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Alfred Tarski
From Investigations into the Sentential Calculus (27 March 1930) by Jan Łukasiewicz and Alfred Tarski
INVESTIGATIONS INTO THE SENTENTIAL
CALCULUS†
In the course of the years 1920-30 investigations were carried out in Warsaw belonging to that part of metamathematics—or better metalogic—which has as its field of study the simplest deductive discipline, namely the sentential calculus. These investigations were initiated by Łukasiewicz ; the first results originated both with him and with Tarski. In the seminar for mathematical logic which was conducted by Łukasiewicz in the University of Warsaw beginning in 1926, most of the results stated below of Lindenbaum, Sobocinski, and Wajsberg were found and discussed. The systematization of all the results and the clarification of the concepts concerned was the work of Tarski.
In the present communication the most important results of these investigations—for the most part not previously published—are collected together.‡
§ 1. GENERAL CONCEPTS
It is our intention to refer our considerations to the conceptual apparatus which was developed in the preceding article (see III). (Etc.)
† BIBLIOGRAPHICAL NOTE. This joint communication of J. Łukasiewicz and A. Tarski was presented (by Łukasiewicz) to the Warsaw Scientific Society on 27 March1930 ; it was published under the title 'Untersuchungen über den Aussagenkalkül' in Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, vol. 23, 1930, cl. iii, pp. 30-50.
‡ To avoid misunderstandings it should be stated that the present article does not contain results discovered by both the authors jointly, but is a compilation of theorems and concepts belonging to five different person.s Each theorem and concept is ascribed to its respective originator. Theorem 3, for instance, is not a theorem of Łukasiewicz and Tarski, but a theorem of Lindenbaum. Nevertheless, some scholars mistakenly referred to both authors, Łukasiewicz and Tarski, the many-valued systems of logic ascribed in the article to Łukasiewicz alone. In spite of a correction which appeared in 1933 in the Journal of Philosophy, vol. 30, p. 364, this mistake persists till today. It clearly follows from § 3 and notes of this article that the idea of a logic different from the ordinary system called by Łukasiewicz the two-valued logic, and the construction of many-valued systems of logic described here, are entirely due to Łukasiewicz alone . . . (etc).
( Tarski 1956, page 38 )
* * *
§ 2. THE ORDINARY (TWO-VALUED) SYSTEM OF THE SENTENTIAL CALCULUS
In the first place we consider the most important of the systems of the sentential calculus, namely the well-known Ordinary system (also called by Łukasiewicz1 the two-valued system), (etc).
1 See (50 a).
* * *
§ 3. MANY-VALUED SYSTEMS OF THE SENTENTIAL CALCULUS
In addition to the ordinary system of the sentential calculus here are many other systems of this calculus which are worthy of investigation. This was first pointed out by Łukasiewicz who has also singled out a specially important class of such systems2 The systems founded by Łukasiewicz are here called n-valued systems of the sentential calculus . . . (etc).
2 What is called the three-valued system of the sentential calculus was constructed by Łukasiewicz in the year 1920 and described in a lecture given to the Polish Philosophical Society in Lwów. A report by the author, giving the content of that lecture fairly thoroughly was published in the journal Ruch Filozoficzny, vol. 5 (1920), p. 170 (in Polish). A short account of the n-valued systems, the discovery of which belongs to the year 1922, is given in Łukasiewicz, J. (51), pp 115 ff. The philosophical implications of n-valued systems of sentential calculus are discussed in the article of Łukasiewicz, 'Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkükuls', Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, vol. 23 (1930), cl. iii, pp. 51-77.
( page 47 )
From Bibliography
(51) J. ŁUKASIEWICZ, Elementy logiki matematycznej (Elements of mathematical logic) ;(1929).
(51a) J. ŁUKASIEWICZ, ' O zaczeniu i potrzebach logiki matematycznej' (On the significance and needs of mathematical logic), Nauka Polska, x. 1929), 604-20.
( page 459 )
Logic, Semantics, Metamathematics, Alfred Tarski.
Oxford 1956.
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 )
† This paper appeared originally under the title 'Philosophische
Bemerkungen zu mehrwertigen Systemen des Aussagenkalküs' 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 Bibliography, Science and Sanity, 1933 by
Alfred
Korzybski
319. ŁUKASIEWICZ, J., and TARSKI, A. Untersuchungen über den Aussagenkalkül.
Ibid.
505. TARSKI, A. Fundamentale Begriffe der Methodologie der deduktiven Wissen-
schaften. I. Monatssh. f. Math, und Phys, B. XXXVII, H. 2. 1930.
506. See Łukasiewicz.
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p. 773.
Tarski's 'semantics'
From The Concept of Truth in Formalized Languages† 1931 by Alfred Tarski
The present article is almost wholly devoted to a single problem—the definition of truth. Its task is to construct—with reference to a given language—a materially adequate and formally correct definition of the term 'true sentence'. (Etc.)
† BBLIOGRAPHICAL NOTE. &nbap; This article was presented (by J. Łukasiewicz) to the Warsaw Scientific society on 21c March 1931. For reasons beyond the author's control, publication was delayed by two years. The article appeared in Polish in Tarski, A. (73). A German translation was published under the title 'Der Wahrhetisbegriff in den formalisierten Sprachen', in Studia Philosophica, vol. 1 (1936), (reprint dated 1935), pp. 261-405 ; it is provided with a Postscript in which some views which had been stated in the Polish original underwent a rather essential revision and modification. The present English version is based upon the German translation. For earlier publications and historical information concerning the results of this work see p. 154, footnote, p. 247, footnote, and the concluding remarks of the Postscript.
( page 152 )
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The investigation of formalized languages naturally demands a knowledge of the principles of modern formal logic. For the construction of the definition of truth certain purely mathematical concepts and methods are necessary, although in a modest degree. I should be happy if this work were to convince the reader that these methods already are necessary tools even for the investigation of purely philosophical problems.1
1 This was communicated to the Society of Sciences in Warsaw by J. Łukasiewicz on 21 March 1931. the results it contains date for the most part from 1929. I have reported on this, among other things, in two lectures which I gave under the title 'On the Concept of Truth in relation to formalized deductive systems' at the logical section of the Philosophical Society in Warsaw (8 October 1930) and the Polish Philosophical Society in Lwów (15 December 1930), a résumé of which appeared in Tarski, A. (73). For reasons unconnected with me the printing of this work was much delayed. his enabled me to supplement the text with some rather important results (cf. p. 247, footnote). In the meantime a résumé of the chief results has appeared in Tarski, A. (76).
( page 154 )
* * *
.. A characteristic feature of the semantical concepts is that they give expression to certain relations between the expressions of language and the objects about which these expression speak, or that by means of such relations they characterize certain classes of expressions or other objects. We could also say (making use of the supposition materialis) that these concepts serve to set up the correlation between the names of expressions and the expressions themselves.
For a long time the semantical concepts have had an evil reputation among specialists i the study of language. They have resisted all attempts to define their meaning exactly, and the properties of these concepts, apparently so clear in their content, have led to paradoxes and antinomies. For that reason the tendency to reduce these concepts to structural descriptive ones must seem quite natural and well-founded. (Etc).
( Page 252 )
From The Establishment of Scientific Semantics† 1935 by Alfred Tarski
The word 'semantics' is used here in narrower sense than usual. We shall understand by semantics the totality of considerations concerning those concepts which, roughly speaking express certain connexions between the expressions of a language and the objects and states of affairs referred to by these expressions. As typical examples of semantical concepts we may mention the concepts of denotation, satisfaction, and definition, which appear, for example, in the following statements:
The expression 'the victor of Jena' denotes Napoleon ; snow satisfied the condition 'xi is white' ; the equation 'x 3 = 2' defines (determines uniquely) the cube root of the number 2.
The concept of truth also—and this is not commonly recognized—is to be included here, at least in its classical interpretation, according to which 'true' signifies the same as 'corresponding with reality'.
Concepts from the domain of semantics have traditionally played a prominent part in the discussions of philosophers, logicians, and philologists. Nevertheless they have long been regarded with a certain scepticism. From the historical point of view this scepticism is well founded ; for, although the content of the semantical concepts, as they occur in colloquial language, is clear enough, yet all attempts to characterize this content more precisely have failed, and various discussions in which these concepts appeared and which were based on quite plausible and seemingly evident premisses, have often led to paradoxes and antinomies. (Etc).
The main source of the difficulties met with seems to lie in the following : it has not always been kept in mind that the semantical concepts have a relative character, that they must always be related to a particular language. people have not been aware that the language about which we speak need by no means coincide with the language in which we speak. They have carried out the semantics of a language in that language itself and, generally speaking, they have proceeded as though there was only one language in the world. the analysis of the antinomies mentioned shows, on the contrary, that the semantical concepts simply have no place in the language to which they relate, that the language which contains its ow semantics, and within which the usual logical laws hold, must inevitably be inconsistent. Only in recent years has attention been given to all these facts (as far as I know Leśniewski was the first to become fully aware of them).
† BIBLIOGRAPHICAL NOTE. This article is a summary of an address which was given at the International congress of Scientific Philosophy in Paris, 1935. The article first appeared in Polish under the title 'O ugruntuwaniu naukowej semantyki', in Przeglad Filozoficzny, vol 39 (1936), pp. 50-57, and later in German under the title 'Grundlegung der wissenschaftlichen Semantik', in Actes du Congrès International de Philosophie Scientifique, vol. 3 (Actualités Scientifiques et Industrielles, vol. 390), Paris, 1936, pp. 1-8.
( Pages 401-2 )
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