LOGICAL REASONING NOTES
LOGICAL REASONING
Reasoning is an important section in aptitude tests and the one the student needs to
master necessarily. To seek accurate explanation, we have to apply logic. Logic is applying
principles of reasoning to obtain valid inferences. Logical reasoning is largely about adopting
complete rational approach to solve a problem, with no chance for ambiguity.
Deductive and inductive reasoning
In deductive reasoning or inferencing, it is asserted that the conclusion is guaranteed
to be true if the premises are true in the deductive inference, the conclusion cannot be more
general than premises(s).
On the contrary, in inductive reasoning or inferencing, the conclusion has a high
probability of being true if the premises are true. Thus, in inductive inference, the conclusion
is more general than the premises(s)
In inductive reasoning, we can generalise beyond the known facts, but we can never
be sure that the generalization is correct. It also means that true premises never guarantee
truth of conclusion. Here, conclusion is just a likelihood.
Deductive Inference
Statement I: All vegetables contain vitamins.
Statement II: Carrot is a vegetable.
Conclusion: So carrot contains vitamins.
Inductive Inference
Statement I: Most vegetables contain vitamins.
Statement II: Carrot is a vegetable
Conclusion: So carrot contains vitamins.
Deductive inferences are further categorized into (i) immediate- where conclusion is
drawn from a single statement and (ii) mediate (where conclusion is drawn from two
statements, called syllogism).
Types of syllogism:
1. Categorical: Here, the relationship between the subject and the predicate is without any
condition.
2. Hypothetical: The relationship between the subject and the predicate is asserted
conditionally. For example, if it rains he will not attend.
3. Disjunctive: I. Either he is courageous or he is strong.
4. Relational : Here the relation between the various terms is shown in an order.
Structure of Arguments
Structure of arguments deals with basic terms, validity of arguments, converting
sentences into their logical form depending on the requirement, and then application of rules
follows so as to arrive at a conclusion.
Proposition
A proposition is a sentence that makes a statement and gives a relation between two
or more terms. In logical reasoning, any statement is termed as a proposition.
A premise is a statement or proposition that is assumed to be true and from which a
conclusion can be drawn.
Quantifier + Subject + Copula + Predicate
Thus, the proposition consists of four parts:
1. Quantifier: All, no, and some. They specify a quantity. ‘All’ and ‘no, are universal
quantifier and ‘some’ is a particular quantifier.
2. Subject (S): About which something is being said.
3. Predicate(P): Something that affirms or denies about the subject.
4. Copula : Relation between subject and predicate.
Examples:
All bats are boys
Some players are doctors.
Classification of Propositions
Propositions are basically of two types, namely, universal and particular. universal
proposition is further divided into two parts:
1. Universal Positive or affirmative (A); It denotes inclusion.
Form: All S is P where S is the subject and P is the predicate. Example: ‘All cats are
animals’. It is basically about inclusion.
Distribution: It distributes the subject only. In the above statement, cats are distributed in
animals.
Predicate is not interchangeable with the subject while maintaining the validity of a
proposition. We cannot say that all animals are cats.
2. Universal Negative (E): It denotes exclusion.
Form: No S is P. Example: ‘No fish are birds’ would be a universal negative.
Distribution: Both subject and predicate. Here, an entire class of predicate term is denied
to the entire class of the subject term.
Particular Propositions: A particular proposition can also be divided into two parts.
1. Particular Positive (I):It denotes ‘partial inclusion’.
Form: Some S is P. Example: Some men are foolish.
Distribution: Neither the subject nor the predicate. In the example, 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.
2. Particular Negative (O):It denotes ‘partial exclusion’.
Form: Some S is not P or Not every S is P.
Example: Some bird are not carnivores’.
Distribution: Only of predicate.
Validity of Arguments
Deductive arguments may be either valid or invalid. If an argument is valid, it is a
valid deduction, and if its premises are true, the conclusion must be true. A valid argument
cannot have true premises and a false conclusion
Types and Main characteristics of Prepositions:
Sign Statement form Examples Quantity Quality Distributed
A All S are P All politicians are liars Universal Positive Only subject
E No S are P No politicians are liars Universal Negative Both subject and
predicate
I Some S are P Some politicians are liars Particular Positive Neither subject
nor predicate
O Some S are not P Some politicians are not liars Particular Negative Only predicate
The conclusion of a valid argument with one or more false premises may be either true
or false.
Logic seeks to discover the valid forms, the forms that make arguments valid A form of
argument is valid if and only if the conclusion is true under all interpretations of that
argument in which the premises are true. Since the validity of an argument depends solely
on its form, an argument can be shown to be invalid by showing that its form is invalid. This
can be done by giving a Counterexample of the same form of argument with premises that
are true under a given interpretation, but a conclusion that is false under that interpretation.
In informal logic, this is called a counterargument.
Certain examples would help in better clarification about validity of arguments.
1. Some Indians are logicians; therefore some logicians are Indians.
Valid argument: It would be self contradictory to admit that some Indians are
logicians but deny that some (any) logicians are Indians.
2. All Indians are human and all humans are mortal; therefore, all Indians are mortal.
Valid argument: If the premises are true, the conclusion must be true.
3. Some Indians are logicians and some logicians are tiresome; therefore, some Indians
are tiresome.
Invalid argument: For example, the tiresome logicians might all be Chinese.
Remember that this does not mean the conclusion has to be true; it is only true if the
premises are true, which they may not be.
The following examples would help to clarify this aspect about structure of arguments:
Premises:
I: Some men are lawyers
II: Some lawyers are rich.
Conclusions: Some men are rich.
This argument is invalid. There is a way where you can determine whether an argument is
valid and give a counterexample with the same argument form.
Note: Logical strength and soundness are properties of statements (or premises or
conclusions).Never say that argument is false’ or that ‘premise is logically strong’.
What is a counterexample? In logic, a counter example is an exception to a proposed general rule.
For example, ’all students are (laziness) holds for all students, even a single example of diligent
student will prove it false. Thus, any hardworking student is a counter-example to’ all students are
lazy’. More precisely, a counterexample is a specific instance of the falsity of a universal
quantification.
Parts of Categorical Propositions
There are three parts of statements in categorical syllogism-major premise, minor
premise, and conclusion, Each of the premise has one term in common with the conclusion.
Parts Example
Major premise All humans are mortal
Minor premise All Greeks are humans
Conclusion All Greeks are mortal
1. Major premise: Predicate of the conclusion is called as the major term. The premise
containing major term is called major premise. In the example, mortal is the major
term.
2. Minor premise: Subject of the conclusion is called minor term. The premise
containing minor term is called minor premise. In the example, Greeks is the minor
term.
3. Middle Term: One term common in both the premises is called middle term. It is not
a part of conclusion. In the example, humans is the middle term.
4. Conclusion: In conclusion statement, first term or (subject) is the subject of the first
proposition and second term (or predicate) is the predicate of the second proposition.
Converting Common Language Statements into their Logical Forms
In logical reasoning or syllogism problems, the common language sentences may have to
be converted into their logical form before we apply logic rules on them to draw a
conclusion.
The rules of reduction can help in solving these types of questions.
1. A-type propositions: Statements starting with words ‘each’, ‘every’, ‘any’ etc. are to be
treated as A-type propositions (starting with all)
Original sentence Logical form
Every man is All men are persons
liable to commit who are liable to
error commit mistakes
Each student All students are persons
participated in the event who participated in the event
Any one of the Indians is laborious All Indians are laborious
Only Indians are students of this college All students of this college are Indians
The honest alone are successful All successful persons are honest.
2. E-type propositions: Sentences with singular term or definite singular term with the sign
of negation are to be treated as E-type propositions. Sentences beginning with the words like
‘no’, ‘never’. and ‘none’ are to be treated as E-type propositions.
‘Never men are perfect’ it ‘No men are perfect’ in its logical form.
3. I-type propositions: Affirmative sentences with words like ‘a few’, ‘certain’, ‘most’, and
‘many’ are to be treated as I-type propositions.
Sentence Logical form
A few men are present Some men are present
Most of the students are laborious Some students are laborious
Few men are not selfish Some men are selfish
Certain books are good Some books are good
Man Indians are religious Some Indians are religious
All students of my class, except a few, are
well prepared
Some students of my class are well
prepared
The poor may be happy Some poor people are happy
4. O- type propositions: A negative can sentence that begins with a word like
‘every’, ‘any, ‘each’, or ‘all’ is to be treated as an O-type proposition.
Sentence Logical form
Every man is not rich Some men are not rich
Certain books are not readable Some books are not readable
Most of the students are not rich Some students are not rich
Some men are not above temptation Few men are above temptation
5. Exclusive proposition
(a) In exclusive propositions, the subject is qualified with words like ‘only’, ‘along’, ‘none
but’, or not one else but’.
(b) Here, the quantity is not explicitly stated.
(c) They can be reduced to A , E or-types by first interchanging the subject and worlds like
‘only’ or ‘alone’ with ‘all’.
1. Exercise:
Statements
1. Intelligent alone are laborious.
2. Most of the girls are intelligent.
These statements should first be converted into logical forms according to the rules
for logical form.
1. All intelligent are laborious. This is in the form B to C.
2. Some girls are intelligent. This is in the form A to B.
Just by changing their order, we can align them. After aligment is done, we move to Step it.
2. Exercise:
Statements
1. Some pens are books.
2. Some stationary are books
As books is the common term, they are in the form A to B and C to B. The first
statement does not require any change. As the second statement is in Particular Positive (I-
type), this can be changed to I-type only according to conversion table given earlier. The
second statement will become, ‘Some books are stationary’.
Now propositions are properly aligned that is, ‘Some pens are books’ and ‘Some
books are stationary’. We now move to Step II.
Rule of Syllogism
Proposition I(A to B) Proposition II(B to C) Conclusion Summarized
form
Universal Positive(A) Universal Positive(A) UniversalPositive (A) A+A=A
Universal Negative(E) Universal Negative(E) A+E=E
Universal Negative(E) Universal Positive(A) Particular negative(O) E+A=O*
Particular Positive(I) Particular Negative(O) E+I=O*
Particular Positive(I) Universal Positive(A) Particular Positive(I) I+A=I
Universal Negative(E) Particular Negative(O) I+E=O
Reasoning is an important section in aptitude tests and the one the student needs to
master necessarily. To seek accurate explanation, we have to apply logic. Logic is applying
principles of reasoning to obtain valid inferences. Logical reasoning is largely about adopting
complete rational approach to solve a problem, with no chance for ambiguity.
Deductive and inductive reasoning
In deductive reasoning or inferencing, it is asserted that the conclusion is guaranteed
to be true if the premises are true in the deductive inference, the conclusion cannot be more
general than premises(s).
On the contrary, in inductive reasoning or inferencing, the conclusion has a high
probability of being true if the premises are true. Thus, in inductive inference, the conclusion
is more general than the premises(s)
In inductive reasoning, we can generalise beyond the known facts, but we can never
be sure that the generalization is correct. It also means that true premises never guarantee
truth of conclusion. Here, conclusion is just a likelihood.
Deductive Inference
Statement I: All vegetables contain vitamins.
Statement II: Carrot is a vegetable.
Conclusion: So carrot contains vitamins.
Inductive Inference
Statement I: Most vegetables contain vitamins.
Statement II: Carrot is a vegetable
Conclusion: So carrot contains vitamins.
Deductive inferences are further categorized into (i) immediate- where conclusion is
drawn from a single statement and (ii) mediate (where conclusion is drawn from two
statements, called syllogism).
Types of syllogism:
1. Categorical: Here, the relationship between the subject and the predicate is without any
condition.
2. Hypothetical: The relationship between the subject and the predicate is asserted
conditionally. For example, if it rains he will not attend.
3. Disjunctive: I. Either he is courageous or he is strong.
4. Relational : Here the relation between the various terms is shown in an order.
Structure of Arguments
Structure of arguments deals with basic terms, validity of arguments, converting
sentences into their logical form depending on the requirement, and then application of rules
follows so as to arrive at a conclusion.
Proposition
A proposition is a sentence that makes a statement and gives a relation between two
or more terms. In logical reasoning, any statement is termed as a proposition.
A premise is a statement or proposition that is assumed to be true and from which a
conclusion can be drawn.
Quantifier + Subject + Copula + Predicate
Thus, the proposition consists of four parts:
1. Quantifier: All, no, and some. They specify a quantity. ‘All’ and ‘no, are universal
quantifier and ‘some’ is a particular quantifier.
2. Subject (S): About which something is being said.
3. Predicate(P): Something that affirms or denies about the subject.
4. Copula : Relation between subject and predicate.
Examples:
All bats are boys
Some players are doctors.
Classification of Propositions
Propositions are basically of two types, namely, universal and particular. universal
proposition is further divided into two parts:
1. Universal Positive or affirmative (A); It denotes inclusion.
Form: All S is P where S is the subject and P is the predicate. Example: ‘All cats are
animals’. It is basically about inclusion.
Distribution: It distributes the subject only. In the above statement, cats are distributed in
animals.
Predicate is not interchangeable with the subject while maintaining the validity of a
proposition. We cannot say that all animals are cats.
2. Universal Negative (E): It denotes exclusion.
Form: No S is P. Example: ‘No fish are birds’ would be a universal negative.
Distribution: Both subject and predicate. Here, an entire class of predicate term is denied
to the entire class of the subject term.
Particular Propositions: A particular proposition can also be divided into two parts.
1. Particular Positive (I):It denotes ‘partial inclusion’.
Form: Some S is P. Example: Some men are foolish.
Distribution: Neither the subject nor the predicate. In the example, 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.
2. Particular Negative (O):It denotes ‘partial exclusion’.
Form: Some S is not P or Not every S is P.
Example: Some bird are not carnivores’.
Distribution: Only of predicate.
Validity of Arguments
Deductive arguments may be either valid or invalid. If an argument is valid, it is a
valid deduction, and if its premises are true, the conclusion must be true. A valid argument
cannot have true premises and a false conclusion
Types and Main characteristics of Prepositions:
Sign Statement form Examples Quantity Quality Distributed
A All S are P All politicians are liars Universal Positive Only subject
E No S are P No politicians are liars Universal Negative Both subject and
predicate
I Some S are P Some politicians are liars Particular Positive Neither subject
nor predicate
O Some S are not P Some politicians are not liars Particular Negative Only predicate
The conclusion of a valid argument with one or more false premises may be either true
or false.
Logic seeks to discover the valid forms, the forms that make arguments valid A form of
argument is valid if and only if the conclusion is true under all interpretations of that
argument in which the premises are true. Since the validity of an argument depends solely
on its form, an argument can be shown to be invalid by showing that its form is invalid. This
can be done by giving a Counterexample of the same form of argument with premises that
are true under a given interpretation, but a conclusion that is false under that interpretation.
In informal logic, this is called a counterargument.
Certain examples would help in better clarification about validity of arguments.
1. Some Indians are logicians; therefore some logicians are Indians.
Valid argument: It would be self contradictory to admit that some Indians are
logicians but deny that some (any) logicians are Indians.
2. All Indians are human and all humans are mortal; therefore, all Indians are mortal.
Valid argument: If the premises are true, the conclusion must be true.
3. Some Indians are logicians and some logicians are tiresome; therefore, some Indians
are tiresome.
Invalid argument: For example, the tiresome logicians might all be Chinese.
Remember that this does not mean the conclusion has to be true; it is only true if the
premises are true, which they may not be.
The following examples would help to clarify this aspect about structure of arguments:
Premises:
I: Some men are lawyers
II: Some lawyers are rich.
Conclusions: Some men are rich.
This argument is invalid. There is a way where you can determine whether an argument is
valid and give a counterexample with the same argument form.
Note: Logical strength and soundness are properties of statements (or premises or
conclusions).Never say that argument is false’ or that ‘premise is logically strong’.
What is a counterexample? In logic, a counter example is an exception to a proposed general rule.
For example, ’all students are (laziness) holds for all students, even a single example of diligent
student will prove it false. Thus, any hardworking student is a counter-example to’ all students are
lazy’. More precisely, a counterexample is a specific instance of the falsity of a universal
quantification.
Parts of Categorical Propositions
There are three parts of statements in categorical syllogism-major premise, minor
premise, and conclusion, Each of the premise has one term in common with the conclusion.
Parts Example
Major premise All humans are mortal
Minor premise All Greeks are humans
Conclusion All Greeks are mortal
1. Major premise: Predicate of the conclusion is called as the major term. The premise
containing major term is called major premise. In the example, mortal is the major
term.
2. Minor premise: Subject of the conclusion is called minor term. The premise
containing minor term is called minor premise. In the example, Greeks is the minor
term.
3. Middle Term: One term common in both the premises is called middle term. It is not
a part of conclusion. In the example, humans is the middle term.
4. Conclusion: In conclusion statement, first term or (subject) is the subject of the first
proposition and second term (or predicate) is the predicate of the second proposition.
Converting Common Language Statements into their Logical Forms
In logical reasoning or syllogism problems, the common language sentences may have to
be converted into their logical form before we apply logic rules on them to draw a
conclusion.
The rules of reduction can help in solving these types of questions.
1. A-type propositions: Statements starting with words ‘each’, ‘every’, ‘any’ etc. are to be
treated as A-type propositions (starting with all)
Original sentence Logical form
Every man is All men are persons
liable to commit who are liable to
error commit mistakes
Each student All students are persons
participated in the event who participated in the event
Any one of the Indians is laborious All Indians are laborious
Only Indians are students of this college All students of this college are Indians
The honest alone are successful All successful persons are honest.
2. E-type propositions: Sentences with singular term or definite singular term with the sign
of negation are to be treated as E-type propositions. Sentences beginning with the words like
‘no’, ‘never’. and ‘none’ are to be treated as E-type propositions.
‘Never men are perfect’ it ‘No men are perfect’ in its logical form.
3. I-type propositions: Affirmative sentences with words like ‘a few’, ‘certain’, ‘most’, and
‘many’ are to be treated as I-type propositions.
Sentence Logical form
A few men are present Some men are present
Most of the students are laborious Some students are laborious
Few men are not selfish Some men are selfish
Certain books are good Some books are good
Man Indians are religious Some Indians are religious
All students of my class, except a few, are
well prepared
Some students of my class are well
prepared
The poor may be happy Some poor people are happy
4. O- type propositions: A negative can sentence that begins with a word like
‘every’, ‘any, ‘each’, or ‘all’ is to be treated as an O-type proposition.
Sentence Logical form
Every man is not rich Some men are not rich
Certain books are not readable Some books are not readable
Most of the students are not rich Some students are not rich
Some men are not above temptation Few men are above temptation
5. Exclusive proposition
(a) In exclusive propositions, the subject is qualified with words like ‘only’, ‘along’, ‘none
but’, or not one else but’.
(b) Here, the quantity is not explicitly stated.
(c) They can be reduced to A , E or-types by first interchanging the subject and worlds like
‘only’ or ‘alone’ with ‘all’.
1. Exercise:
Statements
1. Intelligent alone are laborious.
2. Most of the girls are intelligent.
These statements should first be converted into logical forms according to the rules
for logical form.
1. All intelligent are laborious. This is in the form B to C.
2. Some girls are intelligent. This is in the form A to B.
Just by changing their order, we can align them. After aligment is done, we move to Step it.
2. Exercise:
Statements
1. Some pens are books.
2. Some stationary are books
As books is the common term, they are in the form A to B and C to B. The first
statement does not require any change. As the second statement is in Particular Positive (I-
type), this can be changed to I-type only according to conversion table given earlier. The
second statement will become, ‘Some books are stationary’.
Now propositions are properly aligned that is, ‘Some pens are books’ and ‘Some
books are stationary’. We now move to Step II.
Rule of Syllogism
Proposition I(A to B) Proposition II(B to C) Conclusion Summarized
form
Universal Positive(A) Universal Positive(A) UniversalPositive (A) A+A=A
Universal Negative(E) Universal Negative(E) A+E=E
Universal Negative(E) Universal Positive(A) Particular negative(O) E+A=O*
Particular Positive(I) Particular Negative(O) E+I=O*
Particular Positive(I) Universal Positive(A) Particular Positive(I) I+A=I
Universal Negative(E) Particular Negative(O) I+E=O
