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#1 2013-10-13 10:21:46

Al-Allo
Member
Registered: 2012-08-23
Posts: 293

Reduction ad Absurdum

Hi, I have a little question concerning contradictions :

If I have a statement "A" that I want to prove, and only have the possibility for it to be True or False.

After some manipulations, I arrive at some contradiction. (Here's where my question begins.)

How can we know that a contradiction is enough to be sure at 100 % that a statement is not correct?

Is it because in Mathematics, for a thing to be True or False, it must always be ALWAYS "working" without arriving at some contradiction ? (Mathematical ideas must always work, and not sometimes yes, sometimes no.)

I just want to be sure of thinking of it in the right way, corrections would be greatly appreciated ! Thank you !

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#2 2013-10-13 10:28:03

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,859

Re: Reduction ad Absurdum

A contradiction is by definition a statement that is always false. If by manipulating you find that a statement is equivalent to some contradiction, it must be equivalent to false and thus itself is false.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2013-10-13 19:27:32

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,252

Re: Reduction ad Absurdum

hi Al-Allo

In mathematics, if you start with a TRUE statement and use correct working to obtain another statement, then that statement must also be TRUE.

So, let's assume A is FALSE.  If we can do correct working that leads to a statement that is FALSE, then the assumption must have been incorrect.  That's how we know A is TRUE.

Simple example.  Let's use reductio ad absurdum  to prove that for any integer, n, 2n is EVEN.

A = "2n is EVEN, where n is an integer"

Assume this is FALSE.  ie. 2n is not EVEN.

This means that 2n is ODD.

This means that 2n + 1 is EVEN, so it can be divided by 2, giving an integer result.

(2n + 1)/2 = n + 1/2. This is an integer.

Subtract n (an integer) and the result is also an integer

This means that 1/2 is an integer.

This result is FALSE so the assumption was incorrect.

Therefore 2n is NOT ODD so it is EVEN.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2013-10-14 02:58:25

Al-Allo
Member
Registered: 2012-08-23
Posts: 293

Re: Reduction ad Absurdum

AH thank you ! Last question, a real verity will never make any contradictions, right ? (What ever manipulations you do with it )

Last edited by Al-Allo (2013-10-14 03:00:30)

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#5 2013-10-14 05:10:22

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,252

Re: Reduction ad Absurdum

(What ever manipulations you do with it )

Say:  "What ever correct manipulations you do with it " and I'm happy.  smile

In place of "correct" you could also say "valid".

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2013-10-14 06:31:10

Al-Allo
Member
Registered: 2012-08-23
Posts: 293

Re: Reduction ad Absurdum

bob bundy wrote:

(What ever manipulations you do with it )

Say:  "What ever correct manipulations you do with it " and I'm happy.  smile

In place of "correct" you could also say "valid".

Bob

AH yes, sorry for the mistake. So yes, any valid manipulation !

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