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

On Sat, 01 Dec 2007 18:38:44 +0000, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

But that is just what is wanted: a simple, unremarkable example of everyday mathematical
reasoning in action, using the Law of the Excluded Middle.

Let's get simpler still ...

Prove: 2 is the only even prime.

As far as I can see, it has to be a proof by contradiction.

But all proofs by contradiction are inherently based on the law of the
excluded middle, no?

My guess is that the logics which omit the law of the excluded middle
try to separate out those theorems (in a given theory) which can be
proved without it. But if you plan on doing any real math, the
equivalent concepts of

case analysis (divide and conquer)
proof by contradiction
law of the excluded middle

are essential.

As an even more elementary example, try to prove

x^2 - x = 0 implies x = 0 or x = 1

without using (explicitly or implicitly) the law of the excluded
middle.

Thus, if you insist of rejecting the law of the excluded middle,
you'll have a hard time getting through elementary algebra!

The law of the excluded middle is an axiom of logic that most of us
accept, even from childhood. Don't try to give it the same status as
the Axiom of Choice, or The Continuum Hypothesis.

As an example, if a child (age 2 perhaps) knows that a toy has not
left the room, but has looked everywhere except under the bed, the
child suddenly realizes (by the law of the excluded middle), that it
_must_ be under the bed.

Thus, if you argue against the law of the excluded middle, every
bright 2 year old would regard you as a crank.

quasi
.

Relevant Pages

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