# The Development of Laws of Formal Logic of Aristotle

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1 This paper is dedicated to my unforgettable friend Boris Isaevich Lamdon. The Development of Laws of Formal Logic of Aristotle The essence of formal logic The aim of every science is to discover the laws that could explain one or another phenomenon. Once these laws are discovered, then science proceed to study the other phenomena, which in the nature are of an infinite set. It is interesting to note that in the process of discovering a law, for example in physics, people make thousands of experiments, build proves, among them some experience, or evidence - useful for understanding a certain phenomenon, but other experiments or evidence proved fruitless. But it is found only with hindsight, when the law is already discovered. Therefore, with the discovering of the law it is enough to show 2-3 experiments or prove to verify its correctness. All other experiments were the ways of study and there is no need to repeat them, to understand how the law works. In the exact sciences, it is understood, and therefore the students studies only the information that is necessary to understand specific phenomena. By no means this is the case with the study of formal logic. Formal logic, as opposed to other sciences: physics, chemistry, mathematics, biology and so on, studies not an infinite number of phenomena in nature, but only one how a man thinks, how he learns the world surrounding us, and how people understand each other. In other words, what laws govern the logic of our thinking, i.e., our reasoning and judgments in any science or in everyday life. By the beginning of XVIII century four laws of logic were formulated : the law of identity, Law of Contradiction, the Law of Excluded Middle and the Law of Sufficient Ground. The first three laws were formulated by Aristotle in the 4 th century BC, as the 4th law was introduced by Leibniz at the beginning of 18 century. So far for more than 300 years, none of the philosophers discovered neither the 5th or 6th law of formal logic. During this time, all the "discovery" of formal logic were limited only to it s distortion and confusion. If to consider a formal logic of Aristotle from the point of view of its essence, then its center of gravity is its Laws, that were discovered by Aristotle, based on analysis of the different types of syllogism, which Aristotle classified to track down those Laws. In his research, the syllogisms played the same role as the experiments in physics or chemistry for the discovery of regularities, to explain the process of certain events. Once these logical laws of thinking have been discovered, the syllogisms have fulfilled their role. And it would be foolish to assume that our knowledge in any science, is built only on Aristotle's syllogisms or others discovered later. Whatever syllogisms would not have been discovered since Aristotle, none of them had added something new in the laws of formal logic revealed by Aristotle and Leibniz. But philosophers still continue to analyze Aristotle's syllogisms, a historic mission of which ended more than 2000 years ago. Moreover, after the discovering of 4th of law of formal logic, the law of sufficient ground, the legality of any syllogism is easy checked from the viewpoint of the four laws of

4 formal logic, he can not be called not only as a great philosopher but even as a philosopher, because he did not introduce anything new to the development of formal logic of Aristotle, except of the distortion and perversion. The connection between The Law of Identity and the Law of Sufficient Ground Discovered, long time ago, the laws of formal logic, the law of Identity and the law of sufficient ground, are still interpreted with very limited understanding, as evidenced by their contemporary definitions, which is the reason for the existence, in a logical world, of confusion and doubts in existence of perfect formal logic to know the real world. For Aristotle his logic is not a science, but an instrument (organon) of any science about inference and evidence. Thus four laws of formal logic are the basis of every science and of human reasoning. They are the most common laws of human thinking, but to be such, they should be formulated in a general form. For example, the first and most important law of logic - the law of Identity that Aristotle formulated in the treatise «Metaphysics» read as follows: «... to have more than one meaning - it means not to have any meaning, but if a word has no meaning, then any opportunity is lost to argue with each other, but in reality - and with himself, for it is impossible to think anything, if you do not think anything one». In other words the law of identity states that all concepts, judgments, inferences must be defined uniquely and only on that condition they are meaningful. But the law of identity in the form in which Aristotle formulated it, is not quite complete, because he does not say, first, within what borders these unique concepts, judgments and conclusions remain valid - in other words, the limits of its application - and secondly, that all judgments and conclusions, defined uniquely, should be based on true facts, otherwise, all conclusions would be incorrect. The ancient philosophers took advantage of the first shortcoming of the law and created a series of logical paradoxes that supposedly showed the contradiction of formal logic, i.e. its imperfections. The essence of the paradox is that philosophers in their discussions managed to find a loophole and violate the law of identity described by Aristotle. For example, they determined the judgment as "the truth" in relation to one phenomenon and defined uniquely the same judgment as a "lie" in relation to the other phenomenon. The result is a contradictory conclusions. (A striking example is the Russell's paradox we discussed above). A second drawback of the definition of the law of identity was noticed by Leibniz, who eliminated it in the 17 th century by discovering of the 4th law of formal logic: the law of sufficient ground. This law states that "not a single event can be valid, not a single statement can be truth without sufficient ground, why it is the case, and not otherwise." This definition of the law given by Leibniz also is limited, because it only speaks of the real ground, on which claims are based to be valid, and it does not tell about the role of the sufficient ground as a limit, in which law of identity remains in force. But to refer to this property of sufficient ground it is necessary to be familiar with the dialectical logic, which at that time did not exist. In such form the law of sufficient grounds is interpreted and to this day. Currently, the law is as follows: "The true idea must be sufficiently substantiated" or "it is the law - a general logical principle, according to which a position is considered true only if it can be based on sufficient ground. " (Philosophical Dictionary, ed. Ivan T. Frolov. - 4 th izd.-m.: Politizdat, pp.). Incomplete understanding of the law of sufficient ground by earlier and modern philosophers is

5 reflected in the simple fact that so far neither Aristotle's definition of Law of identity has not been amended to reflect its close relationship with the law of sufficient ground, nor the definition of the law of sufficient ground has not been amended to reflect its impact on the Law of Identity in terms of the limits of its applicability. To eliminate all this shortcomings of the law of sufficient ground, we should formulate it as follows: Law of Sufficient Ground must be not only true but also must be a qualitative basis, in relation to which concepts, judgments and inferences are defined uniquely. As such, the Law includes, on the one hand, the definition given by Leibniz, on the other hand, it sets the boundaries of Law of Identity. Taking this into consideration, the law of Identity formulated as follows: All concepts, judgments and inferences that are used, should be identified uniquely in relation to the same qualitative ground. As such, the Law includes, on the one hand, the definition given by Aristotle, but on the other hand, its direct relationship with the law of sufficient ground, which sets the boundaries of Law of Identity. Such a formulation of laws of formal logic makes it invulnerable to attack of the philosophers who believe it is not always perfect for a full disclosure of truth. The Law of sufficient Ground and its role in science. About how the law of identity and sufficient ground operate in discussions and in science, I explained in my articles "Mystery of Logical paradoxes solved", "Formal and Dialectical Logic as a unity of opposites or the development of classical philosophy", which the reader can find on the Internet, so I will not stop here on this issue. To understand the laws of formal logic, in full volume, i.e. how they work in any science, can not be confined only to the formal logic, it is necessary to see a clear linkage between formal and dialectical logic, while many opponents of dialectical logic diminish or deny its importance, arguing that all the discoveries in science have been made on the basis of formal logic, without any assistance from the dialectical logic. Perversity of this claim just shows that many philosophers and scientists do not understand the simple fact, that formal logic includes in itself the most important element of dialectical logic: the law of sufficient ground. This is sufficient ground is a nothing more than a qualitative foundation on which we construct our knowledge with the help of formal logic, and qualitative foundation is the subject of dialectical logic in terms of its development. In other words, the dialectical logic shows us what the laws govern the development of qualitative ground and its transition into its opposite. But with help of the Laws of the formal logic we discover the content of this qualitative ground in any science. The "development" of formal logic in the form of perversion When Aristotle introduced the variables in his syllogisms then he wanted to emphasize the fact that his syllogisms are common to all the reflections in any science. And in this respect, it turns out that he was not completely correct. For example Russell writes: "The syllogism is only one kind of deductive argument. In mathematics, which is wholly deductive, syllogisms hardly ever occurs. Of course it would be possible to re-write mathematical arguments in syllogistic form, but this would be very artificial and would not make them any more cogent." (p.198) Thus, while Aristotle understood that formal logic is common to all sciences, but this fact he attributed to his

6 syllogisms but not to the laws of formal logic that were discovered on the basis of his analysis of these syllogisms. But the introduction by Aristotle of variables in their ratiocinate is an initial point from which a little misinterpretation and perversion of logic of Aristotle began, although, on the other hand, it is the introduction of these variables that created a good preconditions for the emergence of other formal logics, which are the particular case of the formal logic of Aristotle. Philosophers, based on these variables in syllogisms erroneously concluded that the logic of Aristotle is purely formal, since it deals only with the forms of the type if A is inherent in B, and B is inherent in C, then A is inherent in S. About this, the well-known Russian philosopher V.F. Asmus writes: "Aristotle - the founder of formal logic. He became as such as the result of his fundamental discovery. Exploring the structure of syllogism, he introduced letters instead of terms, i.e., introduced variables in the logic." In this comment Asmus admitted two mistakes. First, Aristotle became the founder of formal logic, not because he used the variables in their ratiocinate, but because on the basis of their study, he discovered three laws of formal logic. Secondly, he did not see, as all other philosophers, that the variables used by Aristotle, contain in themselves the content. When Aristotle said that A characteristic of B, he stressed the fact that the content in the form of A characteristic to the content in the form of B, regardless of in what form this statement stands. The predicate itself "inherent" refers to the essence of forms A and B. Let us take, as the example, a modern interpretation of the law of Identity. In the modern interpretation of the law symbolically is expressed as A is A or A = A. The shortcomings of such a formulation of the law of identity are evident under close consideration. First, in any language and in many sciences, there are words that have multiple values. For example in the Russian language, the word "matter" has two meanings, as the material world, or as a fabric. If we substitute A by the word "matter", then we get the matter is matter. But in this expression is not evident what kind of matter, we are referring to in the first and second instance. This expression reflects formally the equality of the concepts without their relationship to their content. And in this case, the law of Identity, as we see, is not reflected by this expression. The reader can do the same reasoning to the expression of A = A. But if we use the word "inherent" instead of "is" or "=" in the terms, we address the abovedescribed problem. A is inherent in A. If we replace A by word "matter", then we get matter is inherent in matter. In this case, regardless of what we mean by matter, its content should be inherent in matter, i.e. content in both "matter" must match, otherwise it is not inherent. Since expansion of our knowledge comes from the new discoveries in science, then the same concepts in science can have several meanings depending on what grounds they are determined. For example, in geometry Euclid the sum of the angles of triangle equals to 180 degrees, in the geometry of Lobachevsky the sum of the angles of triangle is less than the sum of 180 degrees, and in geometry of Riemann the sum of the angles of triangle more than 180 degrees. Here's the difference in the contents of the same concepts, modern interpretation of the law of Identity A is A or A = A does not captures it. Indeed, if we say that "the sum of internal angles of triangle = the sum of internal angles of triangle ", the equality of statements in terms of their form, is obviously. But the content of the left hand side of equality may be 180 degrees, meanwhile the content in the right hand side of equality may be less than 180 degrees. Here we have obvious violation of the law of Identity, because, as I said earlier, this is a formal equality of terms that completely ignores their content.

7 Therefore, the modern interpretation of the law of Identity deprived concepts, judgments and inferences of their contents. Furthermore, under this approach, the formal logic of Aristotle lost its connection with the dialectical logic, because philosophers took away the contents from their concepts expressed in the variables. What kind of nonsense: form without content. In mathematical logic the expression "A = A" has a well-defined meaning, which no one disputes, but the expression loses its meaning outside of mathematical logic. Conclusion 1. The essence of the formal logic of Aristotle is not in the analysis of classified him syllogisms, but in his discovery of 3 laws of formal logic, based on this analysis: the Law of Identity, Law of Contradiction, the Law of Excluded Middle. Leibniz developed the logic of Aristotle, by discovering the fourth law of sufficient grounds. 2. The Law of Contradiction and the Law Excluded Middle are the consequence of the law of identity. 3. The development of Formal logic of Aristotle is not to find other syllogisms, but rather to expand the content of the law of Identity and the Law of sufficient ground, based on an understanding of the relationship of formal logic of Aristotle and dialectical logic, discovered by me in my article "Formal and Dialectical Logic of unity opposites or development of classical philosophy. " 4. In the form in which I have formulated the Law of Identity and the Law of sufficient Ground, the classical formal logic gets its logical conclusion. 5. The modern interpretation of the law of Identity "A is A" or "A = A" is a gross distortion of the law of Identity, because it eliminates completely the content of the variables and brings a formal logic on the level of mathematical logic, to which was granted a superiority over formal logic of Aristotle. In an addendum to that, when the modern philosophers emphasize in the formal logic of Aristotle, on its syllogisms and not on its laws, they represent artificially classical formal logic as one of the possible forms of thinking, denying its common to all sciences, and thus creating a space for the "discovery" of other types of logic that are not only "successfully" can compete with the formal logic of Aristotle, but also to surpass it in the field of thinking. COPYRIGHT ILYA STAVINSKY. ALL RIGHTS RESERVED ABSTRACT By the beginning of XVIII century four laws of logic were formulated : the law of identity, Law of Contradiction, the Law of Excluded Middle and the Law of Sufficient Ground. The first three laws were formulated by Aristotle in the 4 th century BC, as the 4th law was introduced by Leibniz at the beginning of 18 century. So far for more than 300 years, none of the philosophers

8 discovered neither the 5th or 6 th law of formal logic. During this time, all the "discovery" of formal logic were limited only to it s distortion and confusion. The development of Formal logic of Aristotle is not to find other syllogisms, but rather to expand the content of the law of Identity and the Law of sufficient ground, based on an understanding of the relationship of formal logic of Aristotle and dialectical logic, discovered by me in my article "Formal and Dialectical Logic as unity of opposites or development of classical philosophy. " By doing this we eliminate in the formal logic those loopholes on which all existing paradoxes (in our daily life, physics, math and so on) are based. Key words: formal, dialectical, logic, Aristotle, Hegel, Engels, Marx, Russell, Leibniz, paradox, science, mathematics, physics, law, laws, knowledge

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