The Z operator applied to an unknown and anything but a falsehood yields an unknown. And the Z operator applied to a truth and some other value yields the other value. In addition to these two strong Kleene operators, Peirce defines several other forms of negation, conjunction, and disjunction. The notes also provide some basic properties of some of the operators, such as being symmetric and being associative. Calculus of Relations Building on ideas of De Morgan, Peirce fruitfully applied the concepts of Boolean algebra to relations.
Boolean algebra is concerned with operations on general or class terms.
Peirce presented numerous theorems involving his operations on relative terms. Peirce's calculus of relations has been criticized for remaining unnecessarily tied to previous work on Boolean algebra and the equational paradigm in mathematics. 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.