It is said that there are many kinds of logic, including traditional logic, modern logic, formal logic, mathematical logic and dialectical logic ... which is the most advanced? ! ! ! ! ! ! !

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Logic became a science, which began with Aristotle. I am afraid that few people doubt it. As we know, Aristotle did not call his research "logic", but he clearly pointed out that his research object is "syllogism", which is a science about "inevitably" drawing some conclusions from a real premise. There are two kinds of his syllogism, one is implied syllogism and the other is inductive syllogism. We need not say the former, but the latter is actually a complete induction, so it is deductive. Therefore, "logic" in Aristotle's sense is a science of "rules of inevitable reasoning" or "rules of inevitable proof or argument". Although he mentioned simple enumeration and induction, it was not in the sense of "logic", but in the sense of argument for comparison with "logic".

From the etymological point of view, Heraclitus' earliest use of logos also refers to the "objective order" embodied in language, and is also said in the sense of "inevitability". Therefore, the original meaning of "logic" refers not only to "inference rules" but also to "inevitable inference rules". This is actually the significance of the division between logic and other disciplines. Just as many people in China today accuse economics of not studying "productive forces", it is illogical to insist on using logic to study the authenticity of its contents. If logic can study everything, it should be called "knowledge".

What is inductive logic?

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When Bacon put forward scientific "induction", he did not say that it was logic; Until Mill wrote "induction" into his logical system. However, he did not use the concept of "logic" from the perspective of "inevitable reasoning". His logic refers to "reasoning" based on a set of "procedural rules" It doesn't matter whether using this rule will lead to the inevitable conclusion. He believes that all reasoning has the right to be called logic. It can be seen that even Mill himself thinks that according to the original logical definition, research induction cannot be regarded as logic.

It is worth noting that many masters of modern inductive logic, such as Carnap, don't think Bacon and Mill's inductive method is "logic" at all, but just think it is a "method", and don't think that modern inductive logic originates from them, but from probability theory. The original purpose of studying probability is not to oppose rationalism at all, but to solve the gambling problem. Pascal, the founder of probability theory, is a rationalist himself.

However, the reason why modern inductive logic is called logic is not because it has become a science about "inevitability rules", but because it has been "deduced" itself. However, this does not change that inductive logic is a subject about "probability". It is fundamentally different from the field of logic. Can deductive system be "logic"? Some modern sciences, such as game theory, are also deductive, so can they be called "logic"?

What is dialectical logic?

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We say that modern logic generally distinguishes between induction and inductive logic. Similarly, dialectics and dialectical logic are different. Before Hegel, it should be called dialectical method (not what Yao said in logic), but here in Hegel, it is really necessary to use dialectics as a way of thinking to establish "new logic" Therefore, what he calls dialectics refers to dialectical logic. There are two main ways of thinking: one is to solve the basic problem of logic, that is, to prove his premise with logic (note that it is by no means "inductive" from the outside to prove his premise is true), which is a circular way of thinking, while the previous logic is a linear way of thinking, so it cannot be reversed. Second, logic is not based on identity, but on the law of unity of opposites. We know that in Hegel's era, the fundamental premise of the so-called "formal logic" itself is unproven, so formal logic as a science about "inevitability rules" itself is inevitable. If logic is based on the law of unity of opposites, it can explain the basis of identity, thus making the mutual deduction between logical rules truly "complete" and "inevitable". As far as Hegel is concerned, the dialectical logic he tried to create is indeed more advanced than the traditional formal logic.

As for whether Hegel's thought can really establish his dialectical logic, it can be doubted, discussed and studied. But what is certain is that the logical meaning here is also in terms of inevitability. Hegel said: "Dialectics ... is the only principle to realize the inner connection and inevitability in scientific content." He just wants to elaborate this "the only principle to realize internal connection and inevitability".

So I want to remind other friends who participated in the discussion that the difference between "dialectical logic" and "formal logic" is not the difference between "content" and "form". But what we said above. The so-called "formal logic" means that logic only studies logical constants, and so does dialectical logic. As a science, it is impossible to study those changeable and unpredictable things. Hegel said: "Content has its own form, and it can even be said that only through form can it be alive and full;" Moreover, what is only transformed into one content is the form itself. " Therefore, dialectical logic only studies "dialectical logic constant", that is, the form of logic.

It is nonsense to say that Hegel's dialectical logic studies specific contents, which started with Russell.

The relationship between dialectical logic and induction

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Dialectical logic is also a science about inevitability rules, so it has nothing to do with what Bacon and Mill call induction. Induction and deduction (logic) each have irreplaceable functions. Induction is mainly used for search and discovery, and logic is used for proof; Inductively study the possible process under insufficient conditions and logically study the inevitable process under sufficient conditions. Therefore, dialectical logic can hardly be based on "induction 1 ... deduction 1 ... induction 2 ... deduction 2 ...". If you want to find a formula, let's say: analysis ... synthesis ... The analysis and synthesis here are in a logical sense (for example, Aristotle called his syllogism analysis), not in a methodological sense. The formula in the sense of this method actually exists in Plato's dialectics.

Restrictive logic

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-the organic combination of traditional logic and modern logic (a new field of contemporary logic: restrictive logic)

Two thousand three hundred years ago, Aristotle, a great thinker in ancient Greece (384-322 years earlier than Aristotle), founded traditional formal logic with instrumentalism, which laid the first monument for the history of logic development. /kloc-From the mid-9th century to the early 20th century, through the continuous efforts of British mathematician Boolean, German mathematician Frege, British philosopher and mathematician Rosso, and absorbing the achievements of Leibniz, the modern axiom system of "orthodox mathematical logic" as the theoretical basis of electronic computers was established, which is the second milestone in the history of logic development.

From 65438 to 0968, Lin Bangjin, a director of China Formal Logic Research Association and an engineer from Beijing Kai Guanchang, founded a new logic theory-restricted logic, which challenged the first two monuments. From 65438 to 0978, under the recommendation of Professor Shen Youding, a veteran logician in China, and Professor Wang Hao, a Chinese-American logician, Lin Bangjin published his paper "Introduction to Constraint Logic" in the Digest of the American Mathematical Society. 198565438+In February, Lin Bangjin's monograph Restrictive Logic was officially published in China. Restrictive logic is unique, which shocked the logic circle and attracted the attention of scholars at home and abroad.

Constraint logic is the product of organic combination of traditional formal logic and orthodox mathematical logic (modern logic). It constructs an unorthodox logic constraint system by using the strict and accurate mathematical methods provided by modern logic, which can accurately reflect the profound and correct leading thought of traditional formal logic. Lin Bangjin believes that traditional formal logic closely combines the common thinking of human beings with the reality of natural language, takes the reasoning format from known to unknown as the main research object, and insists on implementing circular argument, which is its profound and correct leading thought. But in theory, it cannot be added to more than a dozen extremely simple inferences; Through analysis, calculus technology is also very rough and outdated, far from meeting the needs of modern times. Orthodox mathematical logic systematically adopts modern mathematical methods, with rigorous argumentation and accurate calculation, but it abandons the essence of non-mathematical logic meaning that plays a decisive role in reasoning format, and regards it as the relationship between truth function and individual truth function, which is far from the dominant idea of traditional formal logic. Lin Bang boldly integrated the advantages of the above two logics and abandoned their shortcomings, and created a new logic system-restricted logic theory, which inherited the correct dominant thinking and effective reasoning format of formal logic and adopted mathematical methods provided by mathematical logic to deal with various logical problems in scientific research and social life. It is a modern development of traditional formal logic with a long history.

The theory of constraint logic points out that the constraint relation is a sufficient conditional relation after being clearly described. In fact, the constraint relationship constitutes the theoretical core of the reasoning format that can be used for acyclic argument in traditional formal logic: the constraint relationship between the antecedent and the afterpart of the reasoning formula must satisfy the universal and effective constraint relationship, and the constraint relationship must also appear in the antecedent or the afterpart. Restrictive logic system consists of semantics, language structure and pragmatics. Constraint logic semantics studies the logical structure and laws of the objective world, with objective constraints and related objective logic laws as the main research objects. The study of logical linguistics defines the mechanical arrangement structure and deformation rules of artificial symbols that describe objective logical structures and laws. The study of logical pragmatics restricts the translation between symbolic language and natural language under the principle of the same reference and predicate. Generally speaking, the research fields of constraint logic are: individuals, sets, univariate or multivariate functions, univariate or multivariate relationships, direct value function relationships among relationships, sufficient conditions (i.e. constraint relationships) among relationships, objective laws of the above relationships, and their reflections in consciousness-concepts (words), propositions and reasoning. Among them, the constraint relation (sufficient condition) is the core of the study.

On the basis of in-depth analysis of the common logical thinking practice of human beings, Lin Bangjin put forward the proposition calculus Cm system and the noun calculus Cn system by using the calculus skills of mathematical logic. There are seven kinds of true and false P and Q in the "constraint" propositions Xi p → q and Cm, and p → q has also been recorded as three true and four false. This is consistent with Strict implication of Louis. But Cm is different from Levi's modal system. Cm system has the following main features: (1) In Cm, the so-called "necessity" is not the nature of some two propositions, but only the relationship between them. P → q means that there is a certain "inevitable" connection between P and Q. (2) In addition to various implication paradoxes such as p → (q → p) that are avoided by general modal systems, Cm also avoids the implication paradoxes such as T p → q that are most difficult to avoid and accommodate by general modal systems. (3) Different from general modal systems, Cn has a formula similar to [p → (q → r)] → [q → (p → r)]. (4) It is equivalent to the theorem of traditional propositional logic reasoning listed in general formal logic books. (5) there is nothing like t (pvq)->; Q: This formula. (6) In traditional formal logic, Cm has a good way to deal with any place that seems to use the binary equivalence theorem of Cm exclusion. The Cn system based on Cm system only extends the formal language (citing eight individual variables, function words and predicates) and has no quantifiers. This can not only avoid many unavoidable troubles in the formal system with quantifiers, but also the process of calculus is propositional calculus in principle, which is closer to the reality of ordinary logical thinking. At the same time, the Cn system will provide a bright future for solving the referee problem.

Lin Bangjin put forward two independent and logical constraint theorems in deductive reasoning, which are called the first independence. The inference theorem with the logical property that the antecedent can be determined to be true without determining the antecedent to be true is called the second independence. For the reasoning formula in argument, "two independences" is the logical essence to ensure that the argument is not circular. This is a profound logical theoretical viewpoint. Some experts and scholars at home and abroad believe that constraint logic is of great significance in academic and scientific practice: (1) it can analyze and deal with a series of controversial and unresolved problems in the history of logic. It is possible to give definite answers to the questions such as the truth of the proposition, the existence of the subject, whether the object GAI and deductive reasoning can introduce new knowledge, whether the proved conclusion has been confirmed, and the paradox that caused the third mathematical crisis in the history of mathematics. (2) The formal system of elementary number theory based on it is n, when Cn. Once Goldbach's judgment problem is solved, it may provide a new idea for finally solving Goldbach's conjecture. This number theory system may also meet the requirements of compatibility and completeness (contrary to Godel's incomplete theorem). (3) It limits the formal axiom system of logic and creates the symbolic language system of computer language. Taking it as the logical theoretical basis of computer science, it can be used to study and design the connotation intelligent machine of the New Lan Dynasty. Software reliability confirmation and program correctness proof provide a new way. (4) Using it to analyze the logical mechanism in scientific theory and scientific creation can help scientists master effective and practical scientific methods.

International logicians and computer scientists are very sensitive to the theory of constrained logic. Shortly after Lin Bangjin's essay An Introduction to Restrictive Logic was published in the United States, universities in the Federal Republic of Germany and Canada actively organized expert seminars for translation and discussion. They believe that Lin Bangjin's "logical system is very important because it is closely related to computers, science, and especially' decision-making procedures'". Dr Livku, secretary general of American mathematical society, recommended the English abstract of constraint logic to the next country. International symposium on logic. The eighth session. Dr. Vaingatner, First Vice-Chairman of the International Symposium on Logic and Professor of Austrian Ransborg University, officially invited Lin Bangjin to attend the International Symposium on Logic held in Moscow from 65438 to 0987 and will make a special speech. In China, Lin Bangjin's restrictive logic has attracted academic attention. The State Science and Technology Commission organized a high-level seminar in Tsinghua University on 1986, and analyzed and discussed restrictive logic.

The criticism of constraint logic is also sharp and intense (Guo Shiming and Dong Yinong: Comments on Several Formal Systems in Constraint Logic, Dialectics of Nature Newsletter 1987, No.3). They think that the Cm system of constraint logic is equivalent to the propositional calculus R system of coherent logic published abroad more than 20 years ago, and R is undecidable, so the Cn system is undecidable (Lin Bangjin thinks that Cm and Cn are determinable). Even if Cn can judge, it is useless to apply Cn's judgment method to number theory system IV, because first-order number theory cannot have limited axiomatization, so it is completely impossible to construct a complete elementary number theory formal system N to solve Goldbach's conjecture and other problems. Cm has no semantics, let alone the reliability and integrity of semantics. Cn cannot define the concepts of "necessity" and "possibility". Cn has no practical value and cannot prove any meaningful inevitable and possible propositions. N system is neither consistent nor expressive enough, of course, it cannot be complete, and there is no set of axioms that can be judged. N system can't define the basic concepts of number theory such as "integer", "prime number" and "subtraction", and can't express propositions such as Goldbach conjecture. Therefore, N system is a rare system plagued by many diseases.

Then, where is the truth and where is the fallacy of limiting logic; How to evaluate its academic status historically; How much will be done; Is it a logical revolution? Whether it can stand the test of social practice; I believe that time will eventually give us a definite answer.

Two methods of logical proof

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First, direct proof.

Direct proof is a proof method that directly deduces the truth value of a topic from the truth value of an argument.

Second, indirect proof.

Indirect proof, also known as reduction to absurdity, is a real proof method to prove the proposition we want to prove by proving the falsity of the counter-proposition.

Indirect proof generally has three steps: (1) establishing a counter-proposition (that is, a proposition that contradicts the proposition we want to prove); (2) Prove that the counter-proposition is false; (3) According to law of excluded middle, the proposition we want to prove is valid. From this feature of indirect proof, indirect proof is essentially the application of negative affirmation of selective reasoning, that is, from the perspective of denying the truth of the antithesis, the truth of the topic we want to prove is deduced. It can be seen that in order to prove indirectly, the most important thing is to prove the falsity of the counter-proposition (that is, to deny the authenticity of the counter-proposition). Therefore, two methods are usually adopted: reduction to absurdity and exhaustive method.

Reduction to absurdity is a method that first assumes that the counter-proposition is true and leads to fallacious reasoning, and then according to the negative formula of hypothetical reasoning, it goes from denying fallacious reasoning to denying the truth of the counter-proposition. Since we deny the truth of the counter-proposition, according to law of excluded middle, it is natural to prove that the proposition we want to prove is true. Another common reduction to absurdity is the exhaustive method. Exhaustion is a method to list all kinds of possible topics other than those we want to prove, and then deny them one by one according to facts or reasoning, thus proving that the topics we want to prove are true. It can be seen that the exhaustive method is essentially a joint application of negative positive reasoning and complete inductive reasoning.

Here's an example:

■ In the Pakistani film On Earth, the heroine Rakiya's husband was wicked and finally shot. The killer was Lakia? Rashia shot. Mansour, an old lawyer, rescued this kind woman from the desperate situation. The honest lawyer proved that Laakia was not the murderer of her husband, and she was innocent. Mansour proved this way:

If Lakiya is the murderer, at least one of the five bullets in her pistol must have hit her husband. Now after on-site inspection, all five bullets in her pistol hit the opposite wall, hit the wall, and of course did not hit her husband. Moreover, if Laakia is the murderer of her husband, then the bullet must have entered her husband's body from the front, because Laakia shot her husband face to face. But after forensic examination, the bullet on the body entered from behind.

In this example, Mansour, an old lawyer, used two negative formulas of hypothetical reasoning with sufficient conditions. Through these two deductive arguments, the thesis that "Laakia is not a murderer" is proved.