Classical Identity	A world in which every individual exemplifies and every haecceity is exemplified is described by adding P10 to the postulates of HI, yielding the sub-theory HI+. P10 xy(y ex X)  (All haecceities are exemplified.)  In HI+, identity is strongly reflexive and indiscernibles are identical. HI+ is thus a classical extension of HI.
	T10* x(x = x)  (Identity is reflexive.)  T11* xy(z(x ex Z  y ex Z)  x = y)  (Indiscernibles are identical.)
Non-Classical Identity	In contrast, a world in which not every individual exemplifies and not every haecceity is exemplified is described by adding P10'13 to the postulates of HI, yielding the sub-theory HI-. P10' ¬xy(y ex X) (Not all haecceities are exemplified.)  T10'* ¬x(x = x)  (Identity is not strongly reflexive.)
	T11'* ¬xy(z (x ex Z y ex Z)  x = y)  (Not all indiscernibles are identical.)  In the resulting sub-theory, identity is nonreflexive and not all indiscernibles are identical. HI- is thus a non-classical extension of HI.