Proving Conditionals

You usually proved a sentence of the type in plane geometry by assuming P and deducing Q. You considered Q the conclusion. In actually, was the conclusion; it was what you were trying to prove.

To prove first assume P to be true. Then using P and all other theorems and axioms try to deduce Q. Once Q is deduced in this manner you have completed a proof of . You have not shown that Q is true; you have only shown that Q is true if P is true. Whether P is true is another question; whether Q is true is another question. What you have shown to be true is .

This technique is called the Rule of Conditional Proof  or the Deduction Theorem . More formally, suppose that are the axioms and previously proved theorems. To prove is to show that

From we can deduce

is a valid argument. To do this temporarily assume P to be an axiom and show that

From we can deduce Q

is a valid argument.

A second technique of proving is by the contrapositive . We can prove by proving . Often the rule of conditional proof is used to prove the contrapositive.