Gelencsér
István
Fuzzy
logic in the science of law
1.
In general we assume that it was the
Megaranians who elaborated the paradoxes aiming at confusing the common,
everyday sense. Indeed: Euclid,=
Eubulid and Stilpon – on the basis of the remaining pieces and refere=
nces
– showed the contradictions of the common sense referring to certain
entities and other contradictions within these entities with examples, and
puzzles, which had been extremely well selected. For this reason even the
contemporaries noted that the Megaranians should not be called σχ=
ολή,
but rather χολή on the basis of their sayings and examp=
les.
Let me mention only two examples, which h=
ave become
well known in the history of logic and philosophy.
On one occasion Menedemos was asked if he=
still
beat his father. It was the method of the Megaranians that they expected si=
mple
yes or no answers to their questions. In case the fellow accepted this
startingpoint they got into a hopeless contradiction. If Menedemos said ye=
s,
it meant that he had beaten his father so far. If the answer was no it meant
that he still beat him.
The other example leads us into the scope=
of
fuzzy logic in a more direct way: as regards the
σωρέιτης it is about a pile of grai=
n of
wheat. Taking out oneone grain of wheat they kept asking whether the grain=
s of
wheat still formed a pile.
The answer was obviously yes, until only =
some, finally
only one grain of wheat remained. Only one grain of wheat cannot form a pil=
e.
Henry Poincar&=
eacute;, one of the greatest mathem=
aticians
of the last century, used the eris=
tic’
examples, when he proved the existence of the paradoxes in mathematics appl=
ying
the same method. Take out oneone grain of sand from a pile of sand until o=
nly
one grain of sand will remain, which is obviously not a pile any longer. Wi=
th a
formula:
A pile of sand=
–
1 a grain of sand =3D a pile of sand.
Appling the complete induction the final resu=
lt will
be as follows:
A pile of sand=
=3D0!
However, the common sense will soon admit=
that
the above examples refer to the abstract ideas of contradiction and quality=
, as
Georg Wilhelm Friedrich Hegel deduced well: as regards the=
contradiction
of the OneMany, QualityQuantity. As long as the grains of wheat form a pi=
le,
we can call a certain number of grains of wheat a pile – this is its
quality. However, after having taken out definite grains of wheat the quali=
ty
of the pile will change, it will mean only x
number of grains of wheat, thus it cannot be called a pile. It is widely kn=
own
as passing of quantity into quality! Anyway, Aristotle’s approach follows similar sequence of ideas as
regards the Sophistic fallacies!
It is peculiar – and it is deeply t=
ypical
of the modern age, but the mathematicians argued differently: they mentioned
the inadequate definition of the pile of sand, which our abstract ideas, wh=
ich
are used in everyday life, cannot be defined by exact methods, and due to t=
his
fact they are of no use for mathematics. It is only true to a certain extent=
; we
have to reverse the idea: mathematics, which is operating with exact abstra=
ct
ideas, cannot handle its abstract ideas, which are obscure and seem ambiguo=
us
for the everyday abstract ideas.
Further on: as regards the example of
ςόρειτες – just like in the c=
ase
of all paradoxes of such nature – it is all about passing of quantity
into quality, i.e.: we have to define, with the help of some methods, how l=
ong it
is about a pile of sand and from which extent do we have to mean a certain =
x number of grain of sands. We can=
not
solve this task with the help of the TrueFalse dichotomy of tradit=
ional
and modern logic. (Consider this: the statement, that it is pile, is true a=
s regards
certain number of grains of sand; however, the same statement is false as
regards x number of grains of sand. Where is the boundary to be drawn?)
Perhaps, the abstract idea of quality is =
even
more important! In fact the quality of the pile of sand is itself – t=
his
is the way we conceive it as an abstract idea. The quality of x number of grains of sand is itse=
lf
again, but in the sense that the entities are themselves. The border is ambiguous between th=
e two
qualities again.
Alfred Tarski =
and the Polish school of log=
ic came
up with the same problem: the solution Tarski applied was that he simply
excluded the everyday abstract ideas from the scope of his logic, he wrote
about object language and meta language, and he did not real=
ize
that with this distinction he hopelessly got into such contradictions, whic=
h could
not be cleared up with those methods, which he had imagined and elaborated,=
not
even defined.
The reference to Tarski leads us to the
abstract basis of logic, in particular the basis of traditional logic and
modern, ambivalent logic.
In fact, since the beginning of mankind&#=
8217;s
awakening to consciousness people have made efforts to get to know the
„world” and to reshape it according to their own image. Knowing the „world̶=
1; has
always meant to be able to cognize and define the actual set of phenomena i=
n a
consistent way with the help of the language, which is the only available t=
ool
for us. However, the history of cognition is full of contradictions. The
attempt to find methods for defining phenomena may be considered heroic, wi=
th
the help of which we can eliminate inconsistency; as you know we have not f=
ound
such method yet.
From the second part of the 19th century =
this
eagerness to consistency increased, when natural and social sciences, as a
result of the industrial revolution, developed dramatically. To be more precise: however, „dramatic development”=
is
the right expression, but it does not express adequately the fact that we w=
ere
able to keep up with the enormous increase of the available material. The
grandiose extent of the factual material enforced the necessity of such dir=
ective
principles, which – again in a specific way –took more and more
simple form. Why not compare =
the
multivarious sensibility of the Greek to problems and/or their method of
approach with ours, and we will be surprised! Let me give you an example.
Stilpon=
of Megara demonstrated a ga=
rden
cabbage. Then he said: this is not the garden cabbage I am demonstrating yo=
u. Because
the cabbage has already existed for many years, thus what I am showing is n=
ot
this cabbage. And in fact: What Stilpon presented was all cabbage or only thi=
s
cabbage. In general: our thinking, our language accordingly, cannot name the
concrete thing, because when I say cabbage, I mean all cabbage in the world
existing at the given moment and not this one or that one. We can perceive =
in
vain this or that object by seeing and touching it; our language cannot exp=
ress
all features of the objects, not even when we try to describe these features
one by one. We abstract ̵=
1; our
language invented this maidenly expression. With this word we refer to the =
fact
that from the endless sea of characteristics, we try to distinguish the
important from the unimportant, and we put the essence of the thing in a
conceptional form. However, this does not affect the basis, i.e. the problem
which Stilpon brought up remained: we do not say, what we think when we name
something with one word, since in our thoughts the object is more multivari=
ous
than what we can express. We =
only
name the general this way, the concrete remains „hidden”. (The
saying is still uptodate: this is such a standingpoint, where our age has
not even arrived!)
Both traditional and ambivalent logic pro=
ceed
from the Aristotelian principles. In particular it is about the commentary =
in
the fourth book of Metaphisics in chapter three! In book four Aristotle dis=
serts
on the science of the existing as existing, and when he examines the most
important principle of science he says that it is impossible that one and t=
he
same verb may belong to and not belong to the one and the same subject at o=
nce
and in the same connection; since by nature this principle is also the basic
principle of all axiom.
After that, as regards logic, it was call=
ed the
law of the excluded middle and the principle of contradiction until the app=
earance
of polyvalent logics.
The two principles of logic phrased in re=
modified
/inverted alternative: all statemen=
ts are
either false or true, but they cannot be both true and false at the same ti=
me!
Particularly, the appearance of the first
significant study about fuzzy logic was not considerable regarded. When in 1965ben Lofti Zadeh, the professor of Berkeley University published his
study the Fuzzy Sets, the philo=
sophers
hardly at all paid any attention to the novelty, which was phrased in the
study: Zadeh neglected the basic principles of Aristotle, and approached fu=
zzy
logic to the everyday way of thinking.&nbs=
p;
Zadeh himself was mainly dealing with the
theory of system. During these researches he realized that it was impossibl=
e to
understand or define systems of great complexity with the help of tradition=
al
logic.
The public opinion of scientists was rath=
er
mixed as regards the settheory, which form the basis of fuzzy logic. Many
people rated it among the scope of the theory of probability, while others
– the majority – rejected it until the 1990’s, because it
seemed to neglect the basic principles of Aristotle, which were considered
sacred. (How much was it about business, or preventing the already establis=
hed
defensive position against the encroachers, because of the fear that the
sponsors might direct the available research funds to somewhere else. Only
those people know who were concerned!)
After all, fuzzy logic gradually gained g=
round,
in the first place as regards the definition of the systems of great
complexity. Professor Mamdani of
London developed the original form of Zadeh to an application, which requir=
ed fewer
calculations.
After 1987 in Japan Sony, Hitachi and the P=
anasonic
National started to distribute consumer products based on fuzzy logic,
which were energy saver with high intelligence, and which are still used in=
our
everyday life. Let’s think about the washing machine, the
airconditioning and video devices, etc., however, the electric shaver and =
the
vacuum cleaner is also based on fuzzy logic. M. Sugeno and T. Takagi=
are worth mentioning here.
The success of the applications in Japan =
and in
the FarEast affected the USA, the homeland of the theory. Until then, the
fuzzy methods had been used in space research and military engineering. As regards the war called the dese=
rt
storm the night target identifying system of the Patriot missiles was based=
on fuzzy
logic.
In Europe the technical applications beca=
me
successful mainly in Germany.
Fuzzy logic became widespread gradually;
however, taking into consideration its new methods it spread rather slowly.=
Thus
in the fields of medical biology and finance; however by now Artificial
Intelligence has also been incorporated. In this field it is mainly applied=
as
regards uncertainty and its treatment, besides nomonotonous and the other
branches of logic.
2.
Traditional and modern logic can only con=
sider
the truefalse dichotomies. However, our cognition, our scientific and
nonscientific orientation in the world does not depend on this dichotomy. =
To
be more precious: we can very rarely say: this and this is true or this and=
this
is false. (Gentle reader, I suggest that you should, for example,
„transform” the basic and atrocious cases of murder, or
unintentional murder into the language of logic; it will take time!)
The introduction of fuzzy logic usually b=
egins
with examples such as the classification of people according to their heigh=
t or
with the abstract idea of speed described by the tools of language. We cann=
ot
define who is considered tall or small, and in consequence we use adjective=
s,
adverbs such as: very tall, tallish, rather small etc. If somebody is 175 c=
entimeters
tall, we considered this person of middle height; however, we might as well
consider him/her: tall, all this is the question of subjective opinion, and=
similarly,
who do we consider old or young.
Classification in these cases, as we see =
it, is
rather ambiguous, dull, uncertain id est fuzzy,
in contrast with the crisp, whi=
ch is
clear, certain and crisp!
It is peculiar and refers to the lack of =
the
philosophical introduction of logic that fuzzy technology is based on logic
defined on the fuzzy sets and not reversely: on fuzzy sets originating from
fuzzy logic.
Then let’s look at the basic concep=
ts of
fuzzy logic and/or fuzzy sets.
The greatest invention of Zadeh was the
introduction of the linguistic vari=
ables.
This new concept for mathematics opened a wide horizon for the diversity of
applications.
The values of the linguistic variables ma=
y be
natural or artificial linguistic expressions. We give these valuables with =
the
fuzzy sets. (The semantics of the fuzzy sets form a different field of logic
accordingly; yet, I am not going to outline that.)
The introduction of the linguistic variab=
les
resulted in unexpected success. Of course it is typical of the AngloSaxons
that the principle of utilitarianism worked here as „the philosophy,
which is directed by the aims”. However, the military utilization was
typical of the North Americans, while the Japanese were rather interested i=
n the
technical innovations. As reg=
ards
us, the linguistic variables have a particularly significant role, since co=
nsidering
the legal facts we can very rarely put them into the category of
truefalse.
The concept of partial membership means to what extent an object is an element=
of
the fuzzy set. We define the degree of adherence to the set by linguistic
variables.
To understand the concept of the membersh=
ip function
we should call forth the concept of characteristic function.
The characteristic function of the crisp =
sets matches
0 and 1 to every element of the basic set. (Where 0 =3D false 1 =3D t=
rue or
reversed.) As regards fuzzy logic the concept of the characteristic function
was extended, and for instance they matched values to the elements of the b=
asic
set from the [0;1] interspace. (I used the word „for instance”,
because a value may be matched to every element of the basic set from other
fixed domain!)
In consequence, to the less or greater ex=
tent
an element is incorporated in the set, the less or greater the value of the
function is as regards the element.
Fuzzy logic and the theory of sets deal w=
ith
two boundaries of the values: 0
– does not at all belong to the set – 1 –belongs complete=
ly to
the set.
In the [0,1] interspace the extent of adh=
erence
is indicated with a number, that is, the grades between nonmember and full
member are given by a membership function, which is a special term of conte=
nt
or fuzzy logic.
Similar levels of uncertainty can also be
encountered elsewhere. The Bayesmo=
del
– which is of great importance in MIresearch – also works with
seemingly similar uncertainties. However, while the Bayesmethod is basical=
ly a
numerical model and is based on probability calculation and welldefined
semantics, the method of fuzzy logic is nonnumerical, and has – to a
certain extent – a wider scope than that of Bayes.
The degree of adherence is marked with
linguistic variables: very high, less high, etc.
The presentation can be done geometricall=
y, or
by providing further functions (Gauss curves are widely used.)
The fuzzy
conclusion is based on the IfThen conclusions that are used in logic. =
The
fuzzy system does not work with real data, but with the values and the
membership functions of the linguistic variables matched to the real data. =
The
result is that care is to be taken for the transformation between linguistic
variables and membership functions in a sense that the real data have to be
fuzzyfied on the input side, and defuzzyfied on the output side.
What does fuzzyfication mean? It means a
transformation that produces fuzzy sets and membership functions from real =
data
on the input side. It is possible here that the various values of the langu=
age
variables can also be matched in a manner described with different membersh=
ip
functions, i.e. several fuzzy rules can be worked out.
Defuzzyification is a result obtained thr=
ough
conclusion  the establishment of the values of the output language variabl=
es
according to a selected method.
After this, let us take a look at a well=
known
example for the above.
The task is to set the washing time on the
washing machine. The degree and the type of contamination is given. The deg=
ree
of contamination is lowmediumstro=
ng,
the type of contamination is not oi=
lyoil
spotsmany oil spots. The washing time (in minutes) can be very short, short, medium length, long,
extended, depending on the above factors.
Here the regulation basis is as follows: =
1/ if the
degree of contamination is =3D medium and the type of contamination is =3D =
many oil
spots, the washing time is =3D long, 2/ if the degree of contamination is =
=3D low
and the type of contamination is =3D many oil spots, the washing time is =
=3D medium
length.
To strongly simplify: if the 45 % degree =
of
contamination and 71 % type of oil spots of the real input data is presente=
d in
a function and is marked on x, the result is 45 minutes.
3.
The Hungarian János Selye wrote down once that certain principles  li=
ke
the definitions of the legal science and mathematics  are rigid laws that =
also
incorporate the related terms and concepts – as against this, a defin=
ition
in physiology is nothing else but a concise summary of the interpretation o=
f a
phenomenon.
Indeed: the terms and concepts that have =
been worked
out by the legal science over millenniums and centuries appear as „ri=
gid
laws” before us. If, however, we take a close look at the question, t=
he
petrified laws of the legal concepts disappear.
What is law?
If we were looking for an answer to this
question, we would get lost in the endless sea of definitions. Obviously, l=
aw
is defined differently in the AngloSaxon and continental systems, as well =
as in
Muslim countries and FarEastern regions.
Here we only try to define law from a log=
ical
viewpoint.
Whatever is the idea or social system that
provides the basis of legal systems, they have one thing in common: namely =
that
something has to happen if certain conditions are fulfilled!
Whatever is a specific legal system, it a=
lways
shows two kinds of logical structure:
1.) if A then B;
2.) B if, and only if, A!
Ad.1/ The first part of the logical sente=
nce
means the occurrence of a specific event or a chain of events. The occurren=
ce
of a specific event or a chain of events is called the facts of the case. T=
he
existence of the facts of the case is the condition to which all laws ̵=
1;
thus also the common law – attach/attached a consequence. If we think=
of
this seemingly simple „facts of the case”, we realise that all
written or unwritten legal norms have the above structure.
Ad.2/ In the science of logic the first
structure is called conditional, and the second is called biconditional.
As can be seen, the biconditional version
contains a strong restriction. A legal consequence can be applied if, and o=
nly
if, the facts of the case fully comply with the strict requirements. Obviou=
sly,
the latter is mainly used in criminal law. If we look at any type of crime =
in
the criminal code, the penalties or measures may only be applied against a
person if his/her action fully complied with all elements of the facts of t=
he case.
If any element of the facts of the case is missing or is not proved, the pe=
rson
cannot be accused of having committed the crime in question. (It is worth
mentioning the neopositivist standpoint for the logical setup of law:
accordingly, all laws are the totality of hypothetical judgements, which can
ultimately be summarised in the If =
A then
it should be B logical sen=
tence.
This „should” evidently refer to the Kantian
origin. In my opinion the usage of the word should raises more
problems than what it solves! In this regard see Hans Kelsen: Reine
Rechtslehre.)
With regard to the application of fuzzy l=
ogic
in law, we immediately encounter the question of the mathematisation of law,
i.e. the question is that the total legal material has to be processed and
translated into the language of mathematics, and thus into the language of
computers.
In this regard, it is worth looking at two
questions: the issues of the axiomatisation and formalisation of law.
However, with regard to the axiomatisatio=
n of
law we can immediately see the contradiction between the social basis of law
and the applicability of the axioms.
The law is the totality of the norms that
organise the social relations. As a totality of norms, law has formal attri=
butes,
as a result, it can be axiomatised; therefore it can also be arithmetised. =
In
fact, Hans Kelsen also came up with a kind of axiomatic legal structure whe=
n he
deducted his clear legal doctrine from a few axioms.
However, the social relations, i.e. the s=
ystem
of relations that forms the basis of law is constantly developing and
transforming. This is why it is so difficult for the legislators to devise
legal regulations for these liquidstatus relations. What exists today will
become nonexistent tomorrow.
In developed societies – that we ca=
ll
capitalism in the lack of a better term – the current form of law has=
to
regulate all social relations without any exceptions. This is a constraint =
that
cannot be avoided by any parliament.
The constant change and/or development pr=
oduces
the consequence that – in logical and mathematical terms – it is
extremely difficult to lead the whole economy back to a few axioms that are
always and everywhere valid. Let us consider: any political, economic or
military event on earth exerts an immediate impact on the developed economi=
es,
and these social changes remain invisible for the legislators for a long ti=
me.
It is not much help if we work with a large number of axioms; this is not
possible anyway due to the complicated calculations.
The constant change in social relations t=
hat
serve as a basis for law against the axiomatisability and/or formalisabilit=
y of
law – this is the contradiction that could not be relieved by the too=
ls
and instruments of either the traditional or the modern – twovalue
– logic. But let us make a more accurate definition: modern logic is/=
was
able to describe the various aspects of law through its own tools, but such=
a
huge apparatus is/was needed to do this that it would render the applicatio=
ns
too complicated or not economical.
Even the law itself also contains linguis=
tic
variables, for example the term of negligence, which is wellknown in all
criminal law systems. One of the basic conditions of negligence is that the
person „carelessly trusts” in the nonoccurrence of a consequen=
ce;
the word „carelessly” is nothing else but a language variable. =
Modelling law is rather difficult. It is =
not by
accident that we rarely meet specific legal programmes apart from
databaseapproach descriptions. This is partly due to what was mentioned ab=
ove,
and partly due to the accuracy or inaccuracy of the legal language.
As is known, the accuracy of the mathemat=
ical
model depends on the accuracy of the applied language. In law, language
accuracy means that we use clearcut, denoted, not vague expressions that
cannot be interpreted in different manners.
Law still does not conform to this latter
requirement. This is also covered by Zadeh’s opinion namely that
strictness is an attribute of humanistic systems, therefore these systems h=
ave
not reached yet the strict rules of mathematical analysis.
Introducing and applying less strict,
„quasi vague terms” in the field of legal logic may enable us to
make a move forward and to build up programmes that are in quasi conformity
with the requirements of mathematical accuracy, i.e. the model, at the same
time these terms are flexible enough to manage the originally essential
feature, i.e. the abovementioned contradiction.
Of course, the scope of application of fu=
zzy
logic in law is rather restricted as – for the time being – we =
are
unable to rewrite all legal references into a mathematical model. The set of
tools of fuzzy logic is also insufficient for this, probably the PAGE (Play=
the
game) logic may offer the way out.
4.
<= o:p>
In consideration of the above, let us tak=
e a
look at a possible field of application. (Of course, I am only describing t=
he
most important conceptual structure for both, as the elaboration will be a =
job
for the programme planning mathematicians and programmers.)
Whatever is the legal system of a society=
, the
court always decides in the form of a single judge or a panel about the giv=
en
facts as well as about the legal consequences to be applied on the basis of=
the
facts. Another important feature of the legal systems is that the help of
laymen is used to a certain extent, and in democracies the laymen element is
built into the legal system. An example for this is the composition of the
panel or the jury.
The same laymen element can also be found=
in
the noncodified, common lawbased legal systems, and from our viewpoint it
does not matter how the legal institution was named.
For the sake of simplicity now we presume=
the
legal system of parliamentary democracies; here we do not make any differen=
ce
between AngloSaxon and continental legal systems. Furthermore, we presume =
the
basic knowledge of criminal, material and penal procedural law.
In both cases the court decides in the ab=
ove
manner. The court makes the decision on the basis of his/her conscience as =
well
as on the basis of the relevant written and case law. The judge’s
decisions are assisted by laymen, the judge is tied to the decision of the =
jury
by virtue of the jury’s resolution. Let us once again presume that the
judge is to decide about guilt or innocence.
T=
oday no
„aid” is available for the decision by the judge (court). Let us
try to find such an aid.
Fuzzy logic manages the guiltyinnocence
dichotomy as extreme values. The related formula is as follows:
&nb=
sp; =
&nb=
sp; 0  guilty =
&nb=
sp; =
&nb=
sp; 1 innocent,
or the other way around  it makes no
difference.
Let us introduce the linguistic variables
between these two limit values, which can be the following: very probably innocent (guilty), probably innocent (guilty), etc. &=
#8211;
the choice of the linguistic variables purely depends on what we wish to re=
late
them to. (Therefore, it is in fact not indifferent what variables we apply:=
the
variables will always be different, depending on whether it is a civil law,
penal or, e.g. labour law case!)
In the {0
; 1} interspace, numerical figures can be matched to the linguistic
variables depending on the level of complexity. If, for example, the numeri=
cal
figure of „very probable” is 0.9,
the figure for less probable is 0.2=
,
and so on!
Another condition to work out our program=
me is
that the judge (court) must possess the data that are indispensable for the
decision, i.e. the facts of the case are to be clarified to an appropriate
extent, that is, the procedure is not forced to be terminated. (The program=
me
may also contain this subcase.)
The decision is based on appropriate data
input, and the available data play the role of „realistic data”,
which is indispensable for fuzzyfication.
Afterwards the programme transforms the d=
ata
for itself through a suitable fuzzyfication procedure, then defuzzifies them
– the programme output for the judge (court) will be a line written in
the form of a language variable.
It is to be highlighted that the items ou=
tlined
above only offer an aid to the decisionmaker, and they in no manner substi=
tute
the conscience of the judge and the laymen! (Although: we may consider the =
idea
of whether conscience can be mathematised? Why not?)
A much more complicated case is when we w=
ish to
fuzzify a trial and a trial series!
In my opinion, the initial steps have jus=
t been
made in order to apply fuzzy logic in law! Obviously, much more complicated
systems – compared to the above outlined one  can also be establishe=
d,
for example, for legislation and for the application of law; trials and tri=
al
series can also be fuzzified in advance – not necessarily in the fiel=
d of
law, but for example in the field of politics, where the level of complexity
may not reach that of the NashHars=
ányitype
mathematical models.
Literature
L.A. Zadeh. Fuzzy sets.
Information and Control, 1965.
L.A.Zadeh. Towards a theo=
ry of
fuzzy systems. In R.E.Kalman and R.N. De Clairis, editors, Aspects of Netwo=
rks
and System Theory, pages 469490. Holt. Rinehard & Winston, New York, 1=
971.
L.A.Zadeh. Outline of a n=
ew
approach to the analysis of complex systems and decision processes. IEEE Tr=
ans.
On SMC, 1973.
L.A.Zadeh. The calculus o=
f fuzzy
if/then rules. AI Expert, 1992.
M.Sugeno and G.K.Park. An approach to linguis=
tic
instruction based learning. Intern. J. of Uncertainity, Fuzziness and
KnowledgeBased Systems, 1993.
M.Sugeno. An introductor=
y survey
of fuzzy control. Information Science, 1985.

8 