Journal of Automated Reasoning

, Volume 37, Issue 4, pp 277–322 | Cite as

The Calculus of Relations as a Foundation for Mathematics

Article

Abstract

A variable-free, equational logic \(\mathcal{L}^\times\) based on the calculus of relations (a theory of binary relations developed by De Morgan, Peirce, and Schröder during the period 1864–1895) is shown to provide an adequate framework for the development of all of mathematics. The expressive and deductive powers of \(\mathcal{L}^\times\) are equivalent to those of a system of first-order logic with just three variables. Therefore, three-variable first-order logic also provides an adequate framework for mathematics. Finally, it is shown that a variant of \(\mathcal{L}^\times\) may be viewed as a subsystem of sentential logic. Hence, there are subsystems of sentential logic that are adequate to the task of formalizing mathematics.

Key words

calculus of relations authomated reasoning algebraic logic set theory mathematical foundation 

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References

  1. 1.
    Andréka, H., Németi, I., Sain, I.: Algebraic logic. In: Gabbay, D.M., Guenther, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 133–247. Second edition, Kluwer, Dordrecht (2001)Google Scholar
  2. 2.
    van Benthem, J.: Exploring Logical Dynamics. Studies in Logic, Language, and Information, xi + 329 pp. CSLI Publications, Stanford, CA (1996)Google Scholar
  3. 3.
    Brink, C., Kahl, W., Schmidt, G. (eds.): Relational Methods in Computer Science. Adv. Comput. Sci. xv + 272 pp. Springer, Berlin Heidelberg New York (1997)Google Scholar
  4. 4.
    Chang, C.C., Keisler, H.J.: Model Theory. Studies in Logic and the Foundations of Mathematics, vol. 73, xvi + 650 pp. Third edition, North-Holland, Amsterdam (1990)Google Scholar
  5. 5.
    Church, A.: Introduction to Mathematical Logic, vol. 1, x + 378 pp. Princeton University Press, Princeton (1956)Google Scholar
  6. 6.
    De Morgan, A.: On the syllogism, no. IV, and on the logic of relations. Trans. Camb. Philo. Soc. 10, 331–358 (1864)Google Scholar
  7. 7.
    Enderton, H.B.: A Mathematical Introduction to Logic, xiii + 295 pp. Academic, New York (1972)MATHGoogle Scholar
  8. 8.
    Enderton, H.B.: Elements of Set Theory, xiv + 279 pp. Academic, New York (1977)MATHGoogle Scholar
  9. 9.
    Gabbay, D.M., Kurucz, Á., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. Studies in Logic and the Foundations of Mathematics, vol. 148, xviii + 747 pp. Elsevier, Amsterdam (2003)Google Scholar
  10. 10.
    Halmos, P.R.: Algebraic Logic, 271 pp. Chelsea, New York (1962)MATHGoogle Scholar
  11. 11.
    Halmos, P.R.: Lectures on Boolean Algebras. Van Nostrand Mathematical Studies, no. 1, ii + 147 pp. D. Van Nostrand Company, Princeton, NJ, (1963) (Reprinted by Springer, Berlin Heidelberg New York (1974))Google Scholar
  12. 12.
    Henkin, L., Monk, D., Tarski, A.: Cylindric Algebras, Part I. Studies in Logic and the Foundations of Mathematics, vol. 64, vi + 508 pp. North-Holland, Amsterdam (1971)Google Scholar
  13. 13.
    Henkin, L., Monk, D., Tarski, A.: Cylindric Algebras, Part II. Studies in Logic and the Foundations of Mathematics, vol. 115, ix + 302 pp. North-Holland, Amsterdam (1985)Google Scholar
  14. 14.
    Hirsch, R., Hodkinson, I., Maddux, R.D.: Relation algebra reducts of cylindric algebras and an application to proof theory. J. Symb. Log. 67, 197–213 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Hirsch, R., Hodkinson, I., Maddux, R.D.: Provability with finitely many variables. Bull. Symb. Log. 8, 348–379 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hirsch, R., Hodkinson, I.: Relation Algebras by Games. Studies in Logic and the Foundations of Mathematics, vol. 147, xvii + 691 pp. Elsevier, Amsterdam (2002)Google Scholar
  17. 17.
    Hungtington, E.V.: New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica. Trans. Amer. Math. Soc. 35, 274–304 (1933)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Hungtington, E.V.: Boolean algebra. A correction. Trans. Amer. Math. Soc. 35, 557–558 (1933)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Math. Ann. 76, 447–470 (1915)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Löwenheim, L.: Einkleidung der Mathematik in Schröderschen Relativkalkul. J. Symb. Log. 5, 1–15 (1940)MATHCrossRefGoogle Scholar
  21. 21.
    Maddux, R.D.: Topics in relation algebras. Doctoral dissertation, University of California, Berkeley, iii + 241 pp. (1978)Google Scholar
  22. 22.
    Maddux, R.D.: Nonfinite axiomatizability results for cylindric and relation algebras. J. Symb. Log. 54, 951–974 (1989)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Maddux, R.D.: Relation algebras of formulas. In: Orlowska, E. (ed.) Logic at Work. Studies in Fuzziness and Soft Computing, vol. 24, pp. 613–636. Springer, Heidelberg and New York (1999)Google Scholar
  24. 24.
    McKenzie, R.N.: The representation of relation algebras, Doctoral dissertation, University of Colorado, Boulder, vii + 128 pp. (1966)Google Scholar
  25. 25.
    McKenzie, R.N.: Representations of integral relation algebras. Mich. Math. J. 17, 279–287 (1970)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Monk, J.D.: On representable relation algebras. Mich. Math. J. 11, 207–210 (1964)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Monk, J.D.: Nonfinitizability of classes of representable cylindric algebras. J. Symb. Log. 34, 331–343 (1969)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Moore, G.H.: A house divided against itself: The emergence of first-order logic as the basis for mathematics. In: Phillips, E. (ed.) Studies in the History of Mathematics. Studies in Mathematics, vol. 26, pp. 98–136. The Mathematical Association of America (1987)Google Scholar
  29. 29.
    Németi, I.: Free algebras and decidability in algebraic logic (in Hungarian). Habilitation, Hungarian Academy of Sciences, Budapest, xviii + 169 pp. (1986) (An English translation is available from the author.)Google Scholar
  30. 30.
    Németi, I., Sain, I. (special eds.): Log. J. IGPL 8, 373–591 (2000)Google Scholar
  31. 31.
    Peirce, C.S.: Note B. The logic of relatives. In: Peirce, C.S. (ed.) Studies in Logic by Members of the Johns Hopkins University, pp. 187–203. Little, Brown, and Company, Boston (1883) (Reprinted by John Benjamins, Amsterdam, 1983.)Google Scholar
  32. 32.
    Russell, B.: The Principles of Mathematics. Cambridge University Press (1903) (Reprinted by Allen & Unwin, London, 1948.)Google Scholar
  33. 33.
    Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik), vol. III, Algebra und Logik der Relative, part 1, 649 pp. Publisher: B. G. Teubner, Leipzig (1895) (Reprinted by Chelsea, New York, 1966.)Google Scholar
  34. 34.
    Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Tarski, A.: A simplified formalization of predicate logic with identity. Arch. Math. Log. Grundl. Forsch. 7, 61–79 (1965)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Tarski, A., Givant, S.: A Formalization of Set Theory without Variables. Colloquium Publications, vol. 41, xxi + 318 pp. Am. Math. Soc. Providence RI (1987)Google Scholar
  37. 37.
    Tarski, A., Mostowski, A., Robinson, R.M.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics, xi + 98 pp. North-Holland, Amsterdam (1953)Google Scholar
  38. 38.
    Vaught, R.L.: On a theorem of Cobham concerning undecidable theories. In: Nagel, E., Suppes, P., Tarski, A. (eds.) Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress, pp. 14–25. Stanford University Press, Stanford (1962)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceMills CollegeOaklandUSA

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