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.
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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)
van Benthem, J.: Exploring Logical Dynamics. Studies in Logic, Language, and Information, xi + 329 pp. CSLI Publications, Stanford, CA (1996)
Brink, C., Kahl, W., Schmidt, G. (eds.): Relational Methods in Computer Science. Adv. Comput. Sci. xv + 272 pp. Springer, Berlin Heidelberg New York (1997)
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)
Church, A.: Introduction to Mathematical Logic, vol. 1, x + 378 pp. Princeton University Press, Princeton (1956)
De Morgan, A.: On the syllogism, no. IV, and on the logic of relations. Trans. Camb. Philo. Soc. 10, 331–358 (1864)
Enderton, H.B.: A Mathematical Introduction to Logic, xiii + 295 pp. Academic, New York (1972)
Enderton, H.B.: Elements of Set Theory, xiv + 279 pp. Academic, New York (1977)
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)
Halmos, P.R.: Algebraic Logic, 271 pp. Chelsea, New York (1962)
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))
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)
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)
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)
Hirsch, R., Hodkinson, I., Maddux, R.D.: Provability with finitely many variables. Bull. Symb. Log. 8, 348–379 (2002)
Hirsch, R., Hodkinson, I.: Relation Algebras by Games. Studies in Logic and the Foundations of Mathematics, vol. 147, xvii + 691 pp. Elsevier, Amsterdam (2002)
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)
Hungtington, E.V.: Boolean algebra. A correction. Trans. Amer. Math. Soc. 35, 557–558 (1933)
Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Math. Ann. 76, 447–470 (1915)
Löwenheim, L.: Einkleidung der Mathematik in Schröderschen Relativkalkul. J. Symb. Log. 5, 1–15 (1940)
Maddux, R.D.: Topics in relation algebras. Doctoral dissertation, University of California, Berkeley, iii + 241 pp. (1978)
Maddux, R.D.: Nonfinite axiomatizability results for cylindric and relation algebras. J. Symb. Log. 54, 951–974 (1989)
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)
McKenzie, R.N.: The representation of relation algebras, Doctoral dissertation, University of Colorado, Boulder, vii + 128 pp. (1966)
McKenzie, R.N.: Representations of integral relation algebras. Mich. Math. J. 17, 279–287 (1970)
Monk, J.D.: On representable relation algebras. Mich. Math. J. 11, 207–210 (1964)
Monk, J.D.: Nonfinitizability of classes of representable cylindric algebras. J. Symb. Log. 34, 331–343 (1969)
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)
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.)
Németi, I., Sain, I. (special eds.): Log. J. IGPL 8, 373–591 (2000)
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.)
Russell, B.: The Principles of Mathematics. Cambridge University Press (1903) (Reprinted by Allen & Unwin, London, 1948.)
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.)
Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)
Tarski, A.: A simplified formalization of predicate logic with identity. Arch. Math. Log. Grundl. Forsch. 7, 61–79 (1965)
Tarski, A., Givant, S.: A Formalization of Set Theory without Variables. Colloquium Publications, vol. 41, xxi + 318 pp. Am. Math. Soc. Providence RI (1987)
Tarski, A., Mostowski, A., Robinson, R.M.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics, xi + 98 pp. North-Holland, Amsterdam (1953)
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)
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This paper is an expanded version of a talk given by the author at the Special Session on Automated Reasoning in Mathematics and Logic, held March 8–10, 2002, at the Georgia Institute of Technology, during the Joint Southeastern Section MAA/Southeast Regional AMS Meeting. The session was organized by Johan G. F. Belinfante.
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Givant, S. The Calculus of Relations as a Foundation for Mathematics. J Autom Reasoning 37, 277–322 (2006). https://doi.org/10.1007/s10817-006-9062-x
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DOI: https://doi.org/10.1007/s10817-006-9062-x