Logic and Its History in the Lvov-Warsaw School

Abstract

We take into account two areas of the logical research of the Lvov-Warsaw School. First, we consider a new approach to research in the history of logic introduced and practiced by Łukasiewicz and some of his followers. In this style of doing history of logic, the knowledge of original philosophical and logical texts was combined with competence in modern logic. This method resulted in many important discoveries both in history and in logic and philosophy. At the same time, we pay attention to contemporary historical research devoted to the logic created by the School itself. In this context, we present an overview of the six papers included in the special issue ‘Logic and its History in the Lvov-Warsaw School’. These papers were presented at an international symposium, ‘History in/of the Lvov-Warsaw School’, held in October 2022 in Warsaw, and they cover various aspects of historical-logical and logical achievements of the School. They concern topics from history of ancient logic, history of modern proof theory, and the issues of applying logic to philosophy and theology in the same way as the one implemented at the Lvov-Warsaw School.

Keywords:

• Lvov-Warsaw School
• Warsaw School of Logic
• logic
• history of logic
• philosophical applications of logic

The history and activities of the Lvov-Warsaw School have been documented in numerous exceptional monographs. Let us cite among them widely recognizable items (Garrido and Wybraniec-Skardowska 2018; Woleński et al. 2017; Jadacki and Paśniczek 2006; Woleński 1989). The unwavering interest in the Lvov-Warsaw School is clearly visible when we take into account the number of publications of various formats appearing in the current international scientific circulation.¹ However, the intensively explored topics related to the School have not yet been exhausted, as evidenced by the papers included in this volume.

The presented texts have been prepared in connection with the international symposium ‘History in/of the Lvov-Warsaw School’, which took place on the 20th to 23th of October 2022 in Warsaw. The event was organized by the University of Warsaw, Kazimierz Wielki University of Bydgoszcz, Cardinal Stefan Wyszyński University in Warsaw, The Lvov-Warsaw School Research Center, Kazimierz Twardowski Philosophical Society of Lviv, and the Polish Academy of Sciences. The collected texts concern topics within the scope of logic and its history in the Lvov-Warsaw School.

Let us say at the beginning that Kazimierz Twardowski, the founder of the Lvov-Warsaw School, did not concentrate his interests on formal logic. However, he considered the logical education of his students to be very important, and he himself was a specialist in semiotics and methodology (Jadacki 2021). Twardowski thought about logic in the old style and sometimes had a skeptical view of the new formal logic, the representatives of which he accused in 1921 of ‘symbolomania and pragmatophobia’ (Twardowski 1921). However, it was to him that Jan Łukasiewicz and Stanisław Leśniewski owed their first inspirations. The two men subsequently educated other outstanding logicians of the Warsaw School of Logic, which was a logical branch of the Lvov-Warsaw School. Members of this intellectual community achieved numerous outcomes, crucial to contemporary formal sciences.

Łukasiewicz wrote in 1929, 425:

Almost all philosophers practicing mathematical logic in Poland are students of Prof. Twardowski; so they belong to the so-called ‘Lvov school of logic’, where they learned to think clearly, conscientiously and methodically.

As he emphasized, Polish mathematical logic, developed at the Warsaw School of Logic, also had its own specificity. Firstly, it was considered an independent scientific discipline, the role of which is not reduced to the control of the correctness of mathematical reasonings. Second, its results often had intended philosophical motivations and/or applications.

Knowledge of the history of classic philosophy (and so also traditional logic) was also included in the educational program implemented in Lvov by Twardowski (cf. memories of Izydora Dąbska in Woleński 1989, 4). As Woleński notes in 1989, 25:

The knowledge of, and importance attached to, the role of the history of philosophy can be treated as a specific characteristic of the Lvov-Warsaw School, which singles it out among other philosophical schools, especially those inclined to an analytic interpretation of philosophy.

Łukasiewicz focused part of his research on the history of logic. Regarding his results in this field, Bolesław Sobociński, one of his closest students, wrote that ‘they are among the most beautiful in his scientific achievements’ (Sobociński 1957, 18). Łukasiewicz claimed that the history of logic ‘must be written anew, by a historian who has fully mastered mathematical logic’ (Łukasiewicz 1970, 197). His style of historical research, which consisted of applying the tools of contemporary logic to the texts of classical logicians, was indeed innovative. Thanks to the combination of thorough historical knowledge and extraordinary logical competence, Łukasiewicz formulated important reconstructions of the history of a few logical ideas and problems and also achieved many breakthrough results for modern logic. His modern method of doing the history of logic was adopted by many of his students (Woleński 1989, 185–190), and today it is used both in the history of logic and in the history of philosophy, and even in theology. The use of formal logic in the history of philosophical problems was also successfully implemented by Łukasiewicz himself, and with his encouragement, this style was practiced by Catholic analysts, gathered in the so-called Krakow Circle, who undertook the reconstruction of historical theological issues (Murawski 2015).

In the papers contained in this volume, the above-mentioned topics are present. There, the reader will find discussions and analyses of Łukasiewicz's achievements in the history of ancient logic, as well as studies on selected problems undertaken within the Warsaw School of Logic. The last two works are devoted to the reconstruction of fragments of the history of philosophy and theology, both made in Łukasiewicz's style.

The first two texts by Z. Rybaříková and M. Tkaczyk refer directly to the famous original reconstructions of Stoic logic and Aristotelian syllogistics by Łukasiewicz. As Łukasiewicz pointed out, the logic of the Stoics was the pre-version of the modern classical propositional logic axiomatized by Frege. Analyses of Aristotle's original texts inspired him to create the first formal system of classical logic of names. These results were already considered groundbreaking in the understanding of the history of formal logic in Łukasiewicz's time.

The role of mentioned results obtained by Łukasiewicz in currently published scientific research is the subject of consideration by Z. Rybaříková in ‘Mathematical Logic as a Research Tool in the History of Logic: Łukasiewicz's Contribution and Its Reception’. The author discusses Łukasiewicz's approach to the history of logic and focuses on reconstructions of the ideas of the Stoics and Aristotle. These considerations are supplemented by an interesting empirical study documenting the current references of scientific publications to Łukasiewicz's results in this area. Rybarikova's query concerns the years 2018–2022 and uses two platforms that index the world's leading scientific literature: Web of Science and Scopus.

M. Tkaczyk in ‘Are Ancient Logics Explosive?’ raises the issue of Łukasiewicz's justification of his view on the relationship between the logical ideas of the Stoics and Aristotle and contemporary classical logic. The author's main interest, however, is the legitimacy of relevant and paraconsistent interpretations of these ideas, that are popular in the literature today. According to the author, the treatment of ancient logics as ‘leaven’ of modern classical logic is perhaps based on an over-interpretation of the source texts. But on the other hand, Tkaczyk also argues that the non-classical interpretations mentioned are not adequate. He shows that one of the laws that is certainly recognized within ancient logical approaches is the classical law of explosion. It is not, however, a law in relevant and paraconsistent logics. In that case, the search for the origins of the above-mentioned logics in the ancient texts is misplaced. In the polemical part, the author points out two arguments by G. Priest, which turn out not to support the paraconsistent interpretation of the logic of the Stoics and syllogistics of Aristotle.

The next two works by A. Indrzejczak and U. Wybraniec-Skardowska present a historical approach to selected results in the field of proof theory achieved by members of the Warsaw School of Logic.

A. Indrzejczak in ‘The Problem of Natural Representation of Reasoning in the Lvov-Warsaw School’ puts forward an interesting hypothesis that both Alfred Tarski and Stanisław Leśniewski knew the first version of natural deduction formulated by Stanisław Jaśkowski much earlier than in 1934, i.e. before the date of Stanisław Jaśkowski's publication of his ‘official’ version of natural deduction. Tarski's conditions for implication given in the general theory of consequence published in 1930 express a deduction theorem. In turn, Leśniewski's proof schemata in the presentation of his theory of groups and theory of Abelian groups from 1929, are, in fact, slightly modified exemplifications of Jaśkowski's system of natural deduction. Taking into account the historical circumstances, Indrzejczak's hypothesis is highly probable. Jaśkowski's first draft ideas on natural deduction date back to Łukasiewicz seminar in 1926, and he made the first announcement about his system public in 1929. The delay in publishing the results was related to his health problems. Both Leśniewski and Tarski, who were in close scientific contact with both Łukasiewicz and Jaśkowski, probably knew about Jaśkowski's results even before 1929. Indrzejczak shows the way of presenting Leśniewski's theories from 1929 in a new light. The author reconstructed the style of displaying proofs of theses from the theory of groups by Leśniewski which closely resembles Jaśkowski's natural deduction proofs.

U. Wybraniec-Skardowska in ‘The Pioneering Proving Methods as Applied in the Warsaw School of Logic: Their Historical and Contemporary Significance’ reviews the results of logicians from the Warsaw School of Logic achieved in the proof theory. She discusses the beginnings of the applications of the axiomatic method, the method of natural deduction and the concept of the rejection method. The first versions of the rejection method were given by Łukasiewicz and it was developed by Jerzy Słupecki, Łukasiewicz's student and one of the significant representatives of the Warsaw School of Logic. The first axiomatic version of Aristotle's syllogistics presented by Łukasiewicz gave an impulse to the construction of numerous systems of classical logic of names, systems of classical propositional logic and many non-classical logics using the axiomatic method. At that time, a new concept for the construction of proofs by means of natural deduction method was developed by Jaśkowski. As is known, in the international logical community it is admitted that Gentzen was the author of the first formalized method of this type. Wybraniec-Skardowska, like Indrzejczak, draws attention to Jaśkowski's achievements before the publication of his results in 1934. It is worth noting that in Polish logical texts (especially in didactic works) natural deduction in the style of Jaśkowski is used much more often than that of Gentzen. Following Aristotle, Łukasiewicz gave a new system of Aristotle's syllogistics, in which the rejection procedure was used for the first time. However, the first deductive-rejective system of Aristotle's syllogistics, on the basis of which every propositional formula is either a syllogistic thesis or is rejected, was formulated by Słupecki. Łukasiewicz's and Słupecki's method of building systems of different logics is also used in a version of natural deduction. The author includes in her work many current applications of the Łukasiewicz-Słupecki method.

A paper by A. Pietruszczak entitled ‘Russell's theories of events and instants from the perspective of point-free ontologies in the tradition of the Lvov-Warsaw School’ contains an original formal reconstruction of B. Russell's theory of events from 1914 and refers to the influence of this concept on the research of Leśniewski, Tarski and Czesław Lejewski. Łukasiewicz's method for historical analysis is perfectly implemented here: up-to-date modern formal tools were used in the formalization. The author compares the obtained formalized theory with the formalized theory of events published by Russell in 1936. It turns out that the axioms of Russell's theory from 1936 are provable in Pietruszczak's new theory, and the axioms of the theory of events by Thomason (1989) are also provable in it. The author shows that the relation of enclosure introduced by Russell can be interpreted as a part relation in the sense of Leśniewski's mereology, and the method of constructing instants in Russell's theory is similar to that used by Tarski when defining the notion of a point in his pointless geometry using the notion of a part relation as the primary one. Lejewski's chronology, on the other hand, was an extension of mereology to include concepts with intended temporal meanings, directly referring to those originally adopted by Russell.

The last paper of this volume ‘Czeżowski's Theory of Reasoning and Mediaeval Biblical Exegesis’ by M. Trepczyński and M. Będkowski, pursues the interests of the members of the Krakow Circle, which we know were directly inspired by Łukasiewicz. The subject of analysis and reconstruction by Trepczyński and Będkowski are selected medieval texts devoted to the interpretation of the Bible. The authors reconstruct some of the reasonings from the analyzed fragments and for this purpose use the conceptual grid of the theory of reasonings by T. Czeżowski, one of the leading representatives of the first generation of the Lvov-Warsaw School. They then point to the specific purpose of the reasonings under study. Reasonings of this type are exegetical, so their main purpose is to justify the adequacy of the selected interpretation of Bible fragments, and they directly refer to these fragments. As the authors note, in exegetical reasoning, in order to justify a conclusion, a number of quotations are usually cited, preceded by the expression ‘unde’ -- an indicator of the connection between the conclusion and the biblical quotation, called by the authors ‘Special Biblical Inference’. The analysis of the nature of a specific type of inference is the subject of the authors' subsequent research.

As editors of this special issue, we thank the Editors of History and Philosophy of Logic for the opportunity to publish it. We would also like to thank the organizers of the symposium ‘History in/of the Lvov-Warsaw School’, and in particular the Head of the Lvov-Warsaw School Research Center of the University of Warsaw, Professor Anna Brożek, for entrusting us with editorial work on the collected material. We also express our gratitude to the reviewers for their insightful reviews of the submitted texts.

Notes

1 For the period 2018–2023, the Google Scholar search engine found in 0.06 seconds 315 Polish-language publications whose titles include the name ‘Szkoła Lwowsko-Warszawska’ and 823 and 238 English-language publications with titles containing the names ‘Lvov-Warsaw School’ and ‘Lviv-Warsaw School’, respectively.