, Volume 37, Issue 4, pp 277-322
Date: 02 Feb 2007

The Calculus of Relations as a Foundation for Mathematics

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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.

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.