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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:
- ,
- ¬
- ¬
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.
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.
But an argument is valid when, 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 that 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.
See also: modus ponens, affirming the consequent, denying the antecedent, falsificationism.
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