Lecture
The laws of double negation allow one to remove and introduce such a negation. They can be expressed as follows: if it is not true that not-A, then A; if A, then it is not true that non-A. For example: “If it is not true that Aristotle did not know the law of double negation, then Aristotle knew this law” and vice versa.
The simplest of all logical laws is, perhaps, the law of identity. He says: if the statement is true, then it is true, "if A, then A". For example, if the Earth rotates, then it rotates, etc. The pure statement of identity seems so empty that it is rarely used by anyone.
The ancient Chinese philosopher Confucius instructed his student: "What you know, consider what you know, what you don't know, consider what you don't know." It’s not just a repetition of the same thing: to know something and to know that you know it is not the same thing.
The law of identity seems supremely simple and obvious. However, his contrivances were misinterpreted. It was declared, for example, that this law states that things always remain the same, identical with themselves. This, of course, is a misunderstanding. The law says nothing about variability or immutability. He asserts only that if a thing changes, then it changes, and if it remains the same, then it remains the same.
"The law of contraposition" is a common name for a number of logical laws that allow, by means of negation, interchange the basis and consequence of a conditional utterance.
One of these laws, sometimes called the law of simple contraposition, is: if the first entails the second, then the negation of the second entails the negation of the first.
For example: "If it is true that a number divisible by six is divisible by three, then it is true that a number not divisible by three is not divisible by six."
Another law of contraposition says: if it is true that if the non-first, then the non-second, then it is true that if the second, then the first.
For example: "If it is true that a manuscript that has not received a positive review is not published, then it is true that the published manuscript has a positive review." Or another example: “If there is no smoke, when there is no fire, then if there is fire, there is smoke”.
Two more laws of contraposition:
• if it is the case that if A is then non-B, then if B, then non-A, for example: “If the square is not a triangle, then the triangle is not a square”;
• if it is true that if not-A, then B, then if not-B, then A; for example: "If the non-obvious is doubtful, then the non-doubt is obvious."
The name of the English logic of the XIX century. A. de Morgana are called logical laws, which connect by means of negation statements formed by using the unions "and" or "or". One of these laws can be expressed as: the negation of the statement “L and B” is equivalent to the statement “not-A or not-B”.
For example: “It is not true that tomorrow it will be cold and tomorrow it will be rainy, if and only if tomorrow is not cold or tomorrow it will not be rainy.”
Another law: it is wrong that Li B is if and only if L is wrong and V. is wrong . For example: “It is not true that the student knows arithmetic or knows geometry, if and only if he does not know either arithmetic or geometry.”
On the basis of these laws, using negation, the bundle “and” can be defined through “or”, and vice versa:
"A and B" means "wrong, that is not-A or not-B",
“A and B” means “wrong, that is not-A and not-B”.
For example: “It is raining and snowing” means “It is not true that there is no rain or no snow”; "Today is cold or damp" means "It is not true that today is not cold and not wet."
The “mode” in logic is a kind of some general form of reasoning. Next will be listed four close to each other modes, known to medieval logics.
Modus ponens, sometimes called a hypothetical syllogism, allows one to proceed from the statement of the conditional statement and the statement of its basis to the statement of the consequence of this statement:
If A, then B: A B
Here the statements “if A, then B” and “A” are premises, saying “B” conclusion. The horizontal bar is in place of the word "therefore." Another entry:
If A, then V. A. Consequently, B.
Thanks to this modus, from the parcel “if A, then B”, using the parcel “A”, we sort of separate the conclusion “B”. For this reason, this modus is sometimes called the “separation rule”. For example:
If a person has diabetes, he is sick. The person has diabetes.
The man is sick.
The reasoning by the rule of separation goes from the statement of the basis of a true conditional statement to the statement of its effect. This logically correct movement of thought is sometimes confused with its similar, but logically incorrect movement from the statement of the consequence of a true conditional statement to the statement of its basis. For example, the inference is correct:
If thallium is metal, it conducts electrical current. Thallium is metal.
Thallium conducts electrical current.
But outwardly similar reasoning with him:
If the electrolyte were metal, it would conduct an electric current.
Electrolyte conducts electrical current.
Electrolyte - metal is logically incorrect. Reasoning according to the last scheme, one can come from true premises to a false conclusion. Against confusing the rule of separation with this incorrect scheme of reasoning, the advice is cautioned: it is permissible to argue from the confirmation of the basis to the confirmation of the investigation, but not from the confirmation of the investigation to the confirmation of the basis.
Modus Tollens is the following reasoning scheme:
If L. then Q: wrong In Wrong L
Here, the statements “if A is, then B” and “incorrectly B” are premises, and the statement “incorrectly A” is a conclusion. Another entry:
If A, then B. Not-in. Therefore, non-A.
Through this scheme, from the statement of the conditional utterance and the negation of its effect, a transition is made to the negation of the base. For example: “If helium is metal, it is electrically conductive. Helium is non-conductive. Therefore, helium is not metal. ”
According to the modus tollens scheme, the process of falsification, the establishment of a falsity of a theory or hypothesis as a result of its empirical testing, is underway. Some empirical statement L is derived from the tested theory T , i.e. establishes the conditional statement "if T, then L". Through empirical methods of knowledge (observation, measurement, or experiment), sentence L is compared with the real state of affairs. It turns out that L is false and true non-A sentence . From the packages “if Tu is A” and “not-A” follows “not-T”, i.e. falsity of theory 7 ".
The outwardly similar inference is often mixed with the modus tollens:
I <•. 1i.-1. then />; wrong l wrong in
In the last inference from the statement of the conditional utterance and the negation of its foundation, a transition is made to the negation of its effect, which is a logically incorrect step. Reasoning according to such a scheme can lead from true premises to a false conclusion. For example:
If clay were metal, it would be plastic. [1o clay is not metal.
It is not true that clay is plastic.
All metals are plastic, and if the clay were metal, it would also be plastic. However, clay is not a metal. But from this it obviously does not follow that the clay is not plastic. In addition to metals, there are other plastic substances, and clay among them.
Against confusing the mode of tollens with this incorrect scheme of reasoning, the advice is cautioned: it is possible to conclude from denying the effect of a conditional statement to denying the basis of this statement, but not from denying the basis to denying the effect.
Affirmatively denying modus are the following reasoning schemes:
Either A. or B: A Wrong B
and
Either A. or B; In Wrong A
Another entry:
Either A or B. A. Consequently, non-B.
Either A or V.V. Therefore, non-A.
Through these schemes, from asserting two mutually exclusive alternatives and establishing which of them takes place, a transition is made to the negation of the second alternative: either the first or the second, but not both; there is the first; hence no second. For example:
Lermontov was born in Moscow or in St. Petersburg. He was born in Moscow.
It is not true that Lermontov was born in Petersburg.
A bundle of “either or”, which is part of the approvingly denying mode, is exclusive, it means: the first is true or the second is true, but not both together. The same reasoning, but with a non-exclusive “or” (the first or second holds, but the first and second are possible) is logically wrong. From true premises it can lead to a false conclusion. For example:
Amundsen was at the South Pole or was Scott. Pa South Pole was Amundsen.
It is not true that Scott was there.
Both premises are true: both Amundsen and Scott reached the South Pole, the conclusion is false. The correct conclusion is:
At the South Pole first was Amundsen or Scott. At this pole, Amundsen was the first.
It is not true that Scott was there first.
The negative-affirmative mode is called the separation-categorical conclusion: the first or the second; not the first; means the second. The first premise is a statement with “or”; the second is a categorical statement denying one of the members of the first complex statement; the conclusion is the second member of this statement.
A or B: incorrect A B
or
A or B; wrong in a
Other form of recording:
A or B. Non-A. Therefore, V.
A or B. Non-B. Therefore, A.
For example:
The set is finite or it is infinite. The set is not finite.
The set is infinite.
Medieval logicians called affirmative-denying modus modus pondo tollens, and denying-affirming modus - modus tollendo ponens.
Dilemmas are arguments, the premises of which are at least two conditional statements (statements with “if, then”) and one separative statement (statement with “or”).
There are the following types of dilemmas.
Simple constructive (approving) dilemma:
If A, then C.
If B, then C.
A or B.
WITH
For example: “If I read Detective Agatha Christie, then I will have a good evening; if I read Detective Georges Simenon, I will also have a good evening; read Detective Christie or read Detective Simenon; so I will have a good evening. ”
The reasoning of this type in mathematics is usually called evidence of cases. However, the number of cases, sorted sequentially in mathematical proof, usually exceeds two, so that the dilemma takes the form:
If the first assumption were true, the theorem would be true; with the validity of the second assumption, the theorem would also be true; under the correct third assumption, the theorem is true; if the fourth assumption is true, the theorem is true; either the first, or the second, or the third, or the fourth assumption holds true.
Hence, the theorem is true. Difficult constructive dilemma:
If A, then B. If C, then O. Lish C. B or O.
For example: “If it rains, we will go to the movies; if it's cold, let's go to the theater; it will rain or be cold; consequently, we will go to the cinema or go to the theater. ” Simple destructive (denying) dilemma:
If A, then B. If A, then C.
Wrong B or Wrong C. Wrong A.
For example: “If a number is divisible by 6, then it is divisible by 3; if the number is divisible by b, then it is divisible by 2; the number in question is not divisible by 2 or not divisible by pa 3; therefore, the number is not divisible by 6 ".
Difficult destructive dilemma:
If A, then B. If C, then D. Non-B or non-P. Not-A or Not-C.
For example: “If I go north, I will go to Tver; if I go south, I will get to Tula; but I will not be in Tver or I will not be in Tula; consequently, I will not go to the north or go to the south. "
This law can be transmitted as follows: if the statement itself derives from the negation of a certain statement, then it is true. Or, in short: a statement arising from its own negation is true.
If it is incorrect that A. then A. A.
For example: if the condition that the machine does not work, is its work, then the machine works.
The law is named after Claudius, a Jesuit scholar who lived in the 16th century, one of the creators of the Gregorian calendar. Clavius drew attention to this law in his commentary on the Euclidean Principles. One of his theorems, Euclid, proved from the assumption that it is false.
Clavius's law is at the heart of the recommendation regarding evidence: if you want to prove A, deduce A from the assumption that non-A is true . For example, you need to prove the statement "Trapezium has four sides." Denial of this statement: "It is not true that the trapezium has four sides." If one succeeds in deriving a statement from this negation, then the latter will be true.
In the novel by I. S. Turgenev “Rudin” there is such a dialogue:
- So, in your opinion, beliefs pet?
- No - and does not exist.
- Is that your belief? -Yes.
- How do you say that they are not? Here you have one for the first time.
The erroneous opinion that there are no convictions is opposed to its denial: there is at least one conviction, namely the conviction that there are no convictions. It follows that beliefs exist.
The law of Clavius is similar in its logical structure to another law that corresponds to the same general scheme: if its negation follows from the statement, then the latter is true. For example, the network condition that the train will arrive on time, will be his late, then the train will be late. The scheme of this reasoning is as follows:
If A. then non-A. Not-A.
This scheme was once used by the ancient Greek philosopher Democritus in a dispute with the sophist Protagor. The latter asserted: "Truly everything that comes into my head." To this, Democritus replied that the statement “Every statement is true” implies truth and its negation: “Not all statements are true.” And this means that this negation, and not the position of Protagoras, is in fact true.
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Logics
Terms: Logics