What is an indirect proof?

In economics, especially in theoretical economics, it is often necessary to formally prove your statements. Meaning to show your statements are correct in a logical way. One possible way of showing that your statements are correct is by providing an indirect proof. The following couple of lines try to explain the concept of indirect proof in a simple way.

Let’s say you want to indirectly prove A \Rightarrow B, i.e. B follows A. In order to do so, the first step is to assume that B is false. The second step is to show that you end up in a logical contradiction because of your assumption that B is false.

The way you formally express this is by adding \bar{B}  to your initial condition A and to show that A \wedge \bar{B} leads to a contradiction. In order to show that A \wedge \bar{B} leads to a contradiction you have to show one of the following three statements:

  1. A \wedge \bar{B} \Rightarrow \bar{A}
  2. A \wedge \bar{B} \Rightarrow B
  3. A \wedge \bar{B} \Rightarrow F

(Where F is an obvious false statement)

If you are able to show one of the above you can conclude that A \wedge \bar{B} is false. This means that if A is correct \bar{B} has to be wrong. Knowing that \bar{B} is false, lets us conclude that B is true. Consequently we know that A \Rightarrow B, i.e. we have indirectly proven that the statement A \Rightarrow B is true. Even though this seems confusing at the beginning it is based on pure logic and often used.

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