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I would like to prove that $P$ is false. The one line proof is:

> Assume that $P$ is true. We already know that $P$ is false. QED

This comes when my friend and I were talking about the following real life situation:

  • I said: I never respect someone who doesn't respect me.
  • He told me: assume that your father doesn't respect you, what to do?
  • I said: My father always respects me.

After that he started saying your reasoning is wrong and not scientific, etc. He said I told you "assume".

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    If you already know $P$ is false why do you need to prove it? – 2017-01-30
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    Because someone asked me to "assume" that $P$ is true. – 2017-01-30
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    I don't quite understand the question. What is $P$? Is it supposed to be an arbitrary statement, or a specific one? Could you give an example related to mathematics? – 2017-01-30
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    I don't think there is anything to prove here. You two just disagree with value of $P$. – 2017-01-30
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    @Mummert $P$ is a property that is false and someone knows that is false but someone else would like to assume that it is true and see what happens. – 2017-01-30
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    I find this question interesting. – 2017-01-30

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No proof of a statement that requires the step "we already know our statement is true" can ever be a correct proof - this is a strategy called begging the question, in which you use the thing you are trying to prove in your proof.

For example, I could suggest the following "proof" of the statement "Every number is even":

  • Suppose that not every number is even.
  • We already know every number is even.
  • Therefore, we have a contradiction. So every number is even.

This argument has exactly the same logical structure as you propose, but "proves" something completely false!

On the other hand, it really doesn't seem like the logical structure you outlined has anything to do with the argument you gave as an example. Your friend isn't trying to prove that your father doesn't respect you; presumably, your friend is trying to demonstrate a consequence of your premise. By saying "Assume your father doesn't respect you", your friend is asking you to consider a universe in which your premise, "I never respect anyone who doesn't respect me" is true, but in which your father doesn't respect you. Unless you think that it is logically impossible for your father to fail to respect you, this isn't an unreasonable universe to consider. By replying with "my father always respects me", you're saying that this hypothetical universe is either impossible or irrelevant.

What your friend seems to be getting at is that if you claim that you "never respect anyone who doesn't respect [you]", then you can't also claim "I will respect my father no matter what" - for example, if your father suddenly stopped respecting you, you would either have to stop respecting him or stop claiming to never respect anyone who doesn't respect you. This is a valid argument, and illustrates an important relationship between the two claims, regardless of whether or not you think it's possible for your father not to respect you.