#1 Nothing exists, means, It is not the case that something exists. (Nothing exists) <-> ~(Something exists) Something exists <-> ~(Nothing exists) Something exists, means, there is an x such that: x exists. Something exists, means, Ex(x exists). (ExE!x) x exists, is defined, there is some y such that: x is equal to y. E!x =df Ey(x=y). Something exists, means, Ex[Ey(x=y)]. Nothing exists, means, ~Ex[Ey(x=y)]. Because, (nothing exists) <-> ~(something exists). But, ExEy(x=y) is a theorem. 1. Ax[x=x] and 2. AxAy[x=y -> (Fx <-> Fy)] are the axioms of identity theory, within first order predicate logic. ExEy(x=y) Proof: 1. Ax(x=x) -> a=a 2. a=a -> Ey(a=y). 3. Ey(a=y) -> Ex[Ey(x=y)]. 4. Ax(x=x) -> ExEy(x=y). 5. ExEy(x=y). By axiom 1, Ax(x=x). If we use the second order Leibnitz-Russell definition of identity, x=y =df AF(Fx <-> Fy), then we can prove that Ax(x=x) is a theorem. x=x means AF(Fx <-> Fx), which is clearly tautologous for all x. Therefore ~ExEy(x=y) is a contradiction. i.e. Nothing exists is a contradiction. ~(Nothing exists), is a theorem of classical logic.