Modus tollens is a common, simple argument form:
- If P, then Q.
- Not Q.
- Therefore, not P.
Consider an example:
- If there is fire here, then there is oxygen here.
- There is no oxygen here.
- Therefore, there is no fire here.
This argument form is valid: indeed, if there is no oxygen here, then we can conclude validly that 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.
Just suppose that the premises are both true. If Lizzy was the murderer, then she really must have owned an axe; and it is a fact that Lizzy does not own an axe. What follows? That she 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; remember to distinguish validity and soundness.
See also: modus ponens.