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On the calculus of relations

  • Alfred Tarski (a1)

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The logical theory which is called the calculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years.

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1 For detailed historical information and thorough bibliographical references, see Lewis, C. I., A survey of symbolic logic, Berkeley, California, 1918.

2 Grundzüge der theoretischen Logik, Berlin 1938 (second edition), p. 45 ff.

3 This part of our axiom system is due essentially to E. V. Huntington; see his paper Sets of independent postulates for the algebra of logic, Transactions of the American Mathematical Society, vol. 5 (1904), p. 292 f. It is easily seen that axioms I–VII are symmetric with respect to the absolute constants ‘+’ and ‘·’, as well as ‘1’ and ‘0’. It would be desirable to have a simple and independent axiom system for the whole calculus of relations which would be symmetric with respect to both absolute and relative constants.

4 Cf. Schröder, op. cit., p. 52 ff.; Kuratowski, C. and Tarski, A., Les opérations logiques et les ensembles projectifs, Fundamenta mathematicae, vol. 17 (1931), p. 240 ff.

5 See Schröder, op. cit., p. 153.

6 Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349360.

7 The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40 (1936), pp. 37111; see in particular p. 106.

8 Postulates for the calculus of binary relations, this Journal, vol. 5 (1940), pp. 8597; see in particular p. 94.

9 A note on the Entscheidungsproblem, this Journal, vol. 1 (1936), pp. 4041 (and Correction, ibid., pp. 101–102).

10 The proof will be given later in a separate paper.

11 Korselt's result was published in the paper of Löwenheim, L., über Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447470.

12 Über die Beschränktheit der Ausdrucksmittel deduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums, no. 7 (1936), pp. 1522. (There are several misprints in that paper. In particular, read ‘metamathematisch’ and ‘Metamathematik’ instead of ‘mathematisch’ and ‘Mathematik.’)

13 I wish to take this opportunity to express my gratitude to Dr. J. C. C. McKinsey for his assistance in preparing this paper.

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