However, people do not agree with the theoretical basis of reduction to absurdity. Now, if you extract a few recently published textbooks, you can see where the differences are.
1. A series of teaching plans for full-time ordinary high schools (PEP) jointly published by People's Education Publishing House and Yanbian Education Publishing House? Line 19 on page 55 of the lesson plan in the first volume of mathematics writes: the theoretical basis of proving the problem by reduction to absurdity is that the original proposition and its negative proposition are both true and false, that is, it is necessary to prove that "if P is Q" is true and "if Q is P" is false, so that "if Q is P" is true (. For the sake of convenience, we will call the viewpoint of the book "the original proposition and its negative proposition are the same as true and false" for short.
2. Shaanxi Normal University Press published the Synchronous Learning Plan for New Textbooks of People's Education Society? Huanggang art of war? On the fifth line of page 69 of senior one mathematics, it is written that reduction to absurdity is an indirect method to prove a proposition, because the negative form of "if P is Q" is "if P is not Q". From the truth table, if it is proved that "if P is not Q" is a false proposition, then "if P is Q" must be a true proposition. It is very different from the negative proposition of proving the original proposition, and its use embodies the dialectical thought of "opposing when there is difficulty" in solving mathematical problems. Therefore, in addition to mastering the steps of applying reduction to absurdity, we should also pay attention to mastering the opportunity of applying reduction to absurdity. Obviously, the book does not agree with the theory of "the same truth and the same falsehood", but thinks that the theoretical basis of reduction to absurdity is to prove that the negation of the original proposition is false. For the time being, we call the book's viewpoint "If the negation of the original proposition is false, there must be the truth of the original proposition", which is referred to as "proposition negation theory" for short.
3. The penultimate line on page 24 of Mathematics Teaching and Testing in Senior One published by Suzhou University Press says that the way to prove "if P is Q" by reduction to absurdity is to prove that "if P is not Q" is false, so it is the characteristic of reduction to absurdity to draw contradictions from "not Q". Obviously, the viewpoint of the book should belong to the "proposition negation theory", but it is slightly different from the narrative in Sun Tzu's Art of War in Huanggang.
The theory of "the same truth and the same falsehood" and the theory of "proposition negation" are tit for tat. Which is right and which is wrong? Is it all right or all wrong? This is exactly the problem to be analyzed in this paper.
Debate on this issue
On page 32 of Senior One Mathematics (People's Education Edition), it is clearly stated that there are three steps to prove a proposition by reduction to absurdity, and there is no objection in this respect. For simplicity, we might as well call these three steps "inversion", "reduction to absurdity" and "conclusion" respectively. "Reduction to absurdity" is not only the core of reduction to absurdity, but also the concrete embodiment of its spiritual essence. I'm afraid the theoretical basis of reduction to absurdity is not consistent, and it also stems from this. In order to distinguish right from wrong, we need to analyze it layer by layer.
1. What is the "starting point" of "returning to absurdity"? Is it a single ┐q or ┐q and P? The author thinks that the general situation is "┐q and p", and only "┐q" is used instead of p in special circumstances. Let's look at the following example first:
Question 1. As we all know, A, B and C are a set of Pythagorean numbers. It is proved that A, B and C cannot all be odd numbers.
Evidence: narrator:
Suppose that A, B and C are all odd numbers and "inverse" (┐q).
Then a2, b2 and c2 are all odd numbers, and the inference is based on "┐q"
When the theme a2+b2 = c2 is set, the condition P is used.
∴a2+b2 = c2 is an even number. Reasoning based on ∴ Q and p "
This contradicts the inference that c2 is an odd number.
So the original proposition holds and draws a conclusion.
This question shows that, generally speaking, the "starting point" of "returning to absurdity" is "┐q and P"
Question 2. Example 3 on page 32 of senior one mathematics textbook (People's Education Edition), disproof proves that if a>b>0, then √ A > √ B.
[Analysis] This question should be changed to: When: a > 0, b>0, if a > B, then √ A > √ B. This change is to change the original proposition of Example 2 on this page to "When a > 0, b>0" is the major premise, "If a > B" is the condition P, then √ A >; √b "is the conclusion Q.
Evidence: narrator:
Suppose √a is not greater than √b "reverse design"
That is √ a.
∵a & gt； 0, b>0 According to the premise, the condition P is not used.
∴√a <√b =√a√a & lt； √b √a
And √ a √ b
√a =√b = a= b
The contradiction between this and the known condition A >:b comes down to "a≤b" and "A >; B "contradiction
∴√a & gt； come to a conclusion
This question shows that under special circumstances, the "starting point" of reduction to absurdity is only "┐q". "
2. What is the "foothold" of "returning to absurdity"? As we all know, the reasoning in Reduction to absurdity must be rigorous and accurate, and the "foothold" of reasoning is to deduce contradictions. Among them, the starting point of returning to absurdity is different, and the ending point of returning to absurdity is also different: everything that starts with ┐q and P often ends up contradicting axioms, theorems, inferences and properties, or contradicting known conditions (such as the above question 65433), and everything that only starts with ┐q ".
3. What's wrong with "propositional negation"? As mentioned above, the core of "proposition negation theory" is: "If it can be proved that' if P is not q' is a false proposition, then' if P is q' must be a true proposition". It should be affirmed that this core itself is a proposition, and it is an indisputable true proposition. However, how to realize the "condition" of this core proposition-"if it can be proved that' if P is not q' is a false proposition"? "Propositional negation" is not specified. In fact, the "reduction to absurdity" part of the reduction to absurdity has nothing in common with the "conditions" for realizing the core proposition; First, the starting point is different. Reduction to absurdity begins with ┐q and p ┐ or ┐q ┐, and the "condition" of the core proposition must begin with p. Second, the foothold is different. The reduction to absurdity is based on "pushing out contradictions", and the "conditions" for realizing the core proposition must be based on "not pushing out ┐q". Obviously, when the "condition" of the core proposition cannot be realized, its "conclusion"-"If P is, Q will be a true proposition"-will also become a "castle in the air".
4. Where is the "theory of the same truth and the same falsehood" incomplete? The core of the theory of identity between truth and falsehood is to prove that if p is q, if q is p is false, and if q is p is true. Obviously, this statement is also correct and achievable. But the realization of the above discussion is limited to "┐q" as the "starting point", that is, from "┐q", we can deduce "┐p" which is contradictory to P. Most reduction to absurdity is based on "┐q and P" and ends with the contradiction between R and ┐ R. Therefore,
The conclusion of the problem
Through the above analysis, the author believes that the theoretical basis of reduction to absurdity is neither "propositional negation" nor "the same truth and the same falsehood", but an indirect affirmation of "negation-reasoning-negation". More specifically, it means "first deny the conclusion of a proposition and affirm its conditions, then make strict reasoning on this premise, and then deduce the contradiction and get a new negation, so as to realize indirect affirmation". (Note 1)
Based on the theory of "indirect positive formula", the premise of "reduction to absurdity" is generally "┐q and P". From then on, correct reasoning is carried out, and the contradiction between R and ┐r is deduced. Especially when r=p, the inverse proposition that contradicts the known conditions and is transformed into the original proposition holds.
[Note Ⅰ] Senior High School Mathematics Competition Tutoring edited by Luo Zengru (published by Shaanxi Normal University Press, p3 1. )