10.3 Types of indirect evidence Consequences contradicting facts Internally contradictory consequences Separating evidence

The course of thought in indirect evidence is determined by the fact that instead of justifying the validity of a thesis, they seek to show the inconsistency of its denial. Depending on how the last problem is solved, there are several types of indirect evidence.

Contradictions

Most often, the antithesis’s falsity can be established by a simple comparison of the consequences arising from it with the facts. This was the case, in particular, in the example of the flu.

A friend of the inventor of the steam engine D. Watt, the Scottish scientist D. Black introduced the concept of the latent heat of melting and evaporation, which is important for understanding the operation of such a machine. Black, observing a common phenomenon - snow melting at the end of winter, reasoned like that. If the snow that had accumulated during the winter melted immediately, as soon as the air temperature was above zero, then devastating floods would be inevitable. And since this does not happen, it means that a certain amount of heat must be spent on melting snow. Her Black and called hidden.

This is indirect evidence. The consequence of the antithesis, which means that he himself is refuted by a reference to the obvious circumstance: at the end of the winter of floods, it is usually petting, the snow melts gradually.

Philosopher R. Descartes argued that animals are not able to reason. His follower L. Racine, the son of the great French playwright, used evidence from the contrary to substantiate this idea. If animals had a soul and the ability to feel and reason, he said, would they have remained indifferent to the unjust public insult inflicted on them by Descartes? Wouldn’t they have revolted in anger against the one who had degraded them so? According to no evidence of a special resentment of animals on Descartes no. Consequently, they are simply unable to ponder his argument and somehow respond to it.

Internally contradictory consequences

According to the logical law of contradiction, one of the two contradictory statements is false. Therefore, if, among the consequences of any provision, both the statement and the negation of the same thing met, one can immediately say that this provision is false.

For example, the position “Square is a circle” is false, since it implies both that the square has corners and that it has no corners.

The position from which an internally contradictory statement or statement about the identity of affirmation and negation is also false will be false.

One of the methods of indirect evidence is the elimination of the antithesis of logical contradiction. If the antithesis contains a contradiction, it is clearly erroneous. Then his negation - the thesis of the proof - is true.

A good example of indirect proof is the well-known proof of Euclid that a series of primes is infinite.

Simple numbers are natural numbers greater than one, divisible only by themselves and by one. Prime numbers are like “primary elements”, into which all integers (more than 1) can be decomposed. It is natural to assume that the series of primes: 2, 3, 5, 7, 11, 13, ... is infinite. To prove this thesis, let us assume that this is not the case, and see what this assumption leads to. If the series of primes is finite, there is the last prime number of the series - A. Further, another number is formed: B = (2 x 3 x x 5 x ... x A) + 1. The number B is greater than A, therefore B cannot be prime number. So B must be divisible by a prime number. But if B is divided into any of the numbers 2, 3, 5, L, then the remainder will be 1. Therefore, B does not divide one of the indicated prime numbers and is thus a simple one. As a result, based on the assumption that there is a last prime number, we come to a contradiction: there is a number at the same time and a simple one, and not a simple one. This means that the assumption made is false and the opposite is true: a series of primes is infinite.

In this indirect proof, a logical contradiction is derived from the antithesis, which directly speaks of the antithesis’s falsity and, accordingly, the thesis’s truth. This kind of evidence is widely used in mathematics.

If only that part of such evidence is meant, in which the erroneousness of any assumption is shown, they are traditionally referred to as the result of absurdity. The fallacy of the assumption is revealed by the fact that absurdity is derived from it , i.e. logical contradiction.

There is another kind of circumstantial evidence when it is not necessary directly to look for false consequences. The fact is that in order to prove the assertion, it suffices to show that it follows logically from its own negation. This technique is based on the logical law of Claudia.

For example, if from the assumption that twice two is five, it is deduced that this is not the case, it is thus proved that twice two does not equal five.

Dividing evidence

In all the considered circumstantial evidence, two alternatives are put forward: thesis and antithesis. Then the falsity of the latter is shown, as a result only the thesis remains.

You can not limit the number of opportunities taken into account only two. This will lead to the so-called dividing indirect evidence, or proof by exception. It is used in cases where it is known that the thesis being proved is among the alternatives that fully exhaust the weight of possible alternatives in this field.

For example, you need to prove that one value is equal to another. It is clear that only three options are possible: either two values ​​are equal, or the first is greater than the second, or, finally, the second is greater than the first. If it was possible to show that none of the quantities exceeds the other, the two options will be discarded and only the third will remain: the values ​​are equal.

The proof follows a simple scheme: one by one, all possibilities are excluded, except for one, which is a provable thesis.

In the separation proof, the mutual incompatibility of possibilities and the fact that they exhaust all conceivable alternatives are determined not by logical, but by factual circumstances. Hence the usual mistake of dividing evidence: not all possibilities are considered.

With the help of dividing evidence, one can try, for example, to show that life in the Solar System is only on Earth. As possible alternatives, we make statements that there is life on Mercury, Venus, Earth, etc., listing all the planets of the Solar System. Then, refuting all the alternatives, except one - speaking about the existence of life on Earth, we obtain the proof of the original statement.

It should be noted that during the proof, assumptions about the existence of life on other planets are considered and refuted. The question of whether life on Earth does not arise at all. The answer is obtained indirectly: by showing that there is no life on any other planet. This proof would, of course, be untenable if, for example, it turned out that, although there is no life on any planet other than Earth, there are living beings on one of the comets or on one of the so-called minor planets, which are also part of Solar system

Finishing the conversation about circumstantial evidence, let us pay attention to their originality, which limits to a certain extent their applicability.

There is no doubt that indirect evidence is an effective means of justification. Having dealt with him, we have to concentrate all the time not on the right position, whose justice must be justified, but on erroneous statements. The very course of the proof consists in the fact that we derive the consequences from the antithesis, which is false, until we arrive at a statement whose fallacy is unquestionable.

Indirect evidence is a good tool for research, but it is not always a good method of presenting the material. It is no coincidence that such paradoxical advice is not uncommon in the practice of teaching: after the indirect evidence has been carried out, it is useful to forget its progress immediately, leaving only the proven position in memory.

There are also more serious objections to indirect evidence. They are connected with the use of the law of the excluded middle in it. As already mentioned, not all, he is recognized as universal, applicable in any and all cases.

It can be noted that the indirect evidence found for some assertion can usually be restructured into a direct proof of the same assertion. Usually, but not always.