It has been frequently claimed that real progress in logic was only realized in the work of Frege and later work of Peirce in which the equational paradigm was dropped and the powerful expressive ability of quantification theory was realized. Nevertheless, Peirce's calculus of relations has remained a topic of interest to this day as an alternative, algebraic approach to the logic of relations. It has been studied by Lowenheim, Tarski and others. Lowenheim's famous theorem was originally a result about the calculus of relations rather than quantification theory, as it is usually presented today. Following his development of quantification theory, Peirce developed a graphical system for analyzing logical reasoning that he felt was superior in analytical power to his algebraic and quantificational notations.
Conjunction is indicated by juxtaposing graphs next to one another.
Juxtaposition and enclosure are the only relevant logical operations. Peirce provides five elegant rules of inference that form a complete set. The rules are Insertion in Odd, Erasure in Even, Iteration, Deiteration, and Double Cut. Odd Any graph can be added to an area enclosed within an odd number of cuts. Erasure in Even Any graph can be erased that occurs within an even number of cuts.