proposition

[mìng tí]

Logical Nouns

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stay Modern philosophy 、 mathematics 、 logic 、 language In science, proposition (judgment) refers to the semantic meaning (actually expressed concept) of a judgment sentence, which is a phenomenon that can be defined and observed. Proposition does not refer to the sentence itself, but to the semantics expressed. When different sentences have the same meaning, they express the same proposition. In mathematics, people usually judge a thing by Declarative sentence It's called a proposition.^[1]

Chinese name: proposition
Foreign name: proposition
Other English translations: problem;issue

Definition 1: A declarative sentence used to judge whether it is true or false
Definition 2: Draft composition topic
Definition 3: Logical Nouns

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Citation explanation

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1. It refers to the theme of the determined poems and articles.

Song Dynasty Wang Yucheng 《 Preface to Farewell Bao Xiucai 》"Ten chapters of the article were published, which is the work of the Jinshi Bao Sheng. The idea of the proposition is almost extraordinary."

2. Draw up a title; Make questions.

bright Wang Ao "Zhenze Changyu Jingzhuan": "The ancients wrote poems, they must set their own questions."

《 Odd Things Witnessed Over Twenty Years; Having Seen Strange Present Conditions for Twenty Years 》The seventh and third times: "Once, Licheng County came to the Academy to set up an examination for the official class in the Academy."

Cao Jinghua Flying Flowers Collection on Prose 》"But my guests are neither like the majestic examiners, who set up their compositions without any rules."

The composition of this college entrance examination is composition with assigned subject 。

3. The questions; Title.

clear Sun Zhiwei The preface to the poem "Giving the Dongzhu Rain Today to Li Jiliu, the Chief Executive of Lu'an": "The Chief Executive who got off the bus and tested the ability of the local people to write poems in different topics, Yu Shi, in his brotherhood office, was glad to hear that he sympathized with the local officials and loved the farmers, and he was positive in his various singing patterns. Seeing the proposition, he did not think about the crudity, but encouraged the local people to write two laws."

《 Xinhua News Digest 》In 1981, Issue 7: "But in terms of ideology and temperament, he is still a prosecutor, so I used the present proposition."

4. Logic terms. Sentences expressing judgment.

Mao Zedong 《 New Democracy 》4： "The Chinese revolution is World revolution This correct proposition was put forward during the first great revolution in China from 1924 to 1927. " One says that the meaning expressed by the declarative sentence is a proposition, and the determined proposition is a judgment.

5. Mathematical concepts

(1) The concept of proposition in junior high school mathematics is: "sentence to judge a thing"; High school textbooks define it as: "sentences that can be used to judge whether they are true or false"

(2) Generally, in mathematics, we call declarative sentences that can be expressed in language, symbol or formula and can judge whether they are true or false proposition 。 The statement judged as true is called True proposition The statement that is judged to be false is called False proposition 。

Classification definition

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Proposition m ì ngt í

(1) [proposition]∶ logic Refers to the language form expressing judgment, which is formed by the connection of subject and object by the copula

(2) [problem;issue]∶ mathematics Or some explanation in physics

Propositional classification

Aristotle's instrumental theory, especially the《 Category 》The different forms of propositions and their relationships are studied, and different types of propositions are classified according to different forms. Aristotle first divided propositions into simple and complex categories, but he Compound proposition No in-depth discussion. He went on to Simple proposition It can be divided into positive and negative propositions according to quality, and full, specific and indefinite propositions according to quantity. He also mentioned the individual proposition, which is equivalent to the later so-called "taking proper names as the main item and taking universal concepts as the predicate Monomorphic proposition of. Aristotle emphatically discussed four propositions represented by A, E, I and O. about modality He discussed the four propositions of necessity, impossibility, possibility and contingency Modal word 。 Aristotle's mode refers to the inevitability possibility Etc.

Logicians after Aristotle, such as Theofrasdo 、 Magala School and Stoic school The logicians in the Middle Ages and the logicians in the Middle Ages also made continuous discussions on the compound propositions containing the propositional connectives "or", "and", "if, then", etc., thus enriching the theory of logic on propositions.

Kantian classification

Kant According to his category theory, he classified judgments, which had a great impact on later generations. Kant's classification of judgment mainly includes four aspects:

① Quantity, including full name, special name and single name;

② Qualities, including affirmation, negation and infinity (all S is or is not P);

③ Relations include straightforward (the relationship between two concepts), hypothetical (the relationship between two judgments), disjunctive (the relationship between several judgments) judgments;

④ modality There are several kinds of judgments: contingent, factual and conclusive. Kant's mode refers to the degree of cognition. He believes that the judgments that constitute hypothetical judgments and disjunctive judgments are all contingent.

Traditional logic classification

In the second half of the 19th century, the classification of propositions in European logic books was different. Generally speaking, according to relation is according to proposition predicate The relationship between Straightforward proposition 、 Hypothetical proposition (The relation between the subject and predicate of the latter is conditional on the former) and Disjunctive proposition (Pairs between predicates Main item There is a selective relationship). From the perspective of quality, there are positive propositions and negative propositions. From the perspective of quantity, there are universal propositions, including simple propositions, universal propositions (where S is P) and Special proposition 。 When discussing disjunctive propositions, these traditional logic books also often discuss Disjunctive proposition , separation proposition (not A and not B), etc. In addition, there is also a kind of analytic proposition that is often mentioned. In this kind of proposition, there is a distinction proposition, whose form is "only S is P"; Another is called exception proposition, which is in the form of "every S is P except S which is M".

Formal analysis

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The analysis of propositional form by modern logic has nothing to do with the specific content of propositions as premise and conclusion, because the validity of reasoning is only related to the form of premise and conclusion. Therefore, in classical binary logic, propositions can only be seen as true (recorded as T) and false (recorded as F), and they are collectively called true values.

Further analysis of the form of propositions should go deep into simple The non propositional component within a proposition. In modern logic, a proposition like "Socrates is a man" is considered to be the simplest proposition. If s represents "Socrates" and M represents "man", such propositions can be recorded as M (s), which means that an individual s has nature R 。 In general, the simplest proposition is F (x) , readable as Field of discourse Individual x in has properties F ； More complex forms can be filled G (x, y), which can be read as the relationship between individuals x, y) in the universe G 。 Here, x, y),... are called individual variables; F , G ,... Is called predicate variable, and F It's one yuan, G It's binary. The form of general full name proposition is wind x（ F x→ Gx ）The form of existential proposition, that is, the so-called special proposition of traditional logic, is ヨ x（ F x∧ Gx )。 All these are the preliminary analysis of propositional form by classical first-order predicate logic in modern logic (see Predicate logic ）。 In addition, adding quantifiers to predicate variables forms Higher-order logic 。 Modal words can also be introduced, or interrogative sentences, imperative sentences, etc. can be analyzed to establish relevant logic theories.^[2]

Form of proposition

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1. For two propositions, if the condition and conclusion They are the conclusions and conditions of the other proposition, so these two propositions are called Reciprocal proposition , one of which is called original proposition Another proposition is called original proposition Inverse proposition 。

2. For two propositions, if the condition and conclusion of one proposition are respectively the negation of the condition and conclusion of the other proposition, then these two propositions are called Reciprocal proposition One of them is called the original proposition, and the other is called the original proposition Negative proposition 。

3. For two propositions, if the condition and conclusion of one proposition are respectively the negation of the conclusion and the negation of the condition of the other proposition, then these two propositions are called mutually negative propositions, one of which is called the original proposition, and the other is called the original proposition Inverse negative proposition 。

interrelation

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1. The interrelationship of the four propositions: the original proposition and the inverse proposition are mutually inverse, the negative proposition and the original proposition are mutually inverse, the original proposition and the inverse negative proposition are mutually inverse, the inverse proposition and the negative proposition are mutually inverse, the inverse proposition and the inverse negative proposition are mutually inverse, and the inverse negative proposition and the negative proposition are mutually inverse.^[3]

2. The relationship between the truth and falsehood of four propositions: (1) Two propositions are mutually negative propositions, and they have the same truth and falsehood. (2) The two propositions are reciprocal propositions or reciprocal negative propositions, and their truth and falseness are not related (the original proposition and the inverse negative proposition are true and false, and the inverse proposition and the negative proposition are true and false)

1. Can judge whether it is true or false Declarative sentence It's called a proposition, and the correct proposition is called True proposition The wrong proposition is called False proposition 。

2. In the proposition of "if p, then q" form, p is called the condition of the proposition, and q is called the conclusion of the proposition.

3. Classification of propositions:

① Original proposition: a proposition itself is called the original proposition. For example, if x>1, f (x)=(x-1) ^ 2 increases monotonically.

② Inverse proposition : a new proposition that inverts the conditions and conclusions of the original proposition, for example, if f (x)=(x-1) ^ 2 monotonically increases, then x>1.

③ Negative proposition : Combine the conditions of the original proposition with conclusion A new proposition that is completely negative, but does not change the order of conditions and conclusions. For example, if x<=1, then f (x)=(x-1) ^ 2 does not increase monotonically.

④ Inverse negative proposition : a new proposition that inverts the conditions and conclusions of the original proposition, and then negates the conditions and conclusions. For example, if f (x)=(x-1) ^ 2 does not increase monotonically, then x<=1.

4． Negation of proposition

The negation of a proposition is a new proposition that only negates the conclusion of the proposition, which is different from the negation proposition.

5.4 Propositions and the relationship between truth and falsehood of negation of propositions

original proposition	Inverse proposition	Negative proposition	Inverse negative proposition
really	really	really	really
really	false	false	really
false	really	really	false
false	false	false	false

(The original proposition is true, and the converse proposition is not necessarily false.)

Propositional condition

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Sufficient and necessary conditions

1. "If p, then q" is a true proposition, which is called deriving q from p, recorded as p=>q, and said that p is a sufficient condition for q, and q is a necessary condition for p.

2. "If p, then q" is a false proposition, which is called that q cannot be inferred from p, and is recorded as p ≠>q, and that p is not a sufficient condition for q (or p is a non sufficient condition for q), and q is not a necessary condition for p (or q is an unnecessary condition for p).

necessary and sufficient condition

If there is both p=>q and q=>p, it is recorded as p<=>q, and p is said to be q Sufficient and necessary conditions (or q is of p Sufficient and necessary conditions ）, short for necessary and sufficient condition , also called p and q equivalence

Connectives

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And

1. Combining p and q with the conjunction "and" is called a new proposition, which is recorded as p ∧ q and pronounced as "p and q".

2. Determination of the truth and falseness of proposition p ∧ q:

When both propositions p and q are True proposition The new proposition p and q are true propositions. If one of the two propositions p and q is False proposition The new proposition p and q are false propositions.^[2]

or

1. Combining p and q with the conjunction "or" is called a new proposition, which is recorded as p ν q and pronounced as "p or q".

2. Determination of the truth and falseness of proposition p ν q:

When one of the two propositions p and q is true, the new proposition p or q formed is true. When two propositions p and q are false propositions, the new propositions p or q formed are false propositions.

wrong

1. For a proposition p, if only its conclusion is negated, a new proposition will be obtained, which will be recorded as "p" and read as "non p".

2. Judgment of proposition p:

In the proposition and his non proposition, there is one and only one is True proposition 。

example

p: Two lines in the plane perpendicular to the same line are parallel, q: two lines in the plane perpendicular to the same line are not parallel.

Where p is a true proposition and q is a false proposition.

Universal quantifier

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1. The words "for all" and "for any" are called in logic Universal quantifier , recorded as“ ∀ ”A proposition containing a full quantifier is called Universal proposition 。^[2]

2. If p (x) holds for any x in M, it is recorded as "∀" x ∈ M, p (x).

3. For the full name proposition p: "∀" x ∈ M, the negation p of p (x) is: "∃" x ∈ M, p (x).

Existential quantifier

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1. The words "there is one" and "at least one" are called in logic Existential quantifier , recorded as "∃", a proposition containing existential quantifiers is called existential proposition.^[4]

2. There is at least one x in M, so that p (x) is true, which is recorded as "∃" x ∈ M, p (x).

Contains a classifier The negation of the proposition of

3. For those with a classifier The special proposition p:: "∃" x ∈ M, the negation p of p (x) is: "∀" x ∈ M, p (x).

Geometric proposition

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Specifically Euclid Of《 Geometric primitives 》The proven propositions in, namely, the following 48 propositions:^[5]

1. Make an equilateral triangle on a known finite line.

2. Making a line segment from a known point (as the endpoint) equals to a known line segment.

3. If two unequal line segments are known, try to intercept one segment from the upper edge of the larger one to make it equal to the other.

4. If two triangles have two sides that are equal to each other, and the angles between these equal lines are equal, then their bottom is equal to the bottom, the triangles are all equal to triangles, and the other angles are equal to the other angles, that is, the angles opposite the equal sides.

5. On an isosceles triangle Middle, the two base angles are equal to each other; Moreover, if two waists are extended downward, the two corners below the bottom are equal to each other.

6. If two angles in a triangle are equal to each other, the sides opposite the equal angles are equal to each other.

7. If two line segments intersect at one point are made on a known line segment (from its two endpoints), it is impossible to make another two line segments intersect at another point on the same side of the line segment (from its two endpoints), so that the two line segments made are equal to the previous two line segments respectively. That is, the line segments from each intersection point to the same endpoint are equal.

8. If one of the two triangles has two sides that are equal to the other, and the bottom of one is equal to the bottom of the other, the angles sandwiched between the equal sides are also equal.

9. An angle can be cut into two equal angles.

10. A line segment can be divided into two equal lines.

11. From a known point on a known straight line, a straight line can be made at right angles to a known straight line.

12. A known point outside the known straight line can be used as the vertical line of the line.

13. The adjacent angle formed by a straight line and another straight line, or two right angles or they are equal to the sum of two right angles.

14. If two straight lines passing through any point on a straight line are not on the same side of the line, and the sum of the adjacent angles to the line is equal to two right angles, then the two straight lines are on the same line.

15. If two straight lines intersect, the opposite vertex angles they intersect are equal.

16. In any triangle, if one side is extended, the external angle is greater than any internal diagonal.

17. In any triangle, the sum of any two angles is less than two right angles.

18. In any triangle, the big side is opposite to the big angle.

19. In any triangle, the big angle is opposite to the big side.

20. In any triangle, the sum of any two sides is greater than the third side.

21. If two line segments intersected in the triangle are made by two endpoints of one side of the triangle, the sum of the line segments from the intersection point to the two ends is less than the sum of the other two sides of the triangle. However, the included angle is greater than the top angle of the triangle.

22. Try to make a triangle from three segments that are equal to three known segments respectively: in such three known segments, the sum of any two segments must be greater than the other segment.

23. On a known straight line and a point above it, make an angle equal to the known angle.

24. If the two sides of one triangle are equal to the two sides of the other, and the included angle of one triangle is greater than the included angle of the other triangle, the side with the larger included angle is also larger.

25. If the two sides of one triangle are equal to the two sides of the other, the angle opposite to the larger third side is also larger.

26. If in two triangles, two angles of one is equal to two angles of the other, and one side is equal to one side of the other. That is, or this side is Equiangular Or equiangular Opposite side 。 Then their other edges are equal to the other edges, and the other corners are equal to the other corners.

27. If the stagger angle formed by the intersection of a straight line and two straight lines is equal to each other, the two straight lines are parallel to each other.

28. If the isometric angles formed by the intersection of a straight line and two straight lines are equal, or the sum of the adjacent internal angles is equal to two right angles, the two straight lines are parallel to each other.

29. If a straight line intersects two parallel straight lines, the resulting stagger angles are equal, the isometric angles are equal, and the sum of the adjacent internal angles is equal to two right angles.

30. If some lines are parallel to the same line, they are also parallel to each other.

31. Make a straight line through a known point and parallel to the known straight line.

32. On arbitrary triangle If one side is extended, the outer angle is equal to the sum of the two nonadjacent inner diagonals, and the sum of the three inner angles of a triangle is equal to two right angles.

33. Connect equal and parallel line segments (endpoints) in the same direction (respectively), and they are also equal and parallel.

34. In a parallelogram patch, the opposite sides are equal, the diagonals are equal, and the diagonal bisects the patch.

35. On the same bottom and on the same two Parallel line Between parallelogram Equal to each other.

36. Parallels on an equal base and between the same two parallel lines are equal to each other.

37. Triangles on the same base and between the same two parallel lines are equal to each other.

38. Triangles on an equal base and between the same two parallel lines are equal to each other.

39. Equal triangles on the same bottom and on the same side of the bottom must be between the same two parallel lines.

40. Equal triangles with equal base and on the same side of the base are also between the same two parallel lines.

41. If a parallelogram and a triangle are both at the same base and between two parallel lines, the parallelogram is twice as large as the triangle.

42. Use a known straight angle to make a parallelogram so that it is equal to a known triangle.

43. In any parallelogram, the complements of parallelograms on both sides of the diagonal are equal to each other.

44. Make a parallelogram with a known line segment and a known straight angle to make it equal to a known triangle.

45. Make a parallelogram with a known straight angle to make it equal to a known straight line.

46. Make a square on a known line segment.

47. In a right triangle, the square on the side opposite the right angle is equal to the sum of the squares on both sides of the right angle.

48. If the square on one side of a triangle is equal to the sum of the squares on the other two sides of the triangle, the angle between the back two sides is a right angle.

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