Modus tollens
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In logic, 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.
An example of an argument relying on modus tollens is:
- If OJ is the murderer, then he must own a glove that fits.
- OJ does not own a glove that fits.
- Therefore, OJ was not the murderer.
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[edit] Explanation
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.
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. If an argument is valid, and if the premises are true, the conclusion must follow.
But suppose we decide that it is not necessarily the case that the murderer owns an axe. For instance, the murderer could have borrowed an axe (and Lizzy could therefore be the murderer, despite not owning an axe). This means that the first premise is false. The argument is nonetheless valid: if the premises were true, the conclusion would follow. It just so happens that in this particular case, not all the premises are true. An argument can be valid even though it has a false premise. Such an argument can reach a false conclusion.
Modus tollens became somewhat legendary when it was used by Karl Popper in his proposed response to the problem of induction, Falsificationism.
[edit] Relation to modus ponens
Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication. For example:
- If P, then Q. (premise -- material implication)
- If Q is false, then P is false. (derived by transposition)
- Q is false. (premise)
- Therefore, P is false. (derived by modus ponens)
Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.
[edit] Formal notation
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.")
Or in natural deduction notation:
[edit] 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)
