A proof is a logical argument that establishes the truth of some claims using a previously proven truths. In this case what you need to prove, is called the thesis, and the arguments and bases are already known truth.

## Proof with truth

Proof «by contradiction» (in Latin «reductio ad absurdum») is characterized in that the process itself is proof of any opinion by refuting opposing opinions. The falsity of the antithesis can be proved, establishing the fact that it is incompatible with true judgment.

Typically, this method is clearly demonstrated using the formula, where A is antithesis, and the truth. If the solution of having variable And leads to results different from V, then it is proved the falsity of A.

## Proof «by contradiction» without using the truth

There is an easier formula proof of the falsity of the «other» — an antithesis. This formula-the rule States: «If the solution variable And the formula have any contradiction And is false». It does not matter whether the antithesis is negative or affirmative judgment. Besides the simpler method of proof by contradiction contains only two facts: the thesis and antithesis, the truth In not in use. In mathematics this greatly simplifies the process of proof.

## Apagoge

In the process of proof by contradiction (which is also called «reduction to absurdity») is used frequently apagoge. This is a logical technique, which aims to prove the infidelity of any judgment so directly in it or in the ensuing consequences have been identified contradiction. The contradiction can be expressed in the identity obviously different objects or as conclusions: the conjunction or equivalence of pairs b and not (true and not true).

The reception of evidence «to the contrary» is often used in mathematics. In many cases, to prove the incorrectness judgments in another way is not possible. In addition to aphogee and there is a paradoxical form of proof by contradiction. This form was used in «Basis» Euclid and represents the following rule: And shall not be contested if it turns out to demonstrate «the truth of falsity» A.

Thus, the process of proof by contradiction (also called indirect proof and apologiesi) as follows. Put forward an opinion opposite to the thesis, antithesis from this investigation are derived, among which is searched false. They find evidence that among the consequences is really a false. It is concluded that the antithesis is wrong, and if wrong, the antithesis, should be a logical conclusion that the truth is contained in the thesis.