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:

The murderer must own an axe.

Lizzy does not own an axe.

Threfore, Lizzy was not the murderer.

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. But it does not mean the argument is invalid, since it remains the case that, if the premises are 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. In logical operator notation:

<math>(P\to Q \and \neg Q) \vdash \neg P</math>

where <math>\vdash</math> represents the logical assertion.

Or in set-theoretic form:

<math>P\subseteq Q</math>

<math>x\notin Q</math>

<math>\therefore x\notin P</math>

("P is a subset of Q. x is not in Q. Therefore, x is not in P.")

1 Explanation
2 Relation to modus ponens
3 See also
4 External links

[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. (Why? If P were true, then Q would be true, by premise 1, but it is not, 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. If an argument is valid, and if the premises are true, the conclusion must follow.

If a modus tollens argument has only true premises, then it is sound.

The argument is unsound

Therefore, it is not a modus tollens argument which has only true premises.

Of course, this particular argument would, if applied to itself, create a paradox.

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.