Lecture 6

 

Reductio ad absurdum

Reductio ad absurdum (Latin for "reduction to the absurd", traceable back to the Greek ἡ εις άτοπον απαγωγη, "reduction to the impossible", often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must have been wrong, since it gave us this absurd result.

This is also known as proof by contradiction. It makes use of the law of non-contradiction — a statement cannot be both true and false. In some cases it may also make use of the law of excluded middle — a statement which is not false, must then be true.

 

Modus Ponens

Modus ponens (Latin, mode that affirms) is a valid, simple argument form:

 

If P, then Q.

P.

Therefore, Q.

 

The argument form has two premises. The first premise is the "if-then" or conditional claim, namely that P implies Q.

The second premise is that P, the antecedent of the conditional claim, is true.

From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well.

Here is an example of an argument that fits the form modus ponens:

If democracy is the best system of government, then everyone should vote.

Democracy is the best system of government.

Therefore, everyone should vote.

The fact that the argument is valid cannot assure us that any of the statements in the argument are true; the validity of modus ponens tells us that the conclusion must be true if all the premises are true. It is wise to recall that a valid argument within which one or more of the premises are not true is called an unsound argument, whereas if all the premises are true, then the argument is sound.

A propositional argument using modus ponens is said to be deductive.

Modus ponens can also be referred to as affirming the antecedent.

 

Modus Tolens

Modus tollens (Latin, mode that denies) is the formal name for indirect proof or proof by contrapositive. It can also be referred to as denying the consequent.

It is a common, simple argument form:

If P, then Q.

Q is false.

Therefore, P is false.

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.

It is important to note that when an argument is valid, if the premises are true, the conclusion must follow. An argument can be valid even though it has a false premise. Such an argument usually reaches a false conclusion.