Proposition is to express the meaning of a sentence, and it is a true or false idea. A proposition is characterized by its truth and falsehood. The proposition that truthfully reflects things is true, and the proposition that does not truthfully reflect things is false.
Types of propositions
① Original proposition: A proposition itself is called the original proposition. For example, if x> 1, then f (x) = (x- 1) 2 monotonically increases.
② Inverse proposition: a new proposition with the opposite conditions and conclusions to the original proposition, such as f (x) = (x- 1) 2 monotonically increasing, then x> 1.
③ No proposition: a new proposition that completely negates the conditions and conclusions of the original proposition, but does not change the order of conditions and conclusions, for example, if X.
④ Negative proposition: a new proposition that reverses the conditions and conclusions of the original proposition and then completely negates it, such as: if f (x) = (x- 1) 2 does not increase monotonically, then x < = 1.
1 declarative sentence
Can you judge whether it is true or not?
For example: "Don't abandon, don't give up" "Isn't an equilateral triangle an isosceles triangle?" The former is a proposition, the latter is not.
"Don't abandon, don't give up" is a declarative sentence. "Isn't an equilateral triangle an isosceles triangle?" This is a rhetorical question. Although the tone of rhetorical question expresses affirmation, it is an irregular propositional concept in mathematics.
Proposition and judgment are two interrelated logical terms. Proposition is to express the meaning of a sentence, and it is a true or false idea. Reasoning consists of propositions. A proposition is characterized by its truth and falsehood. The proposition that truthfully reflects things is true, and the proposition that does not truthfully reflect things is false. Judgment means that the decider asserts whether a proposition is true or false under certain time and space conditions. Straight sentence is the language expression of proposition, and proposition is the ideological content of straight sentence. The same proposition can be expressed in sentences of different national languages. The same sentence can express different propositions, especially direct sentences with pronouns, which can express different propositions in different language environments. Sentences, propositions and judgments belong to three different fields respectively.
Proposition consists of words, and specific propositions contain various words. Some words, such as "or", "and", "if, then", "not", "all" and "you", are often occupied by different specific propositions. These terms are called logical constants, and they don't refer to anything definite. When logical constants are properly matched with other words, it becomes a proposition; This collocation way or structure is the propositional form. For example, in "2 is an even number, 3 is an odd number" and "2 is a positive number and -3 is a negative number", both of them have the same logical constant "AND", and "AND" connects two propositions in both cases (called branching propositions here). The propositional form of these two examples is "... and ...". "..." indicates vacancy, can also be expressed by variables, and can be substituted into specific propositions. If all variables in the propositional form are changed into concrete values, a proposition is obtained. When comparing "2 is an even number, 3 is an odd number" and "3 is an odd number, 2 is an even number", we will find that they not only have the constant "sum", but also the former branch proposition in the former case is the latter branch proposition in the latter case. In order to express this formal connection, different variables or gaps need to be used. For example, the precedent is in the form of "... and ××××", and the latter example is in the form of "××××××× and ……". In the same context, the same variable must be replaced with the same value. In fact, "2 is an even number, 3 is an odd number" and "2 is a positive number and -3 is a negative number" have the same * * *, only that they are both composed of two branch propositions through a constant term "sum" (only once). The premise and conclusion of reasoning are propositions, and the validity of reasoning is only related to the form of premise and conclusion. Therefore, the study of propositional form in formal logic is the basis of reasoning theory.