SYLLOGISM

SYLLOGISM

A syllogism is a logical argument composed of three parts: the major premise, the minor premise, and the conclusion inferred from the premises.

In logic, any categorical statement is termed as the Proposition. A Proposition (or a categorical statement) is a statement that asserts that either a part of, or the whole of, one set of objects - the set identified by the subject term in the sentence expressing that statement - either is included in, or is excluded from, another set - the set identified by the predicate term in that sentence.

The standard form of a proposition is:

Quantifier + Subject + Copula + Predicate

The proposition consists of four parts:

1. Quantifier: The words 'all', 'no' and 'some' are called quantifiers because they specify a quantity.  'All' and 'no' are universal quantifiers while 'some' is a particular quantifier.

2. Subject (S): The subject is that about which something is said.

3. Predicate (P): The predicate is the part of the proposition denoting that which is affirmed or denied about the subject.

4. Copula: The copula is that part of the proposition which denotes the relation between the subject and the predicate.

Four-Fold Classification of Propositions: A proposition is said to have a universal quantity if it begins with a universal quantifier (All/No) and a particular quantity if it begins with a particular quantifier (Some). Besides, propositions which assert something about the inclusion of the whole or a part of one set in the other are said to have affirmative quality, while those which deny the inclusion of the whole or a part of one set in the other are said to have a negative quality. Also, a term is distributed in a proposition if it refers to all members of the set of objects denoted by that term. Otherwise, it is said to be undistributed. Based on the above facts, propositions can be classified into four types:

1. Universal Affirmative Proposition (A): It distributes only the subject i.e. the predicate is not interchangeable with the subject while maintaining the validity of the proposition. e.g., All Books are Pens. This is proposition A since we cannot say 'All Pens are Books'.

2. Universal Negative Proposition (E): It distributes both the subject and the predicate i.e. an entire class of predicate term is denied to the entire class of the subject term, as in the proposition. e.g., No Man is intelligent.

3. Particular Affirmative Proposition (I): It distributes neither the subject nor the predicate. e.g., Some men are Intelligent. Here, the subject term 'men' is used not for all but only for some men and similarly the predicate term 'Intelligent' is affirmed for a part of subject class. So, both are undistributed.

4. Particular Negative Proposition (O): It distributes only the predicate. e.g., Some People are not wise. Here, the subject term 'People' is used only for a part of its class and hence is undistributed while the predicate term 'wise' is denied in entirety to the subject term and hence is distributed. These facts can be summarized as follows:

Statement Form             Quantity                 Quality                        Distributed

(A): All S is P.                 Universal                 Affirmative                 S only

(E): No S is P.                 Universal                 Negative                    Both S and P

(I): Some S is P.              Particular                 Affirmative               Neither S nor P

(O): Some S is not P        Particular                Negative                     P only

What is logical deduction? - The phenomenon of deriving a conclusion from a single proposition or a set of given propositions, is known as logical deduction. The given propositions are also referred to as the premises.

Two Inferential Processes of Deduction

I. Immediate Deductive Inference: Here, conclusion is deduced from one of the given propositions, by any of the three ways -conversion, Obversion and contraposition.

1. Conversion: The Conversion proceeds with interchanging the subject term and the predicate term i.e. the subject term of the premise becomes the predicate term of the conclusion and the predicate term of the premise becomes the subject of the conclusion. The given proposition is called convertend, whereas the conclusion drawn from it is called its converse.

Table of Valid Conversions

Convertend                                         Converse

A: All S is P                                        I: Some P is S

Ex. All pins are tops.                           Some tops are pins.

E: No S is P.                                        E: No P is S.

Ex. No fish is whale.                           No whale is fish.

I: Some S is P.                                     I: Some P is S.

Ex. Some boys are poets.                    Some poets are boys.

O: Some S is not P.                             No valid conversion

In a conversion, the quality remains the same and the quantity may change.

2. Obversion: In Obversion, we change the quality of the proposition and replace the predicate term by its complement.

Table of Valid Obversions

Obvertend                                                       Obverse

A: All birds are mammals.                              E: No birds are non-mammals.

E: No poets are singers.                                  A: All poets are non-singers.

I: Some nurses are doctors.                             O: Some nurses are not non-doctors.

O: some politicians are not statesmen.            I: Some politicians are non-statesmen.

3. Contraposition: To obtain the contrapositive of a statement, we first replace the subject and predicate terms in the proposition and then exchange both these terms with their complements.

Table of Valid Contrapositions

Proposition                                          Contrapositive

A: All birds are mammals.      A: All non-mammals are non-birds.

I: Some birds are mammals.    I: Some non-mammals are non-birds.

The valid converse, obverse or contrapositive of a given proposition always logically follows from the proposition.

II. Mediate Deductive Inference (Syllogism): A Syllogism is a deductive argument in which conclusion has to be drawn from two propositions referred to as the premises.

Example: (1) All lotus are flowers. and (2) All flowers are beautiful. Hence (3) All lotus are beautiful.

Clearly, the propositions 1 and 2 are the premises and the proposition 3, which follows from the first two propositions, is called the conclusion.

Term: In Logic, a term is a word or a combination of words, which by itself can be used as a subject or predicate of a proposition.

Syllogism is concerned with three terms:

1. Major Term: It is the predicate of the conclusion and is denoted by P (first letter of 'Predicate').

2. Minor Term: It is the subject of the conclusion and is denoted by S (first letter of 'Subject').

3. Middle Term: It is the term common to both the premises and is denoted by M (first letter of 'Middle').

Example:

Premises: 1. All dogs are animals. 2. All lions are dogs.

Conclusion: All lions are animals.

Here 'animals' is the predicate of the conclusion and so, it is the major term. P.

'Lions' is the subject of the conclusion and so, it is the minor term, S.

'Dogs' is the term common to both the premises and so, it is the middle term, M.

Major And Minor Premises : Of the two premises, the major premise is that in which the middle term is the subject and the minor premise is that in which the middle term is the predicate.

Rules for deriving conclusion from two given Premises:

1. The conclusion does not contain the middle term.

Example:

Statements: 1. All men are boys. 2. Some boys are students.

Conclusions: 1. All boys are men. 2. Some boys are not students.

Since both the conclusions 1 and 2 contain the middle term 'girls', so neither of them can follow.

2. No term can be distributed in the conclusion unless it is distributed in the premises.

Example:

Statements: 1. Some dogs are cats. 2. All cats are cows.

Conclusions: 1. All cows are cats. 2. Some dogs are cows.

Statement 1 is an I-type proposition which distributes neither the subject nor the predicate.

Statement 2 is an A type proposition which distributes the subject i.e. 'cats' only.

Conclusion 1 is an A-type proposition which distributes the subject 'cow' only since the term 'cows' is distributed in conclusion 1 without being distributed in the premises, so conclusion 1 cannot follow.

           

3. The middle term (M) should he distributed at least once in the premises. Otherwise, the conclusion cannot follow. For the middle term to be distributed in a premise.

(i) M must be the subject if premise is an A proposition.

(ii) M must be subject or predicate if premise is an E proposition.

(iii) M must be predicate if premise is an O proposition.

Note that in an I proposition, which distributes neither the subject nor the predicate, the middle term cannot be distributed.

Example:

Statements: 1. All F’s are W. 2. Some W’s are B.

Conclusions: 1. All W’s are F. 2. Some F’s are B.

In the premises, the middle term is 'W'. Clearly, it is not distributed in the first premise which is an A proposition as it does not form its subject. Also, it is not distributed in the second premise which is an I proposition. Since the middle term is not distributed even once in the premises, so no conclusion follows.

4. No conclusion follows

(a) If both the premises are particular

Example:

Statements: 1. Some books are pens. 2. Some pens are erasers.

Conclusions: 1. All books are erasers. 2. Some erasers are books.

Since both the premises are particular, so no definite conclusion follows.

(b) If both the premises are negative.

Example:

                                                                                                                                 

Statements: 1. No flower is mango. 2. No mango is cherry.

Conclusions: 1. No flower is cherry. 2. Some cherries are mangoes.

Since both the premises are negative, neither conclusion follows.

(c) if the major premise is particular and the minor premise is negative.

Example:

Statements: 1. Some D’s are B.          2. No T are D.

Conclusions: 1. No D are T.                2. Some B are T.

Here, the first premise containing the middle term 'D' as the subject is the major premise and the second premise containing the middle term 'D' as the predicate is the minor premise. Since the major premise is particular and the minor premise is negative, so no conclusion follows.

5. If the middle term is distributed twice, the conclusion cannot be universal.

Example:

Statements: 1. All flowers are chairs. 2. No tables are flowers.

Conclusions: 1. No tables are chairs. 2. Some tables are chairs.

Here, the first premise is an A proposition and so, the middle term 'flowers' forming the subject is distributed. The second premise is an E proposition and so, the middle term 'flowers' forming the predicate is distributed. Since the middle term is distributed twice, so the conclusion cannot be universal.

6. If one premise is negative, the conclusion must be negative.

Example:

Statements: 1. All grasses are trees. 2. No tree is shrub.

Conclusions: 1. No grasses are shrubs. 2. Some shrubs are grasses.

Since one premise is negative, the conclusion must be negative. So, conclusion 2 cannot follow.

      

7. If one premise is particular, the conclusion must be particular.

Example:

Statements: 1. Some boys are intelligent. 2. All intelligent are doctors.

Conclusions: 1. Some boys are doctors. 2. All doctors are boys.

Since one premise is particular, the conclusion must be particular. So, conclusion 2 cannot follow.

8. If both the premises are affirmative, the conclusion must be affirmative.

Example:

Statements: 1. All women are mothers. 2. All mothers are teachers.

Conclusions: 1. All women are teachers. 2. Some women are not teacher.

Since both the premises are affirmative, the conclusion must be affirmative. So, conclusion 2 cannot follow.

9. If both the premises are universal, the conclusion must be universal.

Complementary pair: A pair of contradictory statements i.e. a pair of statements such that if one is true, the other is false and when no definite conclusion can be drawn, either of them is bound to follow, is called a complementary pair. E and I-type propositions together form a complementary pair and usually either of them follows, in a case where we cannot arrive at a definite conclusion, using the rules of syllogism.

Let us study the various possible cases and draw all possible inferences in each case, along with verification through Venn diagrams.

Case 1: All men are boys. All boys are students.

Immediate Deductive Inferences:

The converse of first premise i.e. 'Some boys are men' and the converse of second premise i.e. 'Some students are boys' both hold.

Mediate Deductive Inferences:

Since both the premises are universal and affirmative, the conclusion must be universal affirmative. Also, the conclusion should not contain the middle term. So, it follows that 'All men are students'. The converse of this conclusion i.e. 'Some students are men' also holds.

Case 2: All birds are animals. All fishes are animals.

Immediate Deductive Inferences:

The converse of first premise i.e. 'Some animals are birds' and the converse of second premise i.e. 'Some animals are fishes' both hold.

Mediate Deductive Inferences:

Both, being A-type propositions, distribute subject only. Thus, the middle term 'animals' is not distributed even once in the premises. So, no definite conclusion follows.

Case 3: All puppets are dolls. Some dolls are rattles.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some dolls are puppets' and the converse of the second premise i.e. 'Some rattles are dolls', both hold.

           

Mediate Deductive Inferences: First premise, being an A-type proposition, distributes the subject only while the second premise, being an I-type proposition, distributes neither subject nor predicate. Since the middle term 'dolls' is not distributed even once in the premises, so no definite conclusion can be drawn.

Case 4: Some writers are players. All players are musicians.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some players are writers' and the converse of the second premise i.e. 'Some musicians are players', both hold.

Mediate Deductive Inferences: Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some writers are musicians'. The converse of this conclusion i.e. 'Some musicians are writers' also holds.

Case 5: All boxes are toys. Some boxes are clips.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some toys are boxes' and the converse of the second premise i.e. 'Some clips are boxes', both hold.

Mediate Deductive Inferences: Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some toys are clips'. The converse of this conclusion i.e. 'Some clips are toys' also holds.

           

Case 6: All buses are vans. Some cycles are vans.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some vans are buses' and the converse of the second premise i.e. 'Some vans are cycles', both hold.

Mediate Deductive Inferences: First premise, being an A-type proposition, distributes subject only and the second premise, being an I-type proposition, distributes neither subject nor predicate. So, the middle term 'vans' is not distributed even once in the premises. Hence, no definite conclusion can be drawn.

 Case 7: Some radios are cameras. Some cameras are statues.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some cameras are radios' and the converse of the second premise i.e. 'Some statues are cameras', both hold.

Mediate Deductive Inferences: Since both premises are particular, no definite conclusion follows.

Case 8: All cakes are candies. No candy is pastry.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some candies are cakes' and the converse of the second premise i.e. 'No pastry is candy', both hold.

Mediate Deductive Inferences: Since both premises are universal, the conclusion must be universal. Since one premise is negative, the conclusion must be negative. So, it follows that 'No cake is pastry'. The converse of this conclusion i.e. 'No pastry is cake' also holds.

Case 9: No coin is ring. All rings are bangles.

Immediate Deductive Inferences: The converse of the first premise i.e. 'No ring is coin' and the converse of the second premise i.e.’ Some bangles are rings', both hold.

Mediate Deductive Inferences: First premise, being an E-type proposition, distributes both the subject and the predicate.

Second premise, being an A-type proposition, distributes the subject. Thus, the middle term 'ring' is distributed twice in the premises. So, the conclusion cannot be universal. Also, since one premise is negative, the conclusion must be negative. Thus, the conclusion must be particular negative i.e. O-type. So, it follows that 'some bangles are not coins'.

Case 10: Some lamps are candles. No candle is bulb.

Immediate Deductive Inferences: The converse of the first premise i.e. 'Some candles are lamps' and the converse of the second premise i.e. 'No bulb is candle', both hold.

Mediate Deductive Inferences: Since one premise is particular and the other negative, the conclusion must be particular negative i.e. O-type, so, it follows that 'Some lamps are not bulbs'.

        

Points to Remember: While deriving logical conclusions, always remember that the following conclusions hold:

1. The converse of each of the given premises;

2. The conclusion that directly follows from the given premises in accordance with the rules of syllogism;

3. The converse of the derived conclusions.

Solved Examples

In each question below are given two statements followed by two conclusions numbered I and II. You have to take the given two statements to be true even if they seem to be at variance from commonly known facts. Read the conclusion and then decide which of the given conclusions logically follows from the two given statements, disregarding commonly known facts.

Example1. Statements: All P’s are G. Some B’s are not P.

Conclusions: (I) All G are B. (II) Some B are not G.

 A. Only conclusion I follows             B. Only conclusion II follows           

C. Either I or II follows                      D. Neither I nor II follows

1(D) Clearly, the term 'g' is distributed in both the conclusions without being distributed in any of the premises. So, neither conclusion follows.

Example2. Statements: Many books are trucks. All trucks are trains.

Conclusions: (I) Some books are trains. (II) No truck is books.

 A. Only conclusion I follows             B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

2(A) Since the first premise is particular, the conclusion must be particular and should not contain the middle term. Thus, only I follows.

Example3. Statements: All birds are animals. Some animals are hens.

Conclusions :( I) Some birds are hens. (II) Some hens are animals.

 A. Only conclusion I follows             B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

3(B) Since the middle term 'animals' is not distributed even once in the premises, no definite conclusion follows. However, II is the converse of the second premise and so it holds.

4. Statements: Some dogs are cats. Some horses are cats.

Conclusions :( I) Some cats are horses. (II) Some cats are dogs.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Both I and II follow

4(D) Since both premises are particular, no definite conclusion follows. However, I is the converse of second premise, while II is the converse of the first premise. So, both of them hold.

5. Statements: Some men are educated. Educated persons prefer nuclear families.

Conclusions: (I) All nuclear families are educated. (II)Some men prefer nuclear families.

 A. Only conclusion I follows             B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

5(B) Since one premise is particular; the conclusion must be particular and should not contain the middle term. Thus, only II follows.

                         

6. Statements: Most teachers are boys. Some boys are students.

Conclusions: (I) Some students are boys. (II) Some teachers are students.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

6(A) Since both the premises are particular, no definite conclusion follows. However, I is the converse of the second premise and thus it holds.

7. Statements: Some books are pens. No pen is pencil.

Conclusions: (I) Some books are pencils. (II) No book is pencil.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

7(C) The conclusion must be particular negative and should not contain the middle term. So, it follows that 'Some books are not pencils'. However, I and II involve only the extreme terms and form a complementary pair. Thus, either I or II follows.

8. Statements: No magazine is cap. All caps are cameras.

Conclusions: (I) No camera is magazine. (II) Some cameras are magazines.

 A. Only conclusion I follows             B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

8(C) As discussed above, the conclusion must be particular negative and should not contain the middle term. So, it follows that 'Some cameras are not magazines'. However, I and II involve only the extreme terms and form a complementary pair. Thus, either I or II follows.

9. Statements: Some books are tables. Some tables are mirrors.

Conclusions: (I) Some mirrors are books. (II) No book is mirror.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

 9(C) Since both the premises are particular no definite conclusion follows. However, I and II involve only the extreme terms and form a complementary pair. Thus, either I or II follows.

10. Statements: Raman is always successful. No fool is always successful.

Conclusions: (I) Raman is a fool.  (II) Raman is not a fool.

A. Only conclusion I follows             B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

10(B) Since both the premises are universal and one premise is negative, the conclusion must be universal negative and should not contain the middle term. So, only II follows.

Exercise

1.  Statements: All water is divine. All temples are divine.

Conclusions: (I) All water is temple. (II) All temples are water.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

2.  Statements: Some pastries are toffees. All toffees are chocolates.

Conclusions: (I) Some chocolates are toffees. (II) Some toffees are not pastries.

A. Only conclusion I follows              B. Only conclusion II follows

C. Either I or II follows                      D. Neither I nor II follows

3. Statements: Some bags are pockets. No pocket is a pouch.

Conclusions: (I) No bag is a pouch. (II) Some bags are not pouches. (III) Some pockets are bags. (IV) No pocket is a bag,

A. None follows                                 B. Only I and III follow

C. Only II and III follow                   D. Only either I or IV follows

4. Statements: All fruits are vegetables. All pens are vegetables. All vegetables are rains.

Conclusions: (I) All fruits are rains. (II) All pens are rains. (III) Some rains are vegetables.

A. None follows                                 B. Only I and II follow

C. Only II and III follow                   D. All follow

5.  Statements: Some towels are brushes. No brush is soap. All soaps are rats.

Conclusions: (I) Some rats are brushes. (II) No rat is brush. (III) Some towels are soaps.

A. None follows                                             B. Only either I or II follows

C. Only II follows                                          D. Only I and III follow

6. Statements: Some pictures are frames. Some frames are idols. All idols are curtains.

Conclusions: (I) Some curtains are pictures. (II) Some curtains are frames.

(III)Some idols are frames.

A. Only I and II follow                      B. Only II and III follow

C. Only I and III follow                     D. All follow

             

7. Statements: Some hills are rivers. Some rivers are deserts. All deserts are roads.

Conclusions: (I) Some roads are rivers. (II) Some roads are hills. (III) Some deserts are hills.

A. None follows                                             B. Only I follows

C. Only I and II follow                                  D. Only II and III follow

8. Statements: Some saints are balls. All balls are bats. Some tigers are balls.

Conclusions: (I) Some bats are tigers. (II) Some saints are bats.(III) All bats are balls.

A. Only I and II follow                      B. Only II follows

C. Only I and III follow                     D. Only III follows

9. Statements: Some pens are books. All schools are books. Some colleges are schools.

Conclusions: (I) Some colleges are pens. (II) Some pens are schools. (III) Some colleges are books.

A. Only I and II follow                      B. Only II and III follow

C. Only I and III follow                     D. None of these

10. Statements: All trains are buses. No room is bus. All boats are rooms.

Conclusions: (I) No boat is train. (II) No bus is boat. (III) No train is room.

A. None follows                                 B. Only I and II follow

C. Only II and III follow                   D. All follow

11. Statements: Some mountains are hillocks. Some mountains are rivers. Some mountains are valleys.

Conclusions: (I) All mountains are either hillocks or rivers or valleys. (II) No valley is river. (III) Some river are valleys.

A. None follows                                 B. Only I follows

C. Only either II or III follows           D. Only III follows

12. Statements: Some blades are hammers. Some hammers are knives. Some knives are axes.

Conclusions: (I) Some axes are hammers. (II)  Some knives are blades. (III) Some axes are blades.

A. None follows                                 B. Only I follows

C. Only II follows                              D. Only III follows

13. Statements: Some boxes are hammers. Some hammers are beads. All beads are rings.

Conclusions: (I) Some rings are hammers. (II) Some hammers are boxes. (III) Some rings are boxes.

A. Only I follows

B. Only I and II follow

C. Only II and III follow

D. All follow

14. Statements: Some blankets are beds. Some pillows are blankets. All beds are pillows.

Conclusions: (I) Some blankets are pillows. (II) Some pillows are beds. (III) Some beds are blankets.

A. Only either I or II follows                                     B. Only I and either II or III follow

C. Only III and either I or II follow               D. All I, II and III follow

15. Statements: All dolls are windows. All bottles are windows. All cars are bottles.

Conclusions: (I) All cars are windows. (II) Some cars are dolls. (III) Some windows are cars.

A. Only I and II follow                                  B. Only II and III follow

C. Only I and III follow                                 D. All follow

Directions (Ques nos.16-18): In each of the following questions, examine the given statements carefully and find out which two of the statements cannot be true simultaneously, but can both be false.

16. Examine the following statements regarding a set of balls

            1. All balls are black.

            2. All balls are white.

            3. Only some balls are black.

            4. No balls are black.

Assuming that the balls can only be black or white, which of the two statements given above can both be true, but cannot both be false?

A. 1 and 3       B. 1 and 4       C. 2 and 3       D. 2 and 4

17. 1. All children are inquisitive.

2. Some children are inquisitive.

3. No children are inquisitive.

4. Some children are inquisitive.

    A. 1 and 3   B. 1 and 4       C. 2 and 3       D. 3 and 4

18. Consider the following three statements: (UPSC Pre GS Paper-2, 2011)

1. Only students can participate in the race.
2. Some participants in the race are girls.
3. All girl participants in the race are invited for coaching.
Which one of the following conclusions can be drawn from the above statements?

(a) All participants in the race are invited for coaching.
(b) All students are invited for coaching.
(c) All participants in the race are students.
(d) None of the statements (a), (b) and (c) given above is correct.

Directions (Ques. Nos 19-20): Each of the following two items consists of four statements. Of these four statements, two cannot both be true, but both can be false. Study the statements carefully and identify the two that satisfy the above condition. Select the correct answer using the codes given below each set of statements:

19. Examine the following statements: (UPSC Pre GS Paper-2, 2011)

1. All animals are carnivorous.
2. Some animals are not carnivorous.
3. Animals are not carnivorous.
4. Some animals are carnivorous.

(a) 1 and 3       (b) 1 and 2       (c) 2 and 3       (d) 3 and 4

     20. Examine the following statements:  (UPSC Pre GS Paper-2, 2011)

1. All trains are run by diesel engine.
2. Some trains are run by diesel engine.
3. No train is run by diesel engine.
4. Some trains are not run by diesel engine.

(a) 1 and 2       (b) 2 and 3       (c) 1 and 3       (d) 1 and 4

21. Consider the following argument:  (UPSC Pre GS Paper-2, 2011)

"In order to be a teacher one must graduate from college. All poets are poor. Some Mathematicians are poets. No college graduate is poor." Which one of the following is not a valid conclusion regarding the above argument?

(a) Some Mathematicians are not teachers.
(b) Some teachers are not Mathematicians.
(c) Teachers are not poor,
(d) Poets are not teachers.

                                                Solution

1(D) Since the middle term 'divine' is not distributed even once in the premises, no definite conclusion can be drawn.

2(A) Since one premise is particular; the conclusion must be particular and should not contain the middle term. Thus, it follows that 'Some pastries are chocolates', I is the converse of the second premise and so it holds. Since both the premises are affirmative, the conclusion cannot be negative. Thus, II does not follow.

3(C) Since one premise is particular and the other negative, the conclusion must be particular negative and should not contain the middle term. So, II follows. III is the converse of the first premise and thus it also holds.

4(D) III is the converse of the third premise and so it holds. All fruits are vegetables. All vegetables are rains. The conclusion must be universal affirmative and should not contain the middle term. So, it follows that 'All fruits are rains'. Thus, I follows. All pens are vegetables. All vegetables are rains. Clearly, it follows that 'All pens are rains'. Thus, II follows.

5(B) Some towels are brushes. No brush is soap. Since one premise is particular and the other negative, the conclusion must be particular negative (O-type) and should not contain the middle term. So, it follows that 'Some towels are not soaps'. No brush is soap. All soaps are rats. Since the middle term is distributed twice, the conclusion must be particular. Since one premise is negative, the conclusion must be negative. So, it follows that 'Some brushes are not rats'. Since I and II involve the same terms and form a complementary pair, so either I or II follows.

6(B) III is the converse of the second premise and so it holds. Some pictures are frames. Some frames are idols. Since both the premises are particular, no definite conclusion follows. Some frames are idols. All idols are curtains. Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some frames are curtains'. III is the converse of this conclusion and so it holds. Some pictures are frames. Some frames are curtains. Since both the premises are particular, no definite conclusion can be drawn.

7(B) Some hills are rivers. Some rivers are deserts. Since both the premises are particular, no definite conclusion follows. Some rivers are deserts. All deserts are roads.Since one premise is particular; the conclusion must be particular and shouldn't contain the middle term. So, it follows that 'Some rivers are roads'. I is the converse of this conclusion and so it holds. Some hills are rivers. Some rivers are roads. Again, since both the premises are particular, no definite conclusion follows.

8(A) Some saints are balls. All balls are bats. Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some saints are bats'. Thus, II follows. Some tigers are balls. All balls are bats.

Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some tigers are bats'. I is the converse of this conclusion and so it holds.

9(D) Some pens are books. All schools are books. Since the middle term 'books' is not distributed even once in the premises, so no definite conclusion follows. Some colleges are schools. All schools are books. Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some colleges are books'. Thus, III follows. Some pens are books. Some colleges are books. Since both the premises are particular, no definite conclusion can be drawn. Hence, only III follows.

10(D) All trains are buses. No room is bus. Since both the premises are universal and one premise is negative, the conclusion must be universal negative (E-type) and should not contain the middle term. So, it follows that 'No train is room'. Thus, III follows. All boats are rooms. No room is bus. As discussed above, it follows that 'No boat is bus'. II is the converse of this conclusion and so it holds. All trains are buses. No boat is bus. Again, it follows that 'No train is boat'. I is the converse of this conclusion and so it holds.

11(C) Since each combination of premises shall contain two particular premises, no definite conclusion can be drawn. However, II and III are statements involving the extreme terms of the last two premises and form a complementary pair. Thus, either II or III follows

12(A) Since each combination of premises has two particular premises, so no definite conclusion follows.

13(C) II is the converse of first premise and so it holds. Some boxes are hammers. Some hammers are beads. Since both the premises are particular, no definite conclusion can be drawn. Some hammers are beads. All beads are rings. Since one premise is particular, the conclusion must be particular and should not contain the middle term. So, it follows that 'Some hammers are rings'. I is the converse of this conclusion and so it holds. Some boxes are hammers. Some hammers are rings. Since both the premises are particular, no definite conclusion can be drawn.

14(D) I is the converse of the second premise, II is the converse of the third premise and III is the converse of the first premise and as such, all three of them follow.

15(C) All cars are bottles. All bottles are windows. Since both the premises are universal, the conclusion must be universal and shouldn't contain the middle term, So, it follows that 'All cars are windows'. Thus, I follows. Also, III is the converse of this conclusion and so it holds. All dolls are windows. All bottles are windows. Since the middle term 'windows' is not distributed even once in the premises, no definite conclusion follows.

All cars are windows. All bottles are windows. Again, the middle term 'windows' is not distributed even once in the premises. So, no definite conclusion follows.

16 (B) 1 and 4                         17 (A) 1 and 3

18 (C) From 1 it is clear that all the participants in the race are students.

19 (A) Working with the options, we will realize that Statements “All animals are carnivorous” and “Animals are not carnivorous” cannot be true together; as if one of the statements is true then the other is obviously false. But there is a situation where in both the statements are false that is if some of the animals are carnivorous and some are not.

20(C) Working with the options, we will realize that Statements “All trains are run by diesel engine” and “No trains is run by diesel engine” cannot be true together; as if one of the statements is true then the other is obviously false. But there is a situation where in both the statements are false that is if some of the trains run on diesel engine and some are not.

21(B) (a) is valid since some mathematicians are poets but no teacher is poet. So, some mathematicians are not teachers.
(b) is invalid as it cannot be concluded.
(c) is valid as teachers are not poor and hence not poets.
(d) is valid as none college graduate is poor and hence not poet.