What is an "assertion requiring proof"

Traditionally, it is commonly believed that the founders of geometry as a science are the Greeks, who have adopted from the Egyptians the ability to measure the volumes of different bodies and the earth. The ancient Egyptians, having established over time general patterns, compiled the first evidence. In them, all the propositions were derived logically from a small number of unprovable sentences or axioms. So, if the axiom is a statement that does not need proof, then what is an "assertion requiring proof"? Before you understand this, you need to understand what the term "proof" is.

Interpretation of the concept

Proof (justification) is a logical process of establishing the truth of a certain assertion with the help of other statements, which have already been proved earlier. So, when it is necessary to prove the proposition A, then choose the judgments B, C and D, from which A follows as a logical consequence.

The evidence that is applied in science consists of different kinds of inferences related to each other so that the consequence of one is a prerequisite for the emergence of another and so on.

Proof in Science

The development of any science is determined by the degree of application of evidence in it, with the help of which it is possible to justify the truth of some and the falsity of other statements. It is the evidence that helps to get rid of delusions, opening up the space for scientific creativity. And the connection that is formed with their help between different statements of a particular science makes it possible to determine its logical structure.

In modern times, evidence is widely used in logic and mathematics, they are methods of analysis when there is a need to identify the structure of conclusions.

Mathematics

Many people who comprehend such a science as mathematics, raise the question of what is a statement that requires proof. The answer ("Avatar" testifies to this) is a theorem.

It is a mathematical statement, the veracity of which has already been established through proof. The notion of "theorem" itself developed along with the notion of "mathematical proof". From the point of view of the axiomatic method, the theorem of any theory is those statements that are derived only logically from certain previously fixed statements, called axioms. And since the axiom is true, the theorem must also be true.

Further, an assertion requiring proof (the theorem) was closely intertwined with the notion of "logical consequence". So, over time, the process of logical inference was reduced to the appearance of formulas or mathematical statements that were written in a certain language according to the formulated rules, referring not to the content of the sentence, but to its form. Thus, in theory the proof appears as a sequence of formulas, each of which is an axiom.

In mathematics, a theorem or statement requiring proof is the last formula in the process of proving a theory. This formulation was formed as a result of the use of various mathematical methods. It was also found that the axiomatic theories, which are part of different sections of mathematics, are incomplete. So, there are statements, the plausibility or falsity of which can not be established logically on the basis of axioms. Such theories are unsolvable, do not have one solution method.

Thus, an assertion requiring proof in mathematics Is called a theorem.

Philosophy

Philosophy is a science that studies the system of knowledge about the characteristics and principles of reality and cognition. So, from this standpoint, what is the statement that requires proof? Answer: The "Avatar" says that this thesis.

He in this case is a philosophical or theological position, an assertion that must be proved. In ancient times this term acquired special significance, since then the notion of "antithesis" appeared, which appeared in a contradictory statement or inference. Then Kant drew attention to the fact that it is possible to make contradictory statements with the same plausibility. For example, one can prove that the world is infinite and has arisen by chance, it consists of indivisible atoms, there is freedom in it. Such statements were noted by the philosopher as a totality of thesis and antithesis. Such a contradictory statement, which requires proof, as well as the insolubility of contradictions, is explained by the fact that the mind goes beyond the cognitive abilities of man.

In philosophy, one and the same object of thought is attributed to a property that at the same time is denied. Thus, in order for these components to exist in unity, it is necessary to have three elements: conditions, conditionality (evidence), and concepts.

On the basis of all this Hegel, a dialectical method was derived, based on the transition from the thesis through proof to synthesis. This became an instrument for building metaphysics.

Logics

In logic, an assertion requiring proof is also called a thesis. In this case, he acts as an accurate judgment, which put forward an opponent, which he must justify in the process of proving. The thesis is the main element of the argument.

rules

Throughout the process of argumentation, the thesis must remain the same. If this condition is violated, this leads to the fact that it will be proved not the statement that should be refuted. Here the rule will work: "Whoever proves a lot, he does not prove anything!"

Let's note something else, considering this question: the statement demanding the proof should not be many-valued. This rule protects against the ambiguity of the situation when it proves. For example, very often a person speaks as much, as if something proves, but what exactly, remains unclear, since his thesis is vague. The ambiguity of the statement leads to inconclusive disputes, since each side takes the position to be proved in different ways.

A statement that does not require proof

Aristotle, considering the question of the provability of assertions, put forward the theory of syllogisms. Syllogisms consist of such statements, which contain the words "can" or "should" instead of "is." Such statements are not logically justified, because their prerequisites have not been proven. This raises the question of the starting points for the development of science. According to Aristotle, any science must begin with statements that do not need proof. He called them axioms.

Axiom

An assertion that does not require proof is an axiom. It does not need to be proved in practice, it is only necessary to explain it to make it clear. Speaking of axioms, Aristotle considered geometry, which took the form of systematization. Mathematics was the first science where statements were used that did not need a justification. Then there was astronomy, because to justify the motion of the planets it is necessary to resort to mathematical calculations. As you can see, science was already building up like a hierarchy.

Types of Sciences by Aristotle

Aristotle proposed three types of sciences for the main purposes. Theoretical sciences give knowledge in the foreshortening in which they are opposed to opinions. Mathematics is the most striking example here. This includes physics and metaphysics.

Practical sciences are aimed at learning how to control the behavior of a person in society. This includes, for example, ethics.

Technical sciences are aimed at creating a guide to the creation of objects for their application in life or to admire their artistic beauty.

Logic Aristotle did not refer to any of the groups of sciences. It acts as a general way of operating things, which is indispensable for each of the sciences. Logic is presented as an instrument on which scientific research will be based, as it provides criteria for discrimination and proof.

Analytics

The analyst studies the forms of evidence. It breaks down logical thinking into simple components, and from them they are already moving to complex forms of thinking. Thus, the structure of the proof does not require consideration.

Thus, logic and analytics consider the question of what is a statement that does not require proof. That is, for these industries is characterized by the nomination of axioms. Also for them, the explanation of what is a statement requiring proof is peculiar. Answers to these questions are given in every branch of science, since no scientific research can do without logic and analytics.

Relationship with reality

Having considered the question that such an assertion requiring proof was obvious: the essence of the proof itself lies in the fact that the statement in the statement correlates with the actual state of things or with other facts, the authenticity of which has already been proven earlier. For example, in some cases, the truth of statements can be justified by experiment (physical, biological, chemical), which, by the results of which it becomes clear whether they correspond to the stated judgments or not. In other words, the results of the research will either be a proof of the truth of the statement, or its refutation.

And in other cases, when it is impossible to conduct an experiment, a person resorts to other valid statements, from which he deduces the truth of his judgment. Such evidence is used today in science, where objects are beyond the limits of human ability to observe them. This is especially true in mathematics, where judgments can not be tested experimentally. Therefore, the statement demanding proof, "Avataria" calls a theorem, the only way to establish the truth of which is a proof of inferences based on previously proven true statements.

Results

An assertion that requires proof must be supported by arguments. As such, judgments can be made that earlier axioms, laws, definitions containing statements about facts have been proved earlier. Arguments that are used in the proof, are closely interrelated and represent the form of evidence. They form various kinds of conclusions, which are connected in a chain.

For example, consider an assertion requiring proof: "The metal obtained in the course of the experiment is not sodium." The following arguments are used to prove this statement:

1. All alkali metals decompose water at room temperature.

2. Sodium is an alkali metal. Therefore, it decomposes water.

3. The water formed in the course of the experiment does not decompose water. Therefore, the resulting metal is not sodium.

Apparently, all the arguments used are true, the proof of which occurred as a result of observation, generalization of past experience, syllogistic deduction. The process of proof here is based on two inferences, the effect of one is the prerequisite of the other.