Łukasiewicz, Jan (1878–1956)

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ŁUKASIEWICZ, JAN
(1878–1956)

Jan Łukasiewicz, the Polish philosopher and logician, was born in Lvov. After studying mathematics and philosophy at the University of Lvov he was graduated in 1902 with a PhD in philosophy. Łukasiewicz taught philosophy and logic first at Lvov and from 1915 at the University of Warsaw. In 1918 he interrupted academic work to accept a senior appointment in the Polish ministry of education in Ignacy Paderewski's cabinet. At the end of that year, however, he returned to the university and continued as professor of philosophy until September 1939. During that period he served twice as rector of the university (1922/1923 and 1931/1932). Toward the end of World War II Łukasiewicz left Warsaw. After some time in Münster and then in Brussels, in 1946 he accepted an invitation from the Irish government to go to Dublin as professor of mathematical logic at the Royal Irish Academy, an appointment that he held until his death. Łukasiewicz held honorary degrees from the University of Münster and from Trinity College, Dublin. He was a member of the Polish Academy of Sciences in Kraków, the Society of Arts and Sciences in Lvov, and the Society of Arts and Sciences in Warsaw.

Early Writings

Łukasiewicz studied under Kazimierz Twardowski, who was occupied with conceptual analysis. The rigorous, clear thinking Twardowski advocated is easily recognizable in the first major essays published by Łukasiewicz. Of these works, O zasadzie sprzeczności u Arystotelesa (On the principle of contradiction in Aristotle; Kraków, 1910) was one of the most influential books in the early period of the twentieth-century logical and philosophical revival in Poland. It must have stood high in the author's own estimation, for in 1955 he began translating it into English. The main point of the book is that in Aristotle's work one can distinguish three forms of the principle of contradiction: ontological, logical, and psychological. The ontological principle of contradiction is that the same property cannot both belong and not belong to the same object in the same respect. The logical principle says that two contradictory propositions cannot both be true, and the psychological principle of contradiction holds that no one can, at the same time, entertain two beliefs to which there correspond two contradictory propositions. Łukasiewicz supported his findings with quotations from the writings of Aristotle and then examined the validity of Aristotle's argumentation. One chapter brought to the notice of Polish readers Bertrand Russell's antinomy concerning the class of all classes that are not members of themselves. The appendix contains an elementary exposition of the algebra of logic, as well as an original and interesting methodological classification of the ways of reasoning, a problem with which at least two of Łukasiewicz's early papers were concerned.

Łukasiewicz's writings published before 1918 suggest that until that time he was in quest of topics to which he could devote all his intellectual resources. He found such topics in the logic of propositions and in the logic of the ancient Greeks. From 1918 onward, deviations from this double line of research are few and of little significance.

Logic of Propositions

many-valued logics

The first and perhaps most important result obtained by Łukasiewicz in the logic of propositions was his discovery of three-valued logic in 1917. Our ordinary logic of propositions is two-valued, presupposing only two logical values, truth and falsity, and it tacitly adheres to the principle of bivalence, that a propositional function holds of any propositional argument if it holds of the constant true proposition (usually symbolized by 1) and if it holds of the constant false proposition (represented by 2). If we use δ as a functorial variable that, when followed by a propositional argument, forms a propositional expression, then we can express the principle of bivalence by saying "if δ 1 then if δ 2 then δp," where p is a propositional variable. The meaning of the logical constants forming such expressions as, for instance, Cpq ("if p then q "), Kpq ("p and q "), Apq ("p or q "), and Np ("it is not the case that p ") are, in two-valued logic, conveniently and adequately determined by means of the familiar two-valued truth tables:
C 11 = C 21 = C 22 = 1
C 12 = 2
K 11 = 1
K 12 = K 21 = K 22 = 2
A 11 = A 12 = A 21 = 1
A 22 = 2
N 1 = 2
N 2 = 1

In three-valued logic the principle of bivalence does not hold. It is replaced by the principle of trivalence, which presupposes three logical values: the constant true proposition represented by 1, the constant false proposition by 3, and the constant "possible" proposition by 2. The principle then says "if δ 1 then if δ 2 then if δ 3 then δp. " As a consequence the meanings of implication, conjunction, alternation, and negation have to be readjusted, and the following three-valued truth tables suggest themselves for the purpose:
C 11 = C 21 = C 22 = C 31 = C 32 = C 33 = 1
C 12 = C 23 = 2
C 13 = 3
K 11 = 1
K 12 = K 21 = K 22 = 2
K 13 = K 23 = K 31 = K 32 = K 33 = 3
A 11 = A 12 = A 13 = A 21 = A 31 = 1
A 22 = A 23 = A 32 = 2
A 33 = 3
N 1 = 3
N 2 = 2
N 3 = 1
In this logic alternation and conjunction can be defined as follows: Apq = CCpqq, and Kpq = NANpNq. All expressions involving only C and N and verified by the new truth tables can be constructed into a deductive system based on the axioms CpCqp, CCpqCCqrCpr, CCCpNppp, and CCNpNqCqp. This was shown by Mordchaj Wajsberg, who had studied logic under Łukasiewicz in Warsaw. Wajsberg's system, however, does not enable us to define all the functors available in three-valued logic. In particular the functor T, whose truth table says that T 1 = T 2 = T 3 = 2, cannot be defined in terms of C and N. Jerzy Słupecki, who had also been a pupil of Łukasiewicz, subsequently proved that by adding CTpNTp and CNTpTp to Wajsberg's axioms we get a functionally complete system of three-valued logic, in which any functor can be defined.

The conception of three-valued logic was suggested to Łukasiewicz by certain passages in Aristotle. Purely formal considerations, such as those that led E. L. Post to comparable results, played a subordinate role in Łukasiewicz's thinking. By setting up a system of three-valued logic Łukasiewicz hoped to accommodate the traditional laws of modal logic. He also hoped to overcome philosophical determinism, which he believed was entailed by the acceptance of the bivalence principle and which he had always found repulsive. Interestingly enough, he modified his views in the course of time and saw no incompatibility between indeterminism and two-valued logic.

Once a system of three-valued logic had been constructed, the possibility of four-valued, five-valued, …, n -valued, and, finally, infinitely many-valued logics was obvious. At one time Łukasiewicz believed that the three-valued and the infinitely many-valued logics were of greater philosophical interest than any other many-valued logic, for they appeared to be the least arbitrary. In the end, however, he interpreted Aristotelian modal logic within the framework of a four-valued system.

The philosophical significance of the discovery of many-valued logic can be viewed in the following way: The laws of logic had long enjoyed a privileged status in comparison with the laws propounded by natural sciences. They had been variously described as a priori or analytic, the purpose of such descriptions being to point out that the laws of logic were not related to reality in the same way as were the laws of natural sciences, which had often been corrected or discarded in the light of new observations and experiments. The laws of logic appeared unchallengeable. By discovering many-valued logics Łukasiewicz showed that even at the highest level of generality—within the field of propositional logic—alternatives were possible. By adhering to the principle of bivalence or any other n -valence principle we run the same risk of misrepresenting reality that the scientist does when he offers any of his generalizations.

the classical propositional logic

Although Łukasiewicz contemplated the possibility that a nonclassical logic of propositions applied to reality, he made the classical propositional logic the principal subject of his research. He showed that the axiom systems of the calculus of propositions proposed by Frege, Russell, and Hilbert each contained a different redundant axiom. He proved that all the theses of the CN -calculus could be derived from the three mutually independent axioms CCNppp, CpCNpq, and CCpqCCqrCpr. He solved the problem of the shortest single axiom for the E -calculus and the C -calculus by showing that the E -calculus, whose only functor means "if and only if," with E 11 = E 22 = 1 and E 12 = E 21 = 2 as its truth table, could be based on any of EEpqEErqEpr, EEpqEEprErq, and EEpqEErpEqr and on no shorter thesis and by proving that CCCpqrCCrpCsp is the shortest thesis strong enough to yield the C -calculus. The first single axiom for CN -calculus, consisting of 53 letters, was discovered by Alfred Tarski in 1925. It was soon followed by a series of successive simplifications devised by Łukasiewicz and by Bolesław Sobociński. The latest in this series is a 21-letter axiom, CCCCCpqCNrNsrtCCtpCsp, discovered by C. A. Meredith, Łukasiewicz's Irish colleague. It is likely to prove to be the shortest possible axiom for the CN -calculus.

consistency, completeness, and independence

The metalogical study of deductive systems of the logic of propositions includes the study of consistency and completeness, and, in the case of systems based on several axioms, the mutual independence of the axioms has also to be considered. Independently of Post, Łukasiewicz developed both a method of proving consistency and one of proving the completeness of systems of the calculus of propositions. The completeness proof was based on the idea that if the system under consideration is not complete, there must be independent propositions, that is, propositions not derivable from the axioms of the system which on being adjoined to the axioms lead to no contradiction. If there are independent propositions, then there must be a shortest one among them. Following Łukasiewicz's method, one tries to show that any proposition that is meaningful within the system either is derivable from the axioms or is longer than another proposition inferentially equivalent to it. This method dispenses with the concept of "normal expressions" and is very useful for proving weak completeness of partial systems. Mutual independence of theses is usually established by an appropriate reinterpretation of the constant terms occurring in them. Many such reinterpretations have been provided by Łukasiewicz's many-valued logics. The wealth of metalogical concepts and theorems worked out in Łukasiewicz's logical seminar in Warsaw by Łukasiewicz himself, Tarski, Adolf Lindenbaum, Sobociński, and Wajsberg can best be seen in "Untersuchungen über den Aussagenkalkül," which summarizes the results obtained there between 1920 and 1930.

functorial calculus

In Dublin, Łukasiewicz became interested in a two-valued calculus of propositions involving functorial variables. Since he used only functorial variables requiring one propositional argument to form a propositional expression, his new calculus was only a part of what Stanisław Leśniewski had called protothetic. A very strong rule of substitution invented by Łukasiewicz, together with the usual substitution rules for propositional variables, allows us, for instance, to use a thesis of the form δα to infer not only Nα but also such theses as Cpα, Cαp, CαCNαp, Cαα, and α. By means of the new rule Łukasiewicz was able to base the calculus on the single axiom CδC 22Cδ 2δp. This axiom is identical with the principle of bivalence, because C 22 = 1. Meredith succeeded in showing that Łukasiewicz's axiom could be replaced by Cδδ 2δp or by CδpCδNpδq. He was also able to prove completeness of the system.

Ancient Logic

Concurrently with his investigations of the logic of propositions Łukasiewicz was engaged in a thorough reappraisal of ancient logic. For centuries the logic of the Stoics had been regarded as a sort of appendage to the Aristotelian syllogistic. Łukasiewicz was the first to recognize in it a rudimentary logic of propositions. He found evidence that the main logical functions, such as implication, conjunction, exclusive disjunction, and negation, were known to the Stoics, who, following Philo of Megara, interpreted them as truth-functions, just as we do now. He pointed out that the Stoics, unlike Aristotle, had given their logic the form of schemata of valid inferences. Some of these schemata had been accepted axiomatically and others were rigorously derived from them. He subjected to severe but justified criticism the treatments of Stoic logic by such authorities as Carl Prantl, Eduard Zeller, and Victor Brochard. His preliminary investigations of medieval logic showed beyond doubt that in this field too there was room for fruitful research.

Equally successful was Łukasiewicz's inquiry into Aristotle's syllogistic. No sooner had he mastered the elements of symbolic logic for himself than he realized that the centuries-old traditional treatment of the Aristotelian syllogistic called for revision. A new presentation of the logic of Aristotle was before long included in his regular lectures at the university and then published in Elementy logiki matematycznej (Elements of mathematical logic; Warsaw, 1929). Łukasiewicz completed a detailed monograph on the subject in Polish in the summer of 1939, but the manuscript and all printed copies were lost during the war. Aristotle's Syllogistic (1951) is a painstaking reconstruction undertaken by Łukasiewicz on his arrival in Dublin. The monograph can rightly be called revolutionary. In it Łukasiewicz argued that Aristotelian syllogisms are logical laws rather than schemata of valid inferences, as is taught in traditional textbooks. He put in historical perspective Aristotle's introduction of variables and, referring to a forgotten Greek scholium, gave a plausible explanation of the problem of the so-called Galenian figure. Among more formal results, we owe to Łukasiewicz the first modern axiomatization of syllogistic. The system he set up, based on the axioms Aaa ("every a is a "), Iaa ("some a is a "), CKAbcAabAac, and CKAbcIbalac, seems to be in perfect harmony with Aristotle's own treatment of the subject in the Analytica Priora. The axioms are jointly consistent and mutually independent. Moreover, Słupecki has ingeniously solved the decision problem for the system.

Modal Logic

During the last few years of his life Łukasiewicz devoted much attention to modal logic. The results are presented in "A System of Modal Logic," and in the second edition of Aristotle's Syllogistic (1957) they serve as the basis for a critical examination of Aristotle's theory of modalities. Łukasiewicz's principal idea is that of "basic modal logic," obtained by adding to the classical calculus of propositions the axioms CpMp and EMpMNNp and by axiomatically rejecting CMpp and Mp. In these formulas Mp stands for "it is possible that p. " According to Łukasiewicz any modal system must contain basic modal logic as a part. This condition is fulfilled by the four-valued modal system based on CδpCδNpδq and CpMp as the only axioms, with CMpp and Mp axiomatically rejected.

The logical symbolism used in this entry was worked out by Łukasiewicz in the early 1920s. It requires no punctuation signs, such as brackets or dots, which from the point of view of metalogical investigations is its greatest merit. At the same time Łukasiewicz worked out a simple and perspicuous method of setting out proofs in the logic of propositions and in syllogistic. Both his symbolism and his proof technique have been adopted by many logicians outside Poland.

Łukasiewicz was not only a resourceful and imaginative scholar but also a gifted and inspiring teacher. He was one of the founders, and the life and soul, of the Warsaw school of logic. Tarski, Lindenbaum, Stanisław Jaśkowski, Wajsberg, Father Jan Salamucha, Sobociński, Słupecki, and Meredith have been his most outstanding pupils or collaborators.

Bibliography

principal works by Łukasiewicz

"Analiza i konstrukcja projecia przyczyny" (Analysis and construction of the concept of cause). Przeglqd Filozoficzny 9 (1906): 105–179.

O zasadzie sprzeczności u Arystotelesa. Kraków, 1910.

Die logischen Grundlagen der Wahrscheinlichkeitsrechnung. Kraków: Akademie der Wissenschaften, 1913.

"Opojeciu wielkości" (On the concept of magnitude). Przeglad Filozoficzny 19 (1916): 1–70.

"O logice trójwartościowej" (On three-valued logic). Ruch Filozoficzny 5 (1920): 170–171.

"Logika dwuwartościowa" (Two-valued logic). Przeglqd Filozoficzny 23 (1921): 189–205.

"Démonstration de la compatibilité des axiomes de la théorie de la déduction." Annales de la Société Polonaise de Mathématique 3 (1925): 149.

Elementy logiki matematycznej. Warsaw, 1929; 2nd ed., Warsaw, 1958. Translated by Olgierd Wojtasiewicz as Elements of Mathematical Logic. New York, 1963.

"Untersuchungen über den Aussagenkalkül." Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23 (1930): 30–50. In collaboration with Alfred Tarski. English translation by J. H. Woodger in Logic, Semantics, Metamathematics, by Alfred Tarski, 38–59. Oxford: Clarendon Press, 1956.

"Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls." Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III 23 (1930): 51–77.

"Ein Vollständigkeitsbeweis des zweiwertigen Aussagenkalküls." Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III 24 (1931): 153–183.

"Zur Geschichte des Aussagenlogik." Erkenntnis 5 (1935–1936): 111–131.

"Der Äquivalenzenkalkül." Collectanea Logica 1 (1939): 145–169.

"Die Logik und das Grundlagenproblem." In Les entretiens de Zurich sur les fondements et la méthode des sciences mathématiques 6–9 Décembre 1938, 82–100. Zürich, 1941.

"The Shortest Axiom of the Implicational Calculus of Propositions." Proceedings of the Royal Irish Academy 52, Section A (1948): 25–33.

"On Variable Functors of Propositional Arguments." Proceedings of the Royal Irish Academy 54, Section A (1951): 25–35.

Aristotle's Syllogistic. Oxford, 1951; 2nd ed., Oxford: Clarendon Press, 1957.

"A System of Modal Logic." Journal of Computing Systems 1 (1953): 111–149.

Z zagadnień logiki i filozofü (Problems of logic and philosophy). Edited by Jerzy Słupecki. Warsaw: Panstwowe Wydawn, 1961. Essays. Contains bibliography.

In addition to the above, Łukasiewicz published about twenty papers and over fifty notes and reviews.