The History and Philosophy of Polish Logic

Essays in Honour of Jan Wole?ski

K. Mulligan, K. Kijania-PLacek, T. Placek, K. Mulligan, K. Kijania-PLacek, T. Placek, Kenneth A. Loparo, Katarzyna Kijania-Placek

• Publisher: Palgrave Macmillan
• Year: 2016
• Print ISBN: 9781349440634, 9781137030900
• eBook ISBN: 9781137030894
• Topic: Philosophy
• Subtopic: History & Philosophy of Mathematics

The book presents the state of the art of research into the legacy of interwar Polish analytic philosophy and exemplifies different approaches to the history of philosophy. It contains discussions and reconstructions of aspects of Polish philosophy and logic as well as reactions to and developments of this tradition.

Part I

Logic, Proof and Models

1

Many-valued Logic in Poland: The Golden Age

Alexander S. Karpenko

1.1 Introduction

The origin of many-valued logic was closely connected with the Lvov-Warsaw School (LWS) (see Woleński, 1985). Here is an amazing case of the emergence of one of the most influential scientific schools in the world, which is famous for its research in philosophy and especially in logic.

1895 is considered to be the year of foundation of the LWS, when K. Twardowski (1866–1938) got a chair at Lvov University. Poland as an independent state did not exist at that point in history, for after three partitions it was completely torn into three parts: an Austrian one, a Prussian one and a Russian one (1772–95). The state was restored in 1919 by the Versailles agreement. The second period in the development of the school began just after that restoration. Quite paradoxically, despite the non existence of a Polish state, the conditions for the emergence of the above-mentioned scientific school were in place and thus the emergence of the unique Warsaw school of logic headed by J. Łukasiewicz (1878–1956) and S. Leśniewski (1886–1939) was possible. J. Woleński, the main specialist on the LWS, writes: ‘The Second World War and the events that followed, first of all the presence of Poland in the “socialist camp” where Marxism was the official state philosophy, had led to the disappearance of Lvov-Warsaw school as an organized philosophical movement.’ (Woleński, 2004a, p. 456) Nevertheless it should be noted that the level of development of logic in Poland remains very high. In another work Woleński (2010) writes:

The logical achievements of the LWS became the most famous. Doubtless, the Warsaw school of logic contributed very much to the development of logic in the 20th century.¹

Z. Jordan described the phenomenon of many-valued logic in the following manner (see the chapter V ‘The Discovery of Many-Valued systems of Logic’ in Jordan, 1945/1967, p. 389):

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.

And in Woleński (1985, p. 119), see the whole chapter 6.2 ‘Many-Valued Logics’, we read: ‘The construction of many-valued logical systems is commonly believed to have been one of the major achievements of the Warsaw School, and specifically of Łukasiewicz.’

The early development of many-valued logic in Poland is considered in detail in Woleński (2001) especially from 1910 to 1920. Woleński (2001, p. 196) writes that ‘Kotarbiński² introduced into Poland the idea that there are sentences which are indefinite, that is, neither true nor false’, (see also Woleński, 1990). It is interesting that Kotarbiński didn’t mention it in chapter XXI, ‘Many-Valued Propositional Calculus’ of his book (1957) but criticized Łukasiewicz’s three-valued logics for conceptual reasons.

The first mention of three-valued logic can be found in Łukasiewicz’s lecture delivered in 1918. There, Łukasiewicz says: ‘that new logic [. . .] destroys the former concept of science’; moreover, Łukasiewicz makes a connection between the ‘new logic’ and the ‘struggle for the liberation of the human spirit’ (see Łukasiewicz, 1918/1970, p. 86). One can say that passion for freedom led Łukasiewicz to the discovery of three-valued logic.

Philosophical ideas underlying the third truth-value are discussed in Łukasiewicz’s seminal paper ‘On determinism’ (see Łukasiewicz, 1922).³ There, Łukasiewicz states that Aristotle’s solution of the problem of future contingency (the problem of logical fatalism) destroys one of the main principles of our logic, namely that every proposition is either true or false. Łukasiewicz calls this principle the principle of bivalence. According to him, it is an underlying principle of logic that cannot be proved – one can only believe in it. Łukasiewicz claims that the principle of bivalence does not seem self-evident to him. Therefore, he claims to have the right not to accept it and to stipulate that, along with truth and falsity, there should be at least one more truth-value, which Łukasiewicz considers to be intermediate between the other two. Łukasiewicz concludes that

If this third value is introduced into logic we change its very foundations. A trivalent system of logic [. . .] differs from ordinary bivalent logic, the only one known so far, as much as non-Euclidean systems of geometry differ from Euclidean geometry. (Łukasiewicz, 1922/1970, p. 126)

Similar passages occur in other papers of Łukasiewicz (up to 1951). Those claims foreshadowed a radical revision of the classical logic. Łukasiewicz’s main philosophical conclusion is that determinism could be avoided by rejecting the principle of bivalence.⁴

1.2 Łukasiewicz’s three-valued logic Ł₃

The first system of three-valued logic appeared in Łukasiewicz (1920). It means that for the first time three-valued logical connectives were defined and combined to form a logical system.

Adhering to the classical way of defining implication p → q and negation ~ p wherever their arguments are the classical truth-values 0 and 1, Łukasiewicz defines the meaning of those connectives for the cases featuring his new truth-value in the following way:

The other propositional connectives are defined by means of the primary connectives:

Thus, the truth-tables for the logical connectives look as in Table 1:

A valuation is a function v from the set of formulae S to the set {0, ½,1} of truth-values, ‘compatible’ with the above truth-tables. A formula α is a tautology if and only if v(a) = 1 for every valuation v, where 1 is the designated value. The set of tautologies thus defined is Łukasiewicz’s three-valued logic Ł₃.

Wajsberg (1931) sho...

Table of contents

• Cover
• Title
• Introduction: The History and Philosophy of Polish Logic: Some Basic Thoughts
• Part I Logic, Proof and Models
• Part II Truth and Concepts
• Part III Ontology, Mereology and the Philosophy of Mathematics
• A Selection of Jan Woleński’s Publications
• Index for Names