Syllogisms AIEO Rule All Notes with MCQs

Here I am providing Detailed notes on Syllogism AIEO Rule No Venn Diagram- Logical Syllogisms

The problems based on syllogism are on two parts:
1. Proposition/ Propositions
2. Conclusion/ Conclusions drawn from given proposition

WHAT IS A PROPOSITION?
a statement or assertion that expresses a judgement or opinion.
A proposition is a sentence that makes a statement giving a relation between two terms.
Parts of proposition:
1. Subject
2. Predicate

TYPES OF PROPOSITIONS:

1. CATEGORICAL PROPOSITION

The sentences which are condition free are called as categorical propositions. For example,
“All cats are rats”
“No cat is rat”
“Some cats are rats”
“Some cats are not rats”
In other words a categorical proposition has no condition attached with it and makes direct assertion.

	Positive	Negative
Universal	A All Cars are Rats	E No Cars are Rats
Particular	I Some Cars are Rats	O Some Cars are not Rats

2. NON-CATEGORICAL PROPOSITION

It is different from categorical proposition which has condition attached with it. For example,

“If M then P”

TYPES OF INFERENCES or CONCLUSIONS:
Immediate- An immediate inference is an inference which can be made from only one statement or proposition.
Mediate- Definition of mediate inference. :a logical inference drawn from more than one proposition or premise — compare syllogism. To derive Conclusion one must merge two or more statements

Immediate Inferences are of two types:
Implication- Subject and predicate remain same

If we convert A into I
All Cars are Rats-Some Cars are Rats

If we convert E into O
No Cars are Rats- Some Cars are not Rats
We cannot Implicate I and O

A	I
E	O
I	X
O	X

	Positive	Negative
Universal	A All Cars are Rats	E No Cars are Rats
Particular	I Some Cars are Rats	O Some Cars are not Rats

Conversion- Subject is converted into predicate and predicate is converted into Subject
A into I
All Cars are Rats-Some Rats are Cars

E into E
No Cars are Rats- No Rats are Cars

I into I
Some Cars are Rats- Some Rats are Cars

We cannot convert O

A	I
E	E
I	I
O	X

Valid immediate inferences:
Converse

Given a type E statement, from the traditional square of opposition, "No S are P.", one can make the immediate inference that "No P are S" which is the converse of the given statement.

Given a type I statement, "Some S are P.", one can make the immediate inference that "Some P are S" which is the converse of the given statement.

Obverse

Given a type A statement, "All S are P.", one can make the immediate inference that "No S are non-P" which is the obverse of the given statement.

Given a type E statement, "No S are P.", one can make the immediate inference that "All S are non-P" which is the obverse of the given statement.

Given a type I statement, "Some S are P.", one can make the immediate inference that "Some S are not non-P" which is the obverse of the given statement.

Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some S are non-P" which is the obverse of the given statement.

Contra positive

Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contra positive of the given statement.

Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.

Invalid immediate inferences

Cases of the incorrect application of the contrary, subcontrary and subalternation relations are syllogistic fallacies called illicit contrary, illicit subcontrary, and illicit subalternation. Cases of incorrect application of the contradictory relation are so infrequent, that an "illicit contradictory" fallacy is usually not recognized.

Illicit contrary

It is false that all A are B, therefore no A are B.
It is false that no A are B, therefore all A are B.

Illicit subcontrary

Some A are B, therefore it is false that some A are not B.
Some A are not B, therefore some A are B.

Illicit subalternation (Superalternation)

Some A are not B, therefore no A are B.
It is false that all A are B, therefore it is false that some A are B.

Mediate Inferences
1 Combine two Universal Positive statements will give Universal Positive Conclusion
A+ A= A
Statement:
All Cars are Rats
All Rats are Bats
Conclusion:
All Cars are Bats

2 Combine Universal Positive statement with Universal Negative will give Universal Negative Conclusion
A+ E= E
Statement:
All Cars are Rats
No Rats are Bats
Conclusion:
No Cars are Bats

3 Combine E + A= O Reverse

Statement:
No Cars are Rats
All Rats are Bats
Conclusion:
Some Bats are not Cars

4 E+ I= O Reverse

Statement:
No Cars are Rats
Some Rats are Bats
Conclusion:
Some Bats are not Cars

5 I + A= I

Statement:
Some Cars are Rats
All Rats are Bats
Conclusion:
Some Cars are Bats

6 I + E= O

Statement:
Some Cars are Rats
No Rats are Bats
Conclusion:
Some Cars are not Bats

Rules-
1) If Statement is Positive then Conclusion must be Positive
Example:
Statement- All pens are pencils

2) If Statement is Negative then Conclusion must be Negative
Example:
Statement- No Pens are Pencils

3) +ve plus +ve = positive Conclusion
+ve plus -ve = Negative Conclusion
-ve plus -ve = No Conclusion eg. O+ O= No conclusion

Steps:
1 Choose Statements
If Subject and predicate are in one statements then - Immediate Inference
If Subject and predicate are in Different statements then - Mediate Inference

2 Check alignment:
To merge Questions Common term should be predicate in first statement and Subject in Second statement in this case we change Order or Convert on the bases of IEA or Do Both as per situation

Example: Statement:
All Cars are Rats
All Vans are Cars
Conclusion:
All Vans are Rats

Example: Statement:
Some Cars are Rats
All Vans are Rats

3 Check If conclusion Follows
one statements then - Immediate Inference
More statements then - Mediate Inference

4 Check Complimentary Pairs- Conclusion Either or

Square of Opposition
Example if given is true then which of following is true

Contradictories: A and O, E and I
A and O Both cannot be true or false together means if one is false then other must be true

If A is true then O is false
If O is true then A is false
Should Have same Subject and Predicate

Sub Altern
Truth Downward but False Upward

Sub Contraries I and O Both can be true together but not false together

Contraries A and E Both cannot be true together but can be false together

For mediates:

A+ A= A A+ E= E

E + A= O Reverse

E+ I= O Reverse

I + A= I

I + E= O

December 2010

1 If the statement ‘all students are intelligent’ is true, which of the following statements are false?

(i) No students are intelligent.

(ii) Some students are intelligent.

(iii) Some students are not intelligent.

(A) (i) and (ii)

(B) (i) and (iii)

(C) (ii) and (iii)

(D) (i) only

Answer B

2 Two statements I and II given below are followed by two conclusions (a) and (b). Supposing the statements are true, which of the following conclusions can logically follow ?

Statements:

I. Some flowers are red.

II. Some flowers are blue.

Conclusions:

(a) Some flowers are neither red nor blue.

(b) Some flowers are both red and blue.

(A) Only (a) follows.

(B) Only (b) follows.

(C) Both (a) and (b) follow.

(D) Neither (a) nor (b) follows

Answer D

June 2010

Two statements I and II given below are followed by two conclusions (a) and (b). Supposing the statements are true, which of the following conclusions can logically follow?

I. Some religious people are morally good.

II. Some religious people are rational.

Conclusions:

(a) Rationally religious people are good morally.

(b) Non-rational religious persons are not morally good.

(A) Only (a) follows.

(B) Only (b) follows.

(C) Both (a) and (b) follow.

(D) Neither (a) nor (b) follows.

Answer D

June 2011

If the statement ‘some men are cruel’ is false, which of the following statements/statement are/is true ?

(i) All men are cruel.

(ii) No men are cruel.

(iii) Some men are not cruel.

(A) (i) and (iii)

(B) (i) and (ii)

Answer C

December 2011

If the proposition “Vegetarians are not meat eaters” is false, then which of the following inferences is correct? Choose from the codes given below:

1. “Some vegetarians are meat eaters” is true.

2. “All vegetarians are meat eaters” is doubtful.

3. “Some vegetarians are not meat eaters” is true.

4. “Some vegetarians are not meat eaters” is doubtful.

Codes:

(A) 1, 2 and 3

(B) 2, 3 and 4

(C) 1, 3 and 4

(D) 1, 2 and 4

Answer A

June 2012

Venn diagram is a kind of diagram to

(A) represent and assess the validity of elementary inferences of syllogistic form.

(B) represent but not assess the validity of elementary inferences of syllogistic form.

(C) represent and assess the truth of elementary inferences of syllogistic form.

(D) assess but not represent the truth of elementary inferences of syllogistic form

Answer A प्रतिनिधित्व और syllogistic रूप के प्राथमिक निष्कर्षों की वैधता का आकलन।

If the proposition “All men are not mortal” is true then which of the following inferences is correct ? Choose from the code given below :

1. “All men are mortal” is true.

2. “Some men are mortal” is false.

3. “No men are mortal” is doubtful.

4. “All men are mortal” is false.

Code :

(A) 1, 2 and 3

(B) 2, 3 and 4

(C) 1, 3 and 4

(D) 1 and 3

Answer B

December 2012

By which of the following proposition, the proposition ‘some men are not honest’ is contradicted?

(A) All men are honest.

(B) Some men are honest.

(C) No men are honest.

(D) All of the above.

Answer A

If the statement ‘most of the students are obedient’ is taken to be true, which one of the following pair of statements can be claimed to be true?

I. All obedient persons are students.

II. All students are obedient.

III. Some students are obedient.

IV. Some students are not disobedient.

Codes :

(A) I & II

(B) II & III

(C) III & IV

(D) II & IV

Answer C

June 2013

1. Which of the codes given below contains only the correct statements?

Statements:

(i) Venn diagram is a clear method of notation.

(ii) Venn diagram is the most direct method of testing the validity of categorical syllogisms.

(iii) In Venn diagram method the premises and the conclusion of a categorical syllogism is diagrammed.

(iv) In Venn diagram method the three overlapping circles are drawn for testing a categorical syllogism.

Codes:

(A) (i), (ii) & (iii)

(B) (i), (ii) & (iv)

(C) (ii), (iii) & (iv)

(D) (i), (iii) & (iv)

Answer B

2. If the statement ‘some men are honest’ is false, which among the following statements will be true. Choose the correct code given below:

(i) All men are honest

(ii) No men are honest

(iii) Some men are not honest.

(iv) All men are dishonest.

Codes:

(A) (i), (ii) and (iii)

(B) (ii), (iii) and (iv)

(C) (i), (iii) and (iv)

(D) (ii), (i) and (iv)

Answer B

September 2013

1. If two propositions cannot both be false but may both be true, what is the relation between the two propositions?

(A) Contrary

(B) Sub-contrary

(C) Sub-alternation

(D) Contradictory

Answer B

2. What is equivalent of the statement ‘All atheists are pessimists’ ?

(A) All non-pessimists are non atheists.

(B) All non-atheists are nonpessimists.

(C) All pessimists are atheists.

(D) None of the above.

Answer A

December 2013

1. Given below are two premises. Four conclusions are drawn from those two premises in four codes. Select the code that states the conclusion validly drawn.

Premises:

(i) All saints are religious. (major)

(ii) Some honest persons are saints. (minor)

Codes

(A) All saints are honest

(B) Some saints are honest.

(C) Some honest persons are religious.

(D) All religious persons are honest

Answer C

June 2014

1. If two propositions having the same subject and predicate terms can both be true but cannot both be false, the relation between those two propositions is called

(A) contradictory

(B) contrary

(C) subcontrary

(D) subaltern

Answer C

2. Given below are two premises and four conclusions drawn from those premises. Select the code that expresses conclusion drawn validly from the premises (separately or jointly).

Premises:

(a) All dogs are mammals.

(b) No cats are dogs.

Conclusions:

(i) No cats are mammals

(ii) Some cats are mammals.

(iii) No Dogs are cats

(iv) No dogs are non-mammals.

Codes:

(A) (i) only

(B) (i) and (ii)

(C) (iii) and (iv)

(D) (ii) and (iii)

Answer C

3. Given below is a diagram of three circles A, B & C inter-related with each of Indians. The circle B represents the class of scientists and circle C represents the class of politicians. p,q,r,s... represent different regions. Select the code containing the region that indicates the class of Indian scientists who are not politicians.

(A) q and s only

(B) s only

(C) s and r only

(D) p, q and s only

Answer S

December 2014

1. Given below is a diagram of three circles A, B and C over-lapping each other? The circle A represents the class of honest people, the circle B represent the class of sincere people and circle C represents the class of politicians. p, q, r, s, U, X, Y represent different regions. Select the code that represents the region indicating the class of honest politicians who are not sincere.

(A) X

(B) q

(C) p

(D) s

Answer D

2. By which of the following proposition, the proposition “wise men are hardly afraid of death” is contradicted?

(A) Some wise men are afraid of death.

(B) All wise men are afraid of death.

(C) No wise men is afraid of death.

(D) Some wise men are not afraid of death.

Answer B

3. Namita and Samita are brilliant and studious. Anita and karabi are obedient and irregular. Babita and Namita are irregular but brilliant. Samita and Kabita are regular and obedient. Who among them is/are brilliant, obedient, regular and studious?

(A) Samita alone

(B) Namita and Samita

(C) Kabita alone

(D) Anita alone

Answer A

June 2015

1. Among the following statements two are contradictory to each other. Select the correct code that represents them
(a) All poets are philosophers
(b) Some poets are philosophers
(c) Some poets are not philosophers
(d) No philosopher is a poet

(A) (a) and (b)
(B) (a) and (d)
(C) (a) and (c)
(D) (b) and (b)

Answer C

2. Which of the codes given below contains only the correct statements
(a) Venn diagram represents the arguments graphically
(b) Venn diagram can enhance our understanding
(c) Venn diagram may be called valid or invalid
(d) Venn diagram is clear method of notation

(A) (a), (b) and (c)
(B) (a), (b) and (d)
(C) (b), (c) and (d)
(D) (a), (c) and (d)

Answer B

December 2015

1. Among the following propositions two are related in such a way that they can both be true although they cannot both be false. Which are those propositions? Select the correct code.

Propositions:

(a) Some priests are cunning.

(b) No priest is cunning.

(c) All priests are cunning.

(d) Some priests are not cunning.

Codes:

(A) (a) and (b)

(B) (c) and (d)

(C) (a) and (c)

(D) (a) and (d)

Answer D

2. If the proposition ‘No men are honest’ is taken to be false which of the following proposition/propositions can be claimed certainly to be true?

Propositions:

(A) All men are honest

(B) Some men are honest

(C) Some men are not honest

(D) No honest person is man

Answer B

July 2016

1. Among the following propositions two are related in such a way that one is the denial of the other. Which are those propositions? Select the correct code:

Propositions:

(a) All women are equal to men

(b) Some women are equal to men

(c) Some women are not equal to men

(d) No women are equal to men

Codes:

(A) (a) and (b)

(B) (a) and (d)

(C) (c) and (d)

(D) (a) and (c)

Answer D

2. Given below two premise and four conclusions are drawn from them (taking singly or together). Select the code that states the conclusions validly drawn.

Premises:

(i) All religious persons are emotional.

(ii) Ram is a religious person.

Conclusions:

(a) Ram is emotional.

(b) All emotional persons are religious.

(c) Ram is not a non-religious person.

(d) Some religious persons are not emotional.

Codes:

(A) (a), (b), (c) and (d)

(B) (a) only

(B) (a) and (c) only

(D) (b) and (c) only

Answer C

3. If the proposition ‘All thieves are poor’ is false, which of the following propositions can be claimed certainly to be true?

Propositions:

(A) Some thieves are poor.

(B) Some thieves are not poor.

(C) No thief is poor.

(D) No poor person is a thief.

Answer B

4. Select the code, which is not correct about Venn diagram:

(A) Venn diagram represents propositions as well as classes.

(B) It can provide clear method of notation.

(C) It can be either valid or invalid.

(D) It can provide the direct method of testing the validity.

Answer C

August 2016

1. If two propositions are connected in such a way that they cannot both be false although they may both be true, then their relationship is called

(A) Contrary

(B) Subcontrary

(C) Contradictory

(D) Subalternation

Answer B

2. Given below are two premises, with four conclusions drawn from them (taking singly or together); which conclusions are validly drawn? Select the correct answer from the codes given below:

Premises:

(i) All bats are mammals.

(ii) Birds are not bats.

Conclusions:

(a) Birds are not mammals.

(b) Bats are not birds.

(c) All mammals are bats.

(d) Some mammals are bats.

Codes:

(A) (a), (b) and (d)

(B) (b) and (d)

(C) (a) and (c)

(D) (b), (c) and (d)

Answer B

Jan 2017

1 Given below are two premised (a) and (b). from those two premises four conclusions i, ii, iii, iv are drawn. Select the code that states the conclusions validly drawn from the premises (taking singly or jointly)
Premises:

Untouchability is a curse
All hot pans are untouchable

Conclusions:

All hot pans are curse
some untouchable things are hot pans
All curses are untouchability
Some curses are untouchability

Answer Codes:

(i) and (ii)
(ii) and (iii)
(iii) and (iv)
(ii) and (iv)

Answer A

2 If the statement ‘None but the brave wins the race’ is false, which of the following statements can be claimed to be true?

All brave persons win the race
Some persons who win the race are not brave
Some persons who win the race are brave
No person who wins the race is brave

Answer B

3 If two standard form categorical propositions with the same subject and predicate are related in such a manner that if one is undetermined the other must be undetermined, what is their relation?

Contrary
Subcontrary
Contradictory
Sub-altern

Answer 3

4 Among the following propositions two are related in such a way that they cannot both be true but can be false. Select the code states those two propositions
Propositions:

Every student is attentive (A)
Some students are attentive (I)
Students are never attentive (E)
Some students are not attentive (O)

codes:

(a) and (b)
(a) and (c)
(b) and (c)
(c) and (d)

Answer 2

Nov 2017

Given below are four statements. Among them, two are related in such a way that they can both be true but they cannot both be false. Select the code that indicates those two statements:

Statements: (a) Honest people never suffer.

(b) Almost all honest people do suffer.

(d) Each and every honest person suffers.

Code:

(1) (a) and (c)

(2) (a) and (d)

(3) (b) and (c)

(4) (a) and (b)

3 Sub contrary b and c means I and O

Given below are two premises (a and b), from those two premises four conclusions (i), (ii), (iii) and (iv) are drawn.

Select the code that states the conclusion/conclusions drawn validity (taking the premises singularly or jointly).

Premises:

(a) All bats are mammals

(b) No birds are bats

Conclusion:

(i) No birds are mammals

(ii) Some birds are not mammals

(iii) No bats are birds

(iv) All mammals are bats

2 iii only

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