# Modus tollens

**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
- ¬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.

Another example:

- 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)

Modus tollens became somewhat legendary when it was used by Karl Popper in his proposed response to the problem of induction, Falsificationism.

## See also

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