
general framework for explaining accounting. The foundation he
searches for must be valid for both micro-accounting (business,
governmental, household, and managerial) and macro-accounting
(flow of funds, input-output, balance of payments, and national
income). The core of his theory, within which all These accounting
systems can be explained, is a set of ten basic assumptions, the
most important being "entities" and "economic transactions". To
these are added an additional eight assumptions which are needed to
make the theory operational. A concise, seT-theoretical form for
the first ten assumptions will be used for comparison with the ERT.
MaTTessich's system is actually also a meTaTheory within which
numerous specific assumptions may be made. For example, The use
of historical cost is a possible specification in the system but
not an essential part of it.

Neither Mattessich nor Ijiri make a gleg£_distinction
between syntactics and semantics. Many of their statements have
both a_priori and empirical content. As a result no attempt is
made to make a direct comparison with The syntactics of the ERT.
But since the semantics is a consequence of the syntactics, an

indirect comparison with the syntactics also takes place.

COMPARISON WITH IJIRfS THEORY
Ijiri's theory consists of a set of definitions, axioms,
and measurement rules. The definitions and axioms are:
i) A_subiect is any identifiable Thing That is capable
of owning other things.

ll) Objects are any identifiable things That are capable
of being owned by a subject.

The

53

iii) Time is a real variable; a smaller value of time means
an earlier Time and a larger value a later time.

iv) A_physical measure is a non-negative set function that
is defined on a class of objects and all of its subsets such that
it is countably additive, it takes zero on the empty set, and that
two sets of objects in a same class are substitutable if They are
of a same value of the physical measure. A class with such a
function is called a measurable class.

v) An accounting set is a set of objects that may be
partitioned into a countable collection of measurable classes.

vi) Ownership is a well defined relationship between a
subject and objects at a given time by which for any object it
is uniquely determined whether or not the object belongs to The
subject at The given time.

vii) A property set of a subject at time t is a subset of an
accounting set and consists of all objects That belong to the
subject at time t.

viii) An exchange at time t is a phenomenon at Time t which
results in adding a set of incoming objects (all belonging To a
single class) to the property seT A++ and subtracting a set of
outgoing objects from The property set A+_, where t+ 2:t and T“ Z‘t.

 

 

 

 

Axiom of Quantities: There exists an accounting set U, that is,
a seT of objects That may be partitioned inTo a countable collection
of measurable classes.

Axiom of Ownership; The property set A of a given subject at
any Time t can be uniquely determined at That Time or later.

Axiom of Exchanges: For any object That is added to or
subtracted from the property set A , an exchange that has caused
the addition or subtraction of The Object can be uniquely iden-
tified; and all exchanges that have occurred are identifiable,
countable, and can be ordered completely and uniquely according
to The time of their occurrence.5

 

 

 

measurement rules are:

Measure Allocations: Allocate the u-measure of a nonbasic class

 

to the set of outgoing objects of the class and To the set of
remaining objects in proportion To their physical measures.

Measure Imputations: If incoming objects belong To a non-basic

 

class, assign as the u-measure of The incoming objects The sum

of the u-measures of all out-going objects in the exchange.
Increase the u-measure of the class to which The incoming objects
belong by The u-measure of the incoming objects.

Measure Comparison: If the set of incoming objects is empty or
belongs to the basic class, calculate a measure gain (or loss)

 

 

5‘Ibid., p. 42.

 

54

 

 

 

 

(iii)
Time

 

 

(i)
Subject

km
Object

 

 

 

 

Substi-
Tutability

 

 

 

 

 

u-measu re

 

 

 

(vi)
Ownership

 

 

 

 

 

Axiom of
Ownership

 

 

 

 

 

of

At

 

 

 

 

 

Measurement

 

(viiij
Property
Set

At

 

(

 

iv)
Physical

Measure

 

 

 

 

 

Axiom of
Quantities

 

 

 

 

(

 

 

 

 

 

 

Rules

 

 

 

 

 

u-measure
of

At+l

 

(viii)
Exchange

 

v)
Accounting
Set

 

 

 

 

 

 

 

Property
Set

 

 

At+1

 

 

 

FIGURE 3

 

Axiom of
Exchanges

 

 

STRUCTURE OF IJIRI'S ACCOUNTING THEORY

 

55

by subtracting The sum of the u-measures of outgoing objects
from The u-measure of the incoming objects.

The interrelations between the various elements are more
clearly shown in Figure 3. Each box represents a definition or axiom.
Figure 3 indicates how some of the concepts are dependent on other of
the concepts. For example, Time, subject, object, ownership, and the
axiom of ownership are necessary to identify the property set. In
Terms of the ERT, these concepts serve to select the subject matter to
be recorded in The sequential memory. In addition, the assignment of
quantities to the property set is dependent on physical measure, the
axiom of quantities, and the accounting set. The assignment of a
u-measure (or value) to The property set depends further on exchange,
the axiom of exchanges, and the measurement rules. Each class of the
prOperty set is Thus described by a quantity and a u—measure. In
terms of The ERT, the description space has two dimensions: physical
measure and u-measure. And The sequential memory can be identified
with a sequence of property sets. Within the sequential memory, the
event consists of The physical measure and the u-measure of a
particular class of the property set at a particular time.

In ljiri's Theory there is an example of a selection of one
of each of the kinds of relations identified in Chapter III.
Equivalance relations are represented by The concept of substi-
tutability, which is used to classify the objects of the property

set into equivalance classes.53 Order relations are represented by the

 

52Ipid., p.43

53That the relation "__js substitutable for__fl is an
equivalance relation follows from ljiri's statement in a footnote
on page 37 of his article That The relation is reflexive, sym-
metrical, and Transitive.

56
time concept. Note ThaT under the definITion of time, Ijiri includes
The concepts of "earlier" and "later", which are order relations
because they correspond directly To The order relations "less than"
and "greater than". DefiniTionaI relations are implicit in The
measurement rules. Given the u-measure of any class at a given Time,
The u-measure of That class at a later time would be given by adding
all the u-measures assigned to the incoming objects and subtracting
the u-measures assigned to the outgoing objects. Transfer relations
are represented by The exchange. An exchange is a relation between
a set of incoming and a set of outgoing objects, either of which may
be considered The independent variable.

Classification is dealt with in the axiom of quantities and
in the accounting set, in both of which objects are partitioned
(classified) into measurable classes. The hypotheses of ordering
are present both in the axiom of ownership, where the property set
is given a Time subscript, and in The axiom of exchanges, which
assumes that exchanges can be completely ordered by the time of
their occurrence.

The control hypotheses (error, verification, and evaluation)
are not dealt with in ljiri's theory. However, Ijiri deals with

These concepts elsewhere.54

 

54See, for example, ljiri's Two books The Foundations of
Accounting Measurement (Englewood Cliffs, N. J.: Prentice-Hall,
1967), chap. 6 and Management Goals and Accounting for Control
(Amsterdam: North-Holland Publishing Company, 1965).

 

 

57

COMPARISON WITH MATTESSICH'S THEORY

 

The first ten of MaTTessich's assumptions are given in
his Appendix A in mathematical form. The appendix consists of
Twenty-eight items which are summarized in Figure 4 and labeled
as Ai-A28. In Figure 4 the six primitive (undefined) terms are
shown as circles. An examination of the diagram shows that

entity and transaction are the items around which the rest of The

 

theory is built. MaTTessich's Appendix A or the first ten assump-
tions form the fixed part of the theory, wherein assumptions are
very specific. The remaining eight assumptions form the variable
part of the theory, which creates an outline of The kinds of
further specification needed to make the theory fully operational.
These eight assumptions, labeled 11-18, are as follows:

11. Valuation. There exists a set of hypotheses determining
The value assigned to an accounting transaction.

 

12. Realization. There exists a set of hypotheses, specifying
which of the following three mutually exclusive effects are
exercised by a change (in quantity, value, legal status, etc.)
of an entity's economic object(s). Such a change either:

(I) affects the value assigned to the current income of the
entity; or (2) does not affect the owners' equity of this
entity (within the specified period); or (3) affects The
owners' equity without affecting the current income of the
entity.

 

 

13. Classification. There exists a set of hypotheses required
to establish a chart of accounts.

 

14. Datgrlnpyt. There exists a set of hypotheses required To
determine the form of data input and The level of aggregation
for which accounting transactions are to be formulated.

 

15. Duration. There exists a set of hypotheses about the
expected life of The entity (or entities) under consideration,
and The duration of individual accounting periods or subperiods.

 

16. Extension. There exists a set of hypotheses specifying the
empirical conditions under which two or more accounting systems
can be consolidated and extended To a more comprehensive system.

 

58

17. Materiality. There exists a set of hypotheses (criteria)
determining if and when an economic transaction or related
event is To be reflected by an accounting Transaction.

 

 

 

18. Allocation. There exists a set of hypotheses determining
the allocation of an entity's economic objects or flows of
services to subentities and similar categories.

 

 

 

As seen in Figure 4, each of The primitive terms (circles) as
well as the items Al-A3 serve to select and describe transactions.
A sequence of transactions in MaTTessich's theory corresponds to
the sequential memory of the ERT. The descritpion space for the
Transaction (event in the ERT) consists of four elements: a
negative transactor, a positive transactor, a time instance, and
a value. The transactors represent various categories within the
entity, and the pair of transactors used in a transaction indicates
the two categories between which The transaction (flow) Takes place.
The two categories may belong either to different entities or to
The same entity. Numerous kinds of transactions are defined in
A5-A13. The traditional concept of exchange is covered by A7, a

pair of rquited transactions, where one transaction is the con-

 

sideration of the other.
The description hypotheses of The ERT are illustrated by

MaTTessich's (11) Valuation dealing with the assignment of a value

 

to each Transaction, (12) Realization dealing in part with which

 

transactors are to be assigned to a Transaction, and (14) Data Input
dealing with the format of the raw material for recording
transactions. The classification hypotheses are illustrated by

13) Classification. The classification leads to The establishment

 

55Richard Mattessich, Accounting and Analytical Methods
(Homewood, 111.: Richard D. Irwin, Inc., 1964), pp. 42-45. His
comments on each of The assumptions are omitted in the quotation.

 

59

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60

of a chart of accounts, (A15, A16), from which transactors are

 

selected for each accounting Transaction (A17). Since each

 

Transaction contains two transactors, the Transactions may be
classified on the basis of the classification inherent in the chart
of accounts. The ordering hypotheses are illustrated by part of
(17) Materiality which deals with when a transaction is to be
recognized. The concept of errors is briefly touched on in (14) Data
ippgi_using The idea of statistical discrepancy.

The selection of the various categories of relations are
also amply illustrated in MaTTessich's Theory. Equivalence
relations are represented indirectly by the transactors. Each
transactor is a particular equivalence class, which can be connected
to some equivalance relation. Order relations are represented by
the accountipg period (A14), in which subperiods are introduced as
being "smaller" than an accounting period. The time instances
assigned To the Transactions also serve to order them. DefiniTionaI

relations are plentifully illustrated. (15) Duration deals with the

 

length of an accounting period, which is equivalent To the selection
of a particular a and w for a fundamental relation in The ERT.

(16)Extension is a definitional relation where the independent

 

variables are the trial balances of several entities, and The
dependent variable is the trial balance of The consolidated entity.

The same is True of A27 and A28.Balance of an Account (A18, A19),

 

which corresponds to The fundamental relation of the ERT, also leads
directly to a whole series of conclusions (A20, A21, A22, A23, A27,

and A28) based directly on that concept. (18) Allocation is also a

 

definitional relation where the independent variable may be a

61
transaction and The dependent variables are the allocations of That
transaction To some subentities. Transfer relations are represented

by The transaction (A4), which is a relation between two transactors.

 

Mattessich gives little attention to the control hypotheses
in the theory itself. But like Ijiri, he discusses the concepts

involved in the control hypotheses elsewhere.56

SUMMARY

This chapter has demonstrated how the theories of Mattessich
and Ijiri fit into the framework of the ERT. For a summary, see
Exhibit 3. ljiri's theory is the most specific dealing with The
necessary and sufficient conditions for the explanation of historical
cost accounting. MaTTessich's theory is more general in that it ex-
plains both micro- and macro-accounting. His theory also leaves a
number of elements open to specification. The ERT is on an even
more general level. And in the search for generality, the ERT does
not consider directly some of those things Thought by many To make
accounting unique. Among These are double entry, trial balance,
income statement, balance sheet, expense, revenue, asset, and many
others. These items can now be considered as special cases of the
more general ERT, and their exclusion helps expose the real foun-
dation for accounting. Mattessich used his Theory To sharply outline,
define, and distinguish accounting from other disciplines. With the

more general approach of the ERT, the sharp specific outline of what

 

56For example, on pp. 80-81 he discusses the concept of error,
and in the more recent German edition of his book, he has added a set
of evaluation hypotheses represented by a set of goal assumptions.
Richard Mattessich, Die wissenschaftlichen Grundlagen des
Rechnungswesens (Duesseldorf: Bertlesmann Universitaetsverlag, 1970).

 

62

EXHIBIT 3

COMPARISON OF THREE ACCOUNTING THEORIES

 

EVENT-RELATION

IJIRI

MATTESSICH

 

 

Events

Time, Subject, Object,
Ownership, Axiom of
ownership

Property set

_____ in.-.“

Time, Agents, Objects,
Ownership right, Debt
claim, Transactors,
Entity, Values

—_—— ————_

Transaction

 

Relations:

_. __. _. -._—q

 

 

 

 

 

Equivalence 'Substitutability Transactors
Selec- Chart of accounts
tion
Order Time Time instances
Accounting period
DefiniTionaI Measurement rules Balance of an account,
Duration, Extension,
Trial balance,
Allocation
Transfer Exchange Transaction
Description Physical measure Negative transactors
space u-measure Positive transactors
Set of values
Set of time instances
De-
scrip- Method of Physical measure Valuation
tion associating Axiom of quantities Realization
event with a Accounting set Data input
subset of the Axiom of exchanges Materiality

description
space

 

Measurement rules

 

Classification

Axiom of quantities
Accounting set

Classification
Chart of accounts

 

Ordering

Axiom of ownership
Axiom of exchanges

Materiality

 

Error

Data input

 

Verification

 

 

Evaluation

 

 

 

 

63
constitutes accounting has been lost. But as shown, some present
accounting theories can be considered special cases of the ERT.
The increase in generality has been one step toward the integration
of several related disciplines and maybe even an approach toward an

eventual unified theory.

CHAPTER VI

SUMMARY AND CONCLUSIONS

OVERVIEW OF THE THEORY

 

The event-relation meTaTheory (ERT) looks upon accounting
as The study and communication of a set of events and a set of
relations on those events. In that definition is summarized the
core of the theory, namely events and relations. The exact
accounting nature of an event is left open to definition because of
the generality of the theory. And the relations between events are
only indicated in broad outline. Figure 5 is a diagrammatic presen-
tation of the main elements of the Theory. The diagram displays the
sharp distinction between syntactics (boxes) and semantics (circles).
An event is defined in The syntactics To be an element of a sequential
memory, which is a Three-dimensional concept. Each event consists of
a string of descriptions chosen from a description space. Another
dimension is given by The sequence index, a primitive Time indicator
ordering The events by Time of occurrence. The third dimension
classifies the events into event series, strings of similar events
over Time. Corresponding to These Three dimensions are empirical
hypotheses (description, ordering, classification) that provide a real
world interpretation of the sequential memory. When a set of relations
on the events that make up The sequential memory is added, one obtains

64

65

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>LOE02
_0_+c0:oom
>H_Hzm

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_0L+coo _0L:+03L+m

 

 

 

GLU<J 30(149

66
The concept of The entity. Conventional double-entry accounting
systems are a special case of an entity called a closed system.

Four major categories of relations are presented. Equivalence
relations form the basis for classification of events and relations.
Order relations define the time concept and also play a significant
role in the comparison of events. DefiniTionaI relations combine
events in various ways to form new quantities. And transfer relations
seek to find cause-effect relationships between events that are
classified as independent or dependent variables. In other words,
one set of events is stated as a function of another set of events.

The meTaTheory does not state exactly how the connection with
the real world is to be made--That is the task of a specific theory--
but rather it creates a general framework wherein hypotheses making
these semantical connections may be formed. Of the structural
hypotheses, the hypotheses of description, ordering, and classifi-
cation are a direct consequence of the three-dimensional definition
of the sequential memory. And the selection hypotheses serve to limit
The field of attention. The control hypotheses, consisting of The
hypotheses of verification, error, and evaluation, serve both in
directing the accounting system Toward its optimal use and in
insuring That the results generated are adequate for the purpose

called for.

EVALUATION OF THE METATHEORY

 

The ERT is unique in the fewness of the conventional
assumptions included. Some might think that these are necessary,

but The ERT shows that in a meTaTheory They are not necessary, even

67
though they may appear as special cases of that theory. Among the
conventional ideas That are left out are monetary assumptions, double
entry, income concepts, profit maximization, price-level assumptions,
economic organization, and time period. Instead of taking These
assumptions as basic, the meTaTheory provides the framework within
which such concepts may be formed and applied.

Many accounting theories are object or asset oriented in
their basic assumptions. However, there is a rich background in
accounting thought supporting The idea of events as fundamental.

The emphasis on income determination, which is event oriented, and
the interpretation of assets as service potentials give support to
The idea That it should be possible To have theories with "event"
as a basic Term. The ERT provides a framework for such theories.
But the ERT concept of event is much broader Than the usual
connotation of that word.

Another aspect of the meTaTheory is a redefinition of The
entity. Traditionally, the entity has been thought of as an economic
or legal organization. But the ERT lets The entity be a seT of
events and a set of relations. The events and relations of a
conventional entity can be made a special case of The ERT. Thus the
ERT concept of entity is more general.

The chief aim of the meTaTheory was to attain generality.
This generality was attained at the price of making The definition
of accounting as a discipline less distinct. This lack of precision
can be justified because of The level of generality sought and because
of the potential that it creates for a future unification of accounting

and other disciplines.

68
Most new theories are tentative and This meTaTheory is no
exception. Based on previous attempts to examine the foundations
of accounting, this study Tries To Take one further step. The
slowness with which The concepts of this meTaTheory developed
causes The author To believe That the theory can be improved and
developed further in The years To come. This study should Therefore

not be viewed as a final or complete structure.

IMPLICATIONS FOR ACCOUNTING

 

The implications of The ERT are not immediately apparent.
When one proceeds toward generalization via induction, he gets
farther removed from the real world. And since the ERT is on
a more general level than previous accounting Theories, it is
also farther removed from the real world. In being removed
farther from the real world, it is hopefully closer to a sounder
conceptual foundation for accounting. An examination of recently
published introductory accounting Textbooks indicates that most of
the authors place a strong emphasis upon giving The reader a good
conceptual foundation for accounting. The ERT is one such step
Toward a sound and simple foundation for accounting.

By leaving a large number of items open To specification,
the ERT moves accounting one step closer to mono-purpose accounting.
Mono-purpose accounting exists when an accounting system is
designed specifically with a certain decision Theory, decision
maker, or any special purpose in mind. The ERT is general enough
that it can accommodate numerous different specifications.

The examination of the theories of Ijiri and Mattessich in

the light of The ERT showed a basic unity. The ERT can Thus serve

69
to integrate different accounting theories and to clearly pinpoint
The assumptions in which there may be divergence between theories.
It would be expected That many other accounting Theories could be

fitted into the framework created by The ERT.

FURTHER INVESTIGATION

 

More Than anything else the ERT provides the basic framework
within which full-fledged, sound accounting theories can be developed.
With such a framework, existing theories can also be compared,
criticised, and further developed.

In the construction of the meTaTheory only the bare minimum
of concepts were included and very broad classes of hypotheses
were developed. Because of The generality of the approach, almost
every item within The Theory is subject to future enrichment. And
if the general structure of the meTaTheory can be agreed upon, each
of the elements of both syntactics and semantics provides a fertile
field for both theoretical and empirical study.

The use of mathematical and statistical relations in the
syntactics suggests that it would be useful for accountants to
explore Those fields for the purpose of finding mathematical
formulations of relations that might be of interest in accounting.
Is there a limit to The kind of relations of interest To accountants?

The central problem That emerges from the meTaTheory is
that of limiting The field of inquiry. Using the tools of The
meTaTheory, an almost endless variety of phenomena and relations
may be investigated. The actual field of inquiry is To a large
extent determined by The evaluation hypotheses. These represent one

of the least studied fields of accounting. The other control

7O

hypotheses of error and verification have not been given much
attention in accounting thought either. The ERT thus points
out the need for the thorough study of The control hypotheses.

Accounting has over the years seen a shift in orientation
from merely following rules or uncertain origin and significance
to providing a significant tool for the decision maker--a tool
That is built for the task at hand. The search for a sound
foundation to build upon has intensified in the last decade. This
study represents one step toward such a firmer foundation and

better general understanding of The nature of accounting.

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