 Contemporary logic follows Frege in treating identity, represented by \(=\), as a logical notion. In PM the notion of identity is defined following Leibniz as indiscernibility, namely indiscernible objects are identical. That is, \(\forall \phi ( \phi x \equiv \phi y) \supset x = y\). But since the axiom of reducibility guarantees that if there is any type of function on which x and y differ, they will differ on some predicative function, PM uses the following definition of identity: