It is traditionally believed that the founders of geometry as a science are the Greeks who took over from the Egyptians the ability to measure the volumes of various bodies and the earth. The ancient Egyptians, having established over time the general laws, compiled the first evidence-based works. In them, all the provisions were derived by logical means from a small number of unprovable sentences or axioms. So, if the axiom is a statement that does not need to be proved, then what is a “statement requiring proof”? Before you understand this, you need to understand what the term "evidence" is.
Interpretation of the concept
Proof (justification) is a logical process of establishing the truth of a certain statement with the help of other statements that have already been proved earlier. So, when it is necessary to prove the proposition A, then such judgments B, C and D are selected, from which A follows as a logical consequence.
The evidence that is used in science consists of various types of conclusions, interconnected so that the consequence of one is a prerequisite for the emergence of the other, 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 substantiate the truth of some and the falsity of other statements. It is the evidence that helps get rid of delusions, opening up scope for scientific creativity. And the connection formed between them between the 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 it becomes necessary to identify the structure of conclusions.
Mathematics
Many who comprehend such a science as mathematics have the question of what is a statement that requires proof. The answer (“Avatar” attests to this) is a theorem.
It is a mathematical statement, the veracity of which has already been established through proof. The concept of “theorem” itself developed along with the concept of “mathematical proof”. From the point of view of the axiomatic method, a theorem of a theory is a statement that can only be deduced in a logical way from certain previously fixed statements called axioms. And since the axiom is true, the theorem must also be true.
Further, a statement requiring proof (theorem) was closely intertwined with the concept of “logical consequence”. So, over time, the process of logical inference was reduced to the appearance of formulas or mathematical statements, which were written in a certain language according to the formulated rules, relating not to the content of the sentence, but to its form. Thus, in theory, the proof acts 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 using various mathematical methods. It was also found that axiomatic theories, which are part of different branches of mathematics, are incomplete. So, there are statements whose plausibility or falsity cannot be established in a logical way on the basis of axioms. Such theories are unsolvable, do not have one solution method.
Thus, a statement requiring proof in mathematics 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 perspective, what is a statement that requires proof? Answer: "Avatar" says that this is the thesis.
In this case, it represents a philosophical or theological position, a statement that needs to be proved. In ancient times, this term acquired a special meaning, since then the concept of “antithesis” appeared, which was presented in a contradictory statement or inference. Then Kant drew attention to the fact that contradictory statements can be made with the same plausibility. For example, one can prove that the world is infinite and arose by chance, it consists of indivisible atoms, freedom exists in it. The philosopher noted such statements as a combination of thesis and antithesis. Such a contradictory statement, requiring 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, a property is attributed to the same object of thought, which is at the same time denied. Thus, for these components to exist in unity, the presence of three elements is necessary: conditions, conditionality (evidence) and concepts.
On the basis of all this, Hegel developed the dialectical method, which is based on the transition from the thesis through proof to synthesis. This became an instrument for constructing metaphysics.
Logics
In logic, a statement requiring proof is also called a thesis. In this case, he acts as an exact judgment that the opponent put forward, which he must justify in the process of proof. The thesis is the main element of argumentation.
rules
Throughout the process of argumentation, the thesis must remain the same. If this condition is violated, this leads to the fact that not the statement will be proved, which must be refuted. Here the rule will work: "He who proves a lot, he proves nothing!"
We note one more thing when considering this question: a statement requiring proof does not have to be ambiguous. This rule protects against ambiguity in proving it. For example, very often a person speaks as much as if proving something, but what exactly remains unclear, because his thesis is vague. The ambiguity of the statement leads to futile disputes, since each side differently perceives the proved position.
A statement that does not require proof
Aristotle, considering the question of provability of statements, put forward the theory of syllogisms. Syllogisms consist of statements that contain the words “may” or “should” instead of “eat”. Such statements are not logically justified, because their premises are not proven. This raises the issue of starting points for the development of science. According to Aristotle, any science should begin with statements that do not need proof. He called them axioms.
Axiom
A statement that does not require proof is an axiom. It does not need to be proved in practice, it is only necessary to explain in order to make it clear. Speaking of axioms, Aristotle considered geometry, which took the form of systematization. Mathematics was the first science to use statements that did not need substantiation. Then came astronomy, since to justify the motion of the planets it is necessary to resort to mathematical calculations. As you can see, science was already lining up like a hierarchy.
Types of Sciences by Aristotle
Aristotle put forward three types of sciences for his main goals. Theoretical sciences provide knowledge from the perspective in which they are opposed to opinions. The math here is the clearest example. This also includes physics and metaphysics.
Practical sciences are aimed at learning how to control human behavior in society. This may include, for example, ethics.
Technical sciences are aimed at creating guidelines for the creation of objects for their application in life or in order to admire their artistic beauty.
Aristotle did not attribute logic to any of the groups of sciences. It acts as a general way of handling things, which is mandatory for each of the sciences. Logic is presented as a tool on which scientific research will be based, since it provides criteria for discrimination and proof.
Analytics
Analytics studies the forms of evidence. It decomposes logical thinking into simple components, and from them they are already moving to complex forms of thinking. So, the structure of the evidence does not require consideration.
Thus, logic and analytics consider questions about what is a statement that does not require proof. That is, these industries are characterized by the advancement of axioms. They also have an explanation of what a statement requires proof. The answers to these questions are given in every branch of science, since not a single scientific research is complete without logic and analytics.
Correlation with reality
Having considered the question that such a statement that requires proof, it became obvious: the essence of the proof itself is that the statement in the statement is related to the actual state of things or to other facts, the authenticity of which has already been proved earlier. For example, in some cases the truth of the statements can be justified by an experiment (physical, biological, chemical), according to the results of which it becomes visible whether they correspond to the stated judgments or not. In other words, the research results will be either 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 derives 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 cannot be experimentally verified. Therefore, a statement requiring proof is called an “Avatar” by a theorem, the only way to establish the truth of which is to prove inferences based on previously proved true statements.
Summary
A statement that requires proof must be supported by arguments. They may be judgments that were previously proved, for example, axioms, laws, definitions containing statements about facts. The arguments used in the proof are closely related to each other and represent the form of evidence. They form various kinds of conclusions, which are connected in a chain.
For example, consider a statement that requires proof: "The metal obtained during 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 metal formed during the experiment does not decompose water. Therefore, the resulting metal is not sodium.
As you can see, all the arguments used are true, the proof of which occurred as a result of observation, generalization of past experience, syllogistic inference. The proof process here is based on two conclusions, a consequence of one being the premise of the other.