Archive for History of Exact Sciences

, Volume 46, Issue 4, pp 367–383 | Cite as

An interpretation of Peano's logic

  • Evgeny A. Zaitsev


The problem of an adequate interpretation of G. Peano's logic still remains open, largely due to its striking pecularities:
  1. (i)

    the use of the same notation to designate relations both between classes and propositions;

  2. (ii)

    the lack of precision in his understanding of propositional functions as distinct from propositions;

  3. (iii)

    the systematic use of conditional definitions.


In this paper an attempt is made to explain these apparent shortcomings in the light of Peano's confusion between the logical calculus and its metatheory. In addition, the influence of traditional subject-predicate logic in Peano's writings is analysed. The focus of the study is Peano's representation of singular propositions and relations by means of his newly introduced sign of membership.


Singular Proposition Propositional Function Adequate Interpretation Logical Calculus Conditional Definition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baker, G. P., Hacker, P. M. S. 1984 Frege: Logical excavations. Oxford: Oxford univ. XV, 406 p.Google Scholar
  2. Boole, G. 1951. An investigation of the laws of thought. (1854). New York: Dover. 424 p.Google Scholar
  3. Bottazzini, U. 1985. Dall'analisi matematica al calcolo geometrico: origini delle prime ricerche di logica di Peano. Hist. philos. of logic, 6, 25–52.Google Scholar
  4. Eaton, R. M. 1959. General logic. An introductory survey (1931). New York: Charles Scribner's sons. XII, 630 p.Google Scholar
  5. Frege, G. 1980. Philosophical and mathematical correspondence. Chicago: Univ. of Chicago. XVIII, 214 p.Google Scholar
  6. Frege, G. 1984. Collected papers on mathematics, logic and philosophy. Oxford: Blackwell. VIII, 412 p.Google Scholar
  7. Goldfarb, W. D. 1979. Logic in twenties: the nature of quantifier. J. Symb. Logic, 44, n3, 351–368.Google Scholar
  8. Grattan-Guinness, I. 1977. Dear Russell — Dear Jourdain: A comment on Russell's logic, based on his correspondence with Philip Jourdain. London: Duckworth. (6), 234 p.Google Scholar
  9. Heijenoort, J. van 1967a. From Frege to Gödel. A source book in mathematical logic, 1879–1931. Cambridge (Mass.): Harvard Univ., X, 660 p.Google Scholar
  10. Heijenoort, J. van 1967b. Logic as calculus and logic as language, Synthèse, 17, 324–330.Google Scholar
  11. Jourdain, Ph. 1912. The development of the theories of mathematical logic and the principles of mathematics. Quaterly Journal of Pure and Applied Mathematics, 43, 219–314.Google Scholar
  12. Kneale, W. & Kneale, M. 1962. The development of logic. Oxford: Clarendon, VIII, 761 p.Google Scholar
  13. Lolli, G. 1982. Nel cinquantenario di Peano. Scientia (Milano), 117, 361–367.Google Scholar
  14. Peano, G. 1957. Opere scelte di Giuseppe Peano, v. 1. Roma: Cremonese. VII, 530 p.Google Scholar
  15. Peano, G. 1958. Opere scelte di Giuseppe Peano, v. 2. Roma: Cremonese. VI, 518 p.Google Scholar
  16. Peirce, Ch. S. 1960. Collected papers. 2nd. print. v. 3–4. Exact logic (published papers) and the simplest mathematics, — Cambridge (Mass.): Belknapp press of Harvard Univ. Press. XIV, 433p.; X, 601 p.Google Scholar
  17. Russell, B. 1900–1901. Sur la logique des relations avec des applications à la théorie des séries. Rivista di matematica, 7, 115–148.Google Scholar
  18. Russell, B. 1903. The principles of mathematics. London: Allen & Unwin, 1937. XXXIX, 543 p.Google Scholar
  19. Vailati, G. 1971. Epistolario 1891–1909. Torino: Einaudi. LXII, 767 p.Google Scholar
  20. Vuillemin, J. 1968. Leçons sur la première philosophie de Russell. Paris: Libr. Colin. 354 p.Google Scholar
  21. Zaitsev, E. A. 1988. From the history of mathematical logic at the end of 19th — beginning of 20th centuries: the logical theory of G. Peano. Institute for the History of Science and Technology, n24, 63 p. (Russian; English summary).Google Scholar
  22. Zaitsev, E. A. 1989. G. Peano on the notion of “the” and its elimination from theory. Istoriya i Metodologiya Estestvennyh Nauk, 36, Matematika i Mehanika. Moscow: Univ. of Moscow Press, 50–59. (Russian).Google Scholar
  23. Zaitsev, E. A. 1990. Semantic structure of G. Peano's logic. Istoriko-matematicheskiye Issledovaniya, 32–33. Moscow: Nauka, 146–157. (Russian).Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Evgeny A. Zaitsev
    • 1
  1. 1.Department of the History of MathematicsInstitute for the History of Science & TechnologyMoscow

Personalised recommendations