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ALGEBRAIC MODELS FOR ACCOUNTING SYSTEMS

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Chapter One

Approaches to Accounting

Theory

“Perhaps I am busied with pure numbers and the laws they sym-

bolize: nothing of this sort is present in the world about me, this

world of ‘real fact.’ And yet the world of numbers is also there for

me, as the ﬁeld of objects with which I am arithmetically busied;

while I am thus occupied some numbers or constructions of a nu-

merical kind will be at the focus of vision, girt by an arithmetical

horizon partly deﬁned, partly not; but obviously this being-there-

for-me, like the being there at all, is something very diﬀerent from

this. The arithmetical world is there for me only when and for so

long as I occupy the arithmetical standpoint.”

Edmund Husserl, Ideas p. 94 (italics original)

1.1. Historical Perspectives

Accounting is an ancient human activity. From the time when

men and women ﬁrst engaged in trade, whether for barter or money,

it must have been necessary to keep some kind of record of in-

comings and out-goings, to which the origins of the double entry

bookkeeping system can be traced. Already in the twelfth century

of the Christian Era the Arabic writer Ibn Taymiyyah mentioned in

his book Hisba (literally, “veriﬁcation” or “calculation”) accounting

systems used by Muslims as early as the seventh century. A crit-

ical development in the history of accounting was the publication

in Venice in 1494 of the book “Summa de Arithmetica, Geometria,

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2Chapter 1. Approaches to Accounting Theory

Proportioni et Proportionalita” by the Franciscan monk and math-

ematician Luca Pacioli (1445-1517) – see Pacioli [1963]. This is the

ﬁrst known work to contain a detailed description of the practice

of bookkeeping and the double entry system, “Particularis de Com-

putis et Scripturis”. Today it is widely regarded as the forerunner

of modern bookkeeping practice. It was also Pacioli who introduced

the symbols for plus and minus, which became standard notation in

mathematics during the Renaissance. The ﬁrst book on accounting

in the English language appeared in London in 1543, authored by

John Gouge. An important source for the early history of accounting

is the writings of R. Mattessich ([1998 ], [2000], [2003], [2005b]).

While it seems clear that accounting was considered by Pacioli

and his contemporaries to be part of arithmetic, its relationship with

other parts of mathematics has had to wait much longer for recogni-

tion. The methods of statistics have long been used, almost since the

importance of that branch of applied mathematics was ﬁrst recog-

nized in the seventeenth century. More recently probability theory

and risk analysis have featured in economics. However, algebra has

played little or no role, despite the precision of its language and its

ability to describe complex situations concisely. The purpose of this

monograph is to draw attention to the contribution that abstract

algebra can make to accounting theory. Indeed it is the authors’

contention that, at least in its deterministic form, accounting the-

ory should be considered as a branch of applied algebra.

The book presents and develops a proof-based, algebraic ap-

proach to the study of accounting systems. The analysis provides

a description of single ﬁrms in terms of abstract algebraic objects

such as automata. It concentrates on the process of producing infor-

mation from data provided by the environment through the double-

entry system. This process, although considered by many to be

the core of accounting, has often been ignored in accounting re-

search. In attempting to address this issue, the book adds a level

to the analysis of the information economists through the very act

of exploring the production aspect of accounting information sys-

tems. The motivation is to expose the complexities and subtleties

of information production in this ﬁeld of research. The literature

review which follows reﬂects the rather fragmented nature of the

work which has been done up to this time in axiomatics, natural

languages, formal grammars and information economics. The book

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1.1. Historical Perspectives 3

shows how a basic accounting system can be represented as a formal

algebraic language. The reduction of accounting systems to these

types of languages will lead to a much stronger method of modeling

information systems.

Although much discussion has occurred in the last ﬁfty years

concerning the treatment of accounting as a language and its jus-

tiﬁcation as the language of business, surprisingly little progress

has been made. This is perhaps due to the remarkable diversity

of methods in linguistic research. In pure linguistic research, the

various methods are divided into the natural language and formal

language schools. The natural language schools study naturally oc-

curring human languages as they have arisen from the historic acts of

increasingly complex human communication. The formal language

school arose from this tradition as methodologies were devised to

study natural languages. These methodologies generally tried to re-

duce the complexity of natural language constructs to a ﬁnite system

of grammatical rules. The formal language school became distinct

from the natural language school when it was determined that cer-

tain domains of language, such as parts of mathematics and later

computer science, could be completely speciﬁed by these ﬁnite sys-

tems.

Outside the area of pure linguistics, some applied ﬁelds such as

speech communication and organizational behavior have adopted

certain linguistic approaches and have developed other approaches

independently. Semiotics has been used to determine what signs

employees attend to in their everyday work relationships (Barley

[1983]). Semiotics studies the meanings that people assign to lan-

guage constructs in their search for understanding in their worlds.

Recently, hermeneutics has been used to develop a criticism of the

economics literature (McCloskey [1983]). This method employs the

analysis of texts to identify repetition of linguistic constructs or

changes in constructs over time and to study how the authors of the

texts view their social realities.

With such a diversity of methods available, it is hardly surprising

that the accounting profession has found little success in its search

for a formalization behind the intrinsic meaning of the metaphor

“accounting as the language of business”. It is the contention of

the present writers that the best way to proceed in the issue is to

choose a potential methodological candidate, develop it and make

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4Chapter 1. Approaches to Accounting Theory

a judgement based on its contribution to accounting research. The

method chosen here is a formal, algebraic approach. In order to

present this new approach to accounting in its contemporary setting,

a detailed review of the language studies, both formal and natural,

which have appeared in the accounting literature up to this point,

is given in the sections which follow.

1.2. Algebraic and Proof-Based Approaches

As has been pointed out, the application of abstract algebra to

accounting is something of a novelty. However, it would be wrong to

suggest that nothing has been attempted in this direction. Already

in 1894 the English algebraist Arthur Cayley wrote that “The prin-

ciples of book-keeping by double entry constitute a theory which is

mathematically by no means uninteresting; it is in fact, like Euclid’s

theory of ratios, an absolutely perfect one, and it is only its extreme

simplicity which prevents it from being as interesting as it would

otherwise be” (Cayley [1894]). Even before this time matrices had

been introduced in the framework of accounting theory by Augustus

De Morgan [1846], a route that was not followed by other writers un-

til 100 years later. Indeed matrices reappeared as a topic of research

interest in accounting only in the 1960s and 1970s, when a number of

classic works in accounting theory were published, such as Edwards

and Bell [1961], Chambers [1966], Ijiri [1967] and Mattessich [1964].

Here it should be understood that matrices were considered only as

a tool to describe in a mathematical way the activity of accounting,

and not as an attempt to formalize the concept of an accounting

system. Paton [1922], one of the major personalities in accounting

research in the United States in the 1920’s, seems to have been the

ﬁrst author in formulate some accounting postulates. Nevertheless

at that time fundamental research was not common in this area and

the postulates never became part of a formal system.

Perhaps the most famous axiomatization of accounting was given

by Mattesich [1957, 1964]. The ﬁrst of these publications relies on

a matrix formulation of accounting to provide structure to the ax-

iomatic system. Three axioms are included in this schema: a plu-

rality axiom, a double eﬀect axiom and a period axiom. The ﬁrst

asserts that there exist at least two objects with a common measur-

able property. This provides a basis for the recording of transactions.

The second axiom states the existence of an event which causes an

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1.2. Algebraic and Proof-Based Approaches 5

increase of a property of one object and the corresponding decrease

of the property of another. In eﬀect this is an axiom of double en-

try. The last axiom requires that accounting systems are capable of

being divided into time periods, thus providing a basis for the con-

struction of ﬁnancial statements. In addition to these axioms, the

paper provides numerous deﬁnitions and “requirements”, as well as

several theorems.

The proofs of the theorems in Mattessich’s ﬁrst paper give insight

into the formal relationship between the axioms and the theorems.

The proofs consist of algebraic manipulations of matrices using the

sigma, i.e., summation, notation. While in a sense this does serve

to “demonstrate” the theorems from the deﬁnitions, the proofs do

not consist of formal deductions from the axioms, as would be the

case in a strict deductive system. Thus the axioms do not serve

as a complete basis for the proofs of the theorems. In his second

publication Mattessich shifts from a matrix to a set theoretical ap-

proach. In this work he relies on primitive terms, deﬁnitions stated

using the set notation, and propositions. The theorems which are

proved appeal to the deﬁnitions and propositions and are basically

algebraic in nature. Perhaps the absence of axioms in this second

work was due in part to the diﬃculty noted above, i.e., axioms which

are not used in the proofs of the theorems. Some might argue that

the propositions substitute for axioms in this formulation, but the

propositions here are generally set theoretic deﬁnitions of such con-

cepts as an accounting period or the chart of accounts. Although

they may be invoked as a proof proceeds, the proofs do not begin

with the propositions, nor are the theorems deduced from them.

Again the beginnings of a formal proof-based system can be dis-

cerned here, but it is not coupled with a formal deductive scheme.

This type of scheme may be provided by including the axioms of

the mathematical system – in Mattessich’s case an algebra – as part

of the axiom scheme, thereby speciﬁcally allowing for mathematical

inference within the axiomatized system, as will be seen below.

Ijiri’s [1975] book on accounting measurement also includes three

axioms, but again it lacks any derivation of theorems from the frame-

work of these axioms. He does, however, derive his axioms from the

theoretical structure of the accounting system which he provides

in the book. Therefore it is likely that he sees these axioms more

as general statements about accounting, rather than as a basis for

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6Chapter 1. Approaches to Accounting Theory

any formal deductive system. Indeed he makes no attempt at all

at proving the theorems. One of the contributions of the current

book is that it provides not only an axiomatization of accounting

systems, but also a deductive inference scheme which can operate on

the axioms in a formal way to derive the theorems as consequences.

Tippett [1978] derived axioms of accounting measurement, and

more recently Cooke and Tippett [2000] used a structural matrix

to represent the restrictions imposed in a double-entry bookkeep-

ing system, employing the information in the matrix to predict ﬁ-

nancial ratios. Willett [1987, 1988] demonstrated in two papers the

derivation of axioms of accounting measurement, following Tippett’s

methods. His analysis extended to the stochastic space of account-

ing variables. Gibbons and Willett [1997], building on Willett’s

earlier work, demonstrated that accounting data produced from im-

plemented information systems have a statistical nature due to the

error generated by processing: that statistical nature is shown to

be of value to decision makers under certain conditions. Nehmer

and Robinson [1997] provided an initial description of the algebraic

structure of accounting which is greatly expanded upon in this book.

Nehmer [2010] encodes the algebraic structure in ﬁrst order logic and

derives consequences for the resulting structures.

Aukrust [1955, 1966] made an important contribution to the stan-

dard methodology for national and international accounts, complet-

ing a theoretical discussion of the underlying principles in account-

ing at the national level. He presented some problems of deﬁnition,

classiﬁcation and measurement of national accounts in an axiomatic

way. After stating a set of twenty postulates, he showed that the

structure of a simple system of national accounting can be derived

from them. In this way it is possible to establish algebraic relations

among national accounting concepts. Aukrust concludes: “The set

of twenty postulates used above to derive a national accounting sys-

tem is, of course, not the only one which could be conceived of.

Others are equally feasible. Some would lead to national accounting

systems diﬀerent from the one described here, in much the same

sense as non-Euclidean geometries are diﬀerent from Euclidean ge-

ometry”.

The problem of ﬁnancial statements was dealt with by Arya et

al. [2000], emphasizing the power of the double entry system to

determine all consistent transaction vectors. They showed how a

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1.2. Algebraic and Proof-Based Approaches 7

graphical representation of the accounting system can be used to

obtain the characteristics of the vectors, solving in a simple way

the problems of inverting and selecting the most likely transaction

vector from the set of consistent transaction vectors. Arya et al.

[2004] provided a systematic approach to reconciling diverse ﬁnan-

cial data. Again the key is the ability to represent the double entry

system by a network of ﬂows. Two speciﬁc uses are investigated:

the reconciliation of audit evidence with management by means of

prepared ﬁnancial statements and the creation of transaction level

ﬁnancial ratios.

The ﬁrst collaboration in the area between a philosopher of sci-

ence and a theoretical accountant materialized in Balzer and Mat-

tessich [1991, 2000]. They considered the reconstruction of yield

to be a viable way of capturing the essence and basic structure of

accounting as rigorously as possible. The proposed reconstruction

showed that accounting has the same overall structure as other em-

pirical theories by presenting nine axiomatic principles to establish

the following concepts: economic objects, economic transactions,

state-space for accounting, accounting data systems, accounts, dou-

ble entry accounting systems, accounting morphisms and accounting

systems (in general). By combining these deﬁnitions, they obtain

the kernel of a model for accounting and they claim that all spe-

cial methods and procedures used by accountants can be obtained

from this core model with some appropriate speciﬁcations. All the-

orems are proved, but the authors indicate the need for further de-

velopment of the axiomatic system presented in the paper and they

present details of certain speciﬁcations to appear in future work.

According to Ellerman [1982, 1985, 1986], “Double-entry book-

keeping illustrates one of the most astonishing examples of intellec-

tual insulation between disciplines, in this case, between accounting

and mathematics”. He described a mathematical basis for a treat-

ment of double-entry bookkeeping in terms of the so-called “group

of diﬀerences”, sometimes called the Pacioli group: for details of this

connection see 3.1 below. The possible use of the algebraic concept

of a group in accounting theory is also considered in Brewer [1987]

and Botafogo [2009], but with little progress beyond the formulation

of some deﬁnitions.

There have been many other attempts to formalize accounting

in a scientiﬁc way. Since the present work does not pretend to give

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8Chapter 1. Approaches to Accounting Theory

an exhaustive history of accounting, only some of them have been

mentioned. Details of other attempts can be found in Mattessich

[1995, 1998, 2000, 2003, 2005a, 2005b].

On a ﬁnal note, recently Demski [2007] has tried to answer to the

question “Is accounting an academic discipline?” After analyzing the

meaning of “discipline” and “academic”, his immediate conclusion

was negative. However, Demski was not pleased with this answer

and therefore he preferred to analyze the ten indicators of the ac-

counting as an academic discipline, ending with “... accounting is

not today an academic discipline; it is an ever-narrowing insular

vocational enterprise. But it could and should, in my opinion, be

an academic discipline. Even if you disagree with my assessment,

you should consider whether the state of academic accounting is, in

your view, what it could and should be. The stakes in this game are

enormous and serious”.

1.3. Natural Language Approaches

Research in accounting as a natural language, as opposed to an

proof-based system, has fallen into three broad categories: connota-

tive and denotative meanings, readability of reports and linguis-

tic relativity (McClure [1983]). The connotative and denotative

meanings of language refer to its subjective and objective mean-

ings respectively. The research in this category has emphasized the

interpretation of accounting concepts by diﬀerent groups including

certiﬁed public accountants (CPA’s), users, students and academics.

The results have generally indicated agreement on the connotative

meaning between groups, but there is some evidence of disagree-

ment over denotative meanings (Belkaoui [1980b]). Research into

the readability of ﬁnancial reports has stressed the ability of the re-

ports to communicate information on several levels. Levels of read-

ing ability needed to comprehend the reports have been tested, but

the tests were found to be inappropriate for the analysis of materi-

als in a report format. Lebar [1982] tested several diﬀerent types of

ﬁnancial report along an extentional - intentional axis. Extentional

language is more descriptive and objective, whereas intentional lan-

guage is more general and unqualiﬁed. She found that 10-K reports

(a speciﬁc type of ﬁling that a company makes to the Security and

Exchange Commission) scored well on the extentional components

as compared to the annual reports.

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1.3. Natural Language Approaches 9

The third category of linguistic research in accounting is based

on linguistic relativity (the Sapir-Whorf hypothesis). The two basic

concepts of the hypothesis are that language determines thought and

that consequently individuals with diﬀerent linguistic backgrounds

have diﬀerent world views. Belkaoui [1978, 1980a] used this hy-

pothesis to study disclosure issues in the area of pollution control

costs, with results generally supporting the hypothesis. All three

categories of research in accounting as a language have viewed it as

a natural language and applied natural language techniques to its

study.

Some more recent studies of business communication include Tyr-

vainen et al. [2005], who examine the internal and external commu-

nication of three business units, looking at digital, paper-based and

oral communication. In a series of articles in the accounting area,

Fisher [2004], Fisher and Garnsey [2006] and Garnsey and Fisher

[2008] codify the professional accounting literature. This codiﬁca-

tion is then used to critique the adequacy of the literature (Fisher

[2004]) and to examine amendments to the literature (Fisher and

Garnsey [2006]). Garnsey and Fisher [2008] implement a software

retrieval solution to the professional accounting literature.

An alternative approach is to view accounting as a formal lan-

guage built up from a detailed speciﬁcation of its grammar by ex-

act rules of composition known as production rules. Formal gram-

mars and languages were originally developed for natural language

research and are still used there, especially in computational lin-

guistics research. They have been largely absorbed into computer

science because they are an alternative representation of ﬁnite state

automata. Such automata are used in computer science for the gen-

eral representation of computer languages. The concept is easy to

relate to for anyone who have ever tried to learn a computer language

with its peculiar sentence structure and rules. A good example of

research using the automata approach is Cruz Rambaud and Garc´

ia

P´erez [2005].

Demski et al. [2006], looking for a new language for the treat-

ment of accounting information, examined the nature of quantum

information in order to search for promising conceptual applications

to accounting. They present some important features of quantum

information such as quantum superposition, randomness, entangle-

ment and unbreakable cryptography, and they begin to explore the

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10 Chapter 1. Approaches to Accounting Theory

possible link between quantum information and double-entry infor-

mation which lies in the core of accounting information. The start-

ing point is the work of Cayley [1894] on the parallel between the

Euclid’s theory of ratio and the double entry theory. As a con-

sequence, it is intended to explore the possibility of a hybrid be-

tween accounting information and quantum information, “quantum

double-entry information”. In a second article, Demski et al. [2009]

studied the applications of conceptual topology to quantum informa-

tion and accounting information. The use of topology allows one to

emphasize the qualitative characteristics of accounting information

and to maintain the quantitative ones.

A reasonable and eﬀective mathematization and axiomatization

of the economy, and in particular of accounting, necessarily implies

Diophantine formalisms (Velupillai [2005]), which raises issues of

undecidability and non-computability. In the future there should be

greater freedom for experimental research supported by alternative

mathematical structures. In conclusion Velupillai speaks of “the

notion of a Universal Accounting System, implied by and implying

Universal Turing Machines and universality in cellular automata”.

1.4. A Formal Grammar Approach

One exception to the exclusive use of natural language research

methods in accounting is Stephens, Dillard, and Dennis [1985], here-

after Stephens et al. The article is entitled “Implications of Formal

Grammars for Accounting Policy Development” and it presents a

classiﬁcation scheme for proposed and existing Financial Account-

ing Standards Board (FASB) statements. The examples provided

in the article are partial formal grammars, reﬂecting the accounting

rules promulgated by a speciﬁc standard. The level of analysis is

macro in the sense that it considers the standard for all ﬁrms to

which they apply. As such the analysis focuses on establishing cri-

teria with which to evaluate standards through formal grammars.

The three criteria used are possibility, consistency and resolution.

Possibility refers to the ability to reduce the statement to a formal

grammar. One potential problem here, albeit one which is discussed

in a diﬀerent section of the article, is the diﬃculty in determining

the primitives of the grammar. In the article the example of leases is

cited. The determination of whether a certain economic event should

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1.4. A Formal Grammar Approach 11

be classiﬁed as a rental arrangement or a purchase has become in-

creasingly problematic in accounting. Unless a clear demarcation is

allowed or imposed on the “correct” interpretation of such an event

under every circumstance, the formal grammar will not be capable

of operating in these types of situation.

The second criterion, consistency, refers to the cross-statement

compatibility of the grammars. This compatibility can perhaps

best be addressed in terms of ﬁrst order logic, rather than the for-

mal grammar approach used in the article. The two systems are

equivalent, so the change in approach is warranted. In ﬁrst order

logic consistency is deﬁned in terms of the sentences which can be

proved from the axioms. If both a sentence and its negation are

provable from the axioms, then the system is inconsistent and in

fact any sentence is then provable from the axiom system. In terms

of the article, in order for a formal grammatical analysis to suc-

ceed, a single formal grammar containing all accounting standards

must be demonstrated. Then any proposed new standards could

be appraised in terms of their consistency with the current formal

grammar.

The third criterion, resolution, is an attempt to deal with prob-

lems of inconsistency arising from the diﬀerent rules speciﬁed in the

single formal grammar mentioned above. The article proposes that

uniform ranking rules be included in the grammar in order to re-

move such inconsistencies. It points out that the FASB does provide

such rules in certain situations, but that the rankings so provided

have not been uniform in the past. Stephens et al. classiﬁed the

resulting inconsistencies as being due to one of three situations: ar-

bitrary selection among possible standards, stipulation of standards

without theory and the inability to write a deﬁnitive grammar.

In the ﬁrst situation a choice is made and a particular standard

must be selected, when alternative standards have possible correct

economic interpretations and their own supporters. Stephens et al.

contend that this and the next situation result primarily from lob-

bying by factions of the accounting community. The next situation

occurs when a standard is stipulated which is lacking in theoretical

support; this seems to mean lacking in terms of a justiﬁable eco-

nomic interpretation. The interpretation is usually only provided

a posteriori and may be thought of as imposing a new economic

reality based on the standard. The last situation is the problem of

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12 Chapter 1. Approaches to Accounting Theory

specifying the primitives of the grammar, which was discussed under

“possibility” above.

Stephens et al. divide the economic realm into three parts, the en-

vironment, accounting and decision. The eﬀects of economic events

in the environment are actions which play the role of primitives

subject to the grammatical rules of accounting. The rules produce

accounting results which are used by decision makers to produce

decisions. Stephens et al. restrict their analysis to the accounting

component only, so that the evidence of a transaction occurring is

taken as a given and the use of the output is not analyzed. The

same position is adopted throughout this book.

However, there are several diﬀerences between the article by

Stephens et al. and this book, perhaps the most important be-

ing the level of analysis. The analysis presented here is at a micro

level, as opposed to the macro level of the article. Speciﬁcally the

analysis here pertains to the accounting system of a single ﬁrm.

Secondly, a complete axiom system is developed for the ﬁrm, based

on the double-entry components of the system only. The necessity

of developing such a restricted system is based on the requirement

of demonstrating the existence of such representations of accounting

systems before proceeding with higher level analysis, as is recognized

by the authors of the article.

A contribution of this research is to provide a basic method for

constructing formal proofs in accounting. It interfaces with the ax-

iomatization and formal inference scheme to yield a formal abstract

speciﬁcation; this leads directly to axioms for an accounting system,

as well as to a system of inference which can be used to derive conse-

quences of those axioms. In fact, the analysis of the paper includes

the consideration of information systems as ﬁnite state grammars

(FSG’s) and automata. This representation is the basis of the com-

puter languages which form the structure of any computerized sys-

tem. Therefore ﬁnite state grammars can be used as a general rep-

resentation of the process involved in converting states into signals.

Such FSG’s include relation as well as function operators, thereby

providing a more powerful means of analysis in exploring the pos-

sibilities and limitations of the signal/output generation process of

information systems.

The representation of information systems as FSG’s serves two

purposes in this analysis. First it addresses some problems noted

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1.5. Information Systems 13

below with information economics methodology, i.e., it provides a

speciﬁc formulation of the internal production of information and as-

signs a speciﬁc interpretation to the states recognized by the system

as well as its outputs. Furthermore, it allows for the production of

multiple derivations from the capture of an additional piece of data.

The second use of FSG’s is to provide a convenient bridge between

the representation of accounting systems as FSG’s and their repre-

sentation as proof-based systems. This is accomplished through the

conversion of the production rules of the FSG into axioms of a ﬁrst

order logical system.

1.5. Information Systems in Information

Economics

This book addresses some of the issues in the comparison of infor-

mation systems which occur in the information economics literature,

this being the current standard of comparison of systems in accoun-

tancy. A large body of work has been done in the area using utility

analysis and relying on the results of Blackwell’s “Comparison of

Experiments” (Blackwell [1951]). As the title indicates, Blackwell’s

procedure shows that if an experiment A is a suﬃcient procedure

for a diﬀerent experiment B, then A is more informative than B,

i.e., it provides at least as many statistical measures. Authors such

as Gjesdal [1981] have used the matrix form of Blackwell’s results to

analyze diﬀerent information systems. Demski [1980] and Demski,

Patell and Wolfson [1984] have used the basic matrix framework of

states crossed with signals and in the latter paper relied on Gjesdal’s

information systems comparison result. All of these comparisons of

information systems are founded on the partitioning of the states

of nature, the idea being that diﬀerent information systems will be

able to “recognize” diﬀerent states at various levels of ﬁneness. That

is, a certain information system may produce signal Y1when it rec-

ognizes S1and signal Y2when it recognizes S2, whereas another

information system may not be able to distinguish S1from S2, and

produce the same signal for either realization.

The implication of the states to signals model of information sys-

tems is that there is a set of functions corresponding to the set of

information systems under comparison. Mathematically the conver-

sion of the states to signals is a mapping from the set of possible

states to the set of possible signals. Over the entire state and signal

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14 Chapter 1. Approaches to Accounting Theory

spaces the function family is neither injective nor surjective. In the

ﬁrst place a particular information system function may map two or

more states to the same signal, so the function is not injective. Sec-

ondly, an information system function may not be able to generate

certain signals in the codomain at all, so it is not surjective. Indeed

in Demski’s 1980 examples, it is only in the perfect information case

that the mapping can be bijective, i.e., both injective and surjective.

It is this lack of uniformity in the construction of the state to signal

functions (or information systems) regarding their relevant domains

and codomains which partially explains the failure of Blackwell’s

comparison technique in proof-based systems.

Several topics are important to the present analysis. Firstly,

Blackwell’s result lies in the domain of experimental procedures,

whereas an information system is, in a practical sense, an extant

structure generating outputs from inputs by a formalized system of

rules. As such there are several diﬀerences in the level of analysis

which are apparent. Most importantly the information economics

analysis considers the external or environmental states of nature

only, without considering the internal states of the information sys-

tem itself. It therefore ignores the interaction of the internal com-

ponents of the system in the production of its outputs.

The unspeciﬁed nature of the internal components prevents the

methodology from addressing questions relating to how changes in

the conﬁguration of the system will alter the signal set generated. Of

course, information economists use the term “information system”

in a diﬀerent sense than is used here. But it is the diﬀerence in

representation of the system which allows this additional analysis to

occur. These are important questions for the accounting profession

since they involve the production of information for decision makers

in an organization from the design of the accounting system.

This lack of concern for the internal state of the system also

forces the information economics methodology to ignore explicitly

the problem of data capture versus information production. That

is, a state may occur in the external environment which is captured

or recognized by the system but is not processed in a timely man-

ner. While the techniques of information economics do implicitly

take this into consideration by collecting states into sets based on

the concept of ﬁneness, this does not help in determining why a

particular output is not being generated, i.e., whether the data are

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1.5. Information Systems 15

being processed too slowly or are not available at all.

A second deﬁciency in the statistical analysis of information sys-

tems is its inability to recognize that a particular state may generate

more than one signal. The information economics approach pro-

vides, at best under perfect information, a single state or a single

signal mapping for output production from information systems.

Under imperfect information several diﬀerent states may produce

the same signal, but the reverse situation, of a single state being

mapped onto multiple signals, is not considered. For instance, a

decline in interest rates may cause changes in pension funding re-

quirements, a decline in the mortgage interest rates being paid by

an organization and declines in the dividend rate expected from an

investment in mutual funds. Further, it is possible within an in-

formation systems methodology to develop single states to multiple

signals if a recognition of the interrelationships between states and

signals is provided.

A ﬁnal problem with the current method of analysis is that it does

not provide a convenient way to interpret the states and signals. As

an example consider Gjesdal’s [1981] description of an information

system. Here he reduces the system to merely the speciﬁcation of

the signal’s functional form and proceeds to assert that “the nature

of the signals is of no concern” (p. 212). One can only assume that

the nature of the information system is of no concern as well, yet it

is diﬃcult to comprehend the purpose of comparing objects whose

nature is not the object of comparison.

Generally the matrix representation of information systems and

especially the concept of state (and hence signal) partitioning does

not address the problem of how the signals are generated. This

leaves open the question as to whether and to what extent in the

context of axiomatic information systems, such a partitioning is pos-

sible. This problem is addressed in Chapters 2 and 3 where the

algebraic core of the model is constructed.

Demski’s ([1980]) analysis of information systems diﬀers from an

axiomatic approach in that his complete model consists of a set of

acts, states, state probability functions and utility functions, with

states and acts as parameters. This model is conditioned on the

decision makers’ experience and assumes that the four factors men-

tioned above are correctly speciﬁed. He presents two cases, the

perfect and the imperfect information situations. Under perfect in-

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16 Chapter 1. Approaches to Accounting Theory

formation, the decision maker can directly observe the realization

of the states of nature. Therefore there is no need for the infor-

mation system to produce signals relating to the acts of an agent.

Since the state is known with certainty before an act is chosen, this

situation will match well the derivations of an axiomatized informa-

tion system. If the state is known for certain prior to the act, there

must be some decision procedure which would indicate which state

will occur and such a procedure is axiomatizable. This is the case

because under state certainty the state must already be a fact, in

which case its truth value is known or must be determinable un-

der some formulation which perfectly correlates its own predictions

with the actualization of those predictions. In the latter case, the

decision procedure will be reducible to a ﬁrst order formula, barring

the serious consideration of some form of crystal gazing as providing

perfect information. Of course Demski would agree that this is an

unlikely situation in any complex decision making problem. In fact,

testing which state will occur is likely to involve a great deal of com-

putational complexity in a complex, decision making environment:

the results, if they can be determined with certainty, may not be

produced in a timely manner.

In the imperfect information situation the information system

cannot necessarily distinguish each state uniquely, so the same signal

or output from the system may occur after the realization of diﬀerent

states of nature. When the state is not known with certainty prior

to the act, Demski posits the information system as producing a set

of signals which may, but usually do not, indicate the state which

was achieved or which has transpired. The signal is a function of

the information system with the states as the input and the signals

as the output.

One and only one signal is associated with each state occurrence,

although the same signal may be produced by diﬀerent states. The

state space is partitioned into diﬀerent subsets of the power set of the

set of states, with the information system as the partitioning agent,

i.e., diﬀerent information systems produce diﬀerent partitions. Of

course Demski’s book has a wealth of ideas and constructs which

cannot be explored here, but with this basic framework in mind, we

note that results for algebraic systems are obtainable which diﬀer

from Demski’s conclusions.

In eﬀect the analysis presented here adds a level to the work of

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1.6. Location of the Research 17

the information economists, exploring the possible derivations of,

and treating the information systems used in, their formulations

as information systems: these are representable as systems of ﬁrst

order formulas and consequently are amenable to analysis as struc-

tures in model theory. Whereas the information economics approach

formalizes information systems as collections of functions from the

states to the signals, our approach imposes additional constraints

on the production of the signals themselves by explicitly consider-

ing the language used to express the functional formulation of the

information systems. These additional constraints will be of conse-

quence when the information system is represented as a proof-based

system.

1.6. Location of the Research Justiﬁed

Returning to the article of Stephens et al. [1985] which was dis-

cussed in 1.4, we note their description of accounting interfaces. If

this description is reduced to an individual ﬁrm, then the account-

ing system of the ﬁrm can be seen as a ﬁlter which captures certain

data from the environment, to be processed and presented to de-

cision makers. It is this ﬁlter, the speciﬁcation of which data are

captured, how they are processed and in what general form they are

presented, which locate this book within the accounting process.

In this location an accounting system is constructed as a machine

which follows strict rules, namely the axioms, in converting inputs

to outputs. This procedure is also strictly deﬁned as an inference

scheme, determining how occurrences of inputs combine within the

rules to produce outputs and other secondary rules. These outputs

are the derivates of the accounting system and may also arise from

combining rules only from within the inference scheme. Thus we are

dealing with the construction of a deterministic system.

In addition the book considers how to control accounting sys-

tems which operate under diﬀerent rules. This requires building

on the derivations of rule-based accounting systems. The control

is achieved as follows. The derivations of an axiom system can

be thought of as formal deductions from given premises. In this

case, the formal deductions arise from the inference scheme and the

premises are the axioms and derivations already deduced. The con-

trol of accounting systems under this methodology would then look

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18 Chapter 1. Approaches to Accounting Theory

at the diﬀerences in the sets of consequence of the separate proof-

based accounting systems.

As mentioned previously, the Stephens et al. article describes

accounting as an environment to accommodate accounting informa-

tion systems (AIS) and decision maker ﬂow. The link between the

environment and the AIS and between the AIS and the decision

maker are both areas of considerable research in accounting. The

ﬁrst link contains problem areas involving the recognition of eco-

nomic events as transactions. Research has been concerned with

when and whether an economic event such as a contingency should

be captured by the system and thereafter reported to the decision

maker. The crux of this problem is when an economic event should

be interpreted as being probable. On the other hand, the link be-

tween the AIS and the decision maker involves the interpretation of

whether and under what circumstances data presented by the AIS

change decisions and thereby become information of some value to

decision makers. Thus there are two general types of interpretation

which occur between the AIS and its environment and the AIS and

the decision makers. However, in the context developed in this book

a third and more formal approach to interpretation is employed.

This third type occurs entirely within the AIS and is speciﬁcally

related to the axiomatization of the system. Within axiomatized

systems there is a formal logical interpretation, indeed an interpre-

tation function, between the syntactic components of the system

and their semantic interpretations.

1.7. Accounting and Formal Languages

The axioms and derivations of a formal system are strings of

symbols called sentences or formulas. At the syntactic level these

strings are manipulated by the inference scheme in a purely formal

way, without regard to the meanings which may be attached to

the original or deduced sentences. The syntactic level therefore is

merely concerned with which sentences can be produced by following

the inference scheme. So the only way that a sentence is in essence

“meaningless” in syntax is if it is not derivable from the axioms via a

sequence of inferences. A logical interpretation is a formal map from

the syntactic level to the semantic level which provides meaning or

a translation of the combination of symbols in the sentences.

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1.7. Accounting and Formal Languages 19

As an example, consider the standard rule of inference modus

ponens. According to this rule, if there are two sentences x→P(x)

and x, then P(x) is derivable. Notice that no meaning is attached

to the symbols x,→or P(x), so that modus ponens is a strictly

syntactic construction. Now suppose the interpretation function

maps xto “cash”, →to “implies” and P(x) to “xis a current

asset”. Then at the semantic level the interpretation of this instance

of modus ponens is that “cash implies cash is a current asset”, so

that “cash is a current asset” is derivable.

The syntactic rules are akin to the grammatical rules of a natural

language. In natural languages the meaning of a sentence is based

on an interpretation of its form. This form is regulated by distin-

guishing which sentences are grammatical. However, the distinction

between syntax and semantics in natural language is often a hazy

one because the grammatical rules are not speciﬁed in advance, but

have been deduced from the structure of the language by linguists.

Therefore it may be impossible to describe accurately the syntax of

a language by a ﬁnite number of rules. For example, a basic sen-

tence form in English is subject-verb-object. This rule works well

for “sensible” sentences such as “The computer ran the program.”

Unfortunately, without further rules of grammatical construction,

a naive foreign speaker might deduce the following sentence from

the rule: “The computer walked the program.” The purpose of in-

dicating this type of problem in a natural language is to point out

the close relationship of both syntax and semantics to the interpre-

tation of meaning in these languages. It appears as if the human

mind attends to both syntax and semantics simultaneously through

learned patterns when constructing the meaning of natural language

sentences.

Another type of interpretation known as hermeneutics has been

developed in the naturalistic research methodology. By using this

methodology the researcher attempts to interpret the world as a

text in order to understand the meanings which the actors in the

study attach to objects, to themselves and others, and to actions.

Here the objective is similar to reducing the semantic context of

the world to a somewhat less complex and perhaps hidden syntactic

component. The syntactic component in hermeneutics is seen to be

dynamic, with the actors and their environment constantly interact-

ing to reconstitute meaning and form. This technique is essentially

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20 Chapter 1. Approaches to Accounting Theory

a meta-analysis of sentences which not only looks at sentences in

their own contexts, but also across the contexts of diﬀerent actors

and environments.

In order to relate accounting to the concepts of syntax and se-

mantics, it should be remembered that these concepts are used in

diﬀerent ways in various types of analysis. In the case of natural lan-

guage, the syntactic component of accounting is the systems of rules,

such as the mechanics of double entry bookkeeping, statements of

auditing standards, Financial Accounting Standards Board (FASB)

and International Accounting Standards Board publications, and

Security and Exchange Commission rulings which aﬀect transac-

tions and manipulations of transactions, including disclosure. As

with all natural languages, the syntactic and semantic components

lie very close to one another when accounting is viewed as a natural

language. For example, take a common occurrence when beginning

students are introduced to accounting for merchandizing ﬁrms. A

typical error is for the student to debit inventory and credit accounts

payable when merchandize is purchased on account, instead of deb-

iting purchases. This may happen because the student is confused

about the semantic meaning of the problem of costs of goods sold,

as against the meaning of accounting for inventories.

A further phenomenon which occurs when accounting is viewed as

a natural language is that the interpretational component becomes

closely intertwined with both the syntactic and the semantic compo-

nents. The conceptual framework and the FASB statements which

reﬁne previous interpretations in order to standardize interpretation

of economic events indicate the closeness of this relationship. For

example, FASB statement number 1 is an attempt to standardize

the interpretation of what constitutes an operating lease, as op-

posed to a capital lease for both the lessee and the lessor. This is

similar in form to the hermeneutic concept of interpretation acting

as the meta-rule intermediating between the actors and their envi-

ronment, in this case certiﬁed public accountants, their clients, the

FASB members and the accounting environment.

The explanation of the interrelationships between form, meaning

and interpretation was the original inspiration to the formulation

of formal logics and proof-based systems. The ancient Greeks were

concerned with problems of valid arguments, proceeding from the

development of schools of rhetoric. At the time work was concen-

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1.7. Accounting and Formal Languages 21

trated on developing techniques for identifying correct inferences

and exposing fallacious ones. One of the arguments which arose

was between Diodorus Cronus and his pupil Philo of Megara. The

argument revolved around the correct interpretation of the rule of

inference modus ponens, which is also known as a conditional state-

ment. If the conditional statement is formulated as “a→b”, then a

is termed the antecedent and bthe consequent. Diodorus and Philo

diﬀered as to what would be the conclusion if the antecedent were

false. Diodorus took the position that a false antecedent negated

the conditional, so that the statement is false. Thus the statement

“If the FASB is a governmental agency, then this book is deposited”

is false in Diodorus’ system, since the FASB is not a governmental

agency. Philo took the opposite view, arguing that the only case

where the conditional is false is when the antecedent is true and the

consequent is false.

Philo’s reasoning is important because his position became the

standard one in formal logic. Under his interpretation, a→b, which

semantically might be read as “aimplies b” or “if a, then b”, is

logically equivalent to “not aor b”. In this case, the “or” is in-

terpreted as inclusive, meaning that “not ais true” or “bis true”

or “both are true”. (In the case of an exclusive or, the last case

is disallowed.) Under this interpretation, treating the sentence “If

the FASB is a governmental agency, then this book is deposited” is

equivalent to “either the FASB is not a governmental agency or this

book is deposited”, which is true since the FASB is not currently

a governmental agency. Notice that the second clause “this book

is deposited” can be either true or false and the entire statement

remains true as long as the FASB remains independent. As such,

the Philonian interpretation of false antecedents is often referred to

as the case of trivial truth of the conditional.

Whatever justiﬁcation there may be for the speciﬁc interpreta-

tions that have been given to inference schemes, and there are many

equivalences between rules of inference schemes as well, the point is

that the construction of formal systems requires the speciﬁcation of

exact syntactic rules and speciﬁc interpretive mappings to semantic

meaning. In addition, even after the speciﬁcation of the formal in-

ference scheme, it may be possible to reduce the number of allowed

inferences by eliminating inferences which are logically equivalent

to one another. For example, many of the inferences allowed in

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22 Chapter 1. Approaches to Accounting Theory

formal logics currently used in philosophical and linguistic texts on

the subject were developed in the Middle Ages by the scholastics

in order to match natural language inferences used in disputation

and rhetoric. One such rule of inference is modus tollens, a type of

negated modus ponens. With modus tollens the conclusion “not a”

is deduced from “a→b” and “not b”.

This rule is equivalent to modus ponens, as can be seen when

the conditional is translated into the form “not a” or “b”’. In the

case of modus ponens, given “not b” along with the translated con-

ditional, the only case where “not aor b” is true is when ais false,

since then “not a” is true. This characterizes an important fact

about formal inferences, they preserve truth. This means that if

the premises of the inference, here “a→b” and “not b”, are true,

then the conclusion must be true as well for the inference to be

valid. The disadvantage of eliminating equivalent inferences is that

it moves the logical system, as represented by the sentences, fur-

ther away from natural language. This is true because the natural

language inferences are translated into a reduced set of inferences,

which removes some of the variety from the corresponding formal

language. The variety lost does not entail a loss of content how-

ever, since the reduced system is logically equivalent to the system

with the larger set of inferences. The reduced system does possess

syntactic advantages however, since the number of rules has been

reduced. This allows for simpler analysis at the syntactic level. In

fact, many mathematical logic systems only include modus ponens

in their inference schemes and these are almost always equivalent to

systems which allow a greater number of inference types.

In this work we follow the formal systems of the mathematicians,

rather than the philosophers and linguists, because the reduction in

the number of inference rules reduces the complexity of specifying

the consequences for computation. In order to prepare for the formal

analysis which follows, some of the major concepts of the syntax and

semantics of formal proof-based systems will now be introduced.

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1.8. Proof-Based Systems 23

1.8. Proof-Based Systems

In order to formalize a language, there must be a speciﬁcation

of the signs and symbols of the formal language, as well as a spec-

iﬁcation of the permissible manipulations of the symbols. First an

alphabet for the formal language is needed. The alphabet is divided

into six disjoint subsets, the ﬁrst of which are constants. Constants

are symbols which have a single value such as 0 or 1. The second

subset of the alphabet consists of variables, which can take on a

range of values. Constants and variables are called atomic terms.

The third subset consists of operations or functions. Each func-

tion has a speciﬁed degree 1,2,3,...; their values are called terms.

Multiplication is an example of a function of degree 2 since it has

two arguments. Functions map elements in their domains to ele-

ments in their codomains. They must be well-deﬁned, meaning that

each element in their domain is mapped to a unique element in the

codomain. If fis a function of degree iand t1, t2, ..., tiare terms,

then f(t1, t2,...,ti) is also a term, although not an atomic term.

The next subset of the alphabet consists of predicates, which

also have a speciﬁed degree. In eﬀect a predicate makes a statement

about its arguments. It does this because it is a deﬁned subset of

the domain of discourse or universe of the formal language. The

universe contains all of the object-meanings which are allowed in

the language. For example, if the universe consists of all of the

accounts in an accounting system and a predicate P(a) of degree

1 is deﬁned to be “ais an asset”, then P(a) will be true only if a

represents an asset account. This deﬁnes a mapping in which P(a)

is sent to “true” (or 1) in only those cases where ais in the subset P;

otherwise P(a) is mapped to “false” (or 0). This mapping is called

the characteristic function of the predicate. If Pis a predicate of

degree iand t1, t2,...,tiare terms, then P(t1, t2,...,ti) is an atomic

formula. Notice that it is possible to represent functions of degree i

by predicates of degree i+ 1 by merely adding the codomain of the

function as the (i+1)th object of the subset deﬁned by the predicate.

For example, the binary degree function of addition translates into a

tertiary predicate in which h2,5,7iand h3,8,11iwould be included

in the subset deﬁned by the addition predicate. In general this

predicate would consist of the ordered triplets hx, y, zisuch that

x+y=z.

The ﬁfth subset of the alphabet consists of logical symbols, which

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24 Chapter 1. Approaches to Accounting Theory

are divided into connectives and quantiﬁers. The connectives are →

(implication), ∨(“or” = disjunction), ∧(“and” = conjunction), ¬

(“not” = negation) and ↔(if and only if or logical equivalence).

The quantiﬁers are ∃(there exists) and ∀(for all). The two quan-

tiﬁers are also called the existential and the universal quantiﬁers

respectively. If Fand Gare formulas, then the following are also

formulas:

(F)→(G),(F)∨(G),(F)∧(G),¬(F),(F)↔(G)

and

∃(x)(F),∀(x)(F),

where in the last two formulas xis a variable. The ﬁnal subset

contains punctuation marks, of which only left and right parentheses

and occasional commas are used here.

The rules for forming terms and formulas provide the ability to

recognize well-formed formulas in the language. A formula is well-

formed if and only if it is built up from constants and variables

by repeated application of the rules for forming terms and atomic

formulas. In addition a formula in which all the variables are bound

to quantiﬁers is called a sentence. A variable is bound if it occurs

in a formula Fand in the quantiﬁcation of that formula, i.e., xis

bound in Fby the quantiﬁcations ∃(x)(F) or ∀(x)(F). A variable

which is not bound is considered free.

Next the syntax and semantics of a formal language are con-

structed as follows. Both concepts are founded on the idea of the

truth of formulas, sentences and inferences. Each logical symbol in

the alphabet has a corresponding truth table associated with it. In

the case of implication, the formula is false only when its antecedent

is true and its consequent is false. Likewise, in the case of the inclu-

sive or, the formula is false only when both arguments are false. For

a conjunction, its truth value is true only when both arguments are

true; in all other cases it is false. Negation takes only one argument

and is true if its argument is false and false if its argument is true.

Logical equivalence is true if and only if either both arguments are

true or both are false.

The quantiﬁers “for all” and “there exists” are true in the fol-

lowing cases. “For all x, F (x)” is true only when every symbol of

the alphabet which can be substituted for xin the formula leads to

the formula being true. For “there exists x, F (x)”, the formula is

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1.9. The Scope of the Present Work 25

true if at least one symbol can be substituted for xleading to a true

formula. The assignment of truth values for a complicated formula

begins at the lowest level of atomic terms and atomic formulas and

proceeds to higher levels, in the same manner as the term or formula

was created in its deﬁnition.

It was mentioned earlier that in order for an inference to be valid,

it must preserve truth. This means that it is not valid to deduce a

false conclusion from true premises. The notion of validity is a syn-

tactic one because it involves the construction of formulas through

the application of the rules of inference. Given a set of axiom formu-

las, the formulas which can be validly constructed from the axioms

by repeated inferences are called the consequences of the formulas

and these are said to be deducible or derivable from the axiom for-

mulas.

In terms of the semantic component of a formal language, all

formulas that are true in the language are said to be provable in

the language if the language is complete. Completeness is a seman-

tic concept because it requires that if the meaning of some formula

is true in the sense of the universe of the language, then that for-

mula must be provable. The speciﬁc derivation of the formula does

not have to be given however. Another general concept of formal

languages is consistency. Consistency means that if a formula is

derivable in the language, then its negation is not derivable. This is

an important technical detail since, if both a formula and its nega-

tion can be proved in the language, then any formula in the language

can be proved as well, a situation which certainly adds nothing to

the sum of human knowledge.

1.9. The Scope of the Present Work

After this extended discussion of methodologies in accounting,

the ﬁnal section describes the scope of this book and what the au-

thors believe is accomplished therein. The purpose of the book is

to demonstrate how and under what conditions a basic accounting

system can be reduced to a formal proof-based language. When

this is accomplished, a method for controlling such systems through

their derivations is established which is signiﬁcantly stronger than

methodologies used currently in accounting. The exposition in Chap-

ters 2 through 9 employs deﬁnitions, propositions and proofs to for-

malize the system. The deﬁnitions are intended to represent terms,

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26 Chapter 1. Approaches to Accounting Theory

concepts or constructions currently in use and are carefully stated

in order to avoid confusion as to the precise meaning assigned to

them in the book. Propositions are used to state results which fol-

low logically from the deﬁnitions and are in all cases accompanied

by complete proofs. These proofs are meant to demonstrate the

correctness of the propositions and to illustrate the techniques used

in the algebraic and logical analyses.

Since the location of the research is the accounting system after

an economic event has occurred and been quantiﬁed, but before

the output of the system has been used by decision makers, the

method concentrates on the manipulation and processing of inputs

to outputs. These procedures are reduced to a purely algebraic

system which is capable of receiving transaction data, processing

the data and generating information in the form of summaries of

various types.

What happens when the accounting system is reduced to an alge-

braic system is that the entire range of speech is circumscribed. This

means that all sentences or ideas are known to be true, false or out-

side the particular system. Accounting systems can be thought of as

possessing diﬀerent dialects, some quite similar, others nearly dis-

tinct. The control of accounting systems then takes on the quality of

distinguishing very precisely how the systems diﬀer, i.e., which have

larger vocabularies and which are richer in expressiveness. From a

practical viewpoint this allows accountants as designers to match

the expressive power of particular systems to user needs for more

or less expressive languages. In addition the methodology can pro-

vide a means by which to identify situations of data or information

asymmetry and can therefore act as an indirect guide to action.

It cannot be claimed that this reduction is unique, for there are

many diﬀerent opinions about what constitutes an accounting sys-

tem and consequently many ways to construct a formal system. The

intention here is to provide a method which mirrors a speciﬁed basic

accounting system and which is reasonably comprehensible. It is not

the intention of the book to provide a blueprint for an accounting

system which could be programmed and used in practice. Rather

the concern is to allow the system to recognize and act on the trans-

action data itself. The base level justiﬁcation is to develop a full

formal language for a particular aspect of accounting instead of as-

suming that such a grammar could exist and proceeding with partial

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1.9. The Scope of the Present Work 27

constructions or a higher level analysis. The success of this basic

stage of proof-based research in accounting will furnish researchers

with a secure, well established base for future investigations.

Algebraic concepts employed

It is time to be speciﬁc about the algebraic concepts that have

proved useful in the analysis. There are four principal structures

which are used repeatedly and which appear well suited to applica-

tion in accounting, namely:

•balance vector;

•directed graph (or digraph);

•automaton;

•monoid.

These structures will be familiar to most algebraists. A few words

will be given to elucidate their meaning and to justify the claim of

utility in accounting theory.

Abalance vector is a column vector or column matrix the sum of

whose entries equals zero. In this case the relevance to accounting

will be obvious: the zero sum reﬂects the fundamental property of

any accounting system that it must always be in balance. Math-

ematicians will immediately recognize that balance vectors form a

structure with known algebraic properties; they form a submodule

or hyperplane. Balance vectors are able to represent the state of

an accounting system at any instant. They are also capable of en-

coding the transactions that are applied to the system. There is an

important comment to be made regarding signs: for the entries of

a balance vector can be positive or negative. The great advantage

of using positive and negative signs is that the signs take care of

questions of credit or debit automatically; for example, a credit bal-

ance has a positive sign and a debit balance a negative one. The

theory of balance vectors is developed in Chapters 2 and 3, where

their application to accounting is clearly laid out.

The second useful algebraic notion is that of a directed graph.

This is best thought of geometrically, although its deﬁnition is en-

tirely algebraic. The digraph consists of vertices. i.e., points in the

plane, and edges, or lines with a direction, joining certain vertices.

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28 Chapter 1. Approaches to Accounting Theory

The vertices represent accounts and the edges indicate where there

are ﬂows of value within the system. Thus a digraph gives a pic-

ture of how value can ﬂow around an accounting system. While in

general diﬀerent accounting systems might have the same digraph,

for certain special types the digraph determines the system up to

equivalence.

The third concept, that of an automaton, is frequently used in

information science as a theoretical model of a computer. The au-

tomaton is at any instant in a certain state; it reads a symbol on

an input tape, goes to a another state and then writes a symbol

on an output tape. The applicability to accounting is clear: the

states of the accounting system are the balance vectors, the inputs

are the transactions and the outputs are the new balance vectors.

This simple picture can be made more complex in order to represent

further actions of an accounting system, as is expounded in detail

in Chapters 6 and 9.

The ﬁnal concept of a monoid is the most abstract. Every au-

tomaton has an associated monoid, which is an algebraic structure

with a means of combining its elements subject to suitable rules. An

input to the automaton produces a change in the state of the au-

tomaton and thus determines a function from states to states. The

functions on the set of states form a monoid for which the opera-

tion is functional composition; the associated functions generate a

submonoid of this monoid. Despite their abstraction, monoids pro-

vide useful ways of characterizing accounting systems with special

properties, as is shown in Chapter 7.

With the aid of the concept of a balance vector, the deﬁnition

of an abstract accounting system is laid out in Chapter 4 and its

properties are expounded, with numerous accompanying examples.

Relations between diﬀerent accounting systems are considered in

Chapter 5 by using standard constructions from algebra, namely

quotient systems and homomorphisms. The latter are functions

between diﬀerent accounting systems that relate their structures.

An important topic in algebra is the possible existence of algo-

rithms to perform certain computations or to make decisions: what

is at stake here is the question of what can and cannot be computed,

in principle at least. For example, is it possible to write a program

which is able to test the ﬁnal balance vector of an accounting system

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1.9. The Scope of the Present Work 29

and decide if there have been any irregularities during the account-

ing period? The importance of the question is evident. Chapter 8

contains a full discussion of what one can expect to be able to decide

or compute in an accounting system.

In Chapter 9 all the strands come together to form our ﬁnal

model of an accounting system. In this there are ten parameters, so

the model is referred to as the 10-tuple model. It has the capability

to scan and process incoming transactions, keep track of balances,

generate reports on the system, control access by individuals to the

system, and keep track of frequency of application of transactions. It

is also able to test ﬁnal balances. Our main conclusion in this book is

that the 10-tuple model goes a long way towards representing what

is actually going on during the operation of an accounting system.

The ﬁnal Chapter 10 is intended as a corrective after the many

mathematical considerations of this work. It presents a detailed

example of a small company engaged in trade and it exhibits the

accounting system in the form of a 10-tuple model. The aim of

the example is, of course, to help make the case for the relevance

of the model to accounting practice and to justify the claim that

all connection with reality has not been been eroded through the

process of abstraction.