Whether a proposition is negative is one of the difficulties in logic. In order to break through this difficulty, this paper attempts to make a comprehensive and detailed exposition of it for readers.

First, whether the proposition is negative. Related concepts of the proposition 1 Definition:

Let "if P is Q" be the original proposition, then "if it is not P, it is not Q" is called the negative proposition of the original proposition. Let "P" be a proposition, then "non-P" is called the negation of proposition P, and "non-P" is recorded as "P?"

2. Difference: No proposition denies the conditions and conclusions of the original proposition, which can be true and false, or true and false. The negation of a proposition is (1) to deny the whole proposition without considering the conditions and conclusions of the proposition, so just add "No" before the original proposition. (2) If the conditions and conclusions of the proposition are considered, only the conclusion of the proposition can be drawn.

Second, the keyword analysis in the proposition 1 negation. In the general proposition

"All?" Corresponding to "not all?" Instead of corresponding to "nothing?" ; "All?" Corresponding to "incomplete?" Instead of corresponding to "nothing?" .

"? then what "corresponding"? Still? " ; "? Or? "corresponding"? Then what? " 2. In the full name proposition and the existential proposition.

"Is there?" Corresponding to "yes?" Wait; "Existence?" Corresponding to "all?" Wait a minute.

"At least one" corresponds to "none", and so on; "At most one" corresponds to "at least two" and so on.

Third, rewrite the negative proposition: if the original proposition is "if P is Q" or "if? And then? " If the original proposition is not in the above form, it should be rewritten into the above form, and then its no proposition should be written.

Fourth, whether the proposition is negative.

1. Write the negative proposition "If A and B are both positive numbers, then ABBA 2." Answer: if a and b are not positive numbers, then ABBA 2.

Comment: The negation of "all positive numbers" is "incomplete positive numbers" rather than "no positive numbers". If "A and B are both positive numbers" is understood as "A is positive and B is positive", then its negation can also be written as "A is not positive or B is not positive".

2. Write the negative proposition that "the sum of two odd numbers is even" and the negation of this proposition. Answer: There is no proposition: if two numbers are not all odd, their sum is not even.

The negation of proposition: the sum of two odd numbers is not even.

Comment: (1) "The sum of two odd numbers is even" means "If both numbers are odd, their sum is even".

(2) The negation of "even number" is "not even number" rather than "odd number" (why? ).

3. Write the negative form of the following proposition:

(1) Some constant sequences are not geometric series. (2) The parallelogram is a diamond.

Solution: (1) Any constant series is a geometric series. (2) Parallelograms are not all diamonds.

Comment: Generally speaking, the negation of existential proposition can be full name proposition, and the negation of full name proposition can be existential proposition. So the negation of (1) is a full-name proposition. According to the meaning, "parallelogram is a diamond" is actually a full-name proposition, so "some parallelograms are not diamond" can also be used as an answer. The answer is just a negative conclusion, concise and acceptable.