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In the bibliography of Łukasiewicz' works contained in vol. V ofStudia Logica, this article bears the number 2. In order to inform the reader about a discussed or quoted paper given in the bibliography the respective number will be given in brackets.
Under the titleDie logischen Grundlagen der Wahrscheinlichkeitsrechnung (34).
A note on this theorem is contained in the summary of the lectureO wartościach logicznych (On logical values) [24].
The paperWahrscheinlichkeitslehre was published in 1935.
More details about the notion of logical implication are given by Łukasiewicz in his paperO pojęciu wielkości [39] (On the notion of magnitude). In order to assert that a certain „determinate” sentence is logically implied by other sentences these sentences should — according to Łukasiewicz — be changed into „indeterminate” sentences, hence variables should be put in the place of determinate terms. Łukasiewicz does not explain which are the terms to be replaced. The implication of determinate sentences is thus reduced to the implication of indeterminate sentences. On the other hand, the indeterminate sentenceZ is” a consequence of the indeterminate sentencesP, R, S, ... in other words, it follows from these sentences if there exist no such values of the variables contained in the sentencesP, R, S, ..., Z, which would statisfy the sentencesP, R, S, ... but not satisfy the sentenceZ”. The similarity of this definition to the definition of logical implication as given byA. Tarsk in 1936 is distinct.
InLogika dwuwartościowa [43] (Two-valued logic) Łukasiewicz describes truth not as a true proposition but as an object determined by true propositions and similarly, falsehood not as a false proposition but as an object determined by a false proposition. However, in an article of 1929 [59] Łukasiewicz writes: „I must admit some mistake:in mvTwo-valued Logic I acceptedFrege's „philosophical” notion of „truth” and „falsehood” which today I am inclined to include into mythology”. Let us also mention that in the paperO zasadzie sprzeczn ościu Arystotelesa [22] belonging to the first period of Łukasiewicz' scientific activity we find the remark that abstract objects „do not exist in reality but are only products of the human mind”.
In his farewell lecture [40] delivered in the hall of the Warsaw University in March 1918 Łukasiewicz said: „In 1910 I published a book on Aristotle's principle of contradiction in which I tried to show that this principle is not quite as obvious as it is thought to be. It was already at that thime that I aimed at formulating a non-Aristotelian logic, but without success”.
Łukasiewicz repeated this definition in his lectureZagadnienie prawdy [44] (The problem of truth). In a summary of that lecture we come across the remark that no criterion of truth can be proved but „nevertheless there exist various methods by which truth can be made accessible to human perception”.
In the article in question Łukasiewicz does not use logical symbols.
Zaremba defines as quantitatively comparable those objects to which one of the relations of less, equal or greater is applicable.
Refers to the principles α.
An interesting fact is that at the time Łukasiewicz wrote his article, i. e. in 1916, neither the term „connected relation” nor any other equivalent term had been introduced in Polish mathematical and logical literature. The terms „transitive relation” and „symmetrical relation” were introduced by Łukasiewicz as mentioned in the discussed article.
Farewell lecture [40].
Is one of the sides of a definition.
This rule is formulated as follows: if the expressionsDαDβγ and α are theorems then the expression γ is a theorem.
We call an axiom organic if none of its parts is a theorem.
Meaning the problem of what is cause.
In the bibliography of Łukasiewicz' works numbered: 54, 59, 67, 70, 72, 73, 74.
By philosophical logic Łukasiewicz means logic as pursued by philosophers.
In the paperO pojęciu wielkości Łukasiewicz stands in opposition to abusing in mathematics the term „stipulation” as this may be understood as meaning that „mathematics is based mainly on stipulations” or „conventions”, i. e. has a conventional character. „This view is erroneous”. However, Łukasiewicz has not justified this theory.
Sometimes Łukasiewicz turns from being a prosecutor of philosophy into its brillant and ardent defender. Thus e. g. in his review [49] ofJ. Sleszyński' paperO logice tradycyjnej (On traditional logic) we read: „... philosophy, though containing also scientific elements is as a whole something apart from science. It searches for anwers to problems which we are unable to solve by scientific methods at the given stage of our knowledge. It tries various, often uncertain and new roads and probably loses at times its way. In some of its parts it is a superstructure of already existing sciences, in others — a stage preceding the advent of some science. This is the reason of the variety of its topics which may make the impression of chaos. And yet, how reative is often this chaos!”.
Łukasiewicz is perfectly aware that this postulate is not easy to realize. In his paperO zasadzie sprzeczności u Arystotelesa we read as follows: „Symbolic logic will never be popular with some type of philosophers. It is, of course, a pleasant and gratifying thing to create lofty syntheses expressed in beautifully resounding words. But symbolic logic must be studied; it must be studied just as mathematics with a pencil in hand not omitting any letter, not jumping over any proof. One must want to work and know how to work scientifically. This work is too dry and too tedious for intellects that yearn for the absolute”.
InAristotle's SyllogisticŁukasiewicz deals in more detail with the problem of differentiating analytic and synthetic propositions: „Kant divided all propositions (he calls them „judgements”) into analytic and synthetic according to the relation of the predicate of a proposition to its subject. HisCritique of Pure Reason is chiefly an attempt to explain the problem how true synthetic a priori propositions are possible. Now some Peripatetics, for instance Alexander, were apparently already aware that there exists a large class of propositions having no subject and no predicate, such as implications, disjunctions, conjunctions, and so on. All these may be called functorial propositions, since in all of them there occurs a propositional functor, like „if — then”, „or”, „and”. These functorial propositions are the main stock of every scientific theory, and to them neither Kant's distinction of analytic and synthetic judgements nor the usual criterion of truth is applicable, for propositions without a subject or predicate cannot be immediately compared with facts. Kant's problem loses its importance and must be replaced by a much more important problem: How are true functorial propositions possible? It seems to me that here lies the starting-point for a new philosophy as well as for a new logic”. As regards the question that inference does not extend knowledge Łukasiewicz writes in the articleUwagi o aksjomacie Nicode'a: „So long as the principle dictum de omni was regarted as the foundation of the entire deductive logic it was possible to believe that deduction is an inference from general to particular and that it does not extend our knowledge. However, when the modern „theory of deduction” was formulated and when the Aristotelian dictum de omni principle in form of the rule of substitution, as well as the stoic syllogism modus ponens as the rule od detachment were applied to it, it was found that deductive inference can be just as „creative” as inductive inference and loses nothing of its certainty”.
In the introduction to a series of articlesMyśl katolicka wobec logiki współczesnej (Catholi thought and modern logic). Studia Gnesnensia XV, Poznań 1937. Worth stressing is the fact that Łukasiewicz highly valued the merits of professorJ. Sleszyński for the development of mathematical logic in Poland. At the end of his review [46] ofSleszyński's paperO logice tradycyjnej he writes: „One of the pioneers of this new science (viz. mathematical logic) has been for many years professor Sleszyński. Generally known are his unremitting efforts to arrive at an entirely comprehensible and entirely justified truth. For this idealistic tendency we owe him homage and deep gratitude”.
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Allatum est die 25 Aprili 1957
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Borkowski, L., Sŀupecki, J. The logical works of J. Łukasiewicz. Stud Logica 8, 7–56 (1958). https://doi.org/10.1007/BF02126733
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DOI: https://doi.org/10.1007/BF02126733