What is a special negative proposition in logic?

First, the problems existing in the traditional proposition theory

As a "propositional form", the logical semantics of the traditional outspoken proposition is not clearly defined, and the logical structure is not fully finalized, so there are still various logical theoretical problems, such as:

First, can the subject be empty? This issue is still controversial in the field of traditional formal logic. There is a view that the traditional special proposition requires the existence of the subject S. When S is a function word, the special title is false. However, according to this view, "some solvers of Goldbach's conjecture are from China" and "some solvers of Goldbach's conjecture are not from China" are all false. Whether the traditional full name proposition also requires the existence of the subject S is still controversial. For example, whether "all solvers of Goldbach's conjecture are mathematicians" is true or not is still controversial. Another view is that the traditional outspoken proposition only deals with real nouns, which is meaningless when the subject S is empty, that is, traditional logic cannot judge whether the outspoken proposition like "all solvers of Goldbach conjecture are mathematicians" is true.

Secondly, when the extension of the subject S is infinite, finite or empty, what do the full quantifier "all" and the special quantifier "some" mean? Or "every" or "at least one"? If so, the full-name proposition "Everyone dies", that is, "Everyone dies", can not be proved to be true until everyone in the world dies, while the special proposition "Some tables are heptagonal" (that is, "at least one table is heptagonal") can not be proved to be false so far. The introduction of so-called logical quantifiers in traditional logic theory will inevitably lead to the exploration of each individual in the process of understanding uncountable fields. Obviously, it is beyond the reach of human beings' limited energy and life to ensure that every individual (full-name quantifier) who can't list fields one by one has a certain property that is true, and at least one individual (special quantifier) has a certain property that is false (that is, no individual has a certain property that is true). In this way, if there are logical quantifiers, then we can't determine the real knowledge about uncountable fields. However, there is no logical quantifier (of course, there can be linguistic quantifiers in the language carrier) in the true knowledge of uncountable fields that has been determined for human beings. Naturally, when the subject S thinks of empty sets, "logical quantifiers" are even more absurd.

Third, the meaning of the so-called "copula" is not clear. Traditional logic holds that there are positive logical conjunction and negative logical conjunction in logical structure. Therefore, it is considered that the "outspoken proposition" has both opposition and parallel affirmation and negation. This is caused by the confusion of positive and negative conjunctions in some natural sentences. In terms of logical structure, there is no logical copula not only in the multivariate relation proposition, but also in the so-called "outspoken proposition" (actually a compound proposition based on atomic proposition). Logically speaking, any proposition is affirmative, whether it is an atomic proposition or a compound proposition; Negative proposition (the last conjunction is negative) is a compound proposition (any atomic proposition is not a negative proposition); Without exception, the negative proposition, as a special compound proposition, is of course affirmative in logic. "Negation" is a conjunction of 1 yuan, while "affirmation" is not a conjunction, but a logical property necessary for any proposition (if a proposition has no such property, what is true or false); Therefore, the two are neither opposite nor parallel.

Fourth, can the object of the traditional full-name affirmative proposition A be GAI? Some people think that "GAI can", while others think that "GAI can't". So far, the debate is endless. However, the definitions of "GAI" and "without GAI" are not clear. If "determining the whole extension of the subject (or object)" is GAI and "only determining the true sub-part of the extension of the subject (or object)" is anti-GAI, then the traditional logic "without GAI" means "determining the whole extension of the subject (or object) or determining the true sub-part of its extension", and "without GAI" should be GAI or. In this way, "GAI" is "not a GAI against GAI". Therefore, "GAI is not GAI" and "GAI is GAI sometimes". This is as natural as "Han people are yellow people" and "Yellow people have a large number of Han people".

Second, contemporary formal logic solves the problems of traditional propositional theory.

Contemporary formal logic gives a clear and reasonable solution to the above-mentioned long-debated and unresolved problems. Below, we explain the handling of the above problems by contemporary formal logic in the order of A, E, I and O:

First of all, about proposition a.

The sentence pattern of Proposition A is "All S are P", and the symbolic expression in traditional logic is "sAp". Relative to the logical structure of events considered for it, the following four situations roughly correspond to sAp:

1. The people involved in writing this book are all yellow people.

2. Everyone will die.

All the celestial bodies are in motion.

4. The solver of Goldbach's conjecture is a mathematician.

The fourth example represents four different propositions. Example 1, because "people who participated in writing this book" is a finite set that can be enumerated one by one, and it is an extended proposition whose truth value can be determined. Up to now, people in the world are all finite sets, and it is impossible to enumerate them one by one. "Celestial bodies" are all infinite sets. Therefore, in the traditional outspoken proposition, whether proposition 2 and proposition 3 are true or not depends on the logical semantics of "everything" and "all". If it is an "extended conjunction", it is still uncertain. In the fourth example, the subject is thinking about empty sets. In traditional formal logic, when S thinks about empty sets, how sAp handles it has been unresolved for a long time, so its true value is even more uncertain.

Contemporary formal logic clearly and reasonably solves the above four propositions. Contemporary formal logic divides the traditional outspoken proposition sAp into extensional conjunctive proposition and connotation sufficient condition proposition. According to the logical semantics and empirical content in contemporary formal logic, examples 1, 2, 3 and 4 can all be determined to be true. The type of the proposition 1 is:

p(e 1)∧p(E2)∧…p(ei)∧…p(em)

It belongs to the "closed take proposition with m closed 1 meta atomic propositions as conjunctive branches" in contemporary formal logic, and its sentence pattern is "every countable S is p". The formulas of propositions 2, 3 and 4 are as follows:

s(x)？ P(x) or? (s(x)！ ? p(x))

It belongs to "closed sufficient condition proposition with Kai 1 meta-atomic proposition as antecedent and consequence" or "no proposition with Kai 1 meta-atomic proposition and Kai 1 meta-atomic proposition as approximate closed approximate proposition" in contemporary formal logic, which is referred to as "connotation sufficient condition proposition" for short, and its natural language sentence pattern is "S must be P" or "S must be P"

Second, about the E proposition.

The sentence pattern of E proposition is "All S are not P". The symbolic expression in traditional formal logic is sEp.

Relative to the logical structure of the events considered for it, the following four situations roughly correspond to sEp:

All the people involved in writing this book are not white.

6. Not all orchids are spread by the wind.

7. All celestial bodies are not static.

8. The solver of Goldbach's conjecture is not illiterate.

This fourth example also represents four different propositions. Example 5 is an extended proposition, and its true value can be determined because the people involved in writing this book can list them one by one. The sixth example, orchids can't be listed one by one. So far, although orchids are limited, people don't know how many. The "celestial body" of the seventh example is an infinite set; Therefore, in the traditional formal logic outspoken proposition, whether proposition 6 and proposition 7 are false or true ultimately depends on the logical semantics of "all" and "everything", but whether they are "extension conjunction" is still uncertain. Example 8: In traditional logic, because S considers an empty set, its truth and falsehood cannot be determined.

Contemporary formal logic also has definite and reasonable answers to these four propositions. Contemporary formal logic divides sEp into denotative conjunctive proposition and connotative restrictive proposition. According to the logical semantics and empirical content in contemporary formal logic, the above four types of examples can be determined to be true. The type of proposition 5 is:

p(e 1) ∧？ p(e2) ∧…∧？ p(ei) ∧…∧？ p(em)

It belongs to "taking the negation of the M-closed 1 meta-atomic proposition as a conjunct limb", which is referred to as "the extension conjunct proposition" for short, and its sentence pattern is "any S that can be enumerated one by one is not P". The formulas of propositions 6, 7 and 8 are as follows:

s(x)？ P(x) or? (s(x)！ p(x))

It belongs to "closed sufficient condition proposition with Kai 1 meta-atomic proposition and Kai 1 meta-atomic proposition as antecedents and consequences" or "negative proposition with Kai 1 meta-atomic proposition as approximate limbs" in contemporary formal logic. Its natural language sentence pattern is "s must not be p" or "s cannot be p."

Third, about the I proposition.

The sentence pattern of proposition I is "having S is P", and the symbolic expression in traditional logic is slp. Relative to the logical structure of events considered for it, the following five examples roughly correspond to sIp:

9. Some people involved in writing this book are yellow.

10. Some people involved in writing this book are from Guiyang.

1 1. Some tables are heptagonal.

12. Some celestial bodies are still.

13. There is Goldbach conjecture, and the solver is from China.

This represents five different propositions. Number nine and number nine. 10 is only the referential relationship between subject and object is different. In fact, all the people involved in writing this book are yellow people. In fact, only some people involved in writing this book are Guiyang people. In both cases, "people involved in writing this book" are finite sets that can be enumerated one by one. They are all denotative propositions, and their truth values can be determined. For example, no 1 1 and no 12, because tables are finite sets that cannot be enumerated one by one and celestial bodies are infinite sets, whether "some tables are heptagonal" and "some celestial bodies are static" in traditional formal logic is false depends on the logical semantics of "you". If it is extensional, it is still uncertain. In the case of 13, s is an empty word. True or false, traditional formal logic can't judge. .

Contemporary formal logic also has definite and reasonable answers to the above five propositions. Contemporary formal logic divides sIp into denotative disjunction proposition and connotative reduction proposition. According to the logical semantics and empirical content in contemporary formal logic, the 9th instance 10,1,12, 13 can all be confirmed as true. The types of proposition 9 and proposition 10 are:

p(e 1)∨p(E2)∨p(ei)∨p(em)

It belongs to the closed disjunctive life with m closed 1 atomic propositions as disjunctive limbs in contemporary formal logic.

Topic ",referred to as" extension disjunctive proposition "for short, has the sentence pattern of" at least one of the listed S is P ". The formulas of the proposition 1 1, 12, 13 are as follows.

S(x)！ P(x) or? (s(x)？ p(x))

It belongs to the contemporary formal logic "a closed reduction proposition with Kai 1 meta-atomic proposition as its approximate limb" or "a negative proposition with Kai 1 meta-atomic proposition and Kai 1 meta-atomic proposition as its cause and effect closed sufficient condition proposition". Referred to as "connotation approximate proposition" (approximate proposition can also be called "possible proposition"), its sentence pattern is "S can be P" or "S is not necessarily P".

Fourth, about the O proposition.

The sentence pattern of O proposition is "having S is P". In traditional formal logic, the symbolic expression is sOp. For the logical structure of the events it thinks about, the following five situations roughly correspond to sOp:

14. Some people involved in writing this book are not white.

15. Some people involved in writing this book are not from Guiyang.

16. Some tables are not non-heptagonal.

17. Some celestial bodies are not moving.

18. Some Goldbach conjectures cannot be solved by Americans.

This represents five different propositions. The only difference between case 14 and case 15 lies in the different extended thinking relationship between subject and object. In case 14, in fact, all the people involved in writing this book are not white, while in case 15, only some people involved in writing this book are not Guiyang people: the "people involved in writing this book" in these two cases are all limited sets that can be enumerated one by one, and they are all extensions. The case numbers are 16 and no 17. Because the table is a finite set that cannot be enumerated one by one and the celestial body is an infinite set, whether these two cases are false or not depends on the logical semantics of "some" and "some" in traditional formal logic. If it is extensional, it is still uncertain. In case 18, because the subject word is empty, traditional formal logic cannot judge whether it is true or false.

The definite and reasonable solution of contemporary formal logic to the above five kinds of propositions is to divide them into extensional disjunctive propositions and connotative reduction propositions. According to the logical semantics and empirical content in contemporary formal logic, these five types of examples can be determined to be true. The types of propositions 14 and 15 are:

p(e 1) ∨？ p(E2)∩…？ p(ei)∩…？ p(em)

It belongs to "closed disjunctive proposition with the negation of m closed 1 meta-atomic propositions as disjunctive limbs" in contemporary formal logic, which is referred to as "extension disjunctive proposition" for short. Its sentence pattern is "at least one of the listed S is not P". The formulas of the propositions 16, 17 and 18 are:

s (x)！ ? P (x) or? (s (x)？ p(x))

It belongs to contemporary formal logic "Kai 1 Meta-Atomic Proposition and negation of Kai 1 Meta-Atomic Proposition are closed approximate propositions" or "Kai 1 Meta-Atomic Proposition and negation of Kai 1 Meta-Atomic Proposition are closed sufficient and conditional propositions", and its sentence pattern is "S is not necessarily P" or "S is not necessarily P.

It should be noted that the above examples of 18 may not completely conform to the traditional outspoken proposition. Therefore, we can't simply think that every outspoken proposition is the synthesis of the corresponding extension proposition and connotation proposition. As mentioned above, there are still various logical and theoretical problems in the traditional outspoken proposition as a "propositional form".

Third, the difference between the traditional outspoken proposition and the corresponding extension proposition and connotation proposition.

In order to compare the differences between the outspoken proposition and its corresponding extension proposition and connotation proposition in traditional logic, we take the full name affirmative proposition and its corresponding extension conjunction proposition and connotation sufficient condition proposition as examples to illustrate:

First, the traditional sentence pattern of full-name affirmative proposition is "all S are P" and the symbolic expression is "sAp". The elements of the extension set of subject S may not be listed one by one, but whether the extension of S can be an empty set is still controversial. The sentence pattern of the extended conjunction proposition is "every S that can be enumerated one by one is P", and the symbol is expressed as:

p(e 1)∧p(E2)∧…p(ei)∧…p(em)

The extension set of subject s is S=( e 1, e2, …, ei, …, em), m is a definite natural number greater than zero, from e 1 to em, which can be enumerated one by one; The sentence pattern of the proposition with sufficient connotation is "S must be P", and the symbolic expression is s(x)? P (x), where the extension of S is a finite set, an infinite set or an empty set whose elements cannot be enumerated one by one.

Secondly, in the classification of traditional propositions, the traditional full-name affirmative propositions include simple propositions, outspoken (or nature) propositions, full-name propositions and affirmative propositions; However, the extension conjunction proposition and the connotation sufficient condition proposition are both compound propositions, and it doesn't matter whether they are outspoken (nature), full name, proper name and affirmation.

Thirdly, the traditional full-name affirmative proposition contains full-name quantifiers (confused by linguistic quantifiers); However, there are no quantifiers in the extension conjunction proposition and the connotation sufficient condition proposition (this is the result of focusing on the logical structure of the objective world).

Fourth, the traditional full-name affirmative proposition always carries the affirmative copula "yes", regardless of whether the statement uses the word "yes"; Both the extension conjunctive proposition and the connotation sufficient condition proposition reflect the logical structure with n-ary relationship, and there is no conjunctive word in its scope and conjunctions.

Fifth, GAI, the subject of the traditional full-name affirmative proposition, usually thinks that P is not GAI, but it is still controversial; It doesn't matter whether GAI is GAI or not.

Sixth, in the depth of analysis, the traditional formal logic takes 1 meta-noun as the minimum unit (in essence, atomic proposition as the minimum unit) and does not make further analysis; Contemporary formal logic makes a more in-depth analysis of atomic propositions when analyzing the extension conjunction proposition and the connotation sufficient condition proposition, from which it analyzes N-yuan nouns, N-yuan function words, individual argument words and individual words.

Seventh, in the aspect of formalization, the letter "A" in the traditional full-name affirmative proposition expression is only a kind of code with the nature of "abbreviation" (that is, full-name quantifier and affirmative copula), so the formalization is incomplete; The formalization of propositions in contemporary formal logic adopts all artificial symbols and is written according to strict rules, which can reveal the logical structure of propositions and formalize them thoroughly.

Finally, it should be pointed out that the four kinds of extension propositions and the four kinds of connotation propositions whose subjects can be empty but not contradictory all conform to the traditional reasoning formats such as correspondence. Of course, the four extension propositions only require the subject to be a real noun. However, regarding the four connotation propositions, only the subject is required to be self-contradictory, but it can be an empty noun. Obviously, contradictory nouns must be empty, but empty nouns are not necessarily contradictory. For example, so far, the "solver of Goldbach conjecture" is empty, but it is not contradictory. Generally speaking, logic science cannot determine whether a noun is empty; However, it is the unshirkable responsibility of logic science to determine whether the terms are self-contradictory. Taking the four connotative propositions of the noun "solver of Goldbach conjecture" with empty subject as an example, we show that the four connotative propositions with empty subject but not contradictory completely satisfy all traditional reasoning formats. Its logical phalanx is: