About: Theory of pure equality

In mathematical logic the theory of pure equality is a first-order theory. It has a signature consisting of only the equality relation symbol, and includes no non-logical axioms at all. This theory is consistent but incomplete, as a non-empty set with the usual equality relation provides an interpretation making certain sentences true. It is an example of a decidable theory and is a fragment of more expressive decidable theories, including monadic class of first-order logic (which also admits unary predicates and is, via Skolem normal form, related to set constraints in program analysis) and monadic second-order theory of a pure set (which additionally permits quantification over predicates and whose signature extends to monadic second-order logic of k successors).