6.3. Laws of Identity, Contraposition and De Morgan, Modus Ponens and Modus Tollens, Dilemmas and Clavius's Law

The laws of double negation make it possible 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 not-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 Law of Identity

The law of identity is the principle of constancy, or preservation, of the referential and semantic meanings of propositions (statements) within some known or implied context (in an inference, a proof, a theory) . It is one of the laws of classical logic.

Fig. 1 The law of identity

According to the law of identity, every statement about one and the same object, at one and the same time and in one and the same respect, must be identical to itself, however many times it is reproduced. The law of identity can be expressed by the formula p→p (Fig. 1).

The simplest of all logical laws is, perhaps, the law of identity. It says: if a statement is true, then it is true, "if A, then A". For example, if the Earth rotates, then it rotates, and so on. The pure assertion of identity seems so devoid of content that it is rarely used by anyone.

The ancient Chinese philosopher Confucius taught his pupil: "What you know, consider that you know; what you do not know, consider that you do not know." This is not just a repetition of the same thing: to know something and to know that you know it are not the same.

The law of identity seems extremely simple and obvious. Yet even it has been managed to be interpreted incorrectly. It was claimed, for example, that this law asserts that things always remain unchanged, identical to themselves. This is, of course, a misunderstanding. The law says nothing about changeability or unchangeability. It asserts only that if a thing changes, then it changes, and if it remains one and the same, then it remains the same.

In the process of reasoning, every concept and proposition must be used in one and the same sense. The precondition of this is the possibility of distinguishing and identifying the objects under discussion. . A thought about an object must have a definite, stable content, however many times it is repeated. The most important property of thinking — its definiteness — is expressed by this logical law .

The law of identity was first formulated by Aristotle in the treatise "Metaphysics" as follows:

"...to have not one meaning is to have no meaning; and if words have no meanings, then all possibility of reasoning with one another is lost, and in fact with oneself as well; for it is impossible to think anything if one does not think one thing"

— Aristotle, "Metaphysics"

In formal logic the law of identity is usually expressed by the formula: is , or , where stands for any thought.

Symbolic logic, when constructing propositional calculi, operates with the formulas (read as " implies ") and ≡ (read as " is equivalent to "), where:

• — any statement;
• "" — the sign of implication;
• "≡" — the sign of equivalence.

These formulas correspond to the law of identity.

In predicate logic the law of identity is expressed by the formula , that is, for every it is true that if has the property , then has that property.

Application in Everyday Life

Any acquaintance of ours changes with every year, yet we still distinguish him from other acquaintances and from people unknown to us (there is a possibility of distinguishing), because he retains the main features, which appear as the same throughout the whole life of our acquaintance (there is a possibility of identifying). That is, in accordance with Leibniz's law (which defines the concept of identity) we assert that our acquaintance has changed. However, in accordance with the law of identity we assert that this is one and the same person, since the definition is based on the concept of personality. The law of identity requires that to describe one and the same concept we always use one and the same expression (name). Thus we consider one object (the acquaintance) simultaneously at two different levels of abstraction. The possibility of distinguishing and identifying is determined in accordance with the law of sufficient reason. In this case our sensory perception (see recognition) serves as the sufficient reason.

Application in Formal Logic

By the identity of a thought with itself, formal logic understands the identity of its extension . This means that in place of the logical variable in the formula " is " thoughts of different specific content may be substituted, provided they have one and the same extension. In place of the first in the formula " is " we can substitute the concept "an animal having a soft earlobe", and in place of the second — the concept "an animal having the ability to produce tools" (from the point of view of formal logic, both these thoughts are considered equivalent and indistinguishable, since they have one and the same extension, namely — the features reflected in these concepts pertain only to the class of human beings), and the result is the true proposition "An animal having a soft earlobe is an animal having the ability to produce tools".

Application in Mathematics

In mathematical logic the law of identity is the identically true implication of a logical variable with itself .

In algebra the concept of arithmetic equality of numbers is treated as a special case of the general concept of logical identity. However, there are mathematicians who, in opposition to this view, do not identify the symbol "" found in arithmetic with the symbol of logical identity; they do not consider that equal numbers are necessarily identical, and therefore treat the concept of numerical equality as a specifically arithmetical concept. That is, they hold that the very fact of the presence or absence of a special case of logical identity must be determined within logic.^[10].

Violations of the Law of Identity

When the law of identity is violated involuntarily, through ignorance, logical errors arise that are called paralogisms; but when this law is violated deliberately, with the aim of confusing an interlocutor and proving some false thought to him, errors called sophisms appear .

The following errors are possible when the law of identity is violated:

1. Amphiboly (from Greek ἀμφιβολία — ambiguity, obscurity) — a logical error based on the ambiguity of linguistic expressions. For example: "They rightly say that a tongue will get you to Kyiv. Yesterday I bought a smoked tongue. Now I can boldly go to Kyiv." Another name for this error is "substitution of the thesis".
2. Equivocation (from Latin aequivocatio — equal-voicing, ambiguity) — a logical error in reasoning based on the use of the same word in different meanings. Equivocation is sometimes used as a rhetorical or artistic device. In logic this device is called "substitution of the concept".
3. Logomachy (from Greek λόγος — word and μάχη — battle, fight) — a dispute about words, when in the course of a discussion the participants cannot arrive at a common point of view because they have not clarified the initial concepts.

The Law of Contraposition

"The law of contraposition" is the general name for a group of logical laws that use negation to swap the antecedent and the consequent of a conditional statement.

The law of contraposition is a law of classical logic stating that if some premise A implies some consequence B, then the negation of that consequence (that is, "not B") implies the negation of that premise (that is, "not A").

Like any valid implicative statement, it can also serve as a rule of inference.

As a formula of the algebra of propositions, the law of contraposition has the form . The following similar formulas are also tautologies: , . Substituting arbitrary formulas for also yields tautologies.

The law of contraposition is provable in the propositional calculus, but the formula is not derivable in the intuitionistic propositional calculus, where p, q are propositional variables.

One of these laws, sometimes called the law of simple contraposition, runs as follows: if the first implies the second, then the negation of the second implies 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 not-first, then not-second, then it is true that if second, then 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 a 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 also smoke."

Two more laws of contraposition:

• if it is the case that if A, then not-B, then if B, then not-A, for example: "If a square is not a triangle, then a triangle is not a square";

• if it is true that if not-A, then B, then if not-B, then A; for example: "If what is not obvious is doubtful, then what is not doubtful is obvious."

De Morgan's Laws

De Morgan's laws (De Morgan's rules) are logical rules that connect pairs of logical operations by means of logical negation. They are named after the Scottish mathematician Augustus De Morgan. In brief, they read as follows:

Another law: it is false that A or B if and only if A is false and B is false. For example: "It is false that a student knows arithmetic or knows geometry if and only if he knows neither arithmetic nor geometry."

On the basis of these laws, using negation, the connective "and" can be defined in terms of "or", and vice versa:

"A and B" means "it is false that not-A or not-B",

"A or B" means "it is false that not-A and not-B".

For example: "It is raining and it is snowing" means "It is false that there is no rain or there is no snow"; "Today it is cold or damp" means "It is false that today it is not cold and not damp."

Augustus De Morgan originally noticed that the following relations hold in classical propositional logic:

not (a and b) = (not a) or (not b)

not (a or b) = (not a) and (not b)

In mathematics this looks as follows:

or, alternatively:

In set theory:

or, alternatively:

These rules are also valid for collections of elements (families):

and .

In predicate calculus:

Corollaries:

Using De Morgan's laws, one can express a conjunction through a disjunction and three negations. A disjunction can be expressed similarly:

In the form of a theorem:

If there is a proposition expressed by the operation of logical multiplication of two or more elements, i.e. the operation "and": , then in order to find the inverse of the whole proposition, one must find the inverse of each element and combine them with the operation of logical addition, i.e. the operation "or": . The law works similarly in the opposite direction: .

Applications of De Morgan's Law

De Morgan's laws are used in such important fields as discrete mathematics, electrical engineering, physics and computer science; for example, they are used to optimize digital circuits by replacing some logic gates with others.

Venn diagrams illustrating De Morgan's laws

Representation of De Morgan's rules through logic gates

Modus Ponens and Modus Tollens

A "mood" (modus) in logic is a variety of some general form of reasoning. Below, four closely related moods, already known to medieval logicians, are listed.

Modus ponens ("rule of inference"): if and are derivable formulas, then is also derivable.

Notation: , where are any formulas.

The rule of inference modus ponens, usually called the rule of detachment or the hypothetical syllogism, allows one to pass from the assertion of a conditional statement and the assertion of its antecedent to the assertion of its consequent . For example, metals conduct electric current, zinc is a metal, therefore zinc conducts current. The converse is not always true: nickel and seawater conduct current, but nickel is a metal while seawater is not. In sum, if implies , and is true, then may be either true or false.

Modus ponens is a rule of inference in the propositional calculus. It is a special case of the resolution rule.

Modus ponens, sometimes called the hypothetical syllogism, allows one to pass from the assertion of a conditional statement and the assertion of its antecedent to the assertion of the consequent of that statement:

Here the statements "if A, then B" and "A" are the premises, and the statement "B" is the conclusion.

The horizontal line stands in place of the word "therefore".

Another notation: If A, then B. A. Therefore, B.

Thanks to this mood, from the premise "if A, then B", using the premise "A", we, as it were, detach the conclusion "B". For this reason the mood is sometimes called the "rule of detachment".

For example: If a person has diabetes, he is ill. This person has diabetes. The person is ill.

Reasoning by the rule of detachment proceeds from the assertion of the antecedent of a true conditional statement to the assertion of its consequent. This logically correct movement of thought is sometimes confused with a similar but logically incorrect movement from the assertion of the consequent of a true conditional statement to the assertion of its antecedent.

For example, the following inference is correct:

But an inference that looks similar to it:

If an electrolyte were a metal, it would conduct electric current. An electrolyte conducts electric current.

An electrolyte is a metal, which is logically incorrect. By reasoning according to the latter scheme, one can arrive from true premises at a false conclusion. Against confusing the rule of detachment with this incorrect scheme of reasoning, the following advice warns: it is permissible to reason from the affirmation of the antecedent to the affirmation of the consequent, but not from the affirmation of the consequent to the affirmation of the antecedent.

Modus tollens is reasoning by contradiction (the Latin "modus tollendo tollens" means "the way that denies by denying").

Notation: .

For example, let be "a gold coin" and be "cannot be dented by teeth" ; then modus tollens allows us, from the property "gold coins cannot be dented by teeth", to conclude that if a coin can be dented by teeth, then it is not gold.

Modus tollens is the name given to the following scheme of reasoning:

Here the statements "if A, then B" and "not B" are the premises, and the statement "not A" is the conclusion. Another notation:

If A, then B. Not-B. Therefore, not-A.

By means of this scheme, from the assertion of a conditional statement and the negation of its consequent, one passes to the negation of its antecedent. For example: "If helium is a metal, it is electrically conductive. Helium is not electrically conductive. Therefore, helium is not a metal."

The process of falsification, the establishment of the falsity of a theory or hypothesis as a result of its empirical testing, follows the modus tollens scheme. From the theory T under test, some empirical statement A is derived, i.e. the conditional statement "if T, then A" is established. By means of empirical methods of cognition (observation, measurement or experiment), the proposition A is compared with the actual state of affairs.

It turns out that A is false and the proposition not-A is true. From the premises "if T, then A" and "not-A" it follows that "not-T", i.e. the falsity of the theory T.

Modus tollens is often confused with an outwardly similar inference:

In this last inference, from the assertion of a conditional statement and the negation of its antecedent, one passes to the negation of its consequent, which is a logically incorrect step. Reasoning by such a scheme can lead from true premises to a false conclusion.

For example:

• If clay were a metal, it would be malleable.
• But clay is not a metal.
• It is false that clay is malleable.

All metals are malleable, and if clay were a metal, it too would be malleable. However, clay is not a metal. But it obviously does not follow from this that clay is not malleable. Besides metals, there are other malleable substances, and clay is among them.

Against confusing modus tollens with this incorrect scheme of reasoning, the following advice warns: from the negation of the consequent of a conditional statement one may conclude to the negation of its antecedent, but from the negation of the antecedent to the negation of the consequent one may not.

The Affirming-Denying and Denying-Affirming Moods

The following schemes of reasoning are called the affirming-denying mood:

Either A or B: A

It is false that B

and

Either A or B: B

It is false that A

Another notation:

Either A, or B. A. Therefore, not-B.

Either A, or B. B. Therefore, not-A.

By means of these schemes, from the assertion of two mutually exclusive alternatives and the establishment of which of them holds, one passes to the denial of the second alternative: either the first or the second, but not both together; the first holds; hence the second does not. For example:

Lermontov was born either in Moscow or in Petersburg. He was born in Moscow.

It is false that Lermontov was born in Petersburg.

The connective "either... or" in the affirming-denying mood is exclusive; it means: either the first is true or the second is true, but not both together. The same reasoning with the non-exclusive "or" (the first or the second holds, but it is possible that both hold) is logically incorrect. From true premises it can lead to a false conclusion. For example:

Amundsen was at the South Pole, or Scott was. Amundsen was at the South Pole.

It is false that Scott was there.

Both premises are true: both Amundsen and Scott reached the South Pole, yet the conclusion is false. The following inference is correct:

The first at the South Pole was either Amundsen or Scott. The first at this pole was Amundsen.

It is false that Scott was the first there.

In the denying-affirming mood (modus tollendo ponens) the minor premise denies one disjunct, and the conclusion affirms the other. E.g.: Bonds may be bearer bonds (p) or registered bonds (q). This bond is not a bearer bond (⌉p). This bond is a registered bond (q).

Scheme of the denying-affirming mood:

The affirmative conclusion is obtained by means of negation: by denying one disjunct, we affirm the other.

The conclusion in this mood is always certain if the following rule is observed: the major premise must list all possible propositions, the disjuncts; in other words, the major premise must be a complete (closed) disjunctive statement.

The denying-affirming mood is a disjunctive-categorical inference: the first or the second; not the first; hence the second. The first premise is a statement with "or"; the second is a categorical statement denying one of the members of the first compound statement; the conclusion is the second member of that statement.

A or B: not A

B

or

A or B; not B

A

Another form of notation:

A or B. Not-A. Therefore, B.

A or B. Not-B. Therefore, A.

For example:

A set is finite or it is infinite. The set is not finite.

The set is infinite.

Medieval logicians called the affirming-denying mood modus ponendo tollens, and the denying-affirming mood modus tollendo ponens.

Constructive and Destructive Dilemmas

Dilemmas are arguments whose premises are at least two conditional statements (statements with "if... then") and one disjunctive statement (a statement with "or").

A dilemma is a conditional-categorical inference in which one premise consists of two or more conditional propositions, and the other is a disjunctive proposition, the disjunctive proposition containing two members.

The simple constructive dilemma consists of two premises. The first asserts that one consequent follows from two different antecedents. The second asserts that one or the other of these antecedents is true. The conclusion asserts the consequent. "If I cross by the bridge, I will be noticed. If I cross by the ford, I will be noticed. I can cross by the bridge or by the ford. I will be noticed."

The complex constructive dilemma. There are two antecedents in the first premise; the second premise asserts the truth of one or the other antecedent; the conclusion asserts the truth of one or the other consequent. "If I set off a bomb in the city, I will kill many people. If I set off a bomb in the forest, I will kill only myself. I can set off a bomb in the city or in the forest. I may kill many people or I may kill only myself." The simple destructive dilemma: the first, conditional premise indicates that two different consequents follow from one and the same antecedent, the second premise is the negation of both of these consequents, and the conclusion negates the antecedent. "If a person has tetanus, he will die in one day. After one day the person has not died. This person does not have tetanus."

The complex destructive dilemma: the first premise consists of two conditional propositions with different antecedents and different consequents, and the second premise is the negation of both consequents. "If Petrov is honest, he will do the assignment today, and if Petrov is conscientious, he will do the assignment tomorrow. But Petrov did not do the assignment today and did not do it tomorrow. Petrov is neither honest nor conscientious.

The following varieties of dilemma are distinguished.

The simple constructive (affirming) dilemma:

If A, then C.

If B, then C.

A or B.

C

Reasoning of this type is customarily called proof by cases in mathematics. However, the number of cases examined in turn in a mathematical proof usually exceeds two, so the dilemma takes the form:

If the first assumption were valid, the theorem would be true; if the second assumption were valid, the theorem would also be true; if the third assumption is correct, the theorem is true; if the fourth assumption is correct, the theorem is true; either the first, or the second, or the third, or the fourth assumption is valid.

Therefore, the theorem is true.

The complex constructive dilemma:

If A, then B. If C, then D.

A or C.

B or D.

For example: "If it rains, we will go to the cinema; if it is cold, we will go to the theater; it will rain or it will be cold; therefore, we will go to the cinema or go to the theater."

The simple destructive (denying) dilemma:

If A, then B. If A, then C.

It is false that B or it is false that C. It is false that A.

For example: "If a number is divisible by 6, then it is divisible by 3; if a number is divisible by 6, then it is divisible by 2; the number under consideration is not divisible by 2 or not divisible by 3; therefore, the number is not divisible by 6."

The complex destructive dilemma:

If A, then B. If C, then D. Not-B or not-D. Not-A or not-C.

For example: "If I travel north, I will end up in Tver; if I travel south, I will end up in Tula; but I will not be in Tver or I will not be in Tula; therefore, I will not travel north or I will not travel south."

Clavius's Law

This law can be stated as follows: if a statement follows from its own negation, then it is true. Or, more briefly: a statement that follows from its own negation is true.

If it is false that A, then A.

A.

For example: if the condition for a machine not to work is its working, then the machine works.

The law is named after Clavius, a Jesuit scholar of the sixteenth century, one of the creators of the Gregorian calendar. Clavius drew attention to this law in his commentary on Euclid's Elements. Euclid proved one of his theorems from the assumption that it is false.

Clavius's law underlies a recommendation concerning proof: if you want to prove A, derive A from the assumption that not-A is true. For example, suppose we need to prove the statement "A trapezoid has four sides." The negation of this statement is: "It is false that a trapezoid has four sides." If we manage to derive the statement from this negation, the statement is true.

In I. S. Turgenev's novel "Rudin" there is the following dialogue:

— So, in your opinion, there are no convictions?

— No, and there are none.

— Is that your conviction? — Yes.

— How can you say there are none? Here is one for you already, to start with.

The erroneous opinion that there are no convictions is opposed by its negation: there is at least one conviction, namely the conviction that there are no convictions. It follows that convictions exist.

Another law that fits the same general scheme is close in logical structure to Clavius's law: if a statement's negation follows from the statement itself, then the negation is true. For example, if the condition for a train to arrive on time is its being late, then the train will be late.

The scheme of this reasoning:

If A, then not-A.

Not-A.

This scheme was once used by the ancient Greek philosopher Democritus in a dispute with the sophist Protagoras. The latter asserted:

"Everything that occurs to anyone is true."

To this Democritus replied that from the thesis

"Every statement is true" the truth of its negation also follows: "Not all statements are true."

And so it is this negation, not Protagoras's thesis, that is actually true.

Tests on the Law of Identity:

1. Relying on the law of identity, determine in which case the identity of the propositions is preserved if the highlighted concept is replaced with another: The criminal, fleeing pursuit, turned into a deserted alley.

• 1. blind;
• 2. dark;
• 3. empty;
• 4. narrow.

2. The law of contradiction is violated in the following statement:

• 1. I know only that I know nothing. (Socrates)
• 2. In my childhood I had no childhood. (A. P. Chekhov)
• 3. History teaches only that it teaches nothing. (G. W. F. Hegel)
• 4. In none of these statements.

The law of contradiction states: two statements that stand in the relation of negation cannot both be true at the same time; at least one of them is false. For the law of contradiction to apply, one must reason about one and the same object, at the same time, and in the same respect. The law of contradiction can be expressed by the formula ¬(p∧¬p) (fig. 2)

Fig. 2 The law of non-contradiction

3. In this humorous quatrain

We were walking down Neglinnaya,

We stopped by the boulevard,

They bought us a blue-blue,

Green-ish, red balloon.

(S.V. Mikhalkov)

the following law is deliberately violated

• 1. identity;
• 2. non-contradiction;
• 3. excluded middle;
• 4. sufficient reason.

4. Two contrary propositions about two different objects

• 1. must be true at the same time;
• 2. must be false at the same time;
• 3. must be such that one is true and the other is false;
• 4. may have any truth values whatsoever.

5. Two contradictory propositions about one object must be

• 1. true at the same time;
• 2. false at the same time;
• 3. one true and the other false;
• 4. false at the same time.

6. Relying on the law of non-contradiction and the law of excluded middle, determine which of the pairs of propositions can be false at the same time.

• 1. All children are disobedient. Some children are obedient after all.
• 2. Lewis Carroll is the author of the book "Alice's Adventures in Wonderland". Lewis Carroll is not the author of the book "Alice's Adventures in Wonderland".
• 3. All knowledge is useful. Some knowledge is nevertheless useless.
• 4. Every lie deserves censure. No lie deserves censure.

The law of excluded middle: two mutually contradictory statements cannot be both true or both false at the same time; one of them is true and the other is false, and there is no third possibility. The law of excluded middle is written as a formula as follows: p ∨ ¬p (Fig. 3).

Fig. 3 The law of excluded middle

7. Indicate which of the formal laws of logic is violated in the following reasoning: "This person is not sick, because he does not have a fever."

• 1. the law of identity;
• 2. the law of non-contradiction;
• 3. the law of excluded middle;
• 4. the law of sufficient reason.

Exercises

1. Determine which of the following sentences are statements:

• 1) The sun that has come out from behind the clouds.
• 2) Manuscripts don't burn.
• 3) Who is the author of "War and Peace"?
• 4) No matter how much you feed a wolf, it still longs for the forest.
• 5) It has grown dark.
• 6) The price of commodity X is lower than its value.
• 7) Is it true that Minsk was founded in 1067?
• 8) Who likes trouble? (M. Gorky)
• 9) Drivers, do not violate traffic regulations!
• 10) Atlantis exists.

2. Determine the type of simple statement by the nature of the predicate.

• 1) There is no student who does not experience difficulties in studying logic.
• 2) Anyone can master this course on their own.
• 3) There is no beast more fearsome than the cat.
• 4) Some just actions are advantageous.

3. Write the statements in correct logical form and identify the subject and predicate.

• 1) Among the students there are straight-A students.
• 2) Not all citizens of the Republic of Belarus live in Minsk.
• 3) Most often volcanoes are cone-shaped.
• 4) Some drivers do not smoke while driving.
• 5) There are books that have become widely known.
• 6) In winter people do not pick mushrooms.
• 7) All that glitters is not gold.
• 8) Everything ingenious is simple.
• 9) Ivan has many friends.
• 10) Everything is good in its place.

4. Determine the type of attributive statement by quality and quantity, and identify the subject, predicate, copula and quantifier word.

• 1) He spent all his evenings at home.
• 2) There are no lazy students.
• 3) Many first-year students cannot translate this text without a dictionary.
• 4) Mars revolves around the Sun in a planetary orbit.
• 5) Not all students of our group were admitted to the exam session.
• 6) No one from our group was expelled.
• 7) Some books about Columbus's voyage were in the library.
• 8) This question is considered in most textbooks.
• 9) There are children who love sweets.
• 10) Some agreements are not advantageous to one of the parties.

5. Determine the type of attributive statement by the combined classification, identify the subject and predicate, depict the relations
between the terms by means of circle diagrams, and establish the distribution of the terms in the statement.

• 1) Some people who can read have never studied anywhere.
• 2) Nowadays the seas have turned into a sewer.
• 3) A respectful attitude toward others contributes to success in life.
• 4) Students are sometimes late for classes.
• 5) All ages are subject to love.
• 6) Among the students of BNTU there are future scientists.
• 7) For people engaged in mental work, foods rich in phosphorus (walnuts, peanuts, fish, peas) are beneficial.
• 8) A good blacksmith will shoe even a frog.
• 9) Many students do not live in a dormitory.
• 10) Birds change their plumage every year.

6. Come up with your own examples of universal affirmative, particular affirmative, universal negative and particular negative statements.
7. Compose statements with the given subject and predicate in accordance with the following conditions of distribution of terms:
a) S- P+; b) S+ P+; c) S- P-; d) S+ P-, and say which of the resulting statements are true and which are false:

• 1) S – predator, P – crocodile.
• 2) S – books, P – the best gift.
• 3) S – rhombus, P – equilateral rectangle.
• 4) S – believer, P – Buddhist.

8. What logical connectives are expressed by the following grammatical conjunctions:

• 1) Whoever thinks clearly expresses himself clearly.
• 2) It is not true that Ivanov studied neither at a technical school nor at a university.
• 3) Disabled people and veterans of the Great Patriotic War are received out of turn.
• 4) "War and Peace" was written by L. Tolstoy or F. Dostoevsky.
• 5) I would never have dared to do it had he not been beside me.
• 6) Both lawyers and journalists study logic.
• 7) Only one of the three of them knew about this.
• 8) He read or heard a lot about extraterrestrials.
• 9) He will not get into university, unless he prepares very diligently.
• 10) One of the two knows the other.

9. Translate the compound statements into the language of logic.

• 1) A crisis is inevitable, unless extraordinary political or economic measures are taken.
• 2) People grow rich not by income but by spending.
• 3) This person is a knight, provided he is not lying.
• 4) Passing the exam successfully depends on the quality of independent preparation, work in lectures and the mood of the instructor.
• 5) It is not true that the wind blows if and only if there is no rain.
• 6) Go left and you will lose your horse, and if you don't go, you will perish yourself.

10. Formalize the following reasoning.

• 1) Even if others have harmed you, greet them with a smile when you meet. Out of shame they will lose their resolve or else will ask for forgiveness.
• (E.Kh. Galshiev. The Mirror of Wisdom)
• 2) One who conquers others possesses strength. One who conquers himself becomes strong. (Lao Tzu. Tao Te Ching.)
• 3) Whoever wants to do something finds the means. Whoever does not want to do anything finds an excuse.
• 4) If you grow proud after learning a little arithmetic, you will become an object of ridicule for the wise. (E.Kh. Galshiev. The Mirror of Wisdom.)
• 5) It is reasonable to behave as though another life unconditionally awaits us, and upon entering it the moral
• state in which we ended the present one will be taken into account. (I. Kant)
• 6) In childhood we live by norms that are not merely learned but appropriated, since, although they were not devised by us, we consider them
• our own. (N. Kryshuk)
• 7) ...Ponder the words of those who have upset you; cherish the words of those who have given you hope. (Kh. van Zaichik)
• 8) Never act with unconscious rudeness or your own insolence. If you behave like a maddened elephant, you
• will harm either yourself or others. (E.Kh. Galshiev. The Mirror of Wisdom.)

11. Construct truth tables for the following formulas and determine the type of each formula (tautology, contradiction, contingent):

• 1) (p∧q)↔(¬p∨¬q);
• 2) ((p→q)∧(p→r))→((¬q∨¬r)→¬p);
• 3) (p→q)∨(q→p);
• 4) (¬p→q)↔(¬q→(p∧r));
• 5) (((p→q)→p)→p);
• 6) (¬p∨q)∧(p∨r).

12. Transform the following simple statements with external negation into statements without external negation.

• 1) It is not true that in no branch of industry are there unprofitable enterprises.
• 2) It is not true that there are no incurious children.
• 3) Not all children are curious.
• 4) It is not true that in no library are there books that are consulted very rarely.
• 5) Not every state does without an army.
• 6) Not every student has to retake exams.

13. Statements are given. Relying on the logical square, derive the subaltern, subcontrary and contradictory statements of the original one.
Determine the truth values of the derived statements, if by hypothesis the original statement is true.

• 1) Not all contemporaries of the dinosaurs died out.
• 2) Some students passed the exam session early.
• 3) Many volcanoes are cone-shaped.
• 4) Some doctors are not surgeons.

14. A pair of statements is given.

• 1) Among categorical propositions there are affirmative ones. No categorical proposition is affirmative.
• 2) All adults were once children. Some adults were once children.
• 3) The books of B. Akhmadulina are widely known. All books of B. Akhmadulina are not widely known.
• 4) There are people who have the right to their own point of view. Every person has the right to their own point of view.

Answer the following questions:

• - What is the logical relation between them?
• - What can be said about the truth value of the first statement if the second statement is false?
• - What can be said about the truth value of the second statement if the first is true by hypothesis?
• - What can be said about the truth value of the second statement if the first is false?

15. Check the correctness of the reasoning based on the "logical square", indicate in which examples errors have been made and
what they consist in.

• 1) It is true that some students passed the exam session successfully, therefore it is true that all students passed the exam session successfully.
• 2) It is false that all people are immortal, therefore it is true that some people are immortal.
• 3) It is true that some mushrooms are edible, therefore it is true that all mushrooms are edible.
• 4) It is false that no outstanding mathematician accepted non-Euclidean geometry, therefore it is true that many outstanding
• mathematicians did not accept non-Euclidean geometry.

16. Are the following pairs of statements equivalent (if not, establish the type of relation between their logical forms)?

• 1) If you are afraid, don't do it. If you are doing it, don't be afraid.
• 2) Whoever is not with us is against us. Either someone is with us. Or someone is against us.

17. Find, among the listed compound statements, the contradictory and the equivalent ones.

• 1) Ivanov knows Petrov, but Petrov does not know Ivanov.
• 2) Ivanov and Petrov do not know each other.
• 3) It is not true that Ivanov and Petrov know each other.
• 4) Whereas Petrov knows Ivanov, Ivanov does not know Petrov.
• 5) If Ivanov knows Petrov, then Petrov knows Ivanov.
• 6) It is not true that Petrov knows Ivanov only when Ivanov knows Petrov.
• 7) It is not true that Ivanov knows Petrov or Petrov knows Ivanov.
• 8) Ivanov does not know Petrov or Petrov does not know Ivanov.
• 9) If Ivanov knows Petrov, then Petrov does not know Ivanov.
• 10) It is not true that Ivanov and Petrov do not know each other.
• 11) Ivanov and Petrov know each other.
• 12) Ivanov knows Petrov only on the condition that Petrov knows Ivanov.
• 13) Only one of the two knows the other.

18. Using truth tables, determine whether the following inferences correspond to laws of logic:

• 1) If an electric current flows through a conductor, then a magnetic field forms around the conductor, but no magnetic field forms around the conductor. Therefore, no electric current flows through the conductor.
• 2) If an electric current flows through a conductor, then a magnetic field forms around the conductor, but a magnetic field forms around the conductor. Therefore, an electric current flows through the conductor.
• 3) If an electric current flows through a conductor, then a magnetic field forms around the conductor, but no electric current flows through the conductor. Therefore, no magnetic field forms around the conductor.

19. Which of the basic laws of thought is violated?

• 1) Once, before a battle, the ancient Romans heard a raven croaking on the left side and won the battle; another time they heard the raven croaking on the right side and lost the battle. The matter is clear, the Romans decided: the croaking of a raven on the right side brings doom to the army, while the croaking of a raven on the left side gives it strength.
• 2) Sylvia's appearance was remarkable, one might even say beautiful, if one forgot about the ugly paralyzed legs in braces, the powerful shoulders and the large masculine hands, overdeveloped from constant use of crutches.
• 3) In 1907 the Cadet faction in the Duma resolved, on the question of its attitude toward the government, to express neither confidence nor no confidence in it. Moreover, if there is a resolution of confidence in the government, they were to vote against it. And if there is a resolution of no confidence in the government, they were to vote against it.
• 4) Sasha joyfully tells his mother, who has returned from work, that he has done all the homework assigned for today. But suddenly his little sister Irinka says: "Sasha lied about everything, he told me himself that they were not assigned anything for today." Could Sasha have told the truth to his sister and his mother?
• 5) In one of the articles by young scholars devoted to Eastern medicine, the thought flashed by that Western medicine has outlived itself.
• 6) One late evening someone knocked at a yard and asked: "Do you need firewood?" "No." ...The next morning the owners did not find their firewood in the yard.

20. Jokes are based on violations of the basic laws of logic.
For example:
Students are deciding what to do on the eve of an exam. They decide to toss a coin.

• - If it comes up heads, we'll go to the disco.
• - If it comes up tails, we'll play computer games.
• - If it lands on its edge, we'll write cheat sheets.
• - If it hangs in the air, we'll study.

The requirements of which law are violated in this joke? If you can, give your own examples

See also

• [[b5641]]
• [[b5642]]
• [[b5643]]
• [[b5644]]

• The law of contradiction
• The law of excluded middle
• The law of sufficient reason
• The relation of incompatibility
• The relation of compatibility
• Logical square
• Compound statement
• Negation statements
• Attributive statement
• Statement

• Deductive inference
• Propositional calculus

• Purely conditional inference

• Inclusion-exclusion formula
• Venn diagram

• Latin logical expressions
• Syllogism
• Induction
• Deduction
• The law of non-contradiction
• Argumentation theory

See also

• 6.1 Laws of logic. The law of contradiction (Formulation Imaginary contradictions Implicit contradictions Multiple contradiction problems)
• 6.2 Law of the excluded middle, Some applications of the law, Doubts about the universality of the law, Criticism of the law by Brauer
• 6.3. Laws of Identity, Contraposition and De Morgan, Modus Ponens and Modus Tollens, Dilemmas and Clavius's Law
• 6.4 On the so-called fundamental laws of logic. The interpretation of logical laws in traditional logic. The laws of logic as elements of a logical system