LECTURE No. 21. Argumentation and proof

1. Proof

We cognize the world through the sense organs, and such cognition most often does not need proof, since it is quite obvious. For example, it does not require proof that fire is hot. Just reach out to him.

However, not all phenomena, objects of the surrounding world are so clear that there is no need to prove them. In scientific activity and even in everyday life, one often has to face the need to prove, to defend one's point of view.

Evidence - an important quality of correct thinking.

Theories, proofs and refutations are the means in the hands of man to create new valid knowledge. Proof is necessary in the scientific world, it determines the truth of a phenomenon, judgment, conclusion. Without proof, any hypothesis will forever remain a hypothesis and will not acquire the value of a theory. It's good, because purpose of proof - obtaining true knowledge. Any new phenomenon, conjecture must be proved, whether it be secrets related to outer space or the depths of the ocean, mathematical research, etc.

From these positions, one can define proof as a set of logical methods of substantiating the truth of a proposition with the help of other true and related propositions.

In the ordinary sense, proof is often identified with the belief that it is unacceptable. These two concepts may coincide in part, but they are too different in many ways. So, the proof is based solely on scientifically substantiated facts, research, theories, etc. A belief, on the other hand, often does not depend on whether the asserted is scientifically proven or not. Persuasion is possible in relation to probabilistic or generally false theories.

The structure of the proof is the thesis, arguments and demonstration.

Thesis This is a statement that needs proof.

Arguments are true propositions used in the process of proof.

Demonstration is a way of logical connection between the thesis and arguments.

There are rules for reasoning. Violation of these rules leads to errors related to the thesis being proved, the arguments, or the form of the proof itself.

The proof is either direct or indirect.

Direct proof proceeds from the consideration of arguments to the proof of the thesis, i.e., the truth of the proof is directly substantiated by the arguments.

We can say that with direct proof, true propositions (k, m, l...) necessarily follow from the arguments (a, b, c...), and the thesis q to be proven follows from the latter. This type of evidence is used in judicial practice, in science, and in controversy. Direct evidence is widely used in statistical reports, in various kinds of documents, and in decrees.

With indirect evidence the truth of the put forward judgment is substantiated by proving the falsity of the judgment that excludes it. The use of such a proof is justified when there are no arguments for direct proof.

Depending on the form of the antithesis, two types of indirect evidence can be distinguished - from the opposite and divisive.

proof by contradiction (apagogical) is carried out by establishing the falsity of a judgment that contradicts the thesis. This method is often used in mathematics.

Partition proof produced on the basis of the negation of the antithesis. Provided that all antitheses are listed and their consistent negation (and rejection), one can speak of establishing the truth of the asserted judgment.

2. Argumentation

As has been said, any proof needs arguments. The prover relies on them; they contain information that allows one to speak with certainty about a particular subject. In logic there are several arguments. These include certified individual facts, axioms and postulates, previously proven provisions and definitions.

Certified Facts represent information fixed in any documents, works, databases and on various media. You can define this group of arguments as actual data. Such data include statistics, facts from life, testimonies, documents and documentary chronicles, etc. Such arguments play an important role in the proof process, as they are firm, irrefutable, and have already been proven. They can carry information about the past, which also makes authenticated facts important in terms of knowledge.

Axioms. Many of us, when we hear the word “postulates,” remember school and mathematics lessons. Indeed, axioms are widely used in mathematical constructions, and mathematical logic is often based on them. Confirmed by experience, previously proven facts, and repeated repetition of evidence, these judgments do not require proof and are accepted as arguments.

Statements of laws, theorems, which have been proven in the past, are accepted as arguments of proof, since their truth has already been determined and accepted. This group of arguments reminds us that all arguments underlying the evidence must be proven. The proof of the arguments of this group can be carried out either immediately before the proof of the axiom, or long before that. This group includes scientifically proven laws (for example, nature) and theorems.

The last group of arguments is definitions. They are created within the framework of all sciences regarding the subjects under consideration and reveal the essence of the latter. The proof can be based on definitions accepted and applied in any science. However, we should not forget that many definitions are subject to debate and the proof based on them may not be accepted by the opponent. Here it is necessary to say about the inadmissibility of using unscientific definitions, since the main idea in them may be distorted, and the definitions themselves may be incomplete or even false.

When proving a thesis, you can use several types of arguments - this will lead to greater persuasiveness.

Do not forget also that the main factor in proving the theory is still practical application. If the theory has been confirmed in practice, it does not require other evidence or justification.

Author: Shadrin D.A.

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