Ch-21 :- Syllogism :- Introduction & Formula


The questions of syllogism can be solved with the help of Venn-diagrams and some rules devised with the help of analytical ability. Some people are of the opinion that Venn-diagram can be of great use for solving questions of syllogism. No doubt a few questions can be solved with the help of Venn-diagrams, but Venn-diagrams alone do not help the students to solve variety of questions of syllogism. Even commonsense also will not be of much help in working out certain difficult type of questions. Only the basic concepts and rules, which have a bearing on reasoning faculty could alone help. We have discussed these rules with illustrations throughout the chapter. To proceed further for the study of rules, we must first know some terminology used in syllogism.

Proposition :
A proposition, also known as a premises, is a grammatical sentence which comprises a subject, a predicate and a copula. A subject is that which affirms or denies a fact. predicate is a term which states something about a subject and copula establishes relationship between the subject and the predicate.

Examples:
Capture

Classification of Propositions :
A proposition can mainly be divided into three categories :

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(a) Categorical Proposition : In categorical propositions there exists a relationship between the subject and the predicate without any condition. It means predicate is either affirmation or denial of the subject unconditionally.          Examples:
I. All cups are plates.
II. No Girl is boy.

(b) Hypothetical Proposition: In a hypothetical proposition, relationship between subject and predicate is asserted conditionally.

Examples:
I. If it rains, he will not come.
II. If he comes, I will accompany him.

(c) Disjunctive Proposition: In a disjunctive proposition the assertion is of alteration.
Examples:
I. Either he is honest or he is loyal.
II. Either he is educated or he is illiterate.

            Keeping in view the existing pattern of syllogism in competitive examinations we are concerned only with the categorical  type of proposition.

Types of Categorical Proposition
Categorical propositions have been classified on the basis of quality and quantity of the propositions. Quality denotes whether the proposition is affirmative or negative. Quantity represents whether the proposition is universal or particular.
The four-fold classification of categorical propositions can be summarised as under:

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To draw valid inferences from the statement the candidate is required to have a clear understanding of A, E, I, 0 relationships.

Venn-Diagram Representation of Four Propositions
1. Universal Affirmative (A) : All S are P.

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There are two possibilities to represent the relation between S and P given by universal affirmative proposition “All S are P”. This can also be understood with the help of set theory.

Case I.           S = {a, b, c}                           P = {a, b, c, d}
Case II.          S = {a, b, c}                           P = {a, b, c}
Case I is represented by Fig. (1) and Case II is represented by Fig. (II). In both these cases, we see that every element of set S is also the element of set P. Hence, we can definitely say that the above two figures show “All S are P”.

2. Universal Negative (E) : No S are P

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There is only one possibility of drawing the relationship between Sand P.
S = {a, b, c}                        P = {d, e, f}
From the above two sets, it is clear that none of the elements of S is the element of set P.

3. Particular Affirmative (I) : Some S are P

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There are three possible representations given by Fig. (IV), Fig. (V) and Fig. (VI) depicting particular affirmative proposition “Some S are P”. This can be supported with the help of following sets:
Case I.         S = {a, b, c, d}            P = {c, d, e, f}
Case II.        S = {a, b}                    P = {a, b, c, d}
Case III.       S = {a, b, c, d}            P = {a, b}
In all the three cases we find that some of the elements of S are also the elements of set P.

4. Particular Negative (0) : Some S are not P

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The particular negative proposition “Some S are not P” can be represented with the help of three possible figures given in (VII), (VIII) and (IX).
Venn-diagram representations of the above propositions can be supported by way of following sets:

Case I.            S = {a, b, c}              P = {c, d, e}
Case II.           S = {a, b, c, d}         P = {c, d}
Case III.          S = {a, b, c}             P = {d, e, f}
In all the three cases, we find that there are some elements in set S which are not elements of set P. Hence, all the cases along with the respective figures support the propositioz “Some S are not P”.

Hidden Proposition
The type of proposition we have discussed so far are standard nature. But there are propositions which do not appea:: in standard format, and yet can be classified under any of four types. Let us now discuss the type of such propositions.
(I) A Type Propositions:
(i) All positive propositions beginning with ‘every ‘each’, ‘any’ are ‘A’ type propositions.
Examples:
(a) Every cot is mat.
=> All cots are mats.
(b) Each of students of class V has passed.
=> All students of class V have passed.
(c) Anyone can do this job.
=> All (men) can do this job.

(ii) A positive sentence with a particular person as its subject is always an A-type proposition.
Examples:
(a) He can qualify the CAT written test.
(b) Mahatma Gandhi is known as the father of nation.

(II) E-Type Propositions:
(i) All negative sentences beginning with ‘no one ‘none’, ‘not a single’, etc. are E-type propositions.
Examples:
(a) Not a single student could answer the question.
(b) None can cross the English channel.
(ii) A negative sentence with a very definite exception also of E-type proposition.
Example:
No student except Ram has failed.
(iii) When an interrogative sentence is used to make assertion, this could be reduced to an E-type proposition.
Example:
Is there any person who can scale Mount Everest?
=> None can climb Mount Everest.

(III) I-TypePropositions:
(i) Positive propositions beginning with words such ‘most’, ‘a few’, ‘mostly’, ‘generally’, ‘almost’ , ‘frequently’, ‘often’ are to be reduced to the I-type.
Examples:
(a) Almost all the fruits have been sold.
=> Some fruits have been sold.
(b) Most of the students will qualify in the examination.
=> Some of the student will qualify in the examination.
(c) Girls are frequently physically weak.
=> Some girls are physically weak.
(ii) Negative propositions beginning with words such as ‘few’, ‘seldom’, ‘hardly’, ‘scarcely’, ‘rarely’, ‘little’ etc. are to be reduced to I-type.
Examples:
(a) Seldom players do not take rest.
=> Some players take rest.
(b) Few priests do not tell a lie.
=> Some priests tell a lie.
(c) Rarely IT professionals do not get a good job.
=> Some IT professionals get a good job.
(iii) A positive sentence with an exception which is not definite, is reduced to an I-type proposition.
Examples:
(a) All students except three have passed.
=> Some students have passed.
(b) All innocents except a few are guilty.
=> Some innocents are guilty.

(IV) O-Type Propositions:
(i) All negative propositions beginning with words such as ‘all’, ‘every’, ‘any’, ‘each’ etc. are to be reduced to O-type propositions.
Examples:
(a) All innocents are not guilty.
=> Some innocents are not guilty.
(b) All that glitters is not gold.
=> Some glittering objects are not gold.
(c) Everyone is not present.
=> Some are not present.
(ii) Negative propositions with words as ‘most’, ‘a few’, ‘mostly’, ‘generally’, ‘almost’, ‘frequently’ are to be reduced to the O-type propositions.
Examples:
(a) Girls are usually not physically weak.
=> Some girls are not physically weak.
(b) Priests are not frequently thiefs.
=> Some priests are not thiefs.
(c) Almost all the questions can’t be solved.
=>  Some questions can’t be solved.
(iii) Positive propositions with starting words such as ‘few’, ‘seldom’, ‘hardly’, ‘scarcely’, ‘rarely’, ‘little’, etc. are to be reduced to the O-type.
Examples:
(a) Few girls are intelligent.
=> Some girls are not intelligent.
(b) Seldom are innocents guilty.
=> Some innocent are not guilty.
(iv) A negative sentence with an exception, which is not definite, is to be reduced to the O-type.
Examples:
(a) No student except Ram has passed.
=> Some students have not passed.
(b) No girls except a few are intelligent.
=> Some girls are not intelligent.

Types of Inferences
Inferences drawn from statements can be two step :-
1. Immediate Inference : When inference is drawn from a single statement, then inference is known as immediate inference.
Example:
Statement:
All Books are Pages.
Conclusion: Some Pages are Books.
In the above example, a conclusion is drawn from a single statement and does not require the second statement to be referred, hence the inference is called an immediate inferece

2. Mediate Inference: In mediate inference conclusion is drawn from two given statements. For example:
Statements: All Dogs are Cats.
All Cats are Black.
Conclusion: All Dogs are Black.
In the above example, conclusion is drawn from the two statements or in other words, both the statements are required to draw the conclusion. Hence, the above conclusion is know as mediate inference.

Methods to Draw Inferences
(I) Immediate Inference: There are various methods to draw immediate inferences like conversion, obversion, contraposition, etc. Keeping in view the nature of questions asked in various competitive examinations, we are required to study only two methods- implications and conversion.
(a) Implications (of a given proposition) : Below we shall discuss the implications of all the four types of propositions.
While drawing a conclusion through implication, subject remains the subject and predicate remains the predicate.
A-Type : All boys are blue.
From the above A type proposition, it is very clear that if all boys are blue, then some boys will definitely be blue because some is a part of all. Hence, from A type proposition, we can draw I type conclusion (through implication).
E-Type: No cars are buses.
If no cars are buses, it clearly means that some cars are not buses. Hence, from E-type proposition, O-type conclusion (through implication) can be drawn.
I-Type: Some chairs are tables.
From the above I-type proposition, we can not draw any valid conclusion (through implication).
O-Type : Some Aare not B.
From the above O-type proposition, we can not draw any valid inference (through implication). On first look, it appears that if some A are not B, then conclusion that some A are B must be true but the possibility of this conclusion being true can be overruled with the help of following example:
Case I :         A = {a, b, c}                 B = {d, e, f}
Case II:        A = {a, b, c}                 B = {b, c, d}
The above two cases show the relationship between A and B given by O-type proposition “Some A are not B”.
Now, in case I, none of the element of set A is the element of set’ B. Hence, conclusion “Some A are B” can’t be valid. However, in case (II), elements b, c are common to both sets A and B. Hence, here conclusion “Some A are B” is valid.
But for any conclusion to be true, it should be true for all the cases. Hence, conclusion “Some A are B”is not a valid conclusion drawn from an O-type proposition.
All the results derived for immediate inference through implication can be presented in the table as below:

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(b) Conversion : In conversion, while drawing inference, subject and predicate of a proposition are interchanged, i.e., subject becomes the predicate and predicate becomes the subject but the quality of the proposition remains the same, i.e., the affirmative remains affirmative and the negative remains negative.
In conversion, A-type proposition is converted into I-type. E-type can be converted into E-type, I-type can be converted into I-type and O-type proposition can’t be converted.
Examples:
(I)   Statement: All lamps are mangoes.          (A type)
        Conclusion: Some mangoes are lamps.      (I type)
(II) Statement: No men are wise.                    (E type)
         Conclusion: No wise are men.                    (E type)
(III) Statement: Some chairs are tables.         (I type)
         Conclusion: Some tables are chairs.         (I type)
For making the things clear, we are combining and presenting the results of implication and conversion in the following table. This table will help the candidates, by and large, in drawing immediate inferences from a given proposition quickly.

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It is important to note here that only valid inferences given in the above table can be drawn from the four types 0; propositions. Hence, the candidates are advised to make.. themselves familiar With the types of propositions (A, E, I, O) and to remember the results of immediate inferences as given above for solving questions quickly.

Venn-Diagram Representation
Immediate inferences drawn from each type of propositions (A, E, I, 0), as given in the above table, are based. on the different rules (implication and conversion) as discussed above. The same inferences can also be drawn with the help of Venn-diagrams. But one of the important points to be noted. while drawing inference from Venn-diagrams is that a, possibilities of Venn-diagrams should be taken in account. Le us now discuss each type of proposition in relation to pictorial representation.

1. A-Type-All S are P
It is clear from the A type of proposition that all S are contained in P. Therefore, circle representing S will be either inside or equal to circle representing P. However, in both cases, conclusions (Some Pare S) and (Some S are P) are true. This case can be understood clearly by taking two sets in a, possible ways.

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The above cases show the all the possibilities of two sets S and P showing the relationship between each other as represented by the proposition. All S are P. Now in both the cases we see that set {2, 3} is the part of set S and also of set P. Hence, it is clear that inference (Some S are P) is true from this relationship. Likewise set {2, 3} is the part of set P and also of set S. Therefore, it is also clear that inference (Some P are S) is true. Inference (Some P are not S) is not valid because it is true from case (i) but false from case (ii). Inference (All P are S) not valid because it is true from case (ii) and false from case (i).

2. E-Type-No S is P
There is only one possibility of Venn-diagram representation of E-type proposition. The relationship can also be shown by two sets S = {I, 2, 3} and P = {4, 5, 6}. From these two sets, we see that set {2, 3} is the part of set S but not of set P. It implies that inference (Some S are not P) is true. Similarly, set {5, 6} is the part of set P but not of set S. This means that inference (Some P are not S) is true. Therefore, on the basis of E-type proposition, we can draw following immediate inferences.
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(i) No P is S.
(ii) Some S are not P.
(iii) Some P are not S.
Any other immediate inference drawn from E-type proposition is not valid.

3. I-Type – Some S are P
            This proposition gives rise to many possible representations of Venn-diagrams and hence most of the inferences drawn therefrom are invalid and doubtful. This relationship can be shown by following sets and respective Venn-diagrams.
(i) S = {l, 2, 3, 4}                 P = {3, 4, 5, 6}

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Set {3, 4} is the part of set S as well as set P, hence some S are P
(ii) S = {l, 2, 3, 4}                P = {l, 2}

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Set {l, 2} is the part of set S as well as set P, hence some S are P.
(iii)             S = {l, 2}                   P = {l, 2, 3, 4}

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Set {I} is the part of set S as well as set P, hence some S are P.
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Set {l, 2} is the part of set S as well as set P, hence some S are P.
The above four combinations of sets and respective diagrams show the relationship between S and P as represented by I-type proposition. From all the possible combinations, it is clear that inference (Some P are S) is true. Inference-(Some S are not P) is true from combinations (i) and (ii). But it is not true from combinations (iii) and (iv). Therefore, inference (Some Sare not P) is not a valid inference drawn from the above proposition.

4. O-Type – Some S are not P .
From this proposition no immediate inference can be drawn. Let us discuss this proposition in the light of Venn-diagram representation.

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(i)      S = {l, 2, 3, 4}             P = {3, 4, 5, 6}
Set {l, 2} is the part of set S but-not of set P, hence this shows the relationship represented by the proposition ‘Some S are not P’.

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(ii)      S = {l, 2, 3}                   P = {4, 5, 6}
Set {2, 3} is the part of set S but not of set P, hence this shows the relation represented by the proposition ‘Some S are not P’.

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(iii)      S = {l, 2, 3, 4, 5}            P = {4, 5}
Set {l, 2, 3} is the part of set S but not of set P, hence denotes proposition ‘Some S are not P’. On the basis of all possible combinations showing relationship between S and P, no valid inference can be drawn.Inference – Some S are P is true from case (i) and (iii) but not true from case (ii) and hence it is an invalid inference. Inference – Some P are not S is true from case (i) and (ii) but not true from case (iii) and hence it is an invalid inference.
Students should note that if an inference is true, it has to comply with or follow all the possible pictorial representation of Venn-diagrams.

Mediate Inference
Like immediate inference, mediate inferences can also be drawn with the help of certain rules and Venn diagrams. Some of the techniques of mediate inference has been discussed below:
1. Rules for Mediate Inference
Step I : The first step in drawing mediate inferences is the alignment of statements. As we know, to draw mediate inference we need two statements having one term common to both the statements. By alignment we mean that predicate of the first statement should be the subject of the second statement. Or in other words, we can say that common term is the predicate of the first statement and subject of the second statement. Alignment is required only when we find that common term is either subject in both the statements or predicate in both the statements. If the statements are already aligned, then we can move to the step II. Following examples will make the concept of alignment more clear.

Example 1. Statements:
I. Some girls are boys.
II. All girls are black.
Example 2. Statements :
I. Some mangoes are baskets.
II. No tables are baskets.
In Example 1, we see that common term girls is the subject in both the statements and hence needs alignment. The statements can be aligned in two ways – One by aligning first statement, and the second by aligning second statement and changing the order.

Case I :
Statement I “Some girls are boys” can be converted into “Some boys are girls” using the rule of conversion already discussed before. After conversion of the first statement, both the statements are aligned as below:
Statements :   I. Some boys are girls.
II. All girls are black.

Case II : Statement II “All girls are black” can be converted into “Some black are girls.” After converting the second statement and changing the orders of the statements, the statements are aligned as below:
Statements :  I. Some black are girls.
II. All girls are boys.

Likewise, the statements in Example 2 can be aligned in two ways as below:
Case I : Statements :
I. Some mangoes are baskets.
II. No baskets are tables.

Case II : Statements :
I. No tables are baskets.
II. Some baskets are mangoes.
Since the statements in both the above examples can be aligned in two ways, hence it becomes important to know to which alignment to consider. In this connection, one may note that while aligning, the priority should be given in the following order:
15           i.e., importance for conversion should be given first to I type statement, then E type statement and at last to A type statement.
Using the above priority order in Example 1 and 2, we find that in Example 1, statement I is of I type and statement II is of A type. Hence, statement I should be aligned, being of I type. Hence, alignment as per case I will have to be considered and not the alignment as per case II.
In Example 2, statement I is of I type and statement II is of E type. Hence, statement I should be aligned. As a result proper alignment would be as per case II and ‘not as per’ case I

Step II : After having aligned the statements, if required inference can be drawn using the table as given below:
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          The conclusion will follow the pattern in such a way that subject of the first statement is the subject of the inference and predicate of second statement is the predicate of the inference except the cases (O* as shown in the table) where conclusion will follow the reverse pattern, i.e., subject of the first statement will be the predicate of inference, and predicate of the second statement will be the subject of the inference.
The above table shows the combination of statements which produce definite and valid results. All other combinations will not produce definite inferences. The following examples will make the concept more clear.

Example 1.
Statements:
I. Some books are papers.
II. No copies are papers.
Solution : The statements are not aligned. Hence, first step will be to align the statements. Statement I is of I type and hence will get the priority for alignment. After alignment and reordering, the statements will become
I. No copies are papers.
II. Some papers are books.
(E type can be converted into E type. Refer to the conversion table.)
Now using table for mediate inference, we find that
E + I = O*
Hence, inference will be “Some books are not copies”.

Example 2.
Statements  : I. Some papers are white.
II. No black are papers.
Solution : In above statements, alignment is not required. Simply reordering the sentences will align the statements.
I. No black are papers.
II. Some papers are white.
E + I = 0*. Hence inference will be “Some white are not black.”

Please note again that conclusion of the form O* will follow the reverse pattern, i.e., subject of the first statement will be the predicate of the inference, and predicate of the second statement will be the subject of the inference.

Example 3.
Statements
  : I. All books are clocks.
II. All clocks are pens.
Solution : Both the statements are already aligned. Now using table for mediate inference, we find that A + A = A. Hence, the valid inference will be “All books are pens”. It is important to note that table for mediate inference produces only one valid result from the above statements, and yet we can also convert the conclusion “All books are pens” as “Some pens are books” according to rules for conversion. Hence, the conclusion “Some pens are books” is also valid.

Example 4.
Statements :      All buses are cars.
Some cars are roads.
Conclusions :     I. Some cars are buses.
II. Some buses are roads.
Solution : First we mark the type of conclusions. Conclusion (I) is an immediate inference and, hence will be solved according to the rules for immediate inference. Conclusion (II) is mediate inference and, hence will be solved according to the rules for mediate inference.
Conclusion (I) is immediate inference drawn from first statement. Hence, according to the rules for conversion, A type be converted into I type.
Therefore, “All buses are cars” ⇒ “Some cars are buses”.
Hence, conclusion (I) is valid.
Now, conclusion (II) is a mediate inference for which we find that statements are aligned. Using mediate inference table, we find that (A + I) combination is not available, and therefore no valid mediate inference can be drawn. Hence, conclusion (II) does not follow.

Example5.
Statements :
All gardens are trees.
All trees are plants.

Conclusion :
I. Some gardens are trees.
II. Some plants are gardens.
Solution : Conclusion (I) is an immediate inference drawn from the first statement and conclusion (II) is a mediate inference.
Statement “All gardens are trees.” ⇒ “Some gardens are trees” according to rule of implications for immediate inference. Hence, conclusion (I) is valid.
Now for mediate inference, we find that statements are aligned and A + A → A, i.e., from the above statements, we conclude that “All gardens are plants”, but it is not given in the conclusion. But this conclusion can be converted into “Some plants are gardens” according to rules for conversion. Hence, both the conclusions (I) and (II) are valid.

Example 6.
Statements  :
Some books are papers.
Some papers are copies.
Conclusions :  
I. Some papers are books.
II. Some copies are papers.
Solution : It is clear from the conclusions that none of the conclusions is a mediate inference. Both the conclusions I and II are immediate inferences drawn from Statements I and II respectively. Statement I “Some books are papers” can be converted into “Some papers are books” and statement-II “Some papers are copies.” can be converted into “Some copies are papers.” Hence, both the conclusions are valid.
Step III : Selecting complementary pair of conclusions : In drawing mediate inference from given statements, students are required to be more attentive to select complementary pair of conclusions where neither of the conclusions is definitely true but a combination of both makes a complementary pair. Example 7 describes this situation.

Example 7.
Statements :  Some cars are scooters.
Some scooters are buses.
Conclusions : I. Some cars are buses.
II. No cars are buses.
Solution : Both the statements are properly aligned, therefore, fulfil the first requirement. But both the statements are of I type and as per table for immediate inference, a combination I + I does not produce any valid mediate inference . Hence, no mediate inference can be drawn.
It is important to note here, that conclusion (I) “Some cars are buses” is not valid because there is a possibility of “No cars are buses”. Likewise, conclusion (II) “No cars are buses” is invalid because there is a possibility of “Some cars are buses”.
In other words, both the conclusions are invalid individually. However, we can say that either of conclusions I or II is true. Hence, here, both the conclusions make a complementary pairs of conclusions.

A complementary pair of conclusions must follow the following two conditions:
(I) Both of them must have the same subject and the same predicate and
(II) They are anyone of three types of pairs
(a) I-O type
(b) A-O type
(c) I-E type

The following examples show the complementary pairs:
1. (i) All trees are green.                       (A type)
(ii) Some trees are not green.          (O type)
2. (i) Some trees are green.                  (I type)
(ii) Some trees are not green.          (O type)
3. (i) Some trees-are green.                  (I type)
(ii) No trees are green.                      (E type)
All-the three pairs of conclusions comply with the two conditions given above to form a complimentary pair, hence form a complimentary pairs.

The following examples show the pairs which do not form a complementary pair.
1. (i) All trees are green.                  (A type) .
(ii) Some green are not trees.     (O type)
2: (i) All trees are green.                  (A type)
(ii) No trees are green.                 (E type)
Though the first pair is A O type pair, yet it does not form a complementary pair because subject and predicate of both the propositions are not the same.
Second pair is not a complementary pair because A E type does not form a complementary pair.

Mediate Inference with the help of Venn-Diagrams
Like immediate inference, mediate inference can also be drawn with the help of Venn diagrams. But important point to be noted here is that all the possibilities of diagrams should be drawn.

Example 8.
Statements :   All books are clocks.
All clocks are pens.
Conclusions :  I. All books are pens.
II. Some pens are books.
Solution : There are two ways in which we can show relationship among books; clocks and pens as given in the statement. In both these cases, we find that “All books are pens” and “Some pens are books” are true.

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Hence, both the conclusions follow.

Example 9.
Statements  : Some books are papers.
No copies are papers .
Conclusions : I. Some books are not copies.
II No books are copies.

Solution:

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The above two figures show the relationship among books, papers and copies. Conclusion (I) “Some books are not copies” is valid from both the figures as the shaded portion confirms denial with copies. However, conclusion (II) “No books are copies” is true-from Fig. (I) but false from Fig. (II). Hence, only conclusion (I) is valid.

Example 10.
Statements :
Some cups are plates.
Some plates are jugs.

 Conclusions :
I. Some cups are jugs.
II. All cups are jugs.

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Now let us consider each of the conclusion in the light of the figures given above.
Conclusion (I) “Some cups are jugs” is true from Fig. (II) and it is false from other figures. Hence, it is not valid.
Conclusion (II) “All cups are jugs” is not true from any of these figures. Therefore, it is not valid. Hence, both the conclusions are valid.

Three-Statement Syllogism
In these types of questions, three statements are given in place of two statements. Though the principles involved in solving these qvestions are same yet a slightly different approach is adopted. Following examples will help to understand the pattern of these questions and methods to solve them.

Example 11.
Statements :
1. All bags are caps.
2. Some pens are bags.
3. No caps are desks.
Conclusions :
I. Some pens are caps.
II. No desks are bags.
III. Some pens are desks.
IV. Some caps are bags.
Solution : The first step in solving these questions is to select two statements for each of the conclusions in such a way that subject and predicate of the conclusion should be present in these statements. And, importantly, these statements should be linked with the “common term”.
Conclusion I : For conclusion (I) “Some pens are caps”, we can take Statements (1) and (2) because these two statements have subject “pens” and predicate “caps” of the statements and are liked with the common term “bags”. Now to draw mediate inference these two statements have to be aligned in such a way that predicate of the first statement is the subject of the second statement. Hence, these two statements can be aligned as follows:

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From the table of mediate inference, we know that I + A = I Therefore, conclusion will be “Some pens are caps.” Hence, conclusion I is valid.
Conclusion II : Statements (1) and (3) are relevant for conclusion II “No desks are bags”. These two statements are already aligned.

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We know that A + E = E. Therefore, from these two statements, conclusion “No bags are desks” are valid. But our conclusion is “No desks are bags” We know that E type proposition can be converted into E type proposition, i.e., “No bags are desks” ⇒ “No desks are bags”. Hence, conclusion (II) is valid.
Conclusion III : For this conclusion, we can not choose any two statements because out of three, no two statements containing subject and predicate of the conclusion are liked with the common term. Therefore, all the three statements are relevant to test this inference. For this, we should write the three statements in such a way that the predicate of the first is the subject of second and predicate for second is the subject of the third.
Some pens are bags.               (I type)
All bags are caps.                     (A type)
No caps are desks.                   (E type)
From the combination of all these statements, we find that 1 + A + E = (I + A) + E = I + E = O. Thus, a valid conclusion would be of O type, i.e., “Some pens are not desks” Hence, conclusion III “Some pens are desks” is not valid conclusion.
Conclusion IV : Conclusion IV “Some caps are bags” is immediate inference drawn from statement (1). From the rule of conversion, we know “All bags are caps” ⇒ “Some caps are bags”. Hence, conclusion (IV) is valid.

Example 12.
Statements :
1. All boys are jokers.
2. Some men are boys.
3. All jokers are caps.
Conclusions :
I. Some caps are boys.
II. Some-men are caps.
III. All boys are caps.
IV. Some boys are not caps.
Solution : For conclusions (I), (III) and (IV), subject and predicate are “caps” and “boys”. And for these conclusions, relevant statements would be (1) and (3) because these two statements have “caps” and “boys” and are linked with the common term “jokers”.
Now, as per the rule of mediate inference, we know that
All boys are jokers.             (A type)
All jokers are caps.             (A type)
A + A = A. Therefore, conclusion “All boys are caps” is valid. From the rule of conversion we know that “All boys are caps” ⇒ “Some caps are boys.” Hence, conclusions (I) and (III) are valid and conclusion (IV) is invalid.
For conclusion (II), subject is “men” and predicate is “caps”. These two terms are available in Statements (2) and (3) but these statements are not linked with common term. Therefore, all the three statements are relevant. Now, after aligning all the three statements, we get
22              I + A + A ⇒ (I + A) + A ⇒ I + A = I. Thus, conclusion “Some men are caps.”
Hence, in the above example, conclusion (I), (II) and (III) are valid.