Institute of Philosophy and Sociology, Polish Academy of Sciences

in collaboration with

University of Wrocław

NOTE (added on May 30, 2003). The Polish version of this paper was published in *LOGIKA POLSKA OKRESU POWOJENNEGO Próba rzutu oka wstecz*, Nauka Nr 4 2002, s. 157-175. This version (the translation was done by Jan Zygmunt) is supposed to be a faithful counterpart of the original. None the less we rule out neither corrections nor minor improvements, whenever they are indisputable. Fairly recently, various such corrections and improvements have been suggested to us by Z. Adamowicz, J. M Dunn, Wiktor Marek, Roman Murawski, Zdzisław Pawlak, Johan van Benthem. Many thanks for the assistance. We plan to complete correcting this paper by the end of September.

During the 10th Congress of Logic,
Methodology and Philosophy of Science (Florence, August, 19-25, 1995)
I took part in a discussion panel on the situation of logic in
communist countries. This essay, written to celebrate the 50th
anniversary of the Polish Academy of Sciences, is based mainly on the
paper I presented there.*1* The list of people I stayed in
touch with while writing both this paper as well as its previous
version is fairly long.*2 *By saying „thank you”
to all of them I wish to express my gratitude especially to Wojciech
Buszkowski, Andrzej Grzegorczyk, Witold Marciszewski, Wiktor Marek,
Roman Murawski, Jerzy Tiuryn, and Jan Zygmunt. The comments and
materials sent by them were especially helpful.

The scientific outcome of the postwar Polish logic is rich and tremendously varied. It consists of both logic in its basic meaning (comp. section 2) and numerous applications of logical methods. The knowledge of all the results or even some general orientation in all the branches of logic that are developed in Poland requires the competence to which unfortunately I cannot aspire. That is the reason for which I asked so many people for help while preparing this paper as well as the earlier one. Not all of the comments, suggestions or even critical remarks I decided or even was able to take into consideration. That is why the full responsibility for the final content of this essay is mine. I hope, however, that this paper does not contain any serious errors, which would have been hard to avoid without the assistance I was given. Once again I wish to thank for all the help I received.

The notion „logic”, as a name of the science is understood in various ways. On the answer to the question what logic is depends the answer to the question about the scope of the investigations and obtained results. There is also another reason for dealing with this problem. This paper is both intended for the logicians, who – as I hope – will treat it as an useful attempt to look at their achievements as a whole, and for the specialists in other areas, who would like to become acquainted with Polish logic of the 20th century’s second half. The latter will find in this section the explanations which will make easier to understand the „proper” parts of the paper.

Logic in its basic meaning is the formal theory of reasoning. Thus, it is a theory whose chief concern are the conditions which an argument should satisfy in order to be carried out in a sound way. The soundness of the „rules of inference” is one of the criterions for the correctness of the reasoning. Those rules should ensure that the conclusion is valid if it has been derived from valid premisses. In the course of logical analyses of the reasoning process the premises and the conclusion are sentences. They can be also alternatively regarded as someone’s beliefs or suppositions. A sentence is a syntactic notion. Analysis of the shape of the sentence (logicians also have to deal with it) requires the notion of syntactical rules. On the other hand, the notion of truth is a semantic one. One cannot define it only by syntactical means. It requires the meaning of the expressions we use in sentences; since one constantly needs to refer to the language customs and conventions.

Logical issues are strongly related to and often overlap with linguistic issues, both syntactically and semantically. It is true that logicians limit their linguistic analyses to ideal languages (satisfying conditions that are not satisfied by any natural language), but those languages are not at all „artificial”, as it might be suggested by the often used and unfortunate terminology. The relation between them and natural languages is similar to that between material points and real three-dimensional physical objects in physics.

If logicians, instead of analysing sentences and relations between them, decided to analyse statements that are understood as someone’s beliefs or suppositions, then the terminology used would reveal connections between logic and psychology. From the point of view of the latter, reasoning – regarded by logic as the operation on sentences – is a mental activity of some special kind. Logic, however, is not a part of psychology, as it was thought to be until the beginning of the 20th century. Not because it avoids using the notions of psychology, but because it is a formal discipline. Like every formal discipline it is concerned with some „arbitrarily” chosen set of assumptions that define „logical systems”. Those systems are the subject of logic. „Arbitrarily chosen” does not mean „any possible”. It only means that if logicians define a system, then they have a right to follow their own intuitions concerning the virtues of the system they build – the virtues that are sometimes theoretical, and often practical. Logicians, like other scientists, do not disdain the problem of usefulness of their work.

Usually (not always though) a system of logic is defined as a set of language formulae that are „logical tautologies”, i.e. that are true regardless the way we interpret them. This approach to logic is called „sentential”. Alternative way of defining logic consists in searching for the assumptions that define in the intuitively suitable manner the notion of logical inference. This approach to logic is called „inferential”.

When one is led by different intuitions, while defining a system of logic, one obtains different „logics”. Besides classical logic, which is a formal base for the whole standard mathematics, there were many other logics developed, such as intuitionistic (the one that rejects the law of excluded middle), modal or many-valued logics.

Logic, quite reasonably, is regarded by the
mathematicians as one of the parts of the foundations of mathematics.
Other ones are set theory and theory of algorithms, or more
generally, the recursion theory. Foundations of arithmetics also
belong to the foundations of mathematics, as they are concerned with
the logical analysis of various kinds of formalizations of
arithmetics.This way it is possible to classify as a part of logic
(and it is done so sometimes) the foundations of geometry, the
foundations of algebra, etc. Category theory, universal algebra, and
relatively young discipline called the complexity theory are also
regarded as the branches of modern logic.*3*

Logical foundations of computer science are another important and strongly related to logic discipline. Tremendous importance of logic for computer science has several reasons. Firstly, the structure of computing machines is based on the laws of logic. Secondly, logic provides necessary tools that are needed for design and analysis of programming languages. Thirdly, „cognitive engineering”, that is data bases and expert systems design uses fruitfully logical tools. The language of the predicate calculus and the laws of logical deduction set the general frames for computer languages. Like in case of the foundations of mathematics, the foundations of computer science are also regarded as a part of logic in a general sense.

The foundations of mathematical linguistics and formal foundations of communication theory are two other disciplines which employ the notions and techniques of logic. Also in these two cases it is extremely difficult to draw the border line between logic and those issues that do not belong to logic. Finally, there is another very important neighbour of logic, i.e. the theory of knowledge. If I do not write about it here in more detail, it is because the theory of knowledge is still a conglomerate of many concepts without a well established core.

In this survey I will try to keep a reasonable balance between narrow and wide notion of logic. It is not easy. I have to ask in advance for understanding all those who will think that I mention the results which, according to them, do not belong to logic, and those who will reckon that some important achievements of Polish logicians are not mentioned at all.

The achievements of Polish logicians of the years
1920-1939 have won tremendous worldwide recognition. As a result, the
term „the Polish School of Logic” became popular within
the international community of logicians. Let us note that the work
of Polish logicians overlapped in a very substantial way with the
work of another formation known as the „Lvov-Warsaw School of
Philosophy”.*4*

The Polish school of logic seen from today’s perspective appears to be a period of the achievements that are hard to overestimate. At that time K. Ajdukiewicz formulates his conception of categorial grammar, J. Łukasiewicz inspired by some ideas of T. Kotarbiński developes his idea of many-valued logic, A. Lindenbaum and A. Tarski introduce the method of algebra of language known today as the method of Lindenbaum algebra, A. Tarski publishes his fundamental papers on the conception of truth and deductive systems, S. Leśniewski following T. Kotarbiński’s reism idea makes an attempt to form ontology meant to be a system alternative to set theory, L. Chwistek publishes his works on simplified theory of types, Janina Hosiasson-Lindenbaum examines methodological and logical aspects of probability theory, J. Łukasiewicz and A. Tarski develop an algebraic treatment of logical matrices, S. Jaśkowski (independently of the German logician G. Gentzen) introduces the system of „natural deduction”, that is a system of rules of inference that make the „formalized” inference similar to the „natural” one. This list consists of the most spectacular achievements and obviously is not complete.

Logic
of the interwar period was developed both in the philosophy and in
the mathematics departments. In particular, Leśniewski and
Tarski worked in the mathematics departments. Ajdukiewicz,
Kotarbiński, and Łukasiewicz represented a „philosophical
wing” of logic. But the cooperation between „mathematicians”
and „philosophers” was systematic and quite close.*5*

The Second World War was a disaster for both the Polish logic and the whole Polish science. For three obvious reasons.

Some of the logicians, including those who made a substantial contribution to the development of the discipline, did not survive. Priest Jan Salamucha, historian of logic and a close collaborator of Łukasiewicz and Bocheński, was killed during the Warsaw Uprising in 1944. Because of their Jewish origin Adolf Lindenbaum, his wife Janina Hosiasson-Lindenbaum, Mojżesz Presburger, and Mordechaj Wajsberg (author of the important papers on many-valued logics) were murdered by the Nazis. A. Tarski staying abroad avoided this tragic fate.

When the war was over many Polish logicians who were lucky enough to leave Poland earlier decided not to come back to the country. Thus, for instance, Józef M. Bocheński settled in Switzerland, Czesław Lejewski in England, Jan Łukasiewicz in Ireland, and Henryk Hiż, Bolesław Sobociński, and Alfred Tarski in the USA. A. Ehrenfeucht and J. Mycielski decided to emigrate later on.

And finally, the war destroyed the whole structure of cooperation among scientific institutions. The Jan Kazimierz University of Lvov and the Stefan Batory University of Vilnus ceased to exist. So did numerous research teams. The continuity of teachnig, publishing, and organizational work was broken. The manuscripts of many papers prepared for publication were destroyed. But still, during the war, the illegal „underground university” worked in full swing, and logic was taught or studied there by: K. Ajdukiewicz, Janina and Tadeusz Kotarbiński, A. Mostowski, J. Salamucha, Z. Zawirski, H. Hiż, Z. Czerwiński, A. Grzegorczyk, J. Pelc, R. Suszko, K. Szaniawski, and many others.

Although nobody doubted that both political and social life would undergo dramatic changes, the end of the war brought new hopes. Despite tremendous losses, Polish logic did not cease to exist and quickly started to rebirth. This applies to both logic in philosophy and mathematics (cf. the next section) departments.

Tadeusz Kotarbiński and his wife Janina started their scientific activity in Łódź which became one of the main scientific centres (the Łódź University was founded in 1945) at the time of the rebuilding of Warsaw. Among many young people grouped around them were Jerzy Pelc, Marian Przełęcki, and Klemens Szaniawski. Kazimierz Ajdukiewicz became the Rector of the Poznań University. Besides his rector duties he gathered together a team of people who dealt with various aspects of logic. Under his leadership worked Marcin Czerwiński, Jerzy Giedymin, Seweryna Łuszczewska-Romahnowa, and Roman Suszko. Maria Kokoszyńska-Lutman, well known and recognized for her analyses of Tarski’s notion of truth settled in Wrocław, where also worked Henryk Mehlberg until he decided to emigrate to the USA. In Cracow there was a group of logicians led by Izydora Dąbska and Roman Ingarden. Tadeusz Czeżowski settled in Toruń where Leon Gumański was one of his students.

Except for Ingarden (one of the most eminent Polish philosophers of the 20th century) all the mentioned leaders of the research groups were outstanding representatives of the Lvov-Warsaw School, the formation whose founding father was Kazimierz Twardowski. Ajdukiewicz and Kotarbiński were its most influential members. Not all of the leaders were logicians, even in wide sense of the notion of logic. Izydora Dębska was not close to the subject. All of them, however, appreciated the significance of logic and were ready to support its development. Zygmunt Zawirski was an outstanding example of such an attitude; he was a philosopher and a methodologist, very familiar with the issues of contemporary logic, and was an editor of „Kwartalnik Filozoficzny” („Philosophical Quarterly”), and worked as a professor at the mathematics department of the Jagiellonian University in Cracow. Under his supervision at the J.U. S. Jaśkowski (in 1945), A. Mostowski (in 1945), and J. Słupecki (in 1947) completed their habilitation theses, and R. Suszko (in 1945) received his master degree.

Groups of logicians formed in the mathematics departments usually were labelled as those dealing with the foundations of mathematics or were parts of differently labelled institutions. Such a way of naming the institutions surely had a reasonable justification. At the same time it was a camouflage, since no scientific discipline was perceived by the communist ideologists as neutral, especially logic (as a part of philosophy, which was the main ideological discipline) became the subject of their special attention. In the Soviet Union for instance, the logicians were demanded to supress formal logic by so-called „dialectical logic”, rediscovered from the writings of Hegel and other „classics” of Marxism. The same was expected from other countries of the „block”, especially from Poland.

In Wrocław logic was developed in two groups: one led by professor Czesław Ryll-Nardzewski and the other by professor Jerzy Słupecki. C. Ryll-Nardzewski, who was already the author of a few extremely important papers on logic, grouped around himself several talented people, among them L. Pacholski, B. Węglorz, A. Wojciechowska. J. Słupecki, known for his work with many-valued logics, cooperated with young talents such as Witold A. Pogorzelski, Ludwik Borkowski, Bogusław Iwanuś, Tadeusz Prucnal. One of the most brilliant logicians of the postwar period was Jerzy Łoś, whose talent was discovered by J. Słupecki, although their cooperation did not last long.

Stanisław Jaśkowski began his work in the mathematics department of the Nicolas Copernicus University in Toruń in 1945. Later among his collaborators there were Jerzy Kotas and August Pieczkowski.

Polish postwar logic owes a great deal to professor Andrzej Mostowski, who pursued his academic career in Warsaw. He was such a prominent figure that I shall describe his achievements in a separate section.

There are a few factors that make A. Mostowski an exceptional scholar.

Firstly,
he was an unquestionable scientific authority. Some of his results
concerning the foundations of mathematics were of a breakthrough
significance. His contribution to the set theory was immense. He was
one of those who, together with A. Tarski developed the theory of
decidability (the search for algorithms which enable us to identify
theorems of particular theories). He suggested an algebraic
interpretation of the quantifiers and initiated the research of the
so-called generalized quantifiers.*6* He also made a
substantial contribution to model theory.

Secondly, he was a scientist of a very well established reputation in many international institutions. Thanks to his connections Poland was visited by the most eminent researchers in logic and foundations of mathematics.

Thirdly,
he had a unique skill to look at logic as a stricly mathematical
discipline without losing its deep philosophical content. His
handbook *Logika Matematyczna* („Mathematical Logic”)
published in 1948 is a good example of presenting logical
invetigations showing their technical difficulties and philosophical
significance at the same time. Two other workks by him are of a
similar character: *The Present State of Investigations on the
Foundtations of Mathematics* and *Thirty Years of Foundational
Studies*.*7* The latter will be discussed in section 14.

Of
those who were his students and close collaborators one should
mention the most important names of the postwar logic and foundations
of mathematics: Zofia Adamowicz, A. Ehrenfeucht, A. Grzegorczyk, W.
Guzicki, W. Marek, H. Rasiowa, R. Sikorski, P. Zbierski, and many
others.*8* Thanks to Mostowski research in logic has gained
high prestige in the department of mathematics at the Warsaw
University, where he was the head of the Section of Algebra, the
Section of the Foundations of Mathematics, and of the Institute of
Mathematics at the Polish Academy of Sciences. He was a member of the
Polish Academy of Sciences.

During my talk at the congress in Florence I said the
following: „One might say, not being entirely wrong, that for a
long period of time the chief Polish seminar on logic was held in
Berkeley, California, in the residence of Alfred Tarski who kept very
close cooperation with his friends and colleagues from Poland,
Andrzej Mostowski in particular.” From the 40s Berkeley was
visited by W. Szmielew, A. Mostowski, A. Ehrenfeucht, J. Łoś,
J. Mycielski, L. Szczerba, L. Pacholski, and others, who took part in
various research programmes coordinated by Tarski.*9*

There
is no doubt that Alfred Tarski was one of the most outstanding
logicians of the 20th century. He also influenced immensely the
development of the research in semiotics and philosophy.
Philosophical aspects of his theory of truth (Tarski was not only
fully aware of them, but also put them forwards in his papers
published in philosophical periodicals) are even at present the
subject of heated discussions and analyses.*10* Intuitions
associated with the notion of truth formed a background for his
concept of the consequence operation, as well as (cf. section 15) the
model theory.

The research
center created by Tarski in Berkeley was one of the most influential
centers of the foundations of mathematics in the world. Despite many
difficulties (in gettig the permission to travel abroad) Polish
logicians were able to keep in touch with him and his students (who
are among the most outstanding American logicians nowadays) which was
hard to overestimate for the Polish postwar logic.*11*

In four previous sections I tried to
give some evidence that despite severe war losses Polish logic
continued to develop.*12* But the growing ideological
pressure and some decisions of a political or administrative nature
were a real danger. Those factors had to have consequences in the
situation of logic.

Nowadays, hardly anybody is aware that Jaśkowski’s discussive logic put forward by him in 1948, was an attempt to form a logical system which admits controversies and contradictions in discussions. Jaśkowski did not care to call those controversies „dialectical” or to use the word „dialectical” on whichever occasion in his papers. Not to mention that he never cared to quote any of the so-called „classics” of the Marxism (it was possible in Poland, while in Russia that would be an act of desperate courage).

Jaśkowski’s papers as well as a few more ones published at that time by Maria Kokoszyńska-Lutman, L. S. Rogowski, and T. Kubiński were an attempt to demonstrate implicitly that some ideas of dialectical logic can be stated reasonably and analysed with the use of formal methods. This attempt was bound to fail, since dialectical logic was supposed to be a part of the so-called „Marxist dialectic” and not one of the systems of logic.

The indoctrination of science, along with some institutional changes were to be launched at the First Congress of Polish Science in 1953. Logic was not the main aim of the „ideological offensive” there. The Lvov-Warsaw School was. Party ideologists had to open the process of full eradiction of any politically wrong philosophy (different from „diamat” –dialectical materialism, and „hismat” – historical materialism) from the academic life. Poland could not remain an oasis, free from the rules that were obeyed in other countries of the block.

The public criticism of the Lvov-Warsaw School was directed against its most eminent representatives: K. Ajdukiewicz, T. Kotarbiński, M. Ossowska, and S. Ossowski. Let us note however that as a result of this campaign none of the „bourgeois” philosophers was expelled from the academic life. But there were two precautions taken to prevent them from teaching philosophy and other social sciences. Some of the philosophy professors were employed as the chairs of logic. Others got their jobs in the Polish Academy of Sciences.

Although the future of logic was more and more jeopardized one has to notice that one of the chief Party’s ideologists – Adam Schaff, who opposed Ajdukiewicz in the 1953 debate, supported him in organizing Section of Logic at the Polish Academy of Sciences. That section eventually became a part of the Institute of Philosophy and Sociology of the Academy.

At the same
time K. Ajdukiewicz starts *Studia Logica*, a periodical with
the following editorial board: Kazimierz Ajdukiewicz (editor in
chief), Leszek Kołakowski, Tadeusz Kotarbiński, Andrzej
Mostowski, and Roman Suszko (secretary). The presence of L.
Kołakowski unfortunately does not mean that this today world
famous philosopher began his career as a logician. He simply was the
only person in the editorial board who was a Party member.*13*

Shortly the periodical started to appear regularily, with extended editorial board, and the papers were more and more often published in other languages. The papers of SL were written both by logicians from philosophy and mathematics departments. Mathematical logicians were also the members of the editorial board. From the very beginning A. Mostowski collaborated with SL very closely. After K. Ajdukiewicz’s death in 1963 Jerzy Słupecki becomes the editor in chief. In 1971 Zdzisław Pawlak and in 1978 Helena Rasiowa join the editorial board; Rasiowa takes the position of one of the chief editors in 1979.

In
1976 the periodical undergoes an important change. Thanks to the
efforts of the editors of that time – Stanisław Surma,
Klemens Szaniawski, Ryszard Wójcicki (editor in chief), and
Jan Zygmunt (secretary) – the international editorial board is
established.*14* The process of transforming a periodical in
an international one was not easy and had to be cleared by the State
Security Police.*15*

Since
1953 there have been published over 150 issues of SL. It becomes an
international periodical in 1976 not only because English is the sole
language of publication and not because of the international
editorial board, but also because of the new publisher. For some time
it was published by Ossolineum and North-Holland as a co-publisher,
replaced later by Kluwer Academic Publishers. In 1991 (after losing
some financial support), Kluwer becomes the sole publisher of SL.*16*
It is currently edited by Ryszard Wójcicki and Jacek
Malinowski. Since 2000 it has been published in three volumes a year,
and has been accompanied by the series of books Trends in Logic,
Studia Logica Library since 1995.

*Studia Logica* has not been the
only periodical suitable for papers on logic. Those mathematically
oriented papers have been published in *Fundamenta Mathematicae*,
an eminent journal established in 1920 which quickly became the
leading international publication on the foundations of mathematics.
This journal was a vehicle for a vast number of papers written by the
logicians from all over the world.

Another
important periodical on logic in Poland have been *Reports on
Mathematical Logic*, formed in 1973 by Stanisław Surma. A
newsletter called *Bulletin of the Section of Logic of the
Institute of Philosophy and Sociology of the Polish Academy of
Sciences **17* have been active in promoting both
research and international cooperation. In 1993 Jerzy Perzanowski
started a periodical on philosophical logic called *Logic and
Logical Philosophy*.

Since
1974 *Fundamenta Informaticae* (a journal initiated by Zdzisław Pawlak) has been published under the
editorship of Helena Rasiowa (and Andrzej Skowron, after her death).
This journal covers applications of logic to computer science and is
published by IOS Press.

This
survey of publications on logic would not be complete without
mentioning the series *Biblioteka Myśli Semiotycznej* (*The
Library of Semiotics Ideas*) initiated by Jerzy Pelc. It is hard
to overestimate its importance, also for logic. 46 volumes of this
series gives the notion of richness and variety of the research in
the areas where logic meets linguistics.*18*

An attempt of penetrating only the
most important achievements of the research in the foundations of
mathematics would require much more space than this section.*19*
One can find a highly competent survey of this matter in A.
Mostowski`s work *Thirty years of Foundational Studies *(cf .7),
concerning the development of the subject in 1930-1964. The list of
references in this book consists of 244 items. Here are the names of
Poles that are included there.

A.
Ehrenfeucht is an author of the paper on the methods of game theory
applied to the problem of decidability of the first order theories,
and co-author (together with A. Mostowski) of the paper on model
theory. There are six papers by A. Grzegorczyk mentioned there (one
of them is a joint paper written with A. Mostowski and C.
Ryll-Nardzewski). Grzegorczyk’s papers are, with one exception,
on the problems of decidability and computability. The joint paper is
concerned with the foundations of arithmetic. The decidability theory
is also a subject of the paper by A. Janiczak. The name of Jerzy Łoś
opens a list of six papers, mainly on model theory. One of them
written with C. Ryll-Nardzewski explores the theory of representation
of Boolean algebras and the problem of Stone theorem’s
equivalents. The paper by Łoś and Suszko is concerned with
the operation of summing the models. There is E. Marczewski’s
paper on abstract algebra mentioned there and a large work by S.
Mazur on computational analysis. Mostowski mentiones four of his own
works. H. Rasiowa is an author of the papers on algebraic
representation of some non-classical logics. Algebraic methods are
the subject of a work by H. Rasiowa and R. Sikorski. They are also
the authors of the monograph *The Mathematics of Metamathematics*
(PWN, 1963, position [173] on the list). Two papers of C.
Ryll-Nardzewski are quoted there. One of them presents a
characterization of categorical theories, in the other one the axiom
of induction is examined and it is proved that Peano arithmetic is
not finitely axiomatizable. In the paper by R. Sikorski the notion of
a metric space is applied to the analysis of intuitionistic logic. W.
Szmielew in her paper proves the famous theorem stating the
decidability of the elementary theory abelian groups. Tarski’s
name appears nine times as an author and several times as a
co-author. The monograph *Undecidable Theories* (North-Holland)
is quoted as a work of A. Tarski, A. Mostowski, and R. M. Robinson.
This book set an important direction in research concerned with the
search for general methods of proving the undecidability of
formalized theories in general and various mathematical theories in
particular.

If
one wanted to extend the list of the names and results that have been
important since 1964, one needed to include the following. Z.
Adamowicz (results concerned with „weak” arithmetic and
arithhmetic with open induction; she is a co-author (with P.
Zbierski) of *Logika Matematyczna*, PWN, 1991; translated as
*Logic of Mathematics*. A Modern coures of Classical Logic, John
Willey & Sons, 1997), K.R. Apt (second order arithmetic and
foundations of computer science), A. Ehrenfeucht, A. Grzegorczyk
(author of *Zarys Logiki Matematycznej*, 3rd ed., PWN, 1973;
translated as *An Outline of Mathematical Logic*, Kluwer 1974 ),
W. Guzicki (forcing), M. Krynicki (generalized quantifiers), A.
Mostowski (reflexivity of Peano arithmetic), R. Murawski (expandability of models of Peano arithmetic to models of second order arithmetic), S. Krajewski (nonstandard satsfaction classes), W. Marek and M. Srebrny
(their investigations were concerned with higher-order arithmetic and
set theory, in particular with relations between Zermelo-Frankel
theory and Kelley-Morse class theory), H. Kotlarski (automorphisms of nonstandard models of Peano arithmetic), J. Mycielski (infinite
combinatorics, universal algebra, and the analysis of Hugo
Steinhaus's axiom of determinacy), Z. Ratajczyk, C.
Ryll-Nardzewski, L. Szczerba (foundations of geometry, the notion of
interpretability of theories formulated in various languages), T.
Traczyk (Hilbert spaces, quantuum logic), P. Zbierski (descriptive
set theory), Z. Vetulani (foundations of second and higher-order
arithmetic and artificial intelligence).

It
is obvious that also after 1964 the leading role in the development
of the foundations of mathematics and its important branch –
model theory (comp. section 15) – was played by A. Mostowski
*20* and indirectly by A. Tarski.

This theory has a distinguished position within the foundations of mathematics. Initiated in the 40s and the 50s by the works of L. Henkin, A. Robinson, and A. Tarski, forms an important part of mathematical logic. It deals with the relations between the language in (its formalized version) and its „models”, i.e. the structures which may become what the language expressions (if they are properly interpreted) refer to.

As it was
pointed out in the previous section, many important results in
model theory was obtained by the Poles. Especially significant,
because of its numerous applications, was the Łoś’s
ultraproduct theorem (based on some Skolem’s ideas of the 30s).
Boolean models method of H. Rasiowa and R. Sikorski *21 *was
another important result. It enabled to extend the field of model
theory to non-classical logics. In the 60s and the 70s model theory
was developed by L. Pacholski and B. Węglorz, later by H.
Kotlarski, and quite recently important and widely recognized results
were obtained by Ludomir Newelski.

This branch of logic (overlapping to some
extent with universal algebra) in its history goes back to
Lindenbaum-Tarski’s idea of the algebra of language. To make a
long story short: it deals with the relations between the laws of
logic and theorems that characterize algebraic operations.* 22*

The
basic tools for examining the relations between algebra and logic
(more precisely, between algebraic structures and particular systems
of logic) were established by Alfred Tarski, his collaborators and
followers. The monograph by H. Rasiowa and R. Sikorski *The
Mathematics of Metamathematics*, PWN, 1963 (cf. 20), an
outstanding source of information in this area, was also a source of
inspiration for many authors from all over the world to carry out
research in algebraic logic. In 1974 H. Rasiowa published her
monograph *An Algebraic Approach to Non-Classical Logic* *23*
in which she extended her earlier results to the large class of the
so-called non-implicative logics. This class includes modal logics
among others.

Besides Rasiowa and Sikorski algebraic logic was the research subject of Cecylia Rauszer, who was interested in logics with „constructive falsehood”. A lot of important and difficult results were obtained by A. Wroński and his group of logicians (Section of Logic, Philosophy Department of the Jagiellonian University in Cracow). Their research, carried out partly in cooperation with the Japanese logicians, made important contributions into the theory of pseudo-Boolean algebras, BCK algebras, equivalential algebras and logical systems related to these classes of algebras. In Toruń, algebraic methods were developed by J. Kotas and his group.

The issues of algebraic logics are closely related to those of logical matrices (cf. section 19).

The research in this branch of logic, which is still of growing importance, was initiated by H. Rasiowa and her group (Grażyna Mirkowska, Ewa Orłowska, Cecylia Rauszer, Andrzej Salwicki, Andrzej Skowron, Jerzy Tiuryn, Anita Wasilewska, to mention but a few). The activity of those people was oriented toward developing applied logic such as algorithmic logic, logic of programming (correctness of programs), logic of information (data bases), and logic of knowledge (expert systems). An important role in the development of the foundations of computer science was played by Zdzisław Pawlak („rough sets” theory, Pawlak machines), and also by Andrzej Blikle, Beata Konikowska, Józef Winkowski.

The research in mathematical foundations of computer science initiated in Poland by H. Rasiowa and Zdzisław Pawlak, since the 90s has been developed by the by the group of J. Tiuryn in the Department of Mathematics, Mechanics, and Computer Science at the Warsaw University. This group consists of Damian Niwiński, Jerzy Tyszkiewicz, Paweł Urzyczyn, and Igor Walukiewicz, and deals with numerous aspects of modern computer science: lambda calculus, functional programming, type theory, modal logics, and theory of finite models.

The first postwar papers with reference to
Tarski’s consequence theory were published by J. Łoś,
J. Słupecki and W. A. Pogorzelski. One of the key papers in this
subject was „The algebraic treatment of the methodology of
elementary deductive systems” *24* by J. Łoś.
The research in this area was carried out also by A. Grzegorczyk and
R. Suszko among others. Suszko was the one who coined the notion
„abstract logic”.*25*

There
are two important aspects of the theory of consequence. Firstly, a
methodological aspect. Tarski developed the consequence theory as a
theory of „deductive systems”. This notion is a formal
equivalent of the notion „theory” or even „a system
of beliefs”. Logical analysis of a deductive system has
numerous and significant methodological implications. Secondly, the
notions of consequence lets us see the limitations of the
propositional conception of the system of logic and brings into
prominence the inferential concept of those systems.*26*

Many deep results concerning the so-called structural completeness of logical calculi (the notion was also examined abroad) were obtained by W.A. Pogorzelski (who introduced this notion) and his students: T. Prucnal and P. Wojtylak among others.

The area of the consequence theory also includes the research concerned with the so-called non-monotonic reasoning. This kind of research was carried out by Witold Łukasiewicz, Wiktor Marek and Mirosław Truszczyński.

The notion of logical matrix, understood as a set of „logical values” (e.g. truth and falsehood in the simplest case) that can be taken by propositions, appeared in the beginning of the 20th century. J. Łukasiewicz introduced many-valued matrices while defining many-valued logics. A. Lindenbaum proved that each propositional logic (comp. section 2), i.e. the set of tautologies, has an adequate logical matrix that characterizes it in a unique way. This result as well as one of the papers by A. Tarski and J. Łukasiewicz show that logical matrices can be used as certain generalizations of algebraic methods (the latter were discussed in section 16).

J. Łoś
and R. Suszko tried to find a matrix representation of some
consequence operation. Their result however required some corrections
(given by R. Wójcicki).*27* These two results (the
former is of vital importance) gave a new direction in a very
general, technically difficult, and philosophically interesting field
of research.

This
problem was my special concern in 1970 when I initiated a separate
research project. I started to examine the representations of
consequence operations together with J. Czelakowski, W. Dziobiak, J.
Hawranek, M. Tokarz, T. Prucnal, R. Suszko, J. Zygmunt, A. Wroński,
and P. Wojtylak. Some contributions to this project were made by the
logicians from abroad (e.g. W. Rautenberg), although this subject is
regarded as typically „Polish”.*28* It was
continued and developed later on by J. Czelakowski and W. Dziobiak as
well as by Spanish and American logicians, especially by J.M. Font,
W.J. Block, and D. Pigozzi.*29*

These areas belong to the foundations of mathematical linguistics. They were explored in Poznań by the logicians grouped around Wojciech Buszkowski. The notion „type logics” refers to the set of operations that are used while creating the complex language expressions out of simple ones. „Type logics” belong to the family of substructural logics.

From
the historical point of view this kind of research was initiated by
some of the results of Ajdukiewicz, Bar-Hillel, and refers to the
calculus of Lambek (1958). Buszkowski`s results (as well as those of
M. Kandulski and W. Zielonka) brought a thorough grasp of some of
the fundamental problems of the mathematical linguistics and its
applications. They explored the algebraic and computational
properties of various kinds of languages, especially the so-called
„tree-like languages” generated by categorial grammars,
the algorithms for finding minimal categorial grammars, and the
relations between categorial grammars and Chomsky`s generative
grammars.*30*

In Poznań center the research in logic and linguistics is carried out also by T. Batóg and J. Pogonowski.

In 1934 S. Jaśkowski published an
article *On the Rules of Suppositions in Formal Logic*
(published as the first issue of *Studia Logica* series *31*).
Jaśkowski’s conception and Gentzen’s conception of
natural deduction were discovered independently of each other and
published almost simultaneously.*32* Suppositional system of
logic was the base for the computer program „Mizar”
designed by Andrzej Trybulec (Warsaw Univerity’s branch in
Białystok, currently an independent university). Its function is
well expressed by the title of one of the research projects
(coordinated by W. Marciszewski): „Systems of logic and
algorithms for the computer testing of the correctness of proofs)”.*33*

A very important institution, organized under the auspices of the Polish Academy of Sciences, was the Conference on the History of Logic. It was chaired by T. Czeżowski, J. Słupecki, S. Surma, and since the 80s it has been organized exclusively by A. Wroński, Section of Logic of the Jagiellonian University in Cracow. Substantial contribution to the subject of the history of logic was made by W. Marciszewski, R. Murawski, S. Surma, J. Woleński, and J. Zygmunt among others.

Creation
of logical systems was not a Polish speciality. Besides discussive
logic of Jaśkowski (which was mentioned in section 10) there are
two exceptions: „non-Fregean logic” of R. Suszko –
a system of logic inspired by some intuitions taken from *Tractatus
...* of L. Wittgenstein, and the system
*Grz*
(for Grzegorczyk, its author) – one of the most widely studied
systems of modal logic.

Polish
logicians (J. Słupecki, H. Rasiowa, T. Traczyk, G. Malinowski,
W.A. Pogorzelski, T. Prucnal, M. Tokarz, K. Trzęsicki, R.
Wójcicki, not to mention the eminent precursors of the subject
– J. Łukasiewicz, A. Tarski, and M. Wajsberg) contributed
tremendously to the research in many-valued logics *34*.
Important results were obtained in the field of intuitionistic logic
(H. Rasiowa, R. Sikorski, P. Wojtylak, A. Wroński), modal logic
(A. Grzegorczyk, E. Orłowska, J. Perzanowski, H. Rasiowa, C.
Rauszer, J. Hawranek, and T. Skura), relevance logic (W. Dziobiak, M. Tokarz, K. Swiridowicz).

Quite a
large number of papers were concerned with deontic logic (J.
Gregorowicz, L. Gumański, J. Kalinowski, T. Kubiński, W.
Suchoń, K. Świrydowicz, J. Woleński, Z. Ziemba, Z.
Ziembiński), casual logic (H. Greniewski, A. Pieczkowski, K.
Trzęsicki, and M. Urchs *35*). The latter were mostly
based on some ideas initiated by S. Jaśkowski.

Polish authors also dealt with the logic of questions (erotetic logic). It was initiated by K. Ajdukiewicz and substantially developed by T. Kubiński, L. Koj, and A. Wiśniewski.

The area of logic and theory of communication is explored by M. Tokarz.

1 „The postwar
panorama of logic in Poland”, in: *Logic and Scientific
Methods*, eds. M.L. Dalla Chiara et al., Kluwer 1997, pp. 597-608.

2 Various suggestions to the earlier „Florence” version of this survey were offered by: Janusz Czelakowski, Andrzej Grzegorczyk, Jacek Malinowski, Marcin Mostowski, Roman Murawski, Ewa Orłowska, Witold A. Pogorzelski, Kazimierz Świrydowicz, Max Urchs, Jan Woleński, Andrzej Wójcik, Jan Zygmunt. While preparing this version of the survey I obtained the assistance from: Zofia Adamowicz, Wojciech Buszkowski, Janusz Czelakowski, Witold Marciszewski, Wiktor Marek, Roman Murawski, Jan Mycielski, Mieczysław Omyła, Jerzy Pogonowski, Jerzy Tiuryn, Anita Wasilewski, Andrzej Wiśniewski, Jan Woleński, and Jan Zygmunt.

3 One could think that by extending to the limits the notion of logic the logicians behave like „logical imperialists” who try to invade other branches of mathematics. It is not so. Logic and its methods are the source of inspiration and the basic research tool for mathematics. To call the foundations of mathematics logic enables the specialists of the foundations of mathematics to establish their scientific identity. It also enables them to see their very much varied field of research as a whole which differs from the rest of mathematics.

4 It was formed by the
group of philosophers, logicians, sociologists, and other scientists
who upheld a tradition started by the seminars and papers of the
eminent psychologist and philosopher from Lvov –Kazimierz
Twardowski. Precise analysis and lucid argument were the virtues that
Twardowski considered the base of both the scientific research and
the philosophical writings. Logic was regarded as the basic tool that
helps to accomplish this. It is not odd then that Lvov-Warsaw School
attracted logicians and at the same time the logicians and their
works essentially influenced the School. There is a monograph on the
School by Jan Woleński, *Logic and Philosophy in the
Lvov-Warsaw School*, Kluwer 1989. A brief and informative article
on Polish logic of the interwar period can be found in *The
Routledge Encyclopaedia of Philosophy*, vol 7, Routledge, London
and New York, 1998, pp. 498-500, „Polish Logic” by J.
Zygmunt.

5 Alfred Tarski –
undoubtedly the most significant person of the interwar period in
Polish logic, and one of the greatest logicians of the 20th century.
When he published the collection of his papers *Logic*,*
Semantics, Metamathematics*,* papers from 1923 to 1938*
(Clarendon Press, Oxford, 1956) he dedicated it to T. Kotarbiński,
whom he calls his teacher.

6 This subject was examined in many Polish and foreign centers. In Poland it was examined by: A. Krawczyk, M. Krynicki, L. Szczerba, W. Szmielew, M. Zawadowski, and others. Its computational aspects were analysed by A. Pawlak, H. Rasiowa, and E. Orłowska.

7 A. Mostowski (in
collaboration with A. Grzegorczyk, S. Jaśkowski, J. Łoś,
S. Mazur, H. Rasiowa, and R. Sikorski), „The present State of
Investigations on the Foundations of Mathematics”,
*Dissertationes Mathematicae* 9 (1955), pp. 1-48.

A. Mostowski, „Thirty
Years of Foundational Studies; Lectures on the Development of
Mathematical Logic and the Study of the Foundations of Mathematics in
1930-1964”, *Acta Philosophica Fennica* 17 (1965), 1-180.

8 To the group of close collaborators of A. Mostowski belonged Janusz Onyszkiewicz, Stanisław Krajewski, and Konrad Bieliński. These names are well-known to all who witnessed democratic changes in Poland (J. Onyszkiewicz was the Defence Secretary in two cabinets, S. Krajewski is one of the most eminent members of the Jewish community in Poland, and K. Bieliński was one of the leaders in the underground Solidarity movement). There were also other logicians who were political disidents. One of the most significant role was played by Klemens Szaniawski. Jan Waszkiewicz was especially active.

9 In a letter that I received as a response to my request concerning the remarks to this paper J. Mycielski wrote: „Since there was a close cooperation (exchange of papers and ideas) between Mostowski’s and Tarski’s schools, one can say that there was just one Berkeley-Warsaw school and it is impossible to discuss one without discussing the other.”

10 Not only philosophical
ones. Tarski’s concept was the subject of numerous analyses and
formal generalizations. An interesting survey of this topic can be
found in S. Krajewski’s essay „Prawda” in: *Logika
Formalna*. *Zarys Encyklopedyczny z Zastosowaniami do
informatyki i lingwistyki*, edited by W. Marciszewski, PWN 1987,
pp. 144-156.

11 Tarski’s
contribution to the development of logic is discussed in J. Zygmunt’s
„*Alfred Tarski*” in: *Polska Filozofia Powojenna*,
vol. II, edited by W. Mackiewicz, Agencja Wydawnicza Witmark,
Warszawa, 2001, pp. 342-375.

12 It should be mentioned that after the war it was possible to employ more people in the departments of mathematics. Before the World War II (as I was told by W. Marek) there were just three professors of mathematics at Warsaw University. Karol Borsuk, despite his achievements and international reputation was employed as an assistant. A. Mostowski had a position in the Institute of Meteorology.

13 The note „From
the Editor” says that *Studia Logica* will publish papers
devoted to all areas of logic, including formal logic, mathematical
logic, inductive logic, theory of definition and that of
classification etc., and that SL invite especially works on the
history of Polish logic.

14 In its body there were: N. D. Belnap, Jr and J. M. Dunn (USA), B. I. Dahn (DDR), L. Esakia (USSR, Georgia), D. Follesdal (Norway), R. Gilles (Canada), J. Hintikka (Finland), L. Maksimowa and V. A. Smirnow (USSR, Russia), R. Routley (Australia), I. Ruzsa (Hungary), P.Weingartner (Austria), and P. M. Williams (UK).

15 This „party
vigilance” was not unjustified. Since the scientific
periodicals were not censored, there was a danger that some of the
papers might have been written by the „enemy of the socialism”.
*Studia Logica* committed this kind of crime by publishing the
review of A. A. Zinoviev’s (one of the leading Soviet
disidents) book *Logical Physics* (SL 35, 1976). It was a book
free from ideological issues, but still the members of the Soviet
Academy of Sciences protested.

16 According to the contract, the periodical remains one of the publications of the Institute of Philosophy and Sociology of the Polish Academy of Sciences, and the main Polish libraries receive it at reduced prices.

17 The aim of this
newsletter (founded by the section of logic of Inst. Phil. Soc. Pol.
Ac. Sc. in 1973) was to extend international cooperation and
indirectly to promote *Studia Logica*. Since 1991 it has been
published by Łódź University, and edited by Grzegorz
Malinowski.

18 An important role is
played by *Studia Semiotyczne* (founded by J. Pelc) the
publication of Polskie Towarzystwo Semiotyczne.

19 It requires the competence to which the author of this survey cannot aspire, nor probably can the workers in this discipline, because of its size and scope. On the other hand it would be odd not to mention the foundations of mathematics. The results of this discipline are strongly related to those of logic.

20 The work of Andrzej
Mostowski is discussed in five papers written by: A. Grzegorczyk, W.
Guzicki, W. Marek, L. Pacholski, C. Rauszer, P. Zbierski in: A.
Mostowski, *Foundational Studies. Selected Works*, vol I, PWN,
Warszawa, North-Holland, Amsterdam 1979.

The monograph which
summarizes his metamathematical research in set theory is: A.
Mostowski, *Constructible Sets with Applications*, PWN,
Warszawa, North-Holland, Amsterdam, 1969.

21 H. Rasiowa, R. Sikorski,
*The Mathematics of Metamathematics*, PWN, Warszawa,
1963 (3rd edition, 1970).

22 If by laws of logic one means the laws of classical logic, then they can be characterized by the laws of Boolean algebra. Non-classical logics can be represented by „non-Boolean” algebras. E.g. intuitionistic logic is characterized by pseudo-Boolean algebras.

23 H. Rasiowa, *An
Algebraic Approach to Non-Classical Logic*, PWN, Warszawa,
North-Holland, Amsterdam, 1974.

24 *Studia Logica*
2, 1955, pp. 151-212.

25 This notion was explored also by the Spanish logicians.

26 These two conceptions were discussed in section 2. To what extent they differ from each other can be illustrated by the following. The system of 3-valued Łukasiewicz logic in its inferential meaning has two „non-trivial” extensions, while the same system in its sentential meaning has only one extension. This result (obtained by R. Wójcicki) has been later generalized (G. Malinowski, W. Dziobiak). It makes clearer some of the „paradoxical” results obtained earlier by H. Hiż and Rasiowa, which were concerned with the sentential meaning of the logical system. The distinction between logic understood as a set of tautologies and a logical consequence enables us to show (R. Wójcicki) that we cannot define classical logic by the use of the constants of the intuitionsitic logic in the inferential case (which is possible in the sentential case).

27 The papers in question
here are: J. Łoś, R. Suszko, „Remarks on Sentential
Logic”, *Indagationes Mathematicae* 20, 1958, pp. 177-183,
and R. Wójcicki „Some Remarks on the Consequence
Operation in Sentential Logics”, *Fundamenta Mathematicae*
68, 1970, 269-279.

28 The results mentioned
here as well as the results of other logicians (W. Blok, H. Hiż,
J. Łoś, D. Pigozzi, H. Rasiowa, W.A. Pogorzelski, and
others) are discussed in: R. Wójcicki, *Theory of Logical
Calculi; Basic Theory of Consequence Operations*, Kluwer, 1988.
Some of the results were applied to the theory of automatic theorem
proving (cf. Z. Stachniak, *Resolution Proof System, An Algebraic
Theory*, Kluwer, 1996).

29 This part of research is
discussed in the monograph: J. Czelakowski, *Protoalgebraic Logics*,
Studia Logica Library, Kluwer, 2001.

30 A thorough discussion
of the results with a comparison to other centers’ results can
be found in: W. Buszkowski „Mathematical linguistics and proof
theory”, *Handbook of Logic and Language* (collective work
edited by J. van Benthem and A. ter Meulen, Elsevier and MIT Press).

31 S. Jaśkowski *On
the rules of Suppositions in Formal Logic*, Studia Logica.
Wydawnictwo poświęcone logice i jej historji, nr 1,
Seminarium Filozoficzne Wydz. Matematyczno-Przyrodniczego
Uniwersytetu Warszawskiego, Warszawa, 1934. This series was to be
published under the editorship of J. Łukasiewicz. Unfortunately
only one volume was published. Postwar Studia Logica (cf. section 12)
refered to it but was not its continuation.

32 In order to make S.
Jaśkowski’s system better known J. Słupecki and L.
Borkowski elaborated some version of it (and published it in the book
*Elementy Logiki i Teorii Mnogości*, PWN, 1963; translated
as *Elements of Mathematical Logic and Set Theory*, Pergamon
Press 1967) and presented the possibility of applying it to
mathematical reasoning.

33 This project is
supported by international grants (from EU and NATO, e.g. Ph. D.
scholarships in Germany and Japan) and its main aim is to complete an
encyclopaedia of mathematics „on-line” (a collection of
computer verified proofs of theorems). There is also a quarterly:
*Formalized Mathematics – A Computer Assisted Approach*
published there, edited by R. Matuszewski.

34 A good introduction to
this subject is a book by G. Malinowski *Many-valued Logics*,
Clarendon Press, Oxford 1993.

35 M. Urchs is mentioned here because of his strong connections with Polish logic. He came to Poland as a young man from DDR and started to study in Toruń.