**Technically speaking,**

We have blown up the "root of contradiction" with an _atomic_ bomb.

To the casual philosophers we are, I will show the links below.

To the professional programmer: a crack is not a hack, but a hack is a crack.

Keep up the go(o)d work.

==============================================

Julio Di Egidio

Re: Question about proof by contradiction

Posted: Mar 10, 2008 11:12 AM

http://mathforum.org/kb/message.jspa?messageID=6131405

[...]

BTW, here is an extract from a letter from Wittgenstein to Russel, 1921, which I think sheds some light (I'm

afraid I'll have to traslate, I've got it in Italian):

"I believe our problems track down to _atomic_ propositions. You'll see it if you try to precisely explain

how the Copula is such propositions has meaning. I cannot explain it and I believe that, once an exact answer

is given to this question, the problem of <> and of the apparent variable will be _much_ nearer its

solution, if not solved. Now I think above <> (the good old Socrates!)."

The good old Ludwig!!

Explaining that meaning, by means of the empty set "we" are, is indeed what I have shown (again, until dis-

proved).

Julio

==============================================

Marshall

Re: Question about proof by contradiction

Posted: Mar 10, 2008 5:46 PM

http://mathforum.org/kb/message.jspa?messageID=6131836

> >> Proof by contradiction can be formalized as

>

> >> (P -> (A and not(A))) -> not(P).

[...]

> The proof in question, in fact, does not even use proof by

> contradiction. It has the form

>

> (P -> not(P)) -> not(P).

>

> This is not a proof by contradiction.

[...]

Does anything interesting happen if we transform them somewhat?

(P -> (A and not(A))) -> not(P)

(P -> false) -> not(P).

(not(P) or false) -> not(P)

not(P) -> not(P)

Well, I seem to have destroyed the formula's essential nature

by these manipulations. How did THAT happen? Apparently

truth-value-preserving transformations don't preserve some things

that aren't truth values.

Marshall

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