Page "Free logic" Paragraph 7

Informally, if F is '= y ', G is ' is Pegasus ', and we substitute ' Pegasus ' for y, then ( 4 ) appears to allow us to infer from ' everything identical with Pegasus is Pegasus ' that something is identical with Pegasus.

The problem comes from substituting nondesignating constants for variables: in fact, we cannot do this in standard formulations of first-order logic, since there are no nondesignating constants.

Classically, ∃ x ( x = y ) is deducible from the open equality axiom y = y by particularization ( i. e. ( 3 ) above ).