Modus tollens

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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 logical operator notation:

<math> p \rightarrow q </math>
<math> \not\vdash q, </math>
<math> \not\vdash p </math>

where <math>\vdash</math> represents the logical assertion.

or in set-theoretic form:

<math>P\subseteq Q</math>
<math>x\not\in Q</math>
∴<math>x\not\in P</math>

("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.

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, falsifiability.


References

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