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Modus tollensModus tollens (Latin: mode that denies) is the formal name for indirect proof.
It is a common, simple argument form:
or in set-theoretic form:
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:
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.
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