A simple categorical syllogism
A simple categorical syllogism (PCS) is the conclusion of a categorical proposition from two other categorical judgments that are bound by a common term.
Realism (M) is a clear and sober understanding of reality (P).
"The main quality of the leader (5) - realism (M)
The main quality of the leader (5) is a clear and sober understanding of reality (/').
The PKC is an indirect reasoning having its own structure. In it, the connection between the two concepts (in conclusion) is established by means of the third concept that exists in both premises.
The term contained in both parcels is called the middle term (M). In our example, it is the concept of realism & quot ;.
The terms that are included in the conclusion are called extreme terms. Among the extreme terms are the smaller term (he is the subject of the conclusion) (5) and the larger term (this predicate conclusion) - (P). In our example, the smaller term is the concept of "superior quality of the leader," and the larger term is "a clear and sober understanding of reality."
A premise that contains a larger term is called a larger parcel, and a parcel containing a smaller term is called a smaller parcel. In our example, a large premise first comes up, followed by a smaller one.
but in standard entries a simple categorical syllogism as the first one puts a large parcel, and as the second - smaller. Violation of this requirement makes logical analysis of this kind of reasoning difficult. The formula of the PCS is 5 - М-Р, ie. the subject of imprisonment is connected with the predicate of imprisonment through the middle term. It is no accident that Aristotle (384-322 BC), deeply and comprehensively developed the theory of syllogism, stressed that in syllogism, "the study is conducted for the sake of the average term."
In fact, a simple categorical syllogism is an inference about the relation of the two extreme terms on the basis of their relation to the mean term.
(& pound;) Not a busy person (M) will never enjoy complete happiness (P) (G. Heine).
(L) A loafer (5) is not a busy person (M).
(E) A loafer (5) will never enjoy complete happiness (P).
The scheme shows: if all objects of class 5 enter the volume M, and the class M does not have common elements with P, then 5 has nothing to do with P, which is stated in the conclusion.
Consider another example:
(L) To be able to manage (M) means to be able to choose (F. Panathy).
(L) The main thing for the manager (5) is to be able to manage (M).
(L) The main thing for a leader (5) is to be able to choose (P).
The diagram shows: if all elements of class 5 enter the volume M, and the entire class M - in the volume of the class P, then it is obvious that all elements of class S enter into the volume P. This is stated in the conclusion.
Before us are graphic schemes of the axiom of syllogism:
Everything that is asserted or denied about the class of objects as a whole, is affirmed or denied of either the part or the individual subject of this class. "
The axiom of syllogism is accepted without evidence and is the starting point for the justification of the general rules of a simple categorical syllogism.
The general rules of a simple categorical syllogism are such that each of them is individually a necessary condition of the correctness of the output, and all together they are > sufficient condition for the correctness of the output. A rule is considered necessary if, in the case where it is not satisfied, the reasoning is incorrect. Sufficiency is expressed in the fact that the fulfillment of each of the general rules of the syllogism testifies to the correctness of the reasoning. In other words, the syllogism is correct if all its rules are fulfilled, and is incorrect if at least one of them is not fulfilled. General rules of the syllogism include rules of terms and rules of premises.
Consider the terms of the terms.
• There must be only three terms in the syllogism.
The error that occurs when a rule is violated is called quadrupling terms. It is caused by the fact that the concept, which should be the connecting link between the premises (and this is the role of the middle term), is ambiguous, used in different meanings. In other words, the formula of a simple categorical syllogism is violated: 5 - M-R In this example, an attempt is made to connect the subject and the confinement predicate in two "mean" term: 5 - Ml-M, - R.
(A) Historical Figures (M) are people who have had a significant impact on the development of society (P).
(A) Nozdrev (5) was in some respects a historical person (M) (N. V. Gogol).
(L) Nozdrev (5) in some respects had a significant impact on the development of society (P).
To understand the error that led to a ridiculous conclusion, let us turn to the context of Gogol's phrase: "Nozdryov was in some respects a historical person. Not at one meeting where he was, could not do without history. Some story must have occurred: either the gendarmes would take him out of the hall, or they would be compelled to push out his friends. "
As you can see, the word history in the syllogism is ambiguous: in the first case we mean "social reality in its development", and in the second - "an incident, an adventure, most often unpleasant" ( Fits into the story are said in similar situations).
In other words, here the law of identity is grossly violated in the form of substitution of concepts. In fact, there are not three in the syllogism, but four terms - the middle term, which should be the connecting link between the premises, the peculiar "bridge" for the transition from parcels to imprisonment, is ambiguous. Having discovered this, we see that there is no semantic connection between the premises. Judge for yourself:
Historians are people who have had a significant impact on the development of society. And Nozdrev got into unpleasant situations every time. "
- And what's next? It's like that "In the garden - elder, and in Kiev - uncle". As we see, in the absence of a meaningful connection between the premises, logical reasoning is impossible.
• The middle term should be distributed at least in one of the parcels.
If M is not distributed in both premises, the output is not possible. Error in violation of this rule - Undistributed middle term.
For example, let's take two statements on the topic of identity. The famous Persian poet Saadi (1184-1291) remarked: "The donkey who visited Mecca still remains an ass". And our compatriot, the famous poet GR Derzhavin (1743-1816) expressed this idea in his own way: "The Donkey will remain an ass, though screeching him with stars." Using these statements as premises, we will build a syllogism:
(L) "The donkey who visited Mecca (P +) still remains an ass (M -)".
(L) A donkey sprinkled with stars (5 *), still remains an ass
(m -) ._
(L) A donkey sprinkled with stars (5+) is a donkey that visited Mecca (P ~).
If you like, you can formulate a different conclusion:
"The donkey that visited Mecca is sprinkled with stars", but the essence of the error from this will not change. In the premises, middle terms - the circle of those who always remain an ass, are taken in incomplete volume (in part). And this circumstance turns out to be decisive, since there is no reason (except the game of chance) to believe that in both utterances it is a question of the same subset. In fact, this is an implicit violation of the law of identity.
Formalizing the premises of the syllogism:
All P is M ,
All 5 is M & quot ;, we construct circular schemes:
As you can see, based on the same parcels, you can make four mutually exclusive conclusions.
The diagrams show that there can not be an unambiguous relationship between the terms of the syllogism. This is an indication that the syllogism is wrong.
• The term not distributed in the package should not be distributed in the confinement.
An error occurred in violation of this rule - illegal extension of the extreme term. In other words, having initial information about a part of objects of a set, in the process of reasoning they spread this information to the whole set, which contradicts the logical nature of deduction - and in its traditional understanding (the movement of thought from general to private), and in the modern sense of inference.
For example, let's use a story from ancient Greek mythology about a giant robber named Procrust. He, as is known, forcibly packed the travelers on a couch and those who were larger than his size, cut off his legs, and pulled the short ones up to the size of a bed. Hence the name "Procrustean bed," which in a figurative sense means an artificial measure that does not correspond to the essence of the phenomenon; forcibly imposed on anything. Incidentally, we note that logic also imposes limitations, but has neither direct nor indirect relation to the Procrustes case. So, the syllogism:
(A) State management (M +) is a cruel (P ~) (D. Halifax).
(E) Procrust (5+) did not control the state (M +).
(E) Procruste (5 *) did not deal with the brutal cases (P +).
The signs of discrepancy show that the predicate ("cruel deeds") in the parcel was taken in part of the volume, and in conclusion - in full, which is inadmissible in deductive conclusions.
Formalizing the parcels:
All Vengeance P ,
5 is not M & quot ;, we will construct circular schemes:
Obviously, the information from the parcels is not enough to establish an unambiguous relationship between the terms. Proceeding from the larger premise, we placed all the set in the set P, and, based on the smaller premise, mutually excluded the Mi set 5. But the relation between the extreme terms 5 and P, since 5 may belong to the set P, and may not belong. Both possibilities are equivalent, and the preference of one of them has no relation to the laws of logic.
Consider the rules of the parcels.
• At least one of the parcels must be an affirmative proposition. This means that it is impossible to construct a correct syllogism from two negative judgments.
Example of violation of this rule:
(E) By evil (P +) do not reach the good (M +) (W. Shakespeare).
(E) The game with fire (5+) will not bring to the good (M +).
(L) Playing with fire (5 *) is the path of evil (P ~).
Formalizing the parcels: "None P is not M,"; "None 5 is L /", we construct circular schemes: ___
As we see, there are no single-valued relations between the extreme terms S and P. On the basis of the information contained in the premises, a number of mutually exclusive conclusions can be drawn, namely:
All 5 is P ,
Some 5 are P ,
Some 5 is not a P ,
None 5 is P .
• At least one of the premises must be a general proposition. This means that it is impossible to construct a correct syllogism from two particular judgments. For example:
(/) Posts often (M ~) change their character (P ) & gt; & gt; (Cervantes). (I) Some posts (M-) are vacant (5_). (D) Some jobs (5_) change their character (P).
Already the distribution of terms shows that the right conclusion from these premises is impossible, since the middle term is not distributed in any of them. But this is only a passing remark, referring to a particular case. The essence of the problem is different: if the average terms are taken in part of the volumes, then there is no reason to believe that these are identical parts. And if so, the reasoning crumbles. The situation here is in many respects similar to the quadrupling of terms, only in an implicit form.
Let's analyze the situation in more detail. Suppose there are a lot of students from which some parts (subsets) are taken and some thoughts are expressed in relation to them. It is not excluded that these subsets will be incompatible, and then thoughts will be expressed in relation to different subjects.
(D) Some students take exams on management theory.
(& pound; ') Some students are first graders.
Possible output options: Some first graders pass exams on management theory & quot ;; Some of the examiners on management theory are first-graders & quot ;. In both cases - absurdities. Why? Yes, because the subsets of students are incompatible: in one case it's schoolchildren, in the other - students or graduate students (in any case, not schoolchildren).
Upon returning to the original example.
Formalizing the parcels:
Some M is not P ,
Some M have 5 & quot ;, we will construct circular schemes:
It is clear from the constructions that the volume 5, intersecting with the volume M, turns out in ambiguous relations with the volume Р. Possible output options: All 5 is P & quot ;, None 5 is P & quot ;, Some 5 are P & quot ;.
This indicates that the syllogism is wrong.
• With one negative premise, the conclusion must be negative.
Example of violation of this rule:
(& pound;) The introverts (M) are not sociable (P). (A) I (5) introvert (M).
(A) However, I'm (5) a sociable person (P).
Formalizing the parcels:
None M is not a P ,
5 is M & quot ;, and having constructed the scheme, we obtain the ratio of the extreme terms:
5 is not a P corresponding to the inference rules. However, in violation of these rules, the conclusion is the opposite: "5 is P .
• With one private premise, the conclusion must be private.
Example of violation of this rule:
(A) Disorder (M ~) makes us slaves (P ~) (A. Amiel). (D) Sometimes purity becomes disorder (M).
(A) Purity makes us slaves (P -).
Already in terms of the distribution of terms, a violation is noticeable: the subject not distributed in the package was distributed in the conclusion.
Formalizing the parcels:
All Vengeance R ,
Some 5 have M & quot ;, construct circular schemes:
The ratio of the extreme terms l and P is such that in one case it turns out: "All .9 is P & quot ;, and in another:" Some 5 are P " . Obviously, taking into account the distribution of terms, the second option is acceptable.
For a deeper understanding of the structure of a simple categorical syllogism, one must also take into account the diversity of its shapes and modes.
The figures of a simple categorical syllogism are its varieties depending on the position of the middle term in the premises.
There are four figures of syllogism in total.
The middle term in the first figure plays the role of the subject in the larger premise and the role of the predicate in the smaller premise.
(L) Misrepresentation (M) - a hindrance to success (P)
(Bion of Borisfensky). (L) An exaggerated assessment of one's personality (5) is a doubt (M).
(L) An exaggerated assessment of your personality (N) -a success (P).
The first figure of a simple categorical syllogism is used as a way of spreading some general knowledge, expressed in a larger premise, into special cases. Class 5 is given under the class P, relative to which there is general knowledge.
This is clearly seen in the diagram:
If All 5 is M ,
and All M is P & quot ;,
then "All 5 is P .
The middle term in the second figure plays the role of a predicate of both premises.
(L) Any really effective board (P) for verification turns out to be a dictatorship (M) (Mr. Truman).
(& pound; ) Democracy (5) is not a dictatorship (M).Democracy (5) is not an effective rule.
The second PKC figure is used mainly as a means of refuting wrong substitutions of something under some concept. On the diagram this is also clearly seen: If the "All 5 is M & quot ;, and None M is not P," then "None 5 is P" ;
The middle term in the third figure plays the role of the subject in both assumptions.
(A) Word (M) is the shadow of the business (P) (Democritus).
(A) The word (M) is an act (5) (L. N. Tolstoy).Some actions (5) - the shadow of the case (P).
The third figure is often used as a way to refute unreasonable generalizations. The scheme shows:
If All 5 is M
and All P is M ,
then "Some 5 are P .
In the arguments for the third figure, the crucial point is the quantitative characterization of the conclusion - it must always be private. Following this rule, we avoid unreasonable generalizations.
The middle term in the fourth figure acts as the predicate of the larger and the subject of the smaller parcels.
(& pound;) "Strong words (P) can not be strong evidence (A /) (V. 0. Klyuchevsky). (/) Strong evidence (M) is usually persuasive (5).
(O) Usually, convincing arguments (6 ) do not need strong words (P).
The fourth figure is an artificial construction. Without cognitive value, in practice it is rarely used. If you invoke both parcels, you can get the first figure from the fourth figure.
(E) Strong evidence (N) does not need strong words (P). (D) Usually, convincing arguments (5) are supported by strong evidence (N) .
(O) Usually, convincing arguments (5) do not need strong words (P).
The modes of a simple categorical syllogism are its varieties depending on the quantitative and qualitative characteristics of judgments that make up its composition.
For example, in the last example, all judgments in the syllogism are generally assertive statements, therefore, its modus AAA; strong> so its mode is AAI. Actually, in all the illustrative examples of the PKS figures to the left of the sentences that make up the syllogisms, there are letter symbols, the sequence of which gives us the modes.
Given the four types of categorical judgments (A, E, I, O), , you can calculate that in each figure there are 64 modes, and all of them - 256! But not all of them are the right conclusions. The correct modes are only 24 (6 in each figure). Among them, there are 19 basic (strong) correct modes and 5 weak ones (in them conclusions are private judgments).
Syllogistics in traditional logic has been developed in such detail that all strong right modes have been given special names that, while facilitating memorization, contain all the information about the nature of the components of the mode of judgment. These names were invented by the Byzantine philosopher of the 11th century. by the name of Michael Psell (1018-c. 1096). He wrote a "Compendium for Aristotle's Logic", where he laid out his invention.
Medieval schoolboys in order to easier to learn the strong right modes of a simple categorical syllogism, came up with a poem written by a hexameter. Here it is.
Barbara, Celarent, Darii, Férioquc prions; Cesare, Camestrcs, Festino, Baroko secundae;
Tcrtia, Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fcsapo, Fresison.
Vowels in the names of modes indicate the types of judgments that play the role of a correspondingly larger, smaller premise and conclusion. For example, the mode Felapton means that the large premise is a negative judgment, the smaller premise is generally affirmative, and the conclusion is a privately negative proposition.
Right modes. For the first figure, this is AAA, EAE, AN, EY.
• Modus AAA (Barbara)
(A) Every name means something ... (AF Losev). (A) The word Anna - name.
(A) The word Anna something means.
• The EAE modality (Celarent).
(E) No person can consider himself a smoky person (N. A. Berdyaev). (A) I'm a human being.
(& pound;) I can not consider myself a complete person.
• Modus AN (Darii).
(A) The thought uttered is a lie (FI Tyutchev). (D) Some of the things I have planned are uttered.
(D) Some of what I have planned is false.
• Modus EY (Ferio).
(E) Nothing new is perfect (Cicero). (/) Something in our life is new.
(D) Something in our life is not perfect.
The right modes of the second figure are EAE, AEE, EY, AOO.
The third figure is AAI, IAI, AN, EAO, JSC, EY.
The fourth - AAI, AEE, IAI, EAO, EY.
Especially memorize modes, and even more so their medieval names, there is no need. Correct modes can be easily deduced by a logical way, relying on general and special rules of a simple categorical syllogism (the so-called rules of figures).
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