[FOM] Intuitionists and excluded-middle

Lew Gordeew legor at gmx.de
Tue Oct 11 04:05:34 EDT 2005

On Sun, 9 Oct 2005 09:18:24 +0200, Arnon Avron wrote: 
> On Thu, Oct 06, 2005 at 09:47:10AM +0200, Jeremy Clark wrote:
> > 
> > Please could somebody explain to me what a "logical truth" is and why  
> > the law of excluded middle merits status as such. This seems to me  
> > like another defense of classical logic by bluster. ("Because I say  
> > so!")
> It is usually very unfruitful to argue with  declared intuitionists
> (or declared relevantists, or ... ), because they pretend not to
> understand
> the language we all speak. Still, I'll try to reply this time to
> Jermey Clark, though I have no illusions that I might convince him
> of anything. At best he might see why the gap between us is unbridgeable.
>  So first of all: Classical logic needs no defense. These are its
> attackers who need to justify their rejection of some of its laws.
> Despite of almost 100 years of desperate attempts, they have failed to
> provide any convincing argument for rejecting excluded middle (except
> "Because I say so").

A conventional convincing argument: mathematical proofs using the law of
excluded middle might be "useless". Here is a familiar trivial example
(quoted by A. S. Troelstra, et al).

THEOREM. There exists an irrational real number x such that x^sqrt(2) is

PROOF. Suppose sqrt(2)^sqrt(2) is rational. Then take x:=sqrt(2). Otherwise,
take x:=sqrt(2)^sqrt(2); hence x^sqrt(2) = sqrt(2)^(sqrt(2)*sqrt(2)) =
sqrt(2)^2 = 2 - a rational.
Q.E.D. by the law of excluded middle.

Now obviously this proof says "nothing" about the desired solution x.


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