4.7 Indirect Argument: Contradiction and Contraposition

Argument by Contraposition

Argument by contraposition is a way to prove an argument is true by negating both the hypothesis and the conclusion.

Example 4.7.4 (Includes Proof by contraposition)

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Fun Fact #1

Did you know that the Reductio Ad Absurdum argument is commonly used in modern-day politics and social media debates? It's a way to show the absurdity of an opposing argument by taking it to its extreme and showing why it doesn't make sense.

Proposition 4.7.4

For every integer n, if n^2 is even then n is even.

What is the relation between proof by contradiction and proof by contraposition?

Proof by contradiction and proof by contraposition are both methods of indirect proof. In proof by contradiction, we assume the negation of the statement we want to prove and show that it leads to a contradiction. In proof by contraposition, we prove the contrapositive of the statement instead, which is logically equivalent to the original statement. Both methods are useful for proving statements in mathematics and logic.

Contrapositive

Regular: If P, then Q Contrapositive: If not Q, then not P Regular: For all x in d, if p(x) then q(x) Contrapositive: For all x in d, if not q(x) then not p(x)

Reductio Ad Impossible / Reductio Ad Absurdum

Show that something is true by showing that its opposite is extremely impossible or unlikely to happen.

Supposition

Something that is assumed or taken for granted without conclusive evidence. Something that is guessed which may or may not be true. Starting point for a proof and is tested to see if true or false.

To prove a statement of the form "Vx e D, if P(x) then Q(x)" by contraposition, you suppose that ... and you show that ...

the conclusion is false when x is any particular element the hypothesis is false

To prove a statement by contradiction, you suppose that ... and you show that ...

the statement is false the supposition(guess) leads to a contradiction

A proof by contraposition of a statement of the form "Vx e D, if P(x) then Q(x)" is a direct proof of ...

"Vx e D, if Q(x) is false, then P(x) is false"

What is the Method of Proof by Contradiction?

1. Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true. 2. Show that this supposition leads logically to a contradiction. 3. Conclude that the statement to be proved is true. SSC or Say Something Contridiction