Modus tollens (Latin: the way in which this is denied) is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent.
It is a common, simple argument form:
- If P, then Q.
- Q is false.
- Therefore, P is false.
In logical operator notation:
- P → Q
- ⊢ ¬P
where ⊢ represents the logical assertion.
Or in set-theoretic form:
- P ⊆ Q
- x ∉ Q
- ∴x∉ P
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
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.
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.
It is important to note that when an argument is valid, if the premises are true, the conclusion must follow. Suppose we decide that it is not the case that: if Lizzy was the murderer, then she would have to have owned an axe; Perhaps we have found that she borrowed someone's. This means that the first premise is false. But notice that it does not mean the argument is invalid, since it remains the case that, if the premises are true (and in this case they are not), the conclusion would follow, even though in this particular case the premise is false. An argument can be valid even though it has a false premise. Such an argument usually reaches a false conclusion.
- If a modus tollens argument has true premises, then it is sound.
- The argument is unsound
- Therefore, its premises are false.
(Of course this particular argument applied to itself would be a paradox)