Modus tollensModus tollens
: 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
where represents the logical assertion
or in set-theoretic form:
("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.
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.