Owen Holden
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#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.
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.