Re: The Law of the Excluded Middle again (long)

On Sun, 02 Dec 2007 23:17:36 +0000, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

On Sun, 02 Dec 2007 16:44:23 +0000, I wrote:

This application of the Law of the Excluded Middle [...]
has a characteristic which I didn't notice at the time,
and which no-one else has pointed out either: it is applied only
to propositions containing only free variables.

Inside the scope of quantification, and applied only to propositions
not themselves containing any quantifiers, it seems impossible (i.e.
even more impossible!) to equate (or confuse) truth with provability.

But you managed to do it -- you are confusing it.

There is no question of "proving" that x >= 1, when x is a free
variable not yet subject to quantification.

If the variable x is free, there's no such thing as a valid statement
of the form "x >= 1" unless the statement has an implied
quantification. By default, the implied quantification is "for all",
so the statement x >= 1 is false (and provably so) if the domain for
the variable x is, for example, the set of real numbers.

Quantification of free variables _must_ be specified, either
explicitly or implicitly, otherwise you don't even have a valid
statement.

The distinction between truth and provability, as Goedel's results
clearly show, is a real distinction, but applies only to _valid_
statements. Thus, lack of quantification doesn't yield an example of a
non-provable statement, but rather an invalid statement.

quasi
.

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