Journal of Automated Reasoning

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

The Calculus of Relations as a Foundation for Mathematics



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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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceMills CollegeOaklandUSA

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