Proof by contradictions work by assuming that something is true, and then using logic (along with other assumptions which you know are true) to show that that leads to a contradiction, thus proving that your assumption must be false.

If I can prove that a particular statement P is false via a proof by contradiction, can I always prove that P is false via some other method? (For example, by proving that not P is true without using a proof by contradiction.)

If so, can you provide some proof that every proof by contradiction can be turned into some other form of proof?

If not, is there any well-defined way of describing when a proof by contradiction is the only proof that will work?

Proof by contradictions work by assuming that something is true, and then using logic (along with other assumptions which you know are true) to show that that leads to a contradiction, thus proving that your assumption must be false.

1. If I can prove that a particular statement P is false via a proof by contradiction, can I always prove that P is false via some other method? (For example, by proving that not P is true without using a proof by contradiction.)

2. If so, can you provide some proof that every proof by contradiction can be turned into some other form of proof?

3. If not, is there any well-defined way of describing when a proof by contradiction is the only proof that will work?