10.2 Direct and indirect evidence

Usually, the proof consists of a series of steps. You need to be able to trace every step of the proof, otherwise its parts will lose contact, and it can crumble at any time, like a house of cards. But it is equally important to understand the proof as a whole, as a single structure, each part of which is necessary in its place.

Proof, not understood as a whole, does not convince anything. Even if you learn it by heart, sentence by sentence, this will not add anything to your knowledge of the subject. To follow the evidence and only make sure that each of his subsequent steps is correct - this is tantamount to observing the game of chess, when you notice only that each move is subject to the rules of the game.

The minimum requirement is an understanding of logical inference as a targeted procedure. Only in this case is the intuitive clarity of what we do.

What creates the “unity of evidence” can be presented in the form of a general scheme, covering its main steps, embodying one hundred principle or its final structure. Such a scheme remains in the memory when the details of the proof are forgotten.

Direct evidence

From the point of view of the general movement of thought, all evidence is divided into direct and indirect.

In direct proof, the problem is to find such convincing arguments from which the thesis is obtained by logical rules.

For example, you need to prove that the sum of the angles of a quadrilateral is 360 ° C. From what statements could this thesis be derived? We note that the diagonal divides the quadrilateral into two triangles. This means that the sum of its angles is equal to the sum of the angles of two triangles. It is known that the sum of the angles of a triangle is 180 ° C. From these positions we deduce that the sum of the angles of the quadrilateral is equal to 360 ° C.

In the construction of direct evidence, two interrelated stages can be distinguished: the search for those recognized asserted assertions that can be convincing arguments for a provable position; establishing a logical connection between the found arguments and the thesis. Often, the first stage is considered preparatory, and by evidence is understood deduction, which links the selected arguments and the thesis to be proved.

Another example. It is necessary to prove that space ships are subject to the laws of celestial mechanics. It is known that these laws are universal: all bodies obey them at any point in outer space. It is also obvious that the spacecraft is a space body. Noting this, we build the corresponding deductive reasoning. It is a direct proof of the assertion under consideration.

Indirect evidence

The indirect proof establishes the validity of the thesis by revealing the fallacy of the opposite assumption (antithesis).

As the mathematician D. Poya observes with irony, “indirect evidence bears some resemblance to the tricks of the politician who supports his candidate by discrediting the reputation of the candidate of the other party”. In indirect proof, reasoning proceeds as if in a roundabout way. Instead of directly looking for arguments for deducing from them a provable clause, an antithesis is formulated, a denial of this clause. Further one way or another shows the inconsistency of the antithesis. By the law of the excluded middle, if one of the contradictory statements is wrong, the second should be true. The antithesis is erroneous, which means that the thesis is correct.

Since indirect evidence uses the negation of the provable position, it is, as they say, proof by contradiction.

Suppose you need to build an indirect proof of such a very trivial thesis: "The square is not a circle." An antithesis is being advanced: "The square is a circle." It is necessary to prove the falsity of this statement. For this purpose, we derive the consequences from it. If at least one of them turns out to be false, it will mean that the statement itself, from which the result is derived, is also false. In particular, the following result is incorrect: the square has no corners. Since the antithesis is false, the original thesis must be true.

Another example. The doctor, convincing the patient that he is not sick with the flu, reasons like that. If there really was flu, there would be characteristic symptoms for it: headache, fever, etc. But nothing like this. So there is no flu.

This is again indirect evidence. Instead of a direct substantiation of the thesis, an antithesis is advanced that the patient, in fact, has the flu. Consequences are derived from the antithesis, but they are refuted by objective data. This says that the assumption about the flu is wrong. It follows that the thesis “No flu” is true.

Evidence from the contrary is usual in our reasoning, especially in a dispute. When applied skillfully, they may be particularly persuasive.