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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: - If P, then Q.
- Q is false.
- Therefore, P is false.
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: - 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.
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 See also: modus ponens, affirming the consequent, denying the antecedent, falsificationism. | |||||

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