The concepts of equivalence, equality and identity (Q777979)

This is a nice historical and philosophical essay on the notions of equivalence, equality, identity and indistinguishability. After an etymological and historical discussion of the four notions individually, they are compared. According to the author, equivalence and equality turn out to have the same extension, but not the same intention. As for indistinguishability and identity, identity is an ontological category, while indistinguishability refers to properties that can be expressed in a language. There is a section on Leibniz' views on equality and identity and a section on the notion of identity in Frege's work. Finally, the author shows along the lines of Gödel's completeness theorem that Frege's `equality-axioms' imply everything that can be said about identity in a first order language.