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

On Mon, 03 Dec 2007 04:24:27 -0500, quasi
<quasi@xxxxxxxx> wrote:

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.

Perhaps, but not in the way you think (below).

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

This is what I said, and you seem to be saying exactly the same
thing, perhaps under the impression that I meant something other
than what I said. (Probably this was because, as so often with
me, it just wasn't clear. But my struggles to say things clearly
only seem to result in lengthy verbiage, so I often have to say
things briefly, wait for misunderstandings to occur, and then
correct them - or else breathe a sigh of relief, if, for once,
I have managed to be both clear and brief.)

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.

I'm well aware of all of that.

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.

All true, I'm sure, but irrelevant to my point.

I don't know how to explain my point better. Rather than try
to do so (which would only lead to further possibly unreadable
verbiage), may I simply ask how /you/ think of the meaning of
(for example) the statement "either x > 1 or x <= 1", where x
is a variable, which has been introduced in an informal proof,
and you are still in the middle of the proof? No-one is asking
for this statement to be frozen, quantified, and then assigned a
truth value! That would be ridiculous: which seems to be what
you are saying to me, but is something I already know (as does
everybody else). But in the course of the proof, each of the
statements "x > 1" and "x <= 1" is thought of as having a truth
value (which in a sense is a "variable", just like x). At least,
that is how I seem to think of things (so my unreliable faculty
of introspection tells me). Do you think about it differently?

(It is of course quite possible that I am making some tremendously
ridiculous howling error, but it is not the particular tremendously
ridiculous howling error that you think it is!) :-)

(My whole aim in all of this is to locate precisely where I am
thinking wrongly about mathematics; therefore, I am not offended
in general terms by a suggestion that I have got something very
badly wrong. But I am trying not to be prejudiced as to the
location of my error(s), /and/ I am trying not to automatically
believe someone when they say that some particular belief of
mine is erroneous. So I will argue about particular points -
perhaps quite tenaciously - but always with the overall aim of
eventually conceding defeat, and saying, "Ah, so THAT's it!")
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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