Re: Existence, Self-identity and Uniqueness.

Owen wrote:

Jan Burse wrote:
>>But when Fx has not a unique x, then A(the x:Fx)
>>becomes false.
> Yes. A(the x:Fx) -> E!(the x:Fx), implies,
> ~E!(the x:Fx) -> ~A(the x:Fx).

Not always. It really depends what A is in
the current course. I am using the letter A
here to stand for a formula and not for a
predicate.

That A stands for a formula and not for a
predicate, is also the intention of "x=y
-> (A(x) <-> A(y))". In FOL= this substitution
law for formulas can be derived from the
following more basic substition laws:

x1=y1 & .. & xn=yn -> (R(x1,..,xn)->R(y1,..,yn)) (1)
x1=y1 & .. & xn=yn -> (f(x1,..,xn)=f(y1,..,yn)) (2)

Here in (1) the letter R stands for a predicate.
And in (2) the letter f stands for a function.
But in my discussion the A was standing for
a formula.

For example the following formula A does not behave
as you stated (call it A1):

A1(x) :<=> true.

We do not have:

~A1(the x:Fx)

when Fx is not unique.

Note also that in my post, where I showed that
FOL= gets inconsistent, I am also using a particular
A (call it A0). And you have really to go into
A0 to descide A0(the x:Fx).

Bye
.

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