Jan Łukasiewicz

From On Three-Valued Logic by Jan Łukasiewicz, 1920

" Before the scientific meeting held on 19 June 1920, Professor Jan Łukasiewicz delivered a talk entitled 'On three-valued logic'. The Aristotelian logic assumes that every sentence is either true or false. Therefore it recognizes only two kinds of logical values : truth and falsity. "

" The three-value logic is a system of non-Aristotelian logic, for it assumes that besides true and false sentences there are also sentences which are neither true nor false, thus accepting the existence of a third logical value. We can interpret the third value as 'possibility' "

* * *

The speaker thinks that three-valued logic has, above all, theoretical significance as the first attempt to create a non-Aristotelian logic. What practical significance, if any, the new system of logic will have may be known only after a careful examination, in relation to the new logical laws, of logical phenomena, especially those occurring in the deductive sciences, when it may become possible to compare with experience the consequences (etc).

. . . 'O logice Trójwartosciowej', appeared originally in Ruch Filozoficzny 5 (1920), pp. 169-71. Translated by H. Hiz.

Polish Logic 1920-1939,
ed. Storrs McCall, Oxford 1967, pp. 16-17.

From Philosophical Remarks on Many-Valued Systems of Propositional Logic† 1930 by Jan Łukasiewicz

In the communication 'Untersuchungen über den Aussagenkalkül' (Investigations into the Sentential Calculus) which appeared in this issue under Tarski's and my name, § 3 is devoted to the 'many-valued' systems of propositional logic established by myself. Referring the reader to this communication as far as logical questions are concerned, I here propose to clarify the origin and significance of those systems from a philosophical point of view.

( in McCall 1967, page 40 )

* * *

...     The three-valued system of propositional logic with quantifiers, which owing to the research of Tarski and Wajsberg can be represented axiomatically, is the simplest example of a consistent logical system which is as different from the ordinary two-valued system as any non-Euclidean geometry is from the Euclidean.

I think it may be said that the system mentioned is the first intuitively grounded system differing from the ordinary propositional calculus.   (..)  It is true that Post has investigated many-valued systems of propositional logic from a purely formal point of view, yet he has not been able to interpret them logically.¹ The well-known attempts of Brouwer, who rejects the universal validity of the law of the excluded middle and also repudiates several theses of the ordinary propositional calculus, have so far not led to an intuitively based system. They are merely fragments of a system whose construction and significance are still entirely obscure.²

It would perhaps not be right to call the many-valued systems of propositional logic established by me 'non-Aristotelian' logic, as Aristotle was the first to have thought that the law of bivalence could not be true for certain propositions. Our new-found logic might be rather termed 'non-Chrysippean', since Chrysippus appears to have been the first logician to consciously set up and stubbornly defend the theorem that every proposition is either true or false. This Chrysippean theorem has to the present day formed the most basic foundation of our entire logic.

It is not easy to foresee what influence the discovery of non-Chrysippean systems of logic will exercise on philosophical speculation. However, it seems to me that the philosophical significance of the systems of logic treated here might be at least as great as the significance of non-Euclidean systems of geometry.

    ¹ E. L. Post, 'Introduction to a general theory of elementary propositions', Am. Journ. of Math. 43 (1921), p. 1832 : '. . . . the highest dimensioned intuitional proposition space is two.'
    ² Cf. e.g., L .E. J. Brouwer, 'Intuitionistische Zerlegung mathematischer Grundbegriffe', Jahresber. d. Deutsch. Math.-Vereinigung 33 (1925), pp. 251 ff. ; 'Zur Begründung der intuitionistischen Mathematik. I', Math. Ann. 93 (1925), pp. 244 ff.

( pages 62-3 )

* * *

    † This paper appeared originally under the title 'Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls' in Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Cl. iii, 23 (1930), pp. 51-77. Translated by H. Weber.

Polish Logic 1920-1939, Storrs McCall,
Oxford 1967

From AXIOMATIZATION OF THE THREE-VALUED PROPOSITIONAL CALCULUS,* 1931 by Mordchaj Wajsberg

" The author of the three-valued calculus, and more generally of the n-valued calculus, for every natural number n, as well as of the calculus of denumerably many values, is Professor Jan Łukasiewicz.¹ "

See J. Łukasiewicz and A. Tarski, 'Untersuchungen über den Aussagenkalkül', Comptes rendus des séances de la Société des Sciences et des Lattres de Varsovie, Cl. iii, 23 (1930). See also J. Łukasiewicz, 'Philosophische Bemerkungen zu mehrwertigen Systemen des Auusagenkalküls', ibid.

* This paper originally appeared under the title 'Aksojamtyzacja trójwartosciowego rachunku zdan' in Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Cl. iii, 24 (1931). pp. 126-45. Translated by B. Gruchman and S. McCall.

POLISH LOGIC 1920-1939,
Editor Storrs McCall, Oxford 1967, p. 264

From GESCHICHTE DER LOGIC by Heinrich Scholz 1931 (Eng. transl. 1961)

" Poland has lately become the main country and Warsaw the main bastion of research in symbolic logic by virtue of the work of Jan Łukasiewicz.²³¹ We can only refer to the pertinent treatises . . . in the Fundamenta Mathematicae of which volume 16 appeared in Warszawa during 1930. They are geared to undergirding the foundations of mathematics. Also Leon Chwistek: The Theory of Constructive Types, Principles of Logic and Mathematics (Cracow, University Press, 1925) must at least be alluded to.  "   (page 8)

* * *

" . . . the latest publication of the leading Polish authority on symbolic logic, Jan Łukasiewicz: "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalk�s" (Comptes rendus des s�ances de la Soci�t� des Sciences et des Lettres de Varsovie, Classe III, p. 52-77, a very interesting and considerable piece of labor. Cf.. especially p. 65). "     (page 102)

English version, CONCISE HISTORY OF LOGIC
New York : Philosophical Library 1961.

From A Non-Aristotelian System, (etc), 1931 by Alfred Korzybski

" The Polish school of: (a) 'intuitionist' formalism with Łukasiewicz, Tarski, Lesniewski as representatives, which may be called the non-aristotelian school. Łukasiewicz generalized the A 'logic' to three valued 'logic' which covers modality . . . and . . . finally produced a general many-valued 'logic' of which the two-valued represents only a limiting case.  "

'A Non-Aristotelian System and its Necessity for Rigour in Mathematics and Physics' :
Paper presented before the American Mathematical Society at the New Orleans, Louisiana,
Meeting of the A. A. A.S. December 28, 1931.
In Science and Sanity, 1933, Supplement III,
pp. 747 and 748.

From MIND, 1934

Professor Lukasiewicz is sole author of these systems, having originated the three-valued system in 1920, and n-valued systems in 1922.

Mind XLIII. 104
( per Oxford English Dictionary, 'three' )

From The Development of Mathematical Logic in Poland between the Two Wars 1945 by Zbigniew Jordan

Whatever value may be attached to the above-mentioned results, the discovery by Łukasiewicz of many-valued systems of logic stands out against all of them. Without any doubt it is a discovery of the first order, eclipsing everything done in the field of logical research in Poland.

It is well known from some remarks made by Łukasiewicz in his book The Principle of Non-Contradiction in Aristotle   .  .   .   as well as from some personal statements, that l'ideée-force leading him to the discovery of many-valued systems of logic was the conviction of the indeterminism of future events. The idea of many-valued systems of logic, however, took shape only in 1920 with the introduction of truth matrices. In the same year the three-valued system was originated; that is to say, the truth matrices determined, the definitions of primitive ideas formulated, and the directions of interpretation indicated. The generalized many-valued systems (of any denumerable set of values) came two years later (1922). In the course of time it became apparent that they form a general scheme . . . (etc)

(in McCall 1967, p. 389).

* * *

It seems safe to say that many-valued logic has grown out of the stage during which it was just a new field of research, its results leaving now no room for doubts as to its validity. It seems also very likely that the implications of the discovery concern such widely different problems as the theory of probability, Brouwer's intuitionist logic, quantum mechanics, the discussion about universal strict determinism, etc. But it is difficult to fully realize now all the consequences of Łukasiewicz's discovery. Investigations in this direction have gone little beyond the first shocking conclusions that logical truth has got a multi-form character and that there is a variety of ways in which it may be considered. In some respects the discovery and the foundation of many-valued logic makes us think of the shattering blow dealt by the discovery of the Non-Euclidean geometries to the deeply--rooted conviction that there is one and only one way of constructing the spatial reference frame of our experiences. Similarly it was supposed that a consistent deductive system must follow the Aristotelian pattern, in accordance with his most general 'laws of thought'. In particular it was believed that a statement must be either true or false. In some circles there arose doubts concerning this principle when Brouwer constructed definite examples of mathematical theorems—dealing with 'the infinite' as it occurs in analysis—which are neither true nor false. The construction by Łukasiewicz of a self-consistent deductive system in which the proposition 'a statement is either true or false' no longer holds turned the balance definitely against the Aristotelian assumption.

 

VI.   History of Logic

This sketch of the development of logic in Poland would not be complete without mentioning the awakening of a lively interest in the history of logic. Łukasiewicz once again was the energetic and influential protagonist. He himself for the firs time drew attention to many so far neglected or misunderstood discoveries made in ancient Greece (the School of Megara, the Stoics) and in the Middle Ages (Duns Scotus, Petrus Hispanus).   (Etc.)

* * *

In his researches on the logic of the ancient world Łukasiewicz benefited by the collaboration of several classical scholars. Among them A. Krokiewicz should be especially mentioned ; and it may be added that the help he gave to Łukasiewicz influenced in turn his own work.   (Etc.)  

Łukasiewicz, whose own studies provided a crushing criticism of Prantl's Geschichte der Logik, demanded that the whole history of logic, till now distorted by involuntary ignorance of the subject and consequently limited to the Aristotelian syllogistic, should be rewritten by someone who had mastered equally modern logic and the technique of historical studies. H. Scholz's Geschichte der Logik (Berlin, 1931), the first step in this direction, was made thanks to the lead of, and in collaboration with, Łukasiewciz and his pupils.

(pages 394-5 and 396-7).

POLISH LOGIC 1920-1939,
editor Storrs McCall, Oxford 1967.

From Aristotle's Syllogistic by Jan Łukasiewicz, 1951

I am sincerely grateful to the Royal Irish Academy, which, by giving me a position in Dublin, has enabled me to write this book, and to University College, Dublin, for its kind invitation to deliver lectures on Aristotle�s logic. I am grateful to the Professors of University College, Dublin, Father A. Gwynn, S.J., and Monsignor J. Shine, who were kind enough to lend me the necessary books. I owe a debt to Sir David Ross, who read my typescript and made some suggestions I was glad to accept. My special thanks are due to the late Father A. Little, S.J., who, although already dangerously ill, willingly corrected the English of the first chapter, to Victor Meally in Dublin, and in particular to David Rees of Bangor, who read and corrected the English of the whole work. I am also deeply indebted to the officials of the Clarendon Press for their zeal and courtesy in preparing my typescript for printing. The section on Galen is dedicated to my friend Professor Heinrich Scholz of Münster, Westphalia, who was of great assistance to myself and to my wife during the war, and especially during our stay in Münster in 1944 The whole work I dedicate to my beloved wife, Regina Łukasiewicz née Barwinska, who has sacrificed herself that I might live and work. Without her incessant care during the war, and without her continual encouragement and help in the loneliness of our exile after it, I could never have brought the book to an end.

J. Ł.    

    DUBLIN
7 May 1950

Oxford, 1957, pages viii - ix.

From The Integration of Human Knowledge, 1958 by Oliver Leslie Reiser

Because of his prior familiarity with the work of the Polish logicians . . . Korzybski had a considerable advantage over the American students working in these fields. This is the case, for example, with the Korzybskian espousal of a 'multi-valued' orientation which was presented to replace the outmoded two-valued judgments of Aristotelian logic. It was only later that American students like C. I. Lewis were able to take advantage of the non-Aristotelian logics (etc).

Boston : Porter Sargent 1958, pp. 102-3.

From Encyclopaedia of Ireland, 1968

As in medieval times, so again during and after the second World War, Ireland benefited from an influx of foreign scholars. . . .  Outstanding amongst these were the physicist Erwin Schrödinger, for a time a Director and later Senior Professor of the Institute for Advanced Studies, Jan Łukasiewicz who was Professor of Logic at the Royal Irish Academy and Ludwig Wittgenstein (1889-1951), who wrote volume two of this Philosophical Investigations while living here, between 1947 and 1949.

. . .

Łukasiewicz�s writings, done in Dublin, included Aristotle�s Syllogistic from the standpoint of Modern Formal Logic (Oxford 1951) and many important papers, particularly one on The Intuitionist Theory of Deduction (Amsterdam 1952), in which he showed that �the intuitionistic theory of deduction contains as its proper part the classical theory of deduction.� He concluded that �the intuitionistic theory is richer and consequently more powerful than the classical one . . . among the hitherto known many-valued systems of logic the intuitionistic theory is the most intuitive and elegant�. Using the same Polish notation as Łukasiewicz, Carew Meredith developed, for several logical systems, the shortest single axiom that could, in each case, serve as a foundation. These axioms are surprisingly short, varying from 21 down to 6 letters.

ENCYCLOPAEDIA OF IRELAND, Principal editor Victor Meally
Dublin : Allen Figgis ; New York and Toronto : McGraw-Hill, 1968, page 405.

From MANY-VALUED LOGIC, 1969 by Nicholas Rescher

Łukasiewicz first publicized his 3-valued system of logic in a lecture before the Polish Philosophical Society in Lwów in 1920.² A report written by him (in Polish) giving the content of that lecture was published in the journal Ruch Filozoficzny later that year.³Łukasiewicz's paper of 1920 especially represents a thoroughgoing repudiation of two-valued logic and advocacy of a 3-valued system. The mainstream of the development of many-valued logic proceeded on the basis of elaborations of Łukasiewicz's ideas�especially in their formulation in his later papers in which the 3-valued logic was generalized to the many-valued, indeed even infinite-valued case.⁴ Some of the relevant ideas were given wide circulation in the influential treatise on Symbolic Logic by Charles H. Langford and Clarence Irving Lewis published in 1932.⁵ An important pioneering result was achieved by Mordchaj Wajsberg, who succeeded in 1931 in axiomatizing the 3-valued logic of Łukasiewicz³   (etc).

²However, Łukasiewicz had already published in 1910 his book on The Principle of Non-Contradiction in Aristotle (see Łukasiewicz 1910) in which one principal motivating idea of the later development of his 3-valued logic�viz. the issue of future contingency�is already a major theme. For an account of the contributions of Łukasiewicz to symbolic logic see Jordan (1945), Mostowski (1957), Scholz (1957), and Borkowski and S�upecki (1958).
³Vol. 5 (1920), p. 170.
⁴See specially Łukasiewciz (1930), tr. in S. McCall (ed.), Polish logic : 1920-1939 (Oxford, 1967), pp. 40-65. [Many writers have failed to appreciate that the contribution of Tarski to this paper was limited to one point of detail: ] Łukasiewicz had already made the generalization to many-valued and infinite-valued systems by 1922. See Jordan (1945).
⁵See Lewis and Langford (1932), pp. 213-234.
⁶See Wajsberg (19310, tr. in S. McCall (ed.), Polish Logic : 1920-1939 (Oxford,1967), pp. 264-284.

New York, etc. : McGraw-Hill 1969, p. 8.

From Arthur Prior, Stanford Encyclopedia of Philosophy

"After Findlay, Lukasiewicz was the greatest single influence on Prior's development as a logician. Prior's 1952 review article `Lukasiewicz's Symbolic Logic' is one of the first papers in which he makes extensive use of symbolism. (He discusses Lukasiewicz's book Aristotle's Syllogistic From the Standpoint of Modern Formal Logic (published in 1951) and two articles, `The Shortest Axiom of the Implicational Calculus of Propositions' and `On Variable Functions of Propositional Arguments'.) Prior seems to have first learned of Lukasiewicz's work through Bochenski's writings (Bochenski was a pupil of Lukasiewicz). " etc.

http://www.seop.leeds.ac.uk/archives/sum1999/entries/prior/#Life

Bibliographic

Łukasiewicz, Jan. Title O zasadzie sprzeczności u Arystotelesa; studyum krytyczne. Publisher Krak�w, Akademia Umiejętności, 1910. Description 210 p. Note Bibliographical footnotes.

Łukasiewicz, Jan. Title Elementy logiki matematycznej; skrypt autoryzowany opracowal M. Presburger; z cres̀ciowal subwencji Senatu akademickiego Un�wersytetu warszawskiego. [Microform] Publisher [Warszawa] Nakł. Komisji wydawniczeij Koła matematyczno-fizycznego słuchaczȯw uniwersytetu warszawskiego, 1929. Description viii, 198 p. diagrs.

Łukasiewicz, Jan. Title O nauce. Publisher Lw�w, Polskie Towarzystwo Filozoficzne, 1934. Description 40 p.

Łukasiewicz, Jan. Title Aristotle's syllogistic from the standpoint of modern formal logic. Publisher Oxford, Clarendon Press, 1951. Description x, 141 p. 23 cm. Note Bibliographical footnotes. Language English

Lukasiewicz, Jan, d. 1956. Title Aristotle's syllogistic from the standpoint of modern formal logic. Edition 2d ed., enl. Publisher Oxford, Clarendon Press, 1957. Description xiii, 222 p. 23 cm. Note Includes bibliographical references and index. Language English

Heinrich Scholz Title : "In Memoriam Jan Łukasiewicz," Archiv für mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 3-18.   [Source : Rescher, Many-valued Logic, New York 1969.]

Encyclopaedia of Ireland. Publisher Dublin, A. Figgis; New York, McGraw-Hill, 1968. Description 463 p. illus., maps. 29 cm. Note Includes bibliographies. Language English

Łukasiewicz, Jan. Title Z zagadnień logiki i filozofii; pisma wybrane. Wyboru dokonał, wstępem iprzypisami opatrzył Jerzy Słupecki. Edition [Wyd. 1.] Publisher Warszawa, Państwowe Wydawn. Naukowe, 1961. Description 309 p. 25 cm. Note Errata slip inserted. Note "Bibliografia prac Jana Łukasiewicza": p. [307]-309

Lukasiewicz, Jan Title Aristotle's syllogistic; from the standpoint of modern formal logic Edition 2d ed., enl Publisher Oxford, Clarendon Press [1963, 1957] Description 222 p. 23 cm Note Bibliographical footnotes Language English

Łukasiewicz, Jan Title Elements of mathematical logic. Translated from Polish by Olgierd Wojtasiewicz Publisher Warszawa : PWN-Polish Scientific Publishers; New York, Pergamon Press [1963] Description 124 p. : illus. 23 cm Series International series of monographs in pure and applied mathematics, v. 31 Note Original title: Elementary logiki matematycznej Note Includes bibliography

Łukasiewicz, Jan. Title Elements of mathematical logic. Translated from Polish by Olgierd Wojtasiewicz. Edition [2d ed.] Publisher New York, Macmillan [c1963] Description xi, 124 p. 23 cm.

Łukasiewicz, Jan. Title Selected works. Ed. by L. Borkowski. [Translated from the Polish by O. Wojtasiewicz]. Publisher Amsterdam, North-Holland Pub. Co., 1970. Description xii, 405 p. 23 cm. Series Studies in logic and the foundations of mathematics. Note Bibliography: p. 401-405. ISBN 0720422523 Language English

Conference International Symposium on Multiple-Valued Logic. (9th : 1979 : Bath, Eng.) Title Proceedings : the ninth International Symposium on Multiple-valued Logic, Beaufort Hotel, Bath, England, 1979 / [sponsored by the IEEE Computer Society] Publisher Long Beach, Calif. : IEEE Computer Society Publications Office, c1979 Description 304 p. : ill. ; 28 cm Note "79CH1408-4C." On spine: IEEE 1979 multiple-valued logic This symposium honors Jan Lukasiewicz born in Lwow, Poland, December 21, 1878 Note Includes bibliographical references Language English

Title Zur modernen Deutung der aristotelischen Logik / herausgegeben von Albert Menne und Niels �ffenberger Publisher Hildesheim ; New York : Olms, 1982- Description v. : ill. ; 22 cm Note Some text in English and French Bd. 5 by Jan Łukasiewicz

Łukasiewicz, Jan. Title Aristotle's syllogistic from the standpoint of modern formal logic / Jan Łukasiewicz. Publisher New York : Garland Pub., 1987. Description xiii, 222 p. ; 22 cm. Series Greek & Roman philosophy ;25 Note Reprint. Originally published: Oxford : Clarendon Press, 1957. Note Includes bibliographical references and index. ISBN 0824069242 (alk. paper) :