What is Proof by Contradiction?
- Proof by contradiction aims to prove a point by showing that the opposite isn't true.
- It comes in handy as you only need one contradiction to prove a statement is false.
- For example, to prove the statements "not all swans are white".
- You would assume the opposite statement "all swans are white" and try and find a contradiction to it.
- And if you find a black swan you'll have found a contradiction to the statement, proving that not all swans are white!
- While proof by contradiction is good at proving statements false, it is difficult to prove statements as true, unless they are very generalized.
- For example you can prove the statement "not all swans are white" is true, but anything more specific, such as "swans are only white and black" cannot be proven by contradiction, as you'd need to look at every single swan in the world.
- There could easily be a xanthochromistic swan that would be yellow, proving your statement wrong.
- Sometimes you encounter statements that seem to be true, and for every possible contradiction you consider the statement still holds.
- While this can verify a statement this way, it's not sufficient to prove a statement, as it is possible that there is a contradiction you might not have considered, which would prove the statement wrong.
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