**Modus tollens**(Latin:

*mode that denies*) is the formal name for

**indirect proof**.

It is a common, simple argument form:

- If P, then Q.
- Q is false.
- Therefore, P is false.

or in set-theoretic form:

- ∴

The argument has two premises. The first premise is the conditional "if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false. (Why? If P were true, then Q would be true, by premise 1, but it isn't, by premise 2.)

Consider an example:

- If there is fire here, then there is oxygen here.
- There is no oxygen here.
- Therefore, there is no fire here.

- If Lizzy was the murderer, then she owns an axe.
- Lizzy does not own an axe.
- Therefore, Lizzy was not the murderer.

Suppose one wants to say: the first premise is false. If Lizzy was the murderer, then she would not necessarily have to have owned an axe; maybe she borrowed someone's. That might be a legitimate criticism of the argument, but notice that it does not mean the argument is *invalid*. An argument can be valid even though it has a false premise; one has to distinguish between validity and soundness.

See also: modus ponens, affirming the consequent, denying the antecedent, falsificationism.