Re: Existence, Self-identity and Uniqueness.

Hi

William Elliot wrote:
Ie, when E!x F(x)
So I see not the objection.

The objection is that "x=y -> (A(x)<->A(y))"
should always hold in FOL=, and not only
under some circumstances.

If it holds only under some circustances,
i.e. substituting for x the expression
"the x:Fx", picking the A that I gave (call
it A0), and picking and F that is not
function (call it F0), then we arrive at
a contradiction.

The contraction is seen immediately as
follows. In FOL= we have the following
axiom schema:

FOL= |- forall x forall y(x=y -> (A(x)<->A(y))) (1)

The counterexample from my previous email states:

FOL= |- ~ forall y (the x:F0x -> (A0(the x:F0x) <-> A0(y)) (2)

Now specialize (1) by the x:F0x and A0, and one gets:

FOL= |- forall y(the x:F0x -> (A0(the x:F0x)<->A0(y)) (3)

Thus we have with (2) and (3) that FOL= is inconsistent.
But lets assume that FOL= is not inconsistent, so
the problem is the form "the x:Fx". It renders
FOL= inconsistent.

Usually we want logical forms that do not render our
logic FOL= inconsistent. Because in classical logic,
an inconsistency allows us to derive any formula,
which makes the logic total useless.

Bye





.

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