Proving Statements by Contradiction: Understanding the Logic Behind It

[p3forlife]

Hi, I have a question about proofs by contradiction in general. Without getting into the mathematical details, suppose we had the statement:

For every (condition A), B is true.

If we want to prove this by contradiction, we want to assume the negation of this statement, and then prove it to be false.

My question is, what is the statement we assume when we prove it by contradiction? Is it:

1. There exists a (condition A) such that B is not true.
2. For every (condition A), B is not true.

My guess is 1. But in this case, wouldn't it be hard to prove 1 by contradiction, because you are trying to prove a specific case to be false?

I usually have confusion with logic when "for every" and "there exist" crop up in statements. Then I'm not sure which "for every" and "there exist" to change to prove by contradiction or by contrapositive.

Thanks for your help!

 

[HallsofIvy]

The "inverse" of "for every A, B is true" is "there exist A such that B is not true".

Think about it- if there exist a single condition on A for which B is not true, then "for every A, B is true" is wrong.

 

[HmBe]

Let A be the event it rains. Let B be the event that the ground gets wet.

For all A, B = Whenever it rains, the ground gets wet.

Proof by contradiction?

Assume that there exists a time when it rains and the ground won't get wet. Then quite clearly the ground doesn't always get wet when it rains. Well if we can find a contradiction, clearly our assumption is wrong, so the opposite must be true.

If it rains and the ground doesn't get wet, where has all the water gone? That's just silly. Contradiction, so the ground always gets wet when it rains.