→ Syllogisms formed on the basis of a simple categorical syllogism. Enthymemes and enthymematic sayings

Syllogisms formed on the basis of a simple categorical syllogism. Enthymemes and enthymematic sayings

Complex inferences are those that consist of two or more simple inferences. Most often, complex inferences of this kind, or, as they are also called in logic, chains of inferences, are used in proofs. Let us consider such types of complex inferences as: a) polysyllogism; b) litters; c) epicheyrema.

A polysyllogism is a chain, a chain of syllogisms connected in such a way that the conclusion of the previous syllogism (prasyllogism) becomes one of the premises of the subsequent syllogism (episyllogism).

For example:

No one capable of self-sacrifice is an egoist.

All generous people are capable of self-sacrifice.

No one is generous, not an egoist.

All cowards are selfish.

No coward is generous.

Depending on which premise - greater or lesser - of the episyllogism becomes the conclusion of the prasyllogism, progressive and regressive chains of syllogisms are distinguished, respectively.

The example we have given is a progressive chain of syllogisms. In it our thought moves from the more general to the less general.

Another example of a progressive chain of syllogisms.

All vertebrates have red blood.

All mammals are vertebrates.

All mammals have red blood.

All carnivores are mammals.

All carnivores have red blood.

Tigers are predatory animals.

Tigers have red blood.

In a regressive chain of syllogisms, the conclusion of the prasyllogism becomes the lesser premise of the episyllogism. In such polysyllogism, thought moves from less general to increasingly general knowledge.

For example:

Vertebrates are animals.

Tigers are vertebrates.

Tigers are animals.

Animals are organisms.

Tigers are animals.

Tigers are organisms.

Organisms are destroyed.

Tigers are organisms.

Tigers are destroyed.

In order to check the logical consistency of a pollysyllogism, it is necessary to break it down into simple categorical syllogisms and check the consistency of each of them.

Sorites (from the Greek “heap”) is a complex abbreviated syllogism in which only the last conclusion from a series of premises is given, and intermediate conclusions are not explicitly formulated, but only implied.

Sorites is built according to the following scheme;

All A's are B's.

All B's are C's.

All C's are D's.

Therefore, all A's are D's.

As you can see, the conclusion of the prasyllogism is missing here: “All A is C,” which should also act as a greater premise of the second syllogism - episyllogism.

For example:

Socially dangerous acts are immoral.

A crime is a significantly dangerous act.

Theft is a crime.

Stealing is immoral.

Here the conclusion of the first syllogism (prasyllogism) is missing - “Crime is immoral”, which is the second, lesser premise of the second syllogism (episyllogism). This episyllogism in its entirety would look like this:

Crime is immoral.

Theft is a crime.

Stealing is immoral.

There are two types of sorites - Aristotelian and Hoklenian. They received their name from the authors who first described them.

Aristotle described a sorites in which the conclusion of a prasyllogism is omitted, becoming the lesser premise of an episyllogism:

A horse is a quadruped.

Bucephalus is a horse.

A four-legged animal.

An animal is a substance.

Bucephalus is a substance.

In its full form, this polysyllogism will look like this:

A horse is a four-legged animal.

Bucephalus is a horse.

Bucephalus is a quadruped.

A four-legged animal.

Bucephalus is a quadruped.

Bucephalus is an animal.

An animal is a substance.

Bucephalus is an animal.

Bucephalus is a substance.

Goklenius (prof.. University of Marburg, lived 1547-1628) wrote a description of sorites, in which the conclusion of the prasyllogism is omitted, becoming the first, greater premise of the episyllogism. He cited the following litter:

An animal is a substance.

A four-legged animal.

A horse is a four-legged animal.

Bucephalus horse.

Bucephalus is a substance.

In its full form, this polysyllogism looks like this:

1. An animal is a substance.

A four-legged animal.

Quadruped is a substance.

2. Quadruped - substance.

A horse is a four-legged animal.

A horse is a substance.

3. Horse substance.

Bucephalus is a horse.

Bucephalus is a substance.

Epicheyrema (translated from Greek “attack”, “laying on of hands”) is a syllogism in which each of the premises is an enthymeme.

For example:

All students at the Institute of International Relations study logic because they must think correctly.

We, students of the Institute of International Relations, because we study at this institute.

That's why we deal with logic.

It can be seen that each of the premises of this epicheireme is an abbreviated syllogism - an enthymeme. Thus, the first premise in its entirety will be the following syllogism:

Everyone who should think correctly is engaged in logic.

Everyone, students of the Institute of International Relations must think correctly.

All students at the Institute of International Relations study logic.

We leave the restoration of the second premise to a complete syllogism and the entire chain of syllogisms to the reader.

Epicheyrema We use it quite often in the practice of thinking and in oratory. The Russian logician A. Svetilin noted that epicheyrema is convenient in oratory because it makes it possible to more conveniently arrange a complex conclusion according to its component parts and makes them easily visible, and, consequently, the whole reasoning more conclusive.

Exercise

Determine the type of inference and check its consistency

A. 3 is an odd number.

All odd numbers are natural numbers.

All natural numbers are rational numbers.

All rational numbers are real numbers.

Therefore, 3 is a real number.

B. Everything that improves health is useful.

Sport improves health.

Athletics is a sport.

Running is a type of athletics.

Running is good for you.

B. All organisms are bodies.

All plants are organisms.

All bodies have weight.

All plants are bodies.

All plants have weight.

D. Noble work deserves respect because noble work contributes to the progress of society.

The work of a lawyer is a noble work, as it consists of protecting the legal rights and freedoms of citizens.

Therefore, the work of a lawyer deserves respect.

D, What is good must be desired.

What must be desired must be approved.

And what must be approved is commendable.

Therefore, whatever is good is praiseworthy.

(Example of M.V. Lomonosov)

In the process of reasoning, simple syllogisms appear in logical connection with each other, forming a chain of syllogisms in which the conclusion of the previous syllogism becomes the premise of the subsequent one. The preceding syllogism is called proslogism, the subsequent one is called episyllogism.

A combination of simple syllogisms in which the conclusion of a previous syllogism (prosyllogism) becomes the premise of a subsequent syllogism (episyllogism) is called a complex syllogism, or polysyllogism.

There are progressive and regressive polysyllogisms.

In progressive polysyllogism, the conclusion of the prosyllogism becomes the greater premise of the episyllogism.

For example:

A socially dangerous act (A) is punishable (B) Crime (C) is a socially dangerous act (A)

Crime (C) is punishable (B) Giving a bribe (D) is a crime (C)

Giving a bribe (D) is punishable (B)

In regressive polysyllogism, the conclusion of the prosyllogism becomes the lesser premise of the episyllogism. For example:

Crimes in the economic sphere (A) - socially dangerous acts (B)

Illegal entrepreneurship (C) - a crime in the economic sphere (A)

Illegal entrepreneurship (C) is a socially dangerous act (B)

Socially dangerous acts (B) are punishable (D) Illegal entrepreneurship (C) is a socially dangerous act (B)

Illegal business (C) is punishable (D)

Both examples given are a combination of two simple categorical syllogisms, constructed according to the AAA mode of the 1st figure. However, a polysyllogism can be a combination of a larger number of simple syllogisms, constructed according to different modes of different figures. A chain of syllogisms can include both progressive and regressive connections.

Purely conditional syllogisms that have the following scheme can be complex:

(r->d)l(d->g)A(g-»5)l...l(G1->51)

From the diagram it is clear that, as in a simple purely conditional inference, the conclusion is an implicative connection of the basis of the first premise with the consequence of the last.

In the process of reasoning, polysyllogism usually takes a shortened form;

some of its premises are omitted. A polysyllogism in which some

These premises are called sorites. There are two types of sorites: program polysyllogism with omitted major premises of episyllogisms and per nal polysyllogism with omitted smaller premises. Here is an example of a progressive polysyllogism:

A socially dangerous act (A) is punishable (B) A crime (C) is a socially dangerous act (A) Giving a bribe (D) is a crime (C)

Giving a bribe (D) is punishable (B)

Epicheyrema also belongs to complex abbreviated syllogisms. An epic is called a compound syllogism, both premises of which are;

memes. For example:

1) Dissemination of knowingly false information discrediting the honor and dignity of another person is criminally punishable, since it is slander i.

2) The actions of the accused constitute the spread of

3) The actions of the accused are criminally punishable

Let us expand the premises of the epicheireme into complete syllogisms. To do this, let us restore the complete syllogism first, the 1st enthymeme:

Libel (M) is criminally punishable (R)

Dissemination of deliberately false information discrediting honor

and the dignity of another person (S), is slander (M)

Dissemination of knowingly false information discrediting the honor and dignity of another person (S) is a criminal offense (P)

As we can see, the first premise of the epicheirem consists of a conclusion and a smaller premise of the syllogism.

Now let's restore the 2nd enthymeme.

Deliberate distortion of facts in an application against citizen P. (represents the dissemination of deliberately false information that discredits the honor and dignity of another person (P) The actions of the accused (S) were expressed in deliberate distortion of facts in an application against citizen P. (M)

The actions of the accused (S) constitute the dissemination of deliberately false information discrediting the honor and dignity of another person (P)

From the Greek "heap" (a pile of parcels).

The second premise of the epicheirema also consists of the conclusion and the minor premise of the syllogism.

The conclusion of the epicheirema is derived from the conclusions of the 1st and 2nd syllogisms:

Dissemination of knowingly false information discrediting the honor and dignity of another person (M) is criminally punishable (P) The actions of the accused (S) constitute the dissemination of knowingly false information discrediting the honor and dignity of another person (M)

The actions of the accused (S) are criminally punishable (P)

Expanding the epicheireme into a polysyllogism allows you to check the correctness of the reasoning and avoid logical errors that may go unnoticed in the epicheireme.

This lesson will focus on multi-premise inferences. Just as in the case of single-premise conclusions, all the necessary information in a hidden form will already be present in the premises. However, since there will now be many premises, the methods for extracting them become more complex, and therefore the information obtained in conclusion will not seem trivial. In addition, it should be noted that there are many different types of multi-premise inferences. We will focus only on syllogisms. They differ in that both in the premises and in the conclusion they have categorical attributive statements and, based on the presence or absence of some properties in objects, they allow one to draw a conclusion about the presence or absence of other properties in them.

Simple categorical syllogism

A simple categorical syllogism is one of the simplest and most common conclusions. It consists of two parcels. The first premise speaks about the relationship between the terms A and B, the second - about the relationship between the terms B and C. Based on this, a conclusion is made about the relationship between the terms A and C. This conclusion is possible because both premises contain a common term B, which mediates the relationship between terms A and C.

Let's give an example:

  • All fish cannot live without water.
  • All sharks are fish.
  • Therefore, all sharks cannot live without water.

In this case, the term "fish" is a general term for the two premises, and it helps connect the terms "sharks" and "creatures that can live without water." The common term for two premises is usually called the middle term. The subject of the conclusion (in our example it is “sharks”) is called a lesser term. The predicate of the conclusion (“creatures that can live without water”) is called a major term. Accordingly, the premise containing the minor term is called the minor premise ("All sharks are fish"), and the premise containing the larger term is called the major premise ("All fish cannot live without water").

Naturally, in an argument the premises can be in any order. However, for the convenience of checking the correctness of syllogisms, the larger premise is always put first, and the smaller one - second. Then, depending on the arrangement of terms, all simple categorical syllogisms can be divided into four types. These types are called figures.

A figure is a form of simple categorical syllogism that is determined by the placement of the middle term.

The major premise is at the top, followed by the smaller premise, and below the line is the conclusion. The letter S denotes the smaller term, the letter P the larger term, and the letter M the middle term.

  • Every M is P
  • Every S is M
  • Every S is P
  • No M is P
  • Some M's are S's
  • Some S's are not P's

These different combinations of statements in figures form the so-called modes. Each figure has 64 modes, so there are a total of 256 modes across all four figures. If you think about the whole variety of inferences that have the form of syllogisms, then 256 modes are not so many. In addition, not all modes form correct conclusions, that is, there are modes that, if the premises are true, do not guarantee the truth of the conclusion. Such modes are called irregular. Correct are those modes with the help of which we always obtain a true conclusion from true premises. There are 24 regular modes in total - six for each figure. This means that in all of classical syllogistics, which exhausts the lion's share of reasoning produced by people, there are only 24 types of correct inferences. This is a very small number, so the correct modes are not that hard to remember.

Each of these modes received a special mnemonic name back in the Middle Ages. Each type of categorical attributive statement was designated with just one letter. Statements like “All S are P” are designated by the letter “ A", the first letter of the Latin word "affirmo" ("affirm"), and their spelling became "S a P". Statements of the form “Some S are P” were written using the letter “ i", the second vowel in the word "affirmo", so they looked like "S i P". Statements of the form “No S is a P” are designated by the letter “ e", the first vowel in the Latin word "nego" ("I deny"), they began to be written as "S e P". As you probably already guessed, statements like “Some S are not P” are denoted by the letter “ O", the second vowel in the word "nego", their formal writing was "S o P". Therefore, the modes of regular syllogisms are traditionally denoted using these four letters, which are presented in the form of words for ease of memorization. The table of all correct modes looks like this:

Figure III

For example, the mode of the second figure Cesare (eae) when expanded will look like this:

  • No P is an M
  • All S are M
  • No S is a P

Although 24 modes are not at all a lot and some regularities can be seen in the table (for example, the modes eao and eio are correct for all figures), it is still difficult to remember. Fortunately, this is not at all necessary. You can also use model diagrams to test syllogisms. Only, unlike the diagrams that we built before, they should already contain not two, but three terms: S, P, M.

Let's take the mode of the fourth figure Bramantip (aai) and check it with the help of model diagrams.

  • Every P is M
  • Every M is S
  • Some S's are P's

First you need to find model schemes in which both premises will be true at the same time. There are only four such schemes:




Now, on each of these diagrams, we must check whether the statement “Some S are P,” which represents the conclusion, is true. As a result of checking, we find that in each diagram this statement will be true. Thus, the conclusion based on the mode Bramantip (aai) of the fourth figure is correct. If there were at least one diagram in which this statement was false, then the inference would be incorrect.

The method of testing syllogisms using model diagrams is good because it allows you to visualize the relationships between terms. However, for some premises many schemes may be true at once. As a result, their construction and verification will be a labor-intensive and time-consuming task. Thus, the model circuit method is not always convenient.

Therefore, logicians have developed another method for determining whether a syllogism is correct or not. This method is called syntactic and consists of two lists of rules (rules of terms and rules of premises), subject to which the syllogism will be true.

Rules of terms

  1. A simple categorical syllogism must include only three terms.
  2. The middle term must be distributed in at least one of the premises.
  3. If the greater or lesser term is not distributed in the premise, then it must also be undistributed in the conclusion.

Parcel rules:

  1. At least one of the premises must be affirmative.
  2. If both premises are affirmative, then the conclusion must be affirmative.
  3. If one of the premises is negative, then the conclusion must be negative.

The rules of premises are clear, but the rules of terms require some explanation. Let's start with the rule of three terms. Although it seems obvious, it is quite often violated due to the so-called substitution of terms. Look at the following syllogism:

  • Gold is an element of group 11, the sixth period of D. I. Mendeleev’s periodic table of chemical elements, with atomic number 79.
  • Silence is gold.
  • Silence is an element of group 11, the sixth period of D. I. Mendeleev’s periodic table of chemical elements, with atomic number 79.

First of all, if you remember the figures and the correct modes, you can immediately say that this syllogism is incorrect, since it refers to the second figure and has the mode aaa, which does not belong to the list of correct modes for this figure. But if you don't remember them, you can still detect its falsity, because there are clearly four terms here instead of three. The term "gold" is used in two completely different senses: as a chemical element and as something of value. Let's look at a more complex example:

  • All the books from the collection of the Russian State Library cannot be read in a lifetime.
  • “Fathers and Sons” by Ivan Turgenev is a book from the collection of the Russian State Library.
  • “Fathers and Sons” by Ivan Turgenev cannot be read in a lifetime.

This syllogism seems to correspond to the Barbara mode of the first figure. However, the premises are true and the conclusion is false. The problem is that in this example the terms have again been quadrupled. It seems that this syllogism contains three terms. The smaller term is “Fathers and Sons” of Ivan Turgenev. A larger term is “books that cannot be read in a lifetime.” The middle term is “books from the collection of the Russian State Library.” If you look closely, it will become clear that the subject of the first premise is not the term “books from the collection of the Russian State Library”, but the term “ All books from the collection of the Russian State Library." In this case, “all” is not a quantifier of generality, but a part of the subject, since this word is used not in a divisive sense (each separately), but in a collective sense (all together). If we replaced the word “all” with the words “each individual,” then the first premise would simply become false: “Each individual book from the collection of the Russian State Library cannot be read in a lifetime.” Thus, we get four terms instead of three, and therefore this conclusion is false.

Now let's move on to the rules about the distribution of terms. First, let's explain what this characteristic is. A term is called distributed if the statement refers to all objects included in its scope. Accordingly, the term is not distributed if the statement does not talk about all the objects that make up its scope. Roughly speaking, the term is distributed if we are talking about all objects, and not distributed if we are talking only about some objects, about part of the scope of the term.

Let's take the types of statements and see which terms are distributed in them and which are not. A distributed term is marked with a “+” sign, an undistributed term with a “-” sign.

All S + are P - .

No S+ is P+.

Some S - are P - .

Some S - are not P + .

a + is P - .

a + is not P + .

As you can see, the subject is always distributed in general and individual utterances, but not distributed in private ones. The predicate is always distributed in negative statements, but not distributed in affirmative ones. If we now transfer this to our rules for terms, it turns out that the middle term in at least one of the premises must be taken in its entirety.

  • Penguins are birds.
  • Some birds cannot fly.
  • Penguins can't fly.

Although both the statements above the line and the statement below the line are true, there is no inference as such. There is no logical transition from premises to conclusion. And this can be easily revealed, since the middle term “birds” is never taken in its entirety.

As for the third rule of terms, if in the premises we are talking about only part of the objects from the scope of the terms, then in the conclusion we cannot say anything about all the objects of the scope of the terms. We cannot move from part to whole. By the way, the reverse transition is possible: if we are talking about all elements of the scope of terms, then we can make a conclusion about some of them.

Enthymemes

During real discussions and debates, we quite often omit certain parts of the argument. This leads to the emergence of enthymemes. An enthymeme is a shortened form of inference in which the premises or conclusion are omitted. It is important not to confuse enthymemes with single-premise conclusions. An enthymeme is precisely a multi-premise inference; parts of it are simply omitted for one reason or another. Sometimes such omissions are justified, since both interlocutors are well versed in the problem, and they do not need to spell out all the steps. Meanwhile, unscrupulous interlocutors can deliberately use enthymemes to obscure and confuse their reasoning and hide their true arguments or conclusions. Therefore, it is necessary to be able to distinguish correct enthymemes from incorrect ones. An enthymeme is called correct if it can be restored in the form of the correct mode of a categorical syllogism, and if all the missing premises turn out to be true.

Let's talk about how to restore the enthymeme to a complete syllogism. First of all, you need to understand what exactly is missing. To do this, you need to pay attention to marker words denoting cause-and-effect relationships: “thus,” “therefore,” “since,” “because,” “as a result,” etc. For example, let’s take the argument: “Gold is a precious metal because it practically does not oxidize in air.” Here the conclusion is the statement “Gold is a precious metal.” One of the premises: “Gold practically does not oxidize in air.” Another package missed. It must be said that most often it is one of the parcels that is missed. It is quite strange if the most important thing is missing in the argument - the conclusion.

So, we have established what exactly was missing. In our example, this is a premise. Is this a large package or a smaller one? As you remember, the minor premise contains the subject of the conclusion (“gold”), and the major predicate contains the predicate of the conclusion (“precious metal”). The premise containing the subject of the conclusion is already known to us: “Gold practically does not oxidize in air.” This means that we know the smaller premise, but not the larger one. In addition, thanks to the well-known premise, we can establish the middle term: “metals that practically do not oxidize in air,” a term that is not contained in the conclusion.

Now we place the information we know in the form of a syllogism:

  • 3. Gold is a precious metal.

Or in diagram form:

  • 2.S a M
  • 3.S a P

The major premise must contain a conclusion predicate and a middle term: “precious metals” (P) and “metals that oxidize in air” (M). There are two options here:

  • 1. P M
  • 2.S a M
  • 3.S a P
  • 1. M P
  • 2.S a M
  • 3.S a P

This means that a syllogism of either the second figure or the first figure is possible. Now look at our tablet with the correct modes of syllogisms. In the second figure there are no regular modes at all, where the conclusion would be a statement like A. In the first figure there is only one such mode - Barbara. Let's complete our syllogism:

  • 1M A P
  • 2.S a M
  • 3.S a P
  • 1. All metals that practically do not oxidize in air are precious.
  • 2. Gold practically does not oxidize in air.
  • 3. Gold is a precious metal.

Now we check whether our restored premise is true. In our case it is true, so the enthymeme was correct.

Sorites

Lewis Carroll used the term “sorites” to refer to complex syllogisms that have more than two premises. By and large, sorites is a hybrid of a syllogism and an enthymeme. It is structured as follows: a set of premises are given, from each pair of premises intermediate conclusions are drawn, which are usually omitted, new premises are added to the intermediate conclusions, new intermediate conclusions are drawn from them, to which new premises are again added, and so on until we have gone through everything existing premises and will not reach the final conclusion. In principle, people reason in this way in everyday life. Therefore, it is very important to be able to solve sorites and evaluate whether they are correct or not.

We will give an example of sorites from Lewis Carroll’s book “The Knot Tale”:


2. A man with long hair cannot help but be a poet.
3. Amos Judd never went to prison.

5. There are no other poets in this district except policemen.
6. No one has dinner with our cook except her cousins.

8. Amos Judd likes his lamb cold.

Above the line are the premises, below the line is the conclusion.

How should sorites be solved and verified? We'll give you step-by-step instructions. First, it is necessary to bring all the premises into a more or less standard form:

1. All the policemen from our area have dinner with our cook.
2. All people with long hair are poets.
3. Amos Judd was not in prison.
4. All our cook’s cousins ​​love cold mutton.
5. All poets from our district are policemen.
6. All the people dining with our cook are her cousins.
7. All people with short hair were in prison.

Now you need to take two initial parcels. By and large, it doesn’t matter which premises you start with. The main thing is that your initial premises together contain only three terms. This means we can't take the parcels "Amos Judd wasn't in jail" and "All our cook's cousins ​​like cold mutton." They contain four different terms, and therefore we cannot draw any conclusion from them. I will take premises 7 and 3 as initial ones and draw a conclusion from them according to the rules for simple categorical syllogisms.

  • 1. All people with short hair were in prison.
  • 2. Amos Judd was not in prison.
  • 3. Amos Judd is not a man with short hair.

This syllogism corresponds to the mode Camestres (aee) of the second figure. Now, for convenience, I will restate our intermediate conclusion as follows: “Amos Judd is a man with long hair.” I connect this intermediate output to parcel number 2:

  • 1. All people with long hair are poets.
  • 2. Amos Judd is a man with long hair.
  • 3. Amos Judd is a poet.

This syllogism corresponds to the mode Barbara (aaa) of the first figure. Now I attach this intermediate output to parcel number 5:

  • 1. All poets from our district are policemen.
  • 2. Amos Judd is a poet.
  • 3. Amos Judd is a policeman.

This syllogism again corresponds to the mode Barbara (aaa) of the first figure. We connect the intermediate terminal to parcel number 1:

  • 1. All the policemen from our area have dinner with our cook.
  • 2. Amos Judd is a policeman.
  • 3. Amos Judd is having dinner with our cook.

This syllogism, as you probably already noticed, is also a mode of Barbara (aaa) of the first figure. We attach this conclusion to premise number 6:

  • 1. All the people dining with our cook are her cousins.
  • 2. Amos Judd is having dinner with our cook.
  • 3. Amos Judd is our cook's cousin.

Again Barbara, which is one of the most common modes. We attach the last parcel number 4 to our last intermediate conclusion:

  • 1. All our cook's cousins ​​love cold mutton.
  • 2. Amos Judd is our cook's cousin.
  • 3. Amos Judd likes his lamb cold.

So, with the help of the same Barbara mode, we got our conclusion: “Amos Judd likes cold mutton.” Sorites are thus solved and tested by step-by-step division into simple categorical syllogisms. In our example, sorites turned out to be correct, but the opposite situations are also possible. There are two conditions for the correctness of sorites. First, each sorites must be divided into a sequence of correct modes of syllogisms. Secondly, the conclusion you get when all the premises have been exhausted must coincide with the conclusion of the sorites. This condition applies in cases where you are dealing with someone else's reasoning, in which some kind of conclusion is already present.

So, we examined various multi-premise inferences using the example of simple categorical syllogisms, enthymemes and sorites. By and large, if you know how to deal with them, then you are armed for any discussions with any opponents. The only thing that can currently cause some dissatisfaction is the need to spend a lot of time checking the correctness of conclusions. You shouldn’t be upset about this: it’s better to look like a slow-witted person who thinks correctly than a brilliant demagogue who doesn’t notice his own and others’ mistakes. Moreover, with the accumulation of experience in paying close attention to inferences, you will develop an instinct, an automatic skill that allows you to quickly separate correct reasoning from incorrect ones. Therefore, there will be a lot of exercises for this lesson so that you have the opportunity to improve your skills.

Einstein's problems

This game is our version of the world famous "Einstein's riddle" in which 5 foreigners live on 5 streets, eat 5 types of food, etc. More details about this task are written here. In such tasks, you need to make the correct conclusion based on the existing premises, which, at first glance, are not enough for this.

Exercises

Exercises 1, 2 and 3 are taken from Lewis Carroll’s book “The Knot Story”, M.: Mir, 1973.

Exercise 1

Draw conclusions from the following premises using the rules for a simple categorical syllogism. Remember that a simple categorical syllogism must contain only three terms. Do not forget to reduce statements to standard form.

  • An umbrella is a very necessary thing when traveling.
  • When going on a trip, you should leave everything unnecessary at home.
  • Music that can be heard causes vibrations in the air.
  • Music that cannot be heard is not worth paying money for.
  • No Frenchman likes pudding.
  • All English people love pudding.
  • No old miser is cheerful.
  • Some old curmudgeons are skinny.
  • All non-voracious rabbits are black.
  • No old rabbit is inclined to abstain from food.
  • Nothing sensible has ever baffled me.
  • Logic baffles me.
  • None of the countries explored so far are inhabited by dragons.
  • Unexplored countries captivate the imagination.
  • Some dreams are terrible.
  • Not a single lamb inspires terror.
  • No bald creature needs a comb.
  • Not a single lizard has hair.
  • All eggs can be broken.
  • Some eggs are hard boiled.

Exercise 2

Check if the following reasoning is correct. Try different verification methods. Don't forget to put the big package on the first line.

  • Dictionaries are useful.
  • Useful books are highly valued.
  • Dictionaries are highly valued.
  • Gold is heavy.
  • Nothing but gold can silence him.
  • Nothing easy can silence him.
  • Some ties are tasteless.
  • Anything done with taste delights me.
  • I'm not crazy about some ties.
  • No fossil animal can be unlucky in love.
  • The oyster may be unhappy in love.
  • Oysters are not fossil animals.
  • No hot cake is healthy.
  • All raisin buns are unhealthy.
  • Raisin buns are not baked goods.
  • Some pillows are soft.
  • No poker is soft.
  • Some pokers are not pillows.
  • Boring people are unbearable.
  • No boring person is asked to stay when he is about to leave as a guest.
  • No insufferable person is asked to stay when he is about to leave as a guest.
  • Not a single frog has a poetic appearance.
  • Some ducks look prosaic.
  • Some ducks are not frogs.
  • All intelligent people walk with their feet.
  • All foolish people walk on their heads.
  • No man walks on his head and feet.

Exercise 3

Find the conclusions of the following sorites.

  • Small children are unreasonable.
  • Anyone who can tame crocodiles deserves respect.
  • Unreasonable people do not deserve respect.
  • No duck waltzes.
  • Not a single officer will refuse to dance a waltz.
  • I have no other bird except ducks.
  • Anyone who is of sound mind can practice logic.
  • No lunatic can serve on a juror.
  • Neither of your sons can do logic.
  • There are no pencils in this box.
  • None of my candies are cigars.
  • All my property not in this box consists of cigars.
  • Not a single terrier wanders among the signs of the Zodiac.
  • What does not wander among the signs of the Zodiac cannot be a comet.
  • Only the terrier has a ringed tail.
  • No one will subscribe to The Times unless he has received a good education.
  • No porcupine can read.
  • Those who cannot read have not received a good education.
  • No one who truly appreciates Beethoven will make noise during the performance of the Moonlight Sonata.
  • Guinea pigs are hopelessly ignorant of music.
  • Those who are hopelessly ignorant of music will not remain silent during the performance of the Moonlight Sonata.
  • Items sold on the street do not have much value.
  • Only rubbish can be bought for a penny.
  • Great auk eggs are of great value.
  • Only what is sold on the street is real rubbish.
  • Those who break their promises are not trustworthy.
  • Drinkers are very sociable.
  • A person who keeps his promises is honest.
  • No teetotaler is a moneylender.
  • Someone who is very sociable can always be trusted.
  • Any thought that cannot be expressed in the form of a syllogism is truly ridiculous.
  • My dream of butter buns is not worth writing down on paper.
  • Not a single pipe dream of mine can be expressed in the form of a syllogism.
  • I haven't had a single really funny thought that I wouldn't tell my friend about.
  • All I can dream about is butter buns.
  • I never expressed a single thought to my friend unless it was worth writing down on paper.

Exercise 4

Check the correctness of the following enthymemes.

  1. Barsik is not a law-abiding cat because he stole my sausage.
  2. Mercury is liquid, therefore it cannot be a metal.
  3. No obedient child throws tantrums over trifles. That's why Tolya is a naughty child.
  4. Some women are stupid, which means some men can take advantage of it.
  5. All girls want to get married, because each of them dreams of a fluffy white dress.
  6. No student wants to get a D on an exam, that's why all students are nerds.
  7. Someone stole my wallet, so I had no money left.
  8. Peacocks are narcissistic birds because they have a big beautiful tail.

Test your knowledge

If you want to test your knowledge on the topic of this lesson, you can take a short test consisting of several questions. For each question, only 1 option can be correct. After you select one of the options, the system automatically moves on to the next question. The points you receive are affected by the correctness of your answers and the time spent on completion. Please note that the questions are different each time and the options are mixed.

The opposition to a predicate can be considered as the result of two successive immediate inferences: first, a transformation is made, then the transformation is converted into a judgment.

Categorical syllogism is a type of deductive inference built from two true categorical judgments in which S And P connected by the middle term. The concepts that make up a syllogism are called the terms of the syllogism. A premise containing a predicate of the conclusion (i.e., a major term) is called a major premise. The premise containing the subject of the conclusion (i.e., the minor term) is called the minor premise.

Enthymeme, or abbreviated categorical syllogism, called a syllogism in which one of the premises or conclusion is missing. Enthymemes are used more often than complete categorical syllogisms.

COMPLEX AND COMPLEX SYLLOGISMS (polysyllogisms, sorites, epicheireme)

In thinking there are not only individual complete or abbreviated syllogisms, but also complex syllogisms, consisting of two, three or more simple syllogisms. Chains of syllogisms are called polysyllogisms.

INDUCTIVE INFERENCES

In defining induction in logic, two approaches are identified - the first, carried out in traditional (not mathematical) logic, in which by induction is called an inference from knowledge of a lesser degree of generality to new knowledge of a greater degree of generality (i.e., from individual particular cases we move to a general judgment). With the second approach, inherent in modern mathematical logic, by induction called an inference that gives a probable judgment.

Full induction is called such an inference in which the general conclusion about all elements of the class of consideration of each element of this class is called. In complete induction, all objects of a given class are studied, and single judgments serve as premises. Complete induction gives a reliable conclusion, so it is often used in mathematical and other most rigorous proofs. To use complete induction, the following conditions must be met:

1. Know exactly the number of objects or phenomena to be considered.

2. Make sure that the attribute belongs to each element of this class.

3. The number of elements of the class being studied should be small.

INDUCTIVE METHODS

ESTABLISHING CAUSAL RELATIONS

Cause– a phenomenon or a set of phenomena that directly determine or give rise to another phenomenon (consequence).

Causality is universal, since all phenomena, even random ones, have their own cause. Random phenomena are subject to probabilistic, or statistical, laws.

Causality is necessary, because if there is a cause, the action (effect) will certainly occur. For example, good training and musical ability are the reason that this person will become a good musician. But the reason must not be confused with the conditions. You can create all the conditions for a child: buy an instrument and sheet music, invite a teacher, buy books on music, etc., but if there are no abilities, then the child will not become a good musician. Conditions promote or, conversely, hinder the action of the cause, but the conditions and the cause are not identical.


INTRODUCTION

Logic is one of the oldest sciences. Its eventful history began in Ancient Greece and goes back two and a half thousand years. At the end of the last - beginning of this century, a scientific revolution took place in logic, as a result of which the style of reasoning, methods radically changed, and science seemed to gain a second wind. Now logic is one of the most dynamic sciences, a model of rigor and accuracy even for mathematical theories.

Spontaneously developed skills of logically perfect thinking and the scientific theory of such thinking are completely different things. Logical theory is unique. She expresses about the ordinary - about human thinking - what seems at first glance unusual and needlessly complicated. Hence the difficulty of the first acquaintance with logic: one must look at the familiar and established with new eyes and see the depth behind what was taken for granted.

THE CONCEPT OF PROOF AND ITS STRUCTURE

In logic, proof is understood as a procedure for establishing the truth of a certain statement by citing other statements, the truth of which is already known and from which the first necessarily follows..

The proof differs thesis- a statement that needs to be proven, base(arguments) - those provisions with the help of which the thesis is proven, and logical connection between arguments and thesis. The concept of proof always presupposes, therefore, an indication of the premises on which the thesis is based, and those logical rules by which the transformation of statements is carried out during the proof.

A proof is a correct conclusion with true premises. The logical basis of each proof (its diagram) is logical law.

Proof is always, in a certain sense, coercion.

The task of the proof is to comprehensively establish the validity of the thesis. Since the proof is about complete confirmation, the connection between the argument and thesis should be deductive character.

In its form, proof is a deductive inference or a chain of inferences leading from true premises to the position being proven.

Usually the proof proceeds in a very abbreviated form. Seeing a clear sky, we conclude: “The weather will be fine.” This is proof, but extremely condensed. Omitted is the general statement: “Whenever the sky is clear, the weather will be good.” The “Clear Sky” package was also released. Both of these statements are obvious; there is no need to say them out loud.

Often, the concept of proof is given a broader meaning: proof is understood as any procedure for substantiating a true thesis, including both deduction and inductive reasoning, references to the connection of the position being proven with facts, observations, etc.

As a rule, proof is widely understood in everyday life. To confirm the proposed idea, facts, typical phenomena in a certain respect, etc. are actively used. In this case, of course, there is no deduction; we can only talk about induction. Nevertheless, the proposed justification is often called evidence.

The definition of proof includes two central concepts of logic: the concept truth and concept logical consequence. Both of these concepts are not sufficiently clear, which means that the concept defined through them also cannot be classified as clear.

Many are neither true nor false, i.e. lie outside the “category of truth”. Assessments, norms, advice, declarations, oaths, promises, etc. They do not describe certain situations, but indicate what they should be and in what direction they should be transformed. It is obvious that when using expressions that do not have true meaning, one can and should be both logical and demonstrative. Thus, the question arises of a significant expansion of the concept of evidence, defined in terms of truth. The problem of redefining the proof has not yet been solved logic of assessments, neither deotic(normative) logic.

The model of proof that all sciences strive to follow to one degree or another is mathematical proof. Mathematical proof is the paradigm of proof in general, but even in mathematics the proof is not absolute and final.

DIRECT AND INDIRECT EVIDENCE

All evidence is divided according to its structure, according to the general train of thought into straight And indirect. With direct evidence, the task is to find convincing arguments from which the thesis logically follows. Indirect evidence establishes the validity of the thesis by revealing the fallacy of the assumption opposite to it, antithesis.

For example: All cosmic bodies are subject to the laws of celestial mechanics.

Comets are cosmic bodies.

therefore, comets obey these laws.

In formation direct evidence two interconnected stages can be distinguished: finding those recognized statements that can be convincing arguments for the position being proven; establishing a logical connection between the found arguments and the thesis.

IN indirect evidence the reasoning goes in a roundabout way. Instead of directly finding arguments to deduce from them the position being proven, an antithesis is formulated, a negation of this position. Further, in one way or another, the inconsistency of the antithesis is shown. The antithesis is false, which means the thesis is true.

Since indirect evidence uses the negation of the proposition being proven, it is, proof by contradiction.

For example: If the speech was boring, it would not raise so many questions and heated, meaningful discussion. But it caused such a discussion. So the performance was interesting.

Thus, indirect evidence goes through the following stages: an antithesis is put forward and consequences are derived from it with the intention of finding at least one false one among them; it is established that the antithesis is incorrect; from the falsity of the antithesis, the conclusion is drawn that the thesis is true.

Abbreviated syllogism (enthymeme)- an inference with a missing premise or conclusion. Enthymeme in Greek means “in the mind.”

For example: “Mathematics must then be taught, because it puts the mind in order” (M. Lomonosov).

In the enthymeme, a major premise, as in the above example, may be omitted, as well as a minor premise, or a conclusion. The form of an enthymeme can be taken by a conditionally categorical syllogism, dividing-categorical, or conditionally dividing syllogisms.

For example: “The sum of the digits of a given number is divisible by 3, therefore, the given number is divisible by 3.” The conditional premise “If the sum of the digits of a given number is divisible by 3, then the entire number is divisible by 3” is missing here.

In the conclusion, “An acquittal cannot be rendered in this case. It must be indictable” the dividing premise “The case filed can be either acquitted or convicted” is missing.

proslogism, subsequent – episyllogism polysyllogism.

For example:

For example:


33. Polysyllogisms and sorites, rules of education, examples. The concept of epicheyrema.

In the process of reasoning, simple syllogisms can form a chain of syllogisms in which the conclusion of the previous syllogism becomes the premise of the subsequent one. The preceding syllogism is called proslogism, subsequent – episyllogism. This kind of inference is called polysyllogism.

There are progressive and regressive polysyllogisms.

In progressive polysyllogism the conclusion of the prosyllogism becomes the greater premise of the episyllogism.

For example:

In regressive polysyllogism the conclusion of the preceding syllogism becomes the minor premise of the subsequent one.

For example:

A complex syllogism in which some premises are missing is called sorites(from the Greek "heap"). There are two types of sorites: progressive and regressive.

Progressive sorites is obtained from a progressive polysyllogism by throwing out the conclusions of previous syllogisms and the major premises of subsequent ones. For example:

Progressive sorites scheme:

Regressive sorites is obtained from a regressive polysyllogism by throwing out the conclusions of previous syllogisms and the minor premises of subsequent ones. For example:

Regressive sorites scheme:

Epicheyrema also belongs to complex abbreviated syllogisms. Epicheyrema is a compound syllogism, both premises of which are enthymemes. For example:

The epicheyrema scheme is as follows:

Scheme of the first parcel:

Scheme of the second parcel:


34. Inferences from complex judgments, their types. Purely conditional syllogism, symbolic recording of modes, examples.

Inferences are built not only from simple, but also from complex judgments. The following types of deductive inferences are known, the premises of which are complex judgments: purely conditional, conditionally categorical, dividing-categorical and conditionally dividing syllogisms.

The peculiarity of these inferences is that the derivation of a conclusion from the premises is determined not by the relations between terms, as in a categorical syllogism, but by the nature of the logical connection between judgments. Therefore, when analyzing premises, their subject-predicate structure is not taken into account.

Disjunctive syllogism

Purely conditional syllogism For example:

The outline of this syllogism is as follows:

The conclusion in a purely conditional inference is based on the rule: the consequence of the consequence is the consequence of the reason.


Purely conditional syllogism is an inference whose premises and conclusion are conditional propositions.

Disjunctive syllogism- an inference, the premises and conclusion of which are divisive (disjunctive) judgments.

Conditional disjunctive syllogism- an inference in which one premise is a conditional proposition and the other is a disjunctive one.

Conditional categorical syllogism - an inference in which one of the premises is a conditional proposition, and the other premise and conclusion are categorical judgments. The conditional categorical syllogism has two correct modes:

1) approver,

2) denying.

In the affirmative mode (modus ponens) the categorical premise asserts the truth of the antecedent of the conditional premise, and the conclusion asserts the truth of the consequent. Reasoning is directed from asserting the truth of the reason to asserting the truth of the consequence. His diagram:

For example:

In the negating mode (modus tollens) the categorical premise denies the truth of the consequent, and the conclusion denies the truth of the antecedent. The reasoning is built from the denial of the truth of the consequence to the denial of the truth of the reason. Modus tollens scheme:

For example:

Two more varieties of conditionally categorical syllogism are possible: from the denial of the truth of the reason to the denial of the truth of the consequence:

From asserting the truth of the consequence to asserting the truth of the reason:

However, the conclusion based on these modes will not be reliable, which can be verified using truth tables.

When constructing a conclusion according to the scheme of purely conditional and conditional categorical syllogisms, one should also keep in mind that the truth of the conclusion will be guaranteed only if the conditional premises contain sufficient grounds for the consequences.


Purely conditional syllogism is an inference whose premises and conclusion are conditional propositions.

Conditional disjunctive syllogism- an inference in which one premise is a conditional proposition and the other is a disjunctive one.

Disjunctive syllogism - an inference, the premises and conclusion of which are divisive (disjunctive) judgments. His scheme is as follows:

For example:

This type of inference contains two modes.

I mode– affirmative-denying (modus ponendo tollens). His diagram:

The rule of modus ponendo tollens is that the dividing premise must be an exclusive (strict) disjunction.

II mode– denying-affirming (modus tollendo ponens).

His diagram:

The rule of modus tollendo ponens is that all possible alternatives must be listed in the dividing premise.


37. Conditional disjunctive (lemmatic) inferences. Dilemmas, their types, symbolic notation and examples. The concept of polylemmas.

Purely conditional syllogism is an inference whose premises and conclusion are conditional propositions.

Disjunctive syllogism- an inference, the premises and conclusion of which are divisive (disjunctive) judgments.

Conditional disjunctive syllogism - an inference in which one premise is a conditional proposition and the other is a disjunctive one.

Depending on how many consequences are established in the conditional premise, dilemmas, trilemmas, n - lemmas are distinguished.

Lemma– means sentence in Greek. The conclusion of such a conclusion states an alternative, i.e. the need to choose only one of all possible offers. A dilemma, then, is a conditionally disjunctive conclusion with two alternatives.

There are the following types of dilemmas: simple and complex, constructive and destructive.

Complex destructive dilemma contains one premise consisting of two conditional propositions with different bases and different consequences; the second premise is the disjunction of the negations of both consequences; the conclusion is a disjunction of the negations of both grounds. Her diagram:


38. Induction in logic and its types. Five methods for establishing cause-and-effect relationships. Logic circuits, examples.

Induction is a way of reasoning in which a conclusion, which is a general reasoning, is obtained on the basis of less general knowledge or individual facts.

Incomplete induction– a probabilistic inference in which a conclusion about the belonging of a feature to a whole class of objects is made on the basis of the belonging of this feature to a part of the objects of this class.

The logical structure of incomplete induction can be expressed as follows:

Types of incomplete induction: induction through simple enumeration, statistical induction, induction based on establishing a causal relationship.

Induction via simple enumeration (popular induction)- a type of incomplete induction in which a conclusion about a whole class of homogeneous objects is made on the basis that among the observed cases there was no fact that contradicts the conclusion made.

Induction, based on simple observation, is common in everyday life: swallows fly low - it will rain, if the sun sets red, then tomorrow will be a windy day, etc.

The degree of probability of concluding induction through simple enumeration increases with the number of observed cases. Possible errors associated with the use of this type of inference are called hasty generalization.

Statistical induction– a type of incomplete induction containing information about the frequency distribution of a certain property for a certain class of objects.

This class of objects in statistics is called population, and any population class – sampling.

The degree to which statistical induction is likely to be concluded depends on how well the sample is selected.

Induction based on establishing a causal relationship (scientific)– a type of incomplete induction, in which a conclusion about a whole class of homogeneous objects is made on the basis of knowledge of the necessary ones, i.e. essential features of some items of this class.

 

 
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