Many-Valued Logic through Its History
Angel Garrido
Department of Fundamental Mathematics, Faculty of Sciences UNED, Paseo del Rey, 9, 28040, Madrid, Spain
Keywords: Mathematical Logic, Vagueness, Uncertainty, Many-Valued Logics, Fuzzy Logic.
Abstract: Our purpose is to contribute here to the searching for the origins of many-valued logics and, within them, as
a special case that of “Fuzzy Logic”, also called by different manners, as Diffuse Logic, either Heuristic
Logic, or `logique floue´ (in French), etc. It is also our goal to relate how was welcome to many-valued
logics in our Iberian Peninsula, which is just another province of the world philosophical universe.
1 INTRODUCTION
Searching for the origins could lead too far and
eventually disperse, which, as we know is not very
convenient for a job pretending to be research. So
we will refer to these first signs that appear in the
East (China, India…), and then we may analyze the
problem of “future contingents”, treated by Aristotle
in Peri hermeneias, or De Interpretatione.
That issue would then central in medieval times,
as during the Scholasticism, with William of
Ockham, and Duns Scotus, or Richard of Levenham,
among others, looked at from different point of
views, for his relationship with determinism and
`Divine Foreknowledge´. Then, this issue is taken
up by Spanish Jesuits Luis de Molina or Francisco
Suarez, and even the great polymath G. W. Leibniz
dedicated his time.
Even then there is a dark time for the logic, and
reappearing in the nineteenth century, philosophers
and mathematicians such as George Cantor,
Augustus De Morgan and George Boole, Gottlob
Frege, ... There was born the new set theory, now
called “classic”, but then also had terrible enemies,
as the then almighty Leopold Kronecker, who from
his professorship in Berlin did everything possible to
hinder the work of Cantor, and the rise of those new
ideas.
2 MANY-VALUED LOGICS AND
LWOW-WARSAW SCHOOL
Parallel to this, there arises a new kind of thought
and way of seeing must be the act of philosophizing:
the Polish Lwow-Warsaw School (LWS, by
acronym). This is happening like tributaries of a
great river and sub-tributaries, departing from
Leibniz, from masters to disciples. Start with
Bernard Bolzano, which influence-much about his
disciple, Franz Brentano. This, in turn, greatly
influence on all his subsequent students.
Among these disciples of Franz Brentano will be
one that particularly interested us. This was the
Polish philosopher Kazimierz Twardowski, who
shared many characteristics with his teacher: love
for precision and clarity of ideas, charisma among
those who treated him, preference for the spoken to
the written word, etc… From his chair in the city of
Lvov spread many of the ideas of Franz Brentano,
adding their own. Led to a circle of people, all they
very interested in a compulsory renewal of
philosophical studies, especially from the point of
view of logic. In a certain sense, served a function
similar (but independent) to the Vienna Circle
(Wiener Kreis), or later the Berlin Circle, because
very different and singular characteristics. It's called
Lvov-Warsaw School. Its members took the logical-
philosophical and mathematical studies at Poland to
the forefront of global world research. It was during
the “interbellum”, or period between the two World
Wars, i.e. ranging from 1918-1939. Then, rouse the
Diaspora, after the war and by the strong communist
dictatorship.
Many notable names among the members of this
school of logic, but could cite to Jan Lukasiewicz,
Stanislaw Lesniewski, Alexius Meinong, Kazimierz
Ajdukiewicz, Tadeusz Kotarbinski, Mordechai
Wajsberg, Alfred Tarski, Jerzy Slupecki, ..., or the
170
Garrido, A..
Many-Valued Logic through Its History.
In Proceedings of the 7th International Joint Conference on Computational Intelligence (IJCCI 2015) - Volume 2: FCTA, pages 170-175
ISBN: 978-989-758-157-1
Copyright
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 2015 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

justly remembered, Andrzej Mostowski. Also must
be cited Jan Wolenski (as vindicator of the LWS´
memory), or Helena Rasiowa, Roman Sikorski, etc.
Among them, one of the most interesting must be
Jan Lukasiewicz, the father of many-valued logics
(MVLs, by acronym). Jan Lukasiewicz began
teaching at the University of Lwow, and then at
Warsaw, but after World War II must to continue in
Dublin and then at Manchester.
At first, Jan Lukasiewicz introduced the three-
valued logic and then generalized to the infinite-
valued. That possibility modulation can be expressed
by a membership function, which is to come all the
unit interval [0,1], instead of being reduced to the
dichotomy of classical logic: True vs. False, 0 vs 1,
White vs Black, etc., allowing the treatment of
uncertainty and vagueness, important not only from
the theoretical point of view, but from the
applications.
Lukasiewicz was the mentor of Alfred Tarski,
whereas officially it was Stanislaw Lesniewski. His
biographers, Anita and Solomon Feferman, state
that, “along with his contemporary, Kurt Gödel, he
changed the face of logic in the twentieth century,
especially through his work on the concept of truth
and the theory of models.” Tarski had gone to the
USA to participate in a conference when Nazi troops
invaded his native Poland and could not return to it.
Over time, he was in California most powerful
logical school of his time; in fact, you can consider
continuing the tradition inaugurated by the Lwow-
Warsaw School, outside the continent in ruins
(Europe). Its `Semantic Theory of Truth´ is one of
the greatest achievements of the human thinking of
all time.
But the writings of Jan Lukasiewicz suffered
after a long slumber, which took care to leave an
engineer Azeri, Lotfi Asker Zadeh, who had studied
in Tehran, he prosecuted studies at MIT and
eventually made landfall as a professor at the
University of California, Berkeley. The one who
would see their potential utility in 1965, firstly
obtaining a generalized version of the classical
theory of sets, now denoted by FST, acronym of the
so-called “Fuzzy Set Theory”, and later, its
application to logic, creating the “Fuzzy Logic”,
particularly with the “fuzzy” proposition´s modifiers
and fuzzy rule-based systems, very useful for
instance, on expert systems, such as the Mamdani,
either the Takagi-Sugeno-Kang, or Yatsumoto´s
method.
Another interesting aspect that we must note is
that even in his own age was not the only Zadeh
proposing similar methods for the treatment of
uncertainty. So, we have the case of Max Black or
Dieter Klaua that without getting resonance
therefore proposed similar ideas. Also the same
Bertrand Russell had treated such issue. We must
not forget that Zadeh, an engineer, knew Jan
Lukasiewicz investigations so explained his
colleague and good friend, the brilliant American
logician Stephen Cole Kleene.
But over time, was in Eastern countries where
these ideas came to fruition, creating a powerful
technological “boom”, with new techniques based
on “fuzzy” concepts. This trend was particularly
strong in Japan, and then it spread to other countries
close to the Japanese country, such as South Korea,
China or India.
Because it is surely more in Japan where are
developed to study the applications of Fuzzy Logic.
Thus, in the case of Prof. Michio Sugeno, already in
his Ph. D. thesis proposed the use of fuzzy integrals
and fuzzy measures. Then there have been many
followers of this movement to seek profit fuzzy
logic and fuzzy measures. Such is also the case of
the known Japanese Prof. Kaoru Hirota, among
others.
And much later these ideas, even more
application came to Western countries, both
European and American, and today it will admit,
with brilliant studies both from a mathematical point
of view and its philosophical implications, as always
connected therewith. Some emerging countries, such
as Brazil or Turkey, are currently dumped in the
investigation of all these theories and associated
methods.
One of the most unique cases in the history of
Artificial Intelligence is to Romania, a country
which many consider poor and backward. It is not at
all in regard to science. We have the most landmark
in the mathematician Grigore C. Moisil, who
introduced the computer at their country, and after
he left a very brilliant school of researchers from
Romania devoted to Computer Science.
After World War II, Grigore Moisil started
teaching mathematical logic as he understood that
the new emerging field of computers would have
enormous repercussions for the social fabric of
society. He continued on the ideas of Claude E.
Shannon on Circuits, and fundamentally the Jan
Lukasiewicz advances on Many-Valued Logics,
where eventually derive the Fuzzy Logic.
The so-called Algebras Lukasiewicz-Moisil are
an attempt to semantic consistency Logical n-valued
Lukasiewicz. His study was followed by Gheorghe
Georgescu and Aphrodite Iorgulescu, from
Bucharest; Cristian Calude, Gheorge Paun
Many-Valued Logic through Its History
171

(membrane computing), and many others, in
different areas.
Also worthy of mention and full admiration the
figure of Prof. Solomon Marcus, whose mentor was
once said Prof. Moisil, and has made great
contributions to various fields of mathematics, such
as Mathematical Analysis, or Computational
Linguistics, of which is one of the founders and
principal contributors.
Of course there and increasingly in-out most of
the new publications. But many of the best papers on
Many-Valued Logic proceeds currently of good
European Universities and very active research
groups; for instance, in Warsaw, Prague, Ostrava,
Vienna, Lisbon, Opole, Barcelona, Madrid,
Toulouse, Pamplona, Granada, etc.
Other remarkable researchers multivalued logics
(and in particular the Fuzzy Logic) have created a
solid and consistent basis for these theories. Such
has been the case for the Czech teachers Petr Hájek,
Charles University, Prague, or Vilem Novak,
University of Ostrava. Both have a powerful
research groups and publications are among the most
valued in this field.
And they are not alone, as for example, in France
we have the important task of disclosure and
investigation of Didier Dubois and Henri Prade.
Or in Germany, the cases of Hans-Jürgen
Zimmerman or Siegfried Gottwald, disciple and
follower of the work of Dieter Klaua.
In Poland they follow the great tradition of the
Lwow-Warsaw School of logic, and this is
complemented with contributions to research the
uncertainty topic through the Rough Sets, introduced
by Zdislaw Pawlak, and continued by Andrzej
Skowron, among others.
3 RECEPTION OF MVLS
(MANY-VALUED LOGICS) IN
THE IBERIAN PENINSULA
There are certain groups, mostly centered around a
“hub”, core or accumulation point, from which new
ideas and impulses radiate growing, in the nucleus –
or kernel- of each of these “core engine” is usually a
-more or less-veteran researcher, well connected and
with prestige.
One of the first Hispanic scholars giving notice
of the new currents was Juan David Garcia Bacca,
who in 1936 published his Introduction to modern
logic, a work praised by I. M. Bochenski and
Heinrich Schölz. Later try so eminent teachers,
between them Alfredo Deaño (editor by Spanish
translation of Lukasiewicz´s selected papers),
Miguel Sánchez-Mazas (studying and interpreting in
deep sense the logico-mathematical works and ideas
of Gottfried Whilhelm Leibniz, as –for instance- the
known “characteristic universalis”), either Jesus
Mosterín (historian and philosopher of science), or
Manuel Sacristán (prosecuted in `academia´ due to
its Marxist point of views), all them very often
clashing against a very conservative and nothing
good context to innovative ideas.
But one good initiative has been the creation in
the old mining town of Mieres, and by the
Government of Asturias, named the `Research
Center for Artificial Intelligence and Soft
Computing´, initially around someone well-known
as Enric Trillas, which can be considered the father
of the introduction of Fuzzy Logic in the Spanish
University curricula. This center has attracted many
of the most famous international researchers, such as
well-known Japanese Professor Michio Sugeno. His
topics of research are very broad working, but
revolve around fuzzy methods, as well as
philosophical implications these carry.
Although I have left for last, a name should not
be omitted landmark, from those that appear only
from time to time in Spain. We are referring to the
Father Pablo Domínguez Prieto (1966-2009),
Spanish philosopher and theologian, who wrote the
first major book in Spain on the Lvov-Warsaw
School, starting for that of his doctoral thesis in
Philosophy, who had come to the Complutense
University at Madrid (1993). Such work is called
Indeterminación y Verdad. La polivalencia lógica en
la Escuela de Lvov-Varsovia (Indeterminacy and
Truth), and was published in 1995, with a foreword
by Arch. J. M. Zyzinski, and showing a very strong
influence by Jan Wolenski. Pablo can be considered
as one of the Spanish forerunner in the study of
MVLs, from the philosophical point of view and in
particular of the great Polish contribution (LWS) to
logic and mathematical fields. A romantic `halo´
comes to close its brief existence, because his
passion for the mountain climbing made him want to
do in the snow Moncayo mountain, after giving
lectures to the nuns of the monastery of Tulebras.
And that was his last top headlong he died, leaving
orphans these Spanish studies again. Left this short
comment must be a tribute to his memory.
Another interesting Spanish author who has been
reporting these new streams of logic is Prof. Julián
Velarde, with its paper “Polyvalent Logic”, or his
book Formal Logic, a volume II belonging to its
History of Logic, all them around the University of
FCTA 2015 - 7th International Conference on Fuzzy Computation Theory and Applications
172

Oviedo and its service publications, or later, to the
Editorial Pentalfa. Also of great interest may be his
work Gnoseology of Fuzzy Systems, where analyses
the deep philosophical connections of these issues.
New research groups have been formed in recent
times, as the Spanish institution CSIC (Consejo
Superior de Investigaciones Científicas), centered in
Barcelona (through the `Instituto de Inteligencia
Artificial´; IIA, by acronym), led by Lluis Godo and
Francesc Esteva.
Or the group that belongs to the UPNA (Public
University of Navarra), headed by Humberto
Bustince.
Either in the University of Granada (lead by
Miguel Delgado Calvo-Flores). Also Francisco
Herrera, Serafín Moral, Juan José Acero, etc.
Either can be considered the University of
Zaragoza, with Prof. T. Calvo.
Even, found some valuable researchers in our
own community, Madrid, as the Complutense,
Autonoma, Polytechnic, Carlos III, etc., universities.
But also in Málaga, Santiago de Compostela,
Oviedo, Almería, Albacete, etc.
In Portugal the origins of the study of AI are
linked to the names of Luis Moniz Pereira, Helder
Coelho and Fernando Pereira, who in 1973 created
the LNEC, within which the following year formed a
division of Computer Science.
In 1977 the programming language called DEC-
10 Prolog is designed, and Helder Coelho who
contributes to divulge in Brazil.
In 1984 it is created the Portuguese Association
for AI (ARIA, by acronym), which maintains its
vitality with many publications and also organizing
very interesting conferences, like this FCTA we
attended.
4 CONCLUSIONS
The prospects offered by AI are immense; in
particular the theory and applications of Fuzzy Logic
in Computer. For example, research to try to model
the human brain using supercomputers, hardware
that will become increasingly smaller and cheaper, a
mission that walk after Google and other large
companies such as IBM. After this "brain mapping"
(`cartografía del cerebro´, in Spanish) of its
operation, how to improve, how to age, etc., are the
efforts of scientists and engineer working on
programs such as the European Human Brain
Project, led by Henry Markram, or American
BRAIN, in which Spanish researchers have
participation.
As Ray Kurzweil say (RK, by acronym; the
Google engineering director at San Francisco and
very clever author of many futuristic well-known
works, "The Singularity is Near". Meaning
`Singularity´ the point in history that computers
equal and then surpass human intelligence.
ACKNOWLEDGEMENTS
This work was supported by the MYCINN´s
Research Project and Group of the Spanish UNED
(Universidad Nacional de Educación a Distancia),
entitled “Polemics and Controversies” (El papel de
las polémicas en la producción de las prácticas
teóricas…). Being its former Principal Researcher
Prof. Quintin Racionero, which recently had passed
away, and from then it is directed by our new
Principal Researcher, Prof. Cristina de Peretti, both
from UNED.
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Ibid., 1983, Logic, Semantics, Metamathematics: Papers
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Trillas, E., Alsina, C., and Terricabras, J. M., 1995,
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Many-Valued Logic through Its History
175