logical deduction

LOGICAL DEDUCTION

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

Thus, 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 because they refer to every object in a certain set, while the

quantifier 'some' is a particular quantifier because it refers to at least one existing object in a certain

set.

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

3. Predicate (denoted by '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.

Examples:

Four-Fold Classification of Propositions :

A proposition is said to have a universal quantity if it begins with a universal quantifier, and a

particular quantity if it begins with a particular quantifier. 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 (denoted by 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 snakes are reptiles. This is proposition A since we cannot say 'All reptiles are snakes'.

2. Universal Negative Proposition (denoted by 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 boy is intelligent.

3.Particular Affirmative Proposition (denoted by I): It distributes neither the subject nor the

predicate.

e.g.,Some men are foolish. Here, the subject term 'men' is used not for all but only for some men and

similarly the predicate term 'foolish' is affirmed for a part of subject class. So, both are undistributed.

4. Particular Negative Proposition (denoted by O): It distributes only the predicate. e.g., Some

animals are not wild. Here, the subject term 'animals' is used only for a part of its class and hence is

undistributed while the predicate term 'wild' 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

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

Ex. All pins are tops.

I: Some P is S

Some tops are pins.

E: No S is P.

Ex. No fish is whale.

E: No P is S.

No whale is fish.

I: Some S is P.

Ex. Some boys are poets.

I: Some P is S.

Some poets are boys.

O: Some S is not P. No valid conversion

Note that 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.

Note: The valid converse, obverse or contrapositive of a given proposition always logically follows from

the proposition.

II. Mediate Deductive Inference (SYLLOGISM): First introduced by Aristotle, 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.

2. All flowers are beautiful.

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 tigers are dogs.

Conclusion :

All tigers are animals.

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

'Tigers' 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 girls.

2. Some girls are students.

Conclusions :

1. All girls are men.

2. Some girls 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 goats.

2. All goats are cows.

Conclusions :

1. All cows are goats.

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. 'goats' 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 fans are watches.

2. Some watches are black.

Conclusions :

1. All watches are fans.

2. Some fans are black.

In the premises, the middle term is 'watches'. 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 dogs are bulls.

2. No tigers are dogs.

Conclusions:

1. No dogs are tigers.

2. Some bulls are tigers.

Here, the first premise containing the middle term 'dogs' as the subject is the major premise and the

second premise containing the middle term 'dogs' 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 fans are chairs.

2. No tables are fans.

Conclusions:

1. No tables are chairs.

2. Some tables are chairs.

Here, the first premise is an A proposition and so, the middle term 'fans' forming the subject is

distributed. The second premise is an E proposition and so, the middle term 'fans' 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 thieves.

2. All thieves are dacoits.

Conclusions :

1. Some boys are dacoits.

2. All dacoits 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 sisters.

Conclusions :

1. All women are sisters.

2. Some women are not sisters.

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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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'.

Important 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.

Directions to Solve

1. Statements: All benches are desks. Some desks are roads. All roads are pillars.

Conclusions:

I. Some pillars are benches.

II. Some pillars are desks.

III. Some roads are benches.

IV. No pillar is bench.

A. None follows

B. Only either I or IV, and III follow

C. Only either I or IV follows

D. Only either I or IV, and II follow

E. All follow

Answer & Explanation

Answer: Option D

Explanation:

All benches are desks. Some desks are roads.

Since the middle term 'desks' is not distributed even once in the premises, no definite

conclusion follows.

Some desks are roads. All roads are pillars.

Since one premise is particular, the conclusion must be particular and should not contain

the middle term. So, it follows that 'Some desks are pillars'. II is the converse of this

conclusion and so it holds.

All benches are desks. Some desks are pillars.

Since the middle term 'desks' is not distributed even once in the premises, no definite

conclusion follows. However, I and IV involve the extreme terms and form a

complementary pair. So, either I or IV follows.

2. Statements: Some dogs are rats. All rats are trees. Some trees are not dogs.

Conclusions:

I. Some trees are dogs.

II. All dogs are trees.

III. All rats are dogs.

IV. No tree is dog.

A. None follows

B. Only I follows

C. Only I and II follow

D. Only II and III follow

E. All follow


Answer: Option B

Explanation:

Some dogs are rats. All rats are trees.


the middle term. So, it follows that 'Some dogs are trees'. I is the converse of this


All rats are trees. Some trees are not dogs.

Since the middle term 'trees' is not distributed even once in the premises, no definite

conclusion follows.

3. Statements: Some bricks are trees. All trees are pens. All pens are boats.

Conclusions:

I. Some boats are bricks.

II. Some pens are bricks.

III. Some trees are bricks.

IV. Some bricks are boats.

A. Only I and II follow

B. Only III and IV follow

C. None follows

D. All follow

E. None of these


Answer: Option D

Explanation:

III is the converse of the first premise and so it holds.

Some bricks are trees. All trees are pens.


the middle term. So, it follows that 'Some bricks are pens'. II is the converse of this


All trees are pens. All pens are boats.

Since both the premises are universal and affirmative, the conclusion must be universal

affirmative and should not contain the middle term. So, it follows that 'All trees are

boats'.

Some bricks are trees. All trees are boats.


the middle term. So, it follows that 'Some bricks are boats'. Thus, IV follows. I is the

converse of this conclusion and so it also holds.

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4. Statements: All cups are glasses. Some glasses are bowls. No bowl is a plate.

Conclusions:

I. No cup is a plate.

II. No glass is a plate.

III. Some plates are bowls.

IV. Some cups are not glasses.

A. None follows

B. Only either I or III follows

C. Only II and III follow

D. Only III and IV follow

E. None of these


Answer: Option A

Explanation:

All cups are glasses. Some glasses are bowls.

Since the middle term 'glasses' is not distributed even once in the premises, no definite

conclusion follows.

Some glasses are bowls. No bowl is a plate.

Since one premise is particular and the other negative, the conclusion must be particular

negative and should not contain the middle term. So, it follows that 'Some glasses are

not plates'.


5. Statements: Some trains are roads. No road is jungle. All flowers are jungles.

Conclusions:

I. Some trains are flowers.

II. Some trains are jungles.

III. Some flowers are trains.

IV. No road is flower.

A. None follows

B. Only II follows

C. Only III follows

D. Only IV follows

E. All follow


Answer: Option D

Explanation:

Some trains are roads. No road is jungle.

Since one premise is particular and the other negative, the conclusion must be particular

negative and should not contain the middle term. So, it follows that 'Some trains are not

jungles'.

No road is jungle. All flowers are jungles.

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, it follows that 'No

flower is road'. IV is the converse of this conclusion and so it holds.

Some trains are roads, No flower is road.

As discussed above, it follows that 'Some trains are not flowers'.

6. Statements: All doors are buses. All buses are leaves. No leaf is a flower.

Conclusions:

I. No flower is a door.

II. No flower is a bus.

III. Some leaves are doors.

IV. Some leaves are buses.

A. None follows

B. Only I and II follow


D. Only II, III and IV follow

E. All follow


Answer: Option E

Explanation:

IV is the converse of the second premise and so it holds.

All doors are buses. All buses are leaves.


affirmative and should not contain the middle term. So, it follows that 'All doors are leaves'. III is

the converse of this conclusion and so it holds.

All buses are leaves. No leaf is a flower.

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, it follows that 'No bus is flower'. II

is the converse of this conclusion and so it holds.

All doors are buses. No bus is flower.

As discussed above, it follows that 'No door is flower'. I is the converse of this conclusion and so it

also holds.


7. Statements: All oceans are rivers. Some springs are rivers. All wells are springs.

Conclusions:

I. Some springs are oceans.

II. Some wells are rivers.

III. Some rivers are oceans.

IV. No well is river.

A. None follows

B. Only either I or III, and IV follow

C. Only either II or IV, and III follow

D. All follow

E. Only either II or IV, and I follow


Answer: Option C

Explanation:


All oceans are rivers. Some springs are rivers.

Since the middle term 'rivers' is not distributed even once in the premises, no definite conclusion

follows.

All wells are springs. Some springs are rivers.

Since the middle term 'springs' is not distributed even once in the premises, no definite conclusion

follows. However, II and IV involve the extreme terms and form a complementary pair. Thus,

either II or IV follows.


8. Statements: Some tigers are lions. Some lions are rabbits. Some rabbits are horses.

Conclusions:

I. Some tigers are horses.

II. Some rabbits are tigers.

III. Some horses are lions.

IV. All horses are rabbits.

A. All follow

B. None follows


D. Only II and IV follow

E. Only IV follows


Answer: Option B

Explanation:

Since each combination of premises shall contain two particular premises, no definite conclusion

can be drawn.


9. Statements: Some spoons are bowls. All bowls are knives. All knives are forks.

Conclusions:

I. All spoons are forks.

II. All bowls are forks.

III. Some knives are bowls.

IV. Some forks are spoons.

A. Only II and III follow

B. Only II and IV follow

C. Only III and IV follow

D. All follow

E. None of these


Answer: Option E

Explanation:

III is the converse of the second premise and so it holds.

Some spoons are bowls. All bowls are knives.

Since one premise is particular, the conclusion must be particular and should not contain the

middle term. So, it follows that 'Some spoons are knives'.

All bowls are knives. All knives are forks.


affirmative and should not contain the middle term. So, it follows that.

'All bowls are forks'. Thus, II follows.

Some spoons are knives. All knives are forks.


middle term.

So, it follows that 'Some spoons are forks'. IV is the converse of this conclusion and so it follows.

Hence, II, III and IV follow.


10. Statements: All pencils are birds. All birds are skies. All skies are hills.

Conclusions:

I. All pencils are hills.

II. All hills are birds

III. All skies are pencils.

IV. All birds are hills.


B. Only I and III follow


D. All follow

E. None of these


Answer: Option E

Explanation:

All pencils are birds. All birds are skies.


affirmative (A-type) and should not contain the middle term. So, it follows that 'All pencils are

skies'.

All birds are skies. All skies are hills.

As discussed above, it follows that 'All birds are hills'. Thus, IV follows.

All pencils are skies. All skies are hills.

Clearly, it follows that 'All pencils are hills'. Thus, I follows.

Hence, I and IV follow.

11. Statements: No tree is fruit. All fruits are stones. All stones are rains.

Conclusions:

I. No stone is tree.

II. No rain is tree.

III. Some rains are fruits.

IV. Some rains are trees.

A. Only either II or III, and I follow

B. None follows

C. Only either II or IV, and III follow

D. All follow

E. None of these


Answer: Option C

Explanation:

No tree is fruit. All fruits are stones.

Since the middle term 'fruits' is distributed twice, the conclusion must be particular.

Since one premise is negative, the conclusion must be negative. So, it follows that

'Some stones are not trees'.

All fruits are stones. All stones are rains.

Clearly, it follows that 'All fruits are rains'. III is the converse of this conclusion and so it holds.

No tree is fruit, All fruits are rains.

As discussed above, the conclusion must be particular negative and should not contain the middle

term. So, it follows that 'Some rains are not trees'. However, II and IV involve only the extreme

terms and form a complementary pair. Thus, either II or IV follows.


12. Statements: All players are spectators. Some spectators are theatres. Some theatres are

dramas.

Conclusions:

I. Some dramas are spectators.

II. Some players are dramas.

III. Some theatres are players.

IV. All spectators are players.

A. None follows

B. Only I and III follow

C. Only II follows

D. Only II and IV follow

E. All follow


Answer: Option A

Explanation:

All players are spectators. Some spectators are theatres.

Since the middle term 'spectators' is not distributed even once in the premises, no definite

conclusion follows.

Some spectators are theatres. Some theatres are dramas.

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


13. Statements: All doors are roads. No road is fruit. Some flowers are doors.

Conclusions:

I. Some fruits are doors.

II. Some fruits are flowers.

III. Some roads are flowers.

IV. No fruit is flower.

A. Only either II or III, and IV follow

B. Only either II or IV, and III follow

C. Only either II or IV, and I follow

D. Only either II or IV follows

E. All follow


Answer: Option B

Explanation:

All doors are roads. No road is fruit.


universal negative and should not contain the middle term. So, it follows that 'No door is fruit.'

Some flowers are doors. All doors are roads.


middle term. So, it follows that 'Some flowers are roads'. Ill is the converse of this conclusion and

so it holds.

Some flowers are roads. No road is fruit.

Since one premise is particular and the other negative, the conclusion must be particular negative

and should not contain the middle term. So, it follows that 'Some flowers are not fruits'. II and IV

involve the extreme terms and form a complementary pair. Thus, either II or IV follows.


14. Statements: All needles are threads. All threads are boxes. All trees are boxes.

Conclusions:

I. No needle is tree.

II. Some trees are threads.

III. Some boxes are needles.

IV. Some trees are needles.

A. None follows

B. Only either I or IV follows

C. Only either I or IV, and II follow

D. Only III follows

E. Only either I or IV, and III follow


Answer: Option E

Explanation:

All needles are threads. All threads are boxes.


affirmative (A-type) and should not contain the middle term. So, it follows that 'All needles are

boxes'. III is the converse of this conclusion and so it holds.

All threads are boxes. All trees are boxes.

Since the middle term 'boxes' is not distributed even once in the premises, no definite conclusion

follows.

All needles are boxes. All trees are boxes.

Again, since the middle term 'boxes' is not distributed even once in the premises, no definite

conclusion can be drawn. However, I and IV involve the extreme terms of these two statements

and form a complementary pair. Thus, either I or IV follows.


15. Statements: No house is school. All colleges are schools. All schools are teachers.

Conclusions:

I. No house is teacher.

II. All colleges are teachers.

III. Some teachers are not houses.

IV. No college is house.

A. None follows

B. Only either I or IV follows

C. Only II, III and IV follow

D. All follow

E. Only either I or IV, and III follow


Answer: Option C

Explanation:

All colleges are schools. No house is school.


universal negative and should not contain the middle term. So, it follows that 'No college is

house'. Thus, IV follows.

All colleges are schools. All schools are teachers.

Clearly, it follows that 'All colleges are teachers'. Thus, II follows.

No house is school. All schools are teachers.

Since the middle term 'schools' is distributed twice, the conclusion must be particular.

Since one premise is negative, the conclusion must be negative. So, it follows that 'Some teachers

are not houses'.


16. Statements: Some pearls are stones. Some stones are diamonds. No diamond is a gem.

Conclusions:

I. Some gems are pearls.

II. Some gems are diamonds.

III. No gem is a diamond.

IV. No gem is a pearl.


B. Only III and IV follow

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

D. Only III and either I or IV follow

E. None of these


Answer: Option D

Explanation:

III is the converse of the third premise and so it holds.

Some pearls are stones. Some stones are diamonds.


Some stones are diamonds. No diamond is a gem.


and should not contain the middle term. So, it follows that 'Some stones are not gems'.

However, I and IV involve the extreme terms of the three premises and form a complementary

pair, Thus, either I or IV follows.


17. Statements: All rods are bricks. Some bricks are ropes. All ropes are doors.

Conclusions:

I. Some rods are doors.

II. Some doors are bricks.

III. Some rods are not doors.

IV. All doors are ropes.


B. Only I, II and III follow

C. Only either I or III, and II follow

D. Only either I or III, and IV follow

E. None of these


Answer: Option E

Explanation:

All rods are bricks. Some bricks are ropes.

Since the middle term 'bricks' is not distributed even once in the premises, no definite conclusion

follows.

Some bricks are ropes. All ropes are doors.


middle term. So, it follows that 'Some bricks are doors'. II is the converse of this conclusion and

so it holds.

All rods are bricks. Some bricks are doors.

Since the middle term 'bricks' is not distributed even once in the premises, no definite conclusion

follows.

However, I and III involve the extreme terms. But, since they are not contradictory, they do not

form a complementary pair.

Hence, only II follows.


18. Statements: All myths are fictions. No fiction is novel. All novels are stories.

Conclusions:

I. No myth is novel.

II. Some fictions are novels.

III. Some fictions are myths.

IV. Some myths are novels.

A. Only either I or II and both III and IV follow

B. Only either I or IV and II follow

C. Only either I or IV and both II and III follow

D. All follow

E. None of these


Answer: Option E

Explanation:

III is the converse of first premise and so it holds.

All myths are fictions. No fiction is novel.


universal negative and should not contain the middle term. So, it follows that 'No myth is novel'.

Thus, I follows.

No fiction is novel. All novels are stories.

Since the middle term 'novels' is distributed twice in the premises, the conclusion must be

particular. Since one premise is negative, the conclusion must be negative.

So, it follows that 'Some stories are not fictions'.

Hence, only I and III follow.


19. Statements: No paper is pen. No pen is pencil. All erasers are papers.

Conclusions:

I. Some papers are erasers.

II. No pencil is eraser.

III. No pen is eraser.

IV. All papers are erasers.


B. Only II and III follow

C. Only I, II and III follow

D. All follow

E. None of these


Answer: Option E

Explanation:

I is the converse of the third premise and so it holds.

No paper is pen. No pen is pencil.

Since both the premises are negative, no definite conclusion follows.

All erasers are papers. No paper is pen.


universal negative and should not contain the middle term. So, it follows that 'No eraser is pen'.

III is the converse of this conclusion and so it holds.

Hence, only I and III follow.


20. Statements: No man is sky. No sky is road. Some men are roads.

Conclusions:

I. No road is man.

II. No road is sky.

III. Some skies are men.

IV. All roads are men.

A. None follows

B. Only I follows


D. Only I and III follow

E. None of these


Answer: Option E

Explanation:

II is the converse of the second premise and so it holds.

No man is sky. No sky is road.


No man is sky. Some men are roads.


and should not contain the middle term. So, it follows that 'Some roads are not skies'.

No sky is road. Some men are roads.

As discussed above, it follows that 'Some men are not skies'.

Hence, only II follows.


21. Statements: All buildings are windows. No toys is building. Some tigers are toys.

Conclusions:

I. Some tigers are buildings.

II. Some windows are tigers.

III. All toys are tigers.

IV. Some windows are toys.

A. None follows




E. All follow


Answer: Option A

Explanation:

No toy is building. All buildings are windows.

Since the middle term 'buildings' is distributed twice and one premise is negative, the conclusion

must be particular negative and should not contain the middle term.

So, it follows that 'Some windows are not toys'.

Some tigers are toys. No toy is building.

Since one premise is particular and the other premise is negative, the conclusion must be

particular negative and should not contain the middle term. So, it follows that 'Some tigers are not

buildings'.


22. Statements: Some papers are cats. All cats are bats. No bat is horse.

Conclusions:

I. Some papers are horses.

II. No horse is cat.

III. Some bats are papers.

IV. All papers are bats.




D. Only I and IV follow

E. All follow


Answer: Option B

Explanation:

Some papers are cats. All cats are bats.


middle term. So, it follows that 'Some papers are bats'. III is the converse of this conclusion and

so it holds.

All cats are bats. No bat is horse.


universal negative and should not contain the middle term. So, it follows that 'No cat is horse'. II

is the converse of this conclusion and so it holds.

Some papers are bats. No bat is horse.


and should not contain the middle term. So, it follows that 'Some papers are not horses'.


23. Statements: Some tapes are discs. Some discs are cassettes. Some cassettes are songs.

Conclusions:

I. Some songs are discs.

II. Some cassettes are tapes.

III. Some songs are tapes.

IV. No song is a disc.

A. Only either I or IV follows

B. Only either II or IV follows


D. Only III and either II or IV follows

E. None of these


Answer: Option A

Explanation:


can be drawn. However, I and IV involve the extreme terms of the second and third premises and

form a complementary pair. Thus, either I or IV follows.


24. Statements: No table is fruit. No fruit is window. All windows are chairs.

Conclusions:

I. No window is table.

II. No chair is fruit.

III. No chair is table.

IV. All chairs are windows.

A. None follows



D. All follow

E. None of these


Answer: Option A

Explanation:

No table is fruit. No fruit is window.


No fruit is window. All windows are chairs.

Since the middle term 'windows' is distributed twice and one premise is negative, the conclusion

must be particular negative. So, it follows that 'Some chairs are not fruits'.


25. Statements: All jungles are buses. All books are buses. All fruits are books.

Conclusions:

I. Some fruits are jungles.

II. Some buses are books.

III. Some buses are jungles.

IV. All fruits are buses.

A. Only I, II and III follow

B. Only I, II and IV follow

C. Only II, III and IV follow

D. All follow

E. None of these


Answer: Option C

Explanation:

III is the converse of the first premise and II is the converse of the second premise.

So, both of them hold.

All fruits are books. All books are buses.


affirmative and should not contain the middle term. So, it follows that 'All fruits are buses'. Thus,

IV follows.

All jungles are buses. All books are buses.

Since the middle term 'buses' is not distributed ever once in the premises, no definite conclusion

follows.

All fruits are buses. All books are buses.

As discussed above, no definite conclusion can be drawn.

All jungles are buses. All fruits are buses.

Again, no definite conclusion follows.


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1. 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




E. All follow


Answer: Option E

Explanation:

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.


2. 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


E. None of these


Answer: Option B

Explanation:

Some towels are brushes. No brush is soap.


(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.


3. 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.



C. Only I and III follow

D. All follow

E. None of these


Answer: Option B

Explanation:

III is the converse of the second premise and so it holds.

Some pictures are frames. Some frames are idols.


Some frames are idols. All idols are curtains.


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.


4. 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



E. All follow


Answer: Option B

Explanation:

Some hills are rivers. Some rivers are deserts.


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.


5. 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.


B. Only II follows


D. Only III follows

E. None of these.


Answer: Option A

Explanation:

Some saints are balls. All balls are bats.


middle term. So, it follows that 'Some saints are bats'. Thus, II follows. Some tigers are balls. All

balls are bats.


middle term. So, it follows that 'Some tigers are bats'. I is the converse of this conclusion and so

it holds.


6. 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.




D. All follow

E. None of these


Answer: Option E

Explanation:

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.


middle term.

So, it follows that 'Some colleges are books'. Thus, III follows.

Some pens are books. Some colleges are books.


Hence, only III follows.


7. 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




E. All follow


Answer: Option E

Explanation:

All trains are buses. No room is bus.


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.


8. 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

E. None of these


Answer: Option C

Explanation:


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.


9. 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

E. None of these


Answer: Option A

Explanation:

Since each combination of premises has two particular premises, so no definite conclusion follows.


10. 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. None follows

B. Only I follows



E. All follow


Answer: Option C

Explanation:

II is the converse of first premise and so it holds.

Some boxes are hammers. Some hammers are beads.


Some hammers are beads. All beads are rings.


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.



11. 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

E. None of these


Answer: Option D

Explanation:

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.


12. 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.




D. All follow

E. None of these


Answer: Option C

Explanation:

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.


13. Statements: All tigers are lions. No cow is lion. Some camels are cows.

Conclusions:

I. Some lions are camels.

II. No camel- is tiger.

III. Some tigers are cows.

A. None follows

B. Only I follows

C. Only II follows

D. Only III follows

E. Either I or II follows


Answer: Option A

Explanation:

All tigers are lions. No cow is lion.


universal negative (E-type) and shouldn't contain the middle term. So, it follows that 'No tiger is

cow'.

Some camels are cows. No cow is lion.


(O-type) and should not contain the middle term. So, it follows that 'Some camels are not lions'.

Some camels are cows. No tiger is cow.


(O-type) and should not contain the middle term. So, it follows that 'Some camels are not tigers'.


14. Statements: All flowers are toys. Some toys are trees. Some angels are trees.

Conclusions:

I. Some angels are toys.

II. Some trees are flowers.

III. Some flowers are angels.

A. None follows

B. Only I follows

C. Only II follows

D. Only III follows

E. Only I and III follow


Answer: Option A

Explanation:

All flowers are toys. Some toys are trees.

Since the middle term 'toys' is not distributed even once in the premises, no definite conclusion

follows.

Some toys are trees. Some angels are trees.



15. Statements: Some rats are cats. Some cats are dogs. No dog is cow.

Conclusions:

I. No cow is cat.

II. No dog is rat.

III. Some cats are rats.

A. None follows



D. Only III follows

E. All I, II and III follow


Answer: Option D

Explanation:


Some rats are cats. Some cats are dogs.


Some cats are dogs. No dog is cow.


(O-type) and should not contain the middle term. So, it follows that 'Some cats are not cows'.


16. Statements: All tigers are jungles. No jungle is bird. Some birds are rains.

Conclusions:

I. No rain is jungle.

II. Some rains are jungles.

III. No bird is tiger.


B. Only III follows

C. Only either I or II, and III follow

D. All follow

E. None of these


Answer: Option C

Explanation:

All tigers are jungles. No jungle is bird.


universal negative (E-type) and should not contain the middle term.

So, it follows that 'No tiger is bird'. III is the converse of this conclusion and so it holds.

No jungle is bird. Some birds are rains.


(O-type) and should not contain the middle term. So, it follows that 'Some jungles are not rains'.

Since I and II also involve the same terms and form a complementary pair, so either I or II

follows.


17. Statements: All snakes are trees. Some trees are roads. All roads are mountains.

Conclusions:

I. Some mountains are snakes.

II. Some roads are snakes.

III. Some mountains are trees.

A. Only I follows

B. Only II follows

C. Only III follows

D. Both I and II follow

E. None follows


Answer: Option C

Explanation:

All snakes are trees. Some trees are roads.

Since the middle term is not distributed even once in the premises, so no definite conclusion

follows.

Some trees are roads. All roads are mountains.


middle term. So, it follows that 'Some trees are mountains'. III is the converse of this conclusion

and so it holds.

All snakes are trees. Some trees are mountains.

Since the middle term is not distributed even once in the premises, so no definite conclusion

follows.


18. Statements: All trees are flowers. No flower is fruit. All branches are fruits.

Conclusions:

I. Some branches are trees.

II. No fruit is tree.

III. No tree is branch.

A. None follows


C. Only II follows

D. Only either I or III, and II follow

E. None of these


Answer: Option E

Explanation:

All trees are flowers. No flower is fruit.


universal negative (E-type) and should not contain the middle term. So, it follows that 'No tree is

fruit'. II is the converse of this conclusion and so it follows.

All branches are fruits. No flower is fruit.


universal negative (E-type) and should not contain the middle term. So, it follows that 'No branch

is flower'.

All trees are flowers. No branch is tree.

As discussed above, it follows that 'No tree is branch'. So, III follows.

Hence, both II and III follow.


19. Statements: Some uniforms are covers. All covers are papers. All papers are bags.

Conclusions:

I. All covers are bags.

II. Some bags are covers, papers and uniforms.

III. Some uniforms are not papers.

A. Only I follows


C. Only III follows

D. All I, II and III follow

E. None of these


Answer: Option B

Explanation:

Some uniforms are covers. All covers are papers.


middle term. So, it follows that 'Some uniforms are papers'. All covers are papers. All papers are

bags.


affirmative (A-type) and should not contain the middle term. So, it follows that 'All covers are

bags'. Thus, I follows. The converse of this conclusion i.e. 'Some bags are covers' also holds.

Some uniforms are covers. All covers are bags.


middle term. So, it follows that 'Some uniforms are bags', The converse of this conclusion i.e.

'Some bags are uniforms' also holds.

Further, the converse of the third premise i.e. 'Some bags are papers' holds.

Now, II is the cumulative result of the conclusions 'Some bags are covers', 'Some bags are papers'

and 'Some bags are uniforms'. Thus, II follows.


20. Statements: No rabbit is lion. Some horses are lions. All rabbits are tables.

Conclusions:

I. Some tables are lions.

II. Some horses are rabbits.

III. No lion is table.

A. None follows



D. Only III follows

E. None of these


Answer: Option B

Explanation:

Some horses are lions. No rabbit is lion.


(O-type) and should not contain the middle term.

So, it follows that 'Some horses are not rabbits'.

All rabbits are tables. No rabbit is lion.

Since the middle term 'rabbits' is distributed twice, the conclusion must be particular.

Since one premise is negative, the conclusion must be negative. So, it follows that 'Some tables

are not lions'. Since I and III involve the same terms and form a complementary pair, so either I

or III follows.

logical deduction

Documents

subject term men

subject class

subject term animals

predicate term wild

predicate term foolish

universal negative proposition

entire class of predicate

particular quantifier