Modus tollens
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Modus tollens (Latin for "mode that denies") is the formal name for indirect proof or proof by contrapositive (contrapositive inference), often abbreviated to MT. It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming the consequent or denying the antecedent).
Modus tollens has the following argument form:
- If P, then Q.
- Q is false.
- Therefore, P is false.
- P → Q
- ¬Q
- ⊢ ¬P
Or in set-theoretic form:
- P ⊆ Q
- x ∉ Q
- ∴x∉ 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.
It is important to note that when an argument is valid, 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 the 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. Such an argument can reach a false conclusion.
- If a modus tollens argument has true premises, then it is sound.
- The argument is unsound
- Therefore, its premises are false.
Modus tollens became somewhat legendary when it was used by Karl Popper in his proposed response to the problem of induction, Falsificationism.
See also
- Affirming the consequent
- Denying the antecedent
- Falsificationism
- Modus ponens - "one man's modus ponens is another man's modus tollens" (Dretske 1995)
- Non sequitur (logic)
External links
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