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

On Tue, 04 Dec 2007 04:13:40 +0000, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

On Mon, 03 Dec 2007 21:33:41 -0500, quasi <quasi@xxxxxxxx> wrote:

Do you ever plan to teach?

Of course not.

Thus, if you assume temporarily that x >= 0, then, within the scope of
that assumption, x _is_ greater or equal to zero. There's no need to
prove it -- it's regarded as an additional hypothesis, valid for the
duration of the assumption. Thus, in the scope of that assumption, x
= 0 is true. But while proving it would be silly, the proof would
only take one line: x >= 0 (by temporary hypothesis). So it's
incorrect to say that "x >= 0" can't be proved. Rather, it's true and
provably so within the scope of the assumption.

Recall the context of the argument. When the LEM is applied, as in
"(x >= 0) or not (x >= 0)", /if/ there is to be an objection on the
grounds that truth means provability, then the truth of a statement
certainly cannot be identified with its provability using itself as
a hypothesis, because such an identification would make every state-
ment true.

You lost me.

How in the world would it make any statement true?

When you make an assumption within a proof, anything you prove based
on that assumption is conditional on that assumption (together with
the original hypothesis, and any other enclosing assumptions in
force).

Thus, if you assume x >= 0 within a proof, then within the scope of
that assumption, x >= 0 is true, and provably so (but it would be
silly to prove it, since you already have it as a temporary
hypothesis). Still, if challenged, the proof is trivial ...

x >= 0 (reason: temporary hypothesis)

Therefore x >= 0 (reason: prior line)

You appear to be claiming that allowing such assumptions would make
every statement true. But that's silly. You can only prove those
things that are provable from the hypotheses (including any temporary
ones in force), and any conclusions obtained are conditional on those
hypotheses. Thus, I don't get your point.

quasi
.

Relevant Pages

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