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III SEMESTER CORE COURSE BA PHILOSOPHY PROGRAMME SYMBOLIC LOGIC AND INFORMATICS

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III SEMESTER CORE COURSE BA PHILOSOPHY PROGRAMME SYMBOLIC LOGIC AND INFORMATICS
BA PHILOSOPHY PROGRAMME
III SEMESTER
CORE COURSE
SYMBOLIC LOGIC AND INFORMATICS
( 2014 Admission onwards)
(CU-CBCSS)
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
Calicut university P.O, Malappuram Kerala, India 673 635.
School of Distance Education
CONTENTS
1. Preface
2. Objectives, Highlights and References
3. Pattern of Question Paper
4. Module I - Introduction
5. Module II - Truth functional Connectives
6. Module III - Statement Forms and Argument Forms
7. Module IV - Formal Proof of Validity
8. Module V - Informatics
9. Model Question Paper
10. Question Bank for Internal Assessment
11. Team of Teachers
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PREFACE
You are already familiar with the study materials for the Core Course PHL2B02 - Logic and Scientific Method of the II
Semester B. A. Philosophy Programme. The study materials in this section are for the Core Course IV - PHL3B04
Symbolic Logic and Informatics of the III Semester B. A. Philosophy Programme in the SDE stream. This course is
designed to make the learners familiar with the more advanced level of logic in which we use symbolic language and
signs for making logical operations more precise and clear. Hence, it is necessary to keep in mind the points of
continuity between the two courses on logic - PHL2B02 and PHL3B04. Informatics is another key component of this
course, and this will enable you to grasp the correlation between logic and computer language or the binary logic of
computer language. A keen study of these courses on Logic will definitely guide you to detect the common ambiguities
and fallacies in reasoning so that one can express and analyze thoughts and arguments in their most legible and
precise form.
The modules in the syllabus of this course contain problem-solving exercises in addition to the descriptive topics.
Hence, the learner is required to do much deskwork on the basis of the theoretical lessons in the modules. The content
of each module is prepared according to the approved model question paper. The Question Bank for conducting the
Internal Evaluation also forms a part of the course materials. The Part A of the question paper for examination contains
MCQs in the model of those given in the Question Bank. With all the best wishes for your careful and confident
performance in the examinations,
Dr. M. Ramakrishnan
Coordinator
(Chairperson, Board of Studies in Philosophy)
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OBJECTIVES
Module I
 To introduce the main differences between traditional and symbolic logic.
 To grasp and experience the advantages of symbolization.
 To learn the different symbols for various logical functions.
Module II




To delineate the distinction between simple and compound statements.
To learn how to construct the truth tables for different types of compound statements.
To practise problem solving exercises involving truth tables.
To get an overview of De-Morgan’s theorem.
Module III
 To familiarize with the distinction between argument and argument form and between
statement and statement form.
 To introduce the distinction between validity and invalidity.
 To familiarize with the logical classification of statements.
Module IV
 To grasp and analyze the characteristics of formal proof of validity.
 To familiarize with the nine rules of inference.
 To practise the technique of proving the validity of deductive arguments by applying the
nine rules.
Module V




To introduce the definition and scope of informatics.
To define the terms like data, information and knowledge in the domain of informatics.
To make an analytic study of the emerging ethical issues in the cyber world.
To introduce logic gates as the meeting point between symbolic logic and computer
language.
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HIGHLIGHTS
MODULE 1 - Introduction
1.1 Traditional logic and symbolic logic- Differences
1.2 Advantages of symbolization.
1.3 The symbols for Conjunction, Negation and Disjunction.
MODULE 2 - Truth functional connectives
2.1. Compound statements
a) Difference between simple and compound statements
b) Truth functional compound statement
2.2. Truth tables for conjunction and negation
a) Finding truth values of statements containing conjunction and negation
2.3. Disjunction
a) Truth table for disjunction
b) Finding truth values of statements containing disjunction, conjunction and
negation
2.4. Implication
a) Truth table for implication
b) Finding truth values of statements containing implication, disjunction,
conjunction and negation
2.5. Equivalence
a) Material equivalence
b) Biconditional
c) Logical equivalence- truth table for De-Morgan’s theorem
MODULE 3 - Statement Forms and argument forms
3.1. Argument form- Definition, validity and invalidity
3.2. Substitution instance and specific form- Definitions
3.3. Statement forms and statements
a) Definitions
b) Classification of statements into tautology, contradictory and contingent
MODULE 4 - Formal proof of validity
4.1. Definition
4.2. Nine rules of inference
MODULE 5 - Informatics
5.1.
5.2.
5.3.
5.4.
Etymology and definition
Data, information and knowledge
Issues in cyber ethics - reduced privacy, cyber addiction and information overload
Logic Gates
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References:
1.
2.
3.
4.
5.
Symbolic Logic, IM Copi (Module 1-4)
Wikipedia Online Encyclopaedia (Section 5.1)
Informatics, Siny G Benjamin (Section 5.2 and 5.3)
Philosophy and Computing: An Introduction, Luciano Floridi (Section 5.4)
Alan Evans et.al. Informatics:Technology in Action. Delhi: Pearson, 2012.
Websites:
1. http://english.answers.com/definitions/the-definition-of-logic-and-its-uses
2. http://johnmacfarlane.net/dissertation.pdf
3. http://www.classical-homeschooling.org/dialectic/logic.html
4. http://www.fecundity.com/logic
5. Digital Logic design by: Dr. Wa’el Al Qassas,
Al Albayt University http://web2.aabu.edu.jo:8080/tool/course_file/
901220_lectures.pdf
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Pattern of Question Paper
Duration
3Hrs
Section
A
Pattern
Total number
of
questions
Questions
to be
answered
Marks
for each
question
Total
marks for
each
section
10
10
½
10 x ½ = 5
8
5
3
5 x 3 = 15
9
6
5
6 x 5 = 30
4
2
15
Objective Type
Multiple choice
questions
B
Short Answer
questions
C
Paragraph Answer
questions
D
Essay questions
2 x 15 = 30
TOTAL = 80
Time: Three Hours
Maximum: 80 marks
PART - A - Multiple-choice questions
Answer all questions. Each question carries ½ marks. (10 x ½ = 5marks)
PART - B - Short answer questions
Answer any five out of the eight questions.
Each question carries 3 marks. (5 x 3 = 15 marks)
PART - C - Paragraph answer questions
Answer any six out of the nine questions. Answer should not exceed 100 words.
Each question carries 5 marks. (6x5 =30marks)
PART - D - Essay questions
Answer any two out of the four questions. Answer should not exceed 1000 words.
Each question carries 15 marks. (2 x 15 = 30marks)
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Scrutinized by:
Dr. M. Ramakrishnan
Former Head of the Dept. of Philosophy
Govt. Brennen College, Thalassery
(Chairperson, Board of Studies in
Philosophy, University of Calicut)
Prepared by:
Dr. K. Syamala
Head of the Dept. of Philosophy
Sree Sankaracharya University of
Sanskrit, Regional Center
Module 1
Edat, Payyannur, Kannur (Dt.)
Module II
Dr. Sirajul Muneer. C
Assistant Professor of Philosophy
Sree Sankaracharya University of
Sanskrit, Regional Center
Edat, Payyannur, Kannur (Dt.)
Module III
Ms. Priya
Assistant Professor of Philosophy
Govt. Brennen College
Thalassery - 670 106
Module IV
Dr. Smitha T. M.
Assistant Professor of Philosophy
Maharaja’s College, Ernakulum
Cochin - 682 011
[email protected]
Layout:
SYMBOLIC LOGIC AND INFORMATICS
Computer Section, SDE
©
Reserved
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SYMBOLIC LOGIC AND INFORMATICS
CONTENTS
PAGES
MODULE 1
10
MODULE 1I
17
MODULE III
25
MODULE 1V
33
MODULE V
43
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MODULE 1
INTRODUCTION
PART - A - Multiple-choice questions
For model questions, see the Question Bank.
PART - B - Short answer questions
1. Summarize the main benefits of using symbolic language in logic.
Logicians make use of constant and variable symbols mainly to avoid the linguistic defects
in making and examining arguments. By replacing linguistic expressions with symbols, an
argument becomes more explicit without ambiguity. Symbolic language helps logicians to
determine the validity of an argument more easily and accurately. With symbolic language,
we can also maintain economy and precision of arguments.
2. What are the difficulties faced by logicians while presenting an argument in ordinary
language?
While presenting an argument in natural languages, the words used may be vague or
equivocal. In ordinary language expressions, metaphors and idioms may often create
confusion. Ordinary language expressions are not always free from emotional appeal, but
logicians are not concerned with such expressions. These difficulties can be avoided by
using symbolic language in logic.
3. Define conjunction.
A compound statement formed by inserting the word ‘and’ between the two component
statements is called a conjunction. The two statements so combined are called conjuncts. In
symbolic logic, the dot ‘∙’ is the special symbol for conjunction. E.g. p . q.
4. What do you mean by a disjunctive proposition?
Two simple propositions when combined by ‘either -- or’ forms a disjunctive proposition.
The two component propositions so combined are called disjuncts. The symbol for
disjunction is ‘∨’. For example, ‘Either you can go to the park or you can watch the
TV’. Its symbolic representation is ‘P v T’.
5. Define negation.
Negation is the logical function of denying a fact. A negative proposition shows that ‘It is
not the case that p’ and the symbolic form is ~p.
6. Distinguish between simple proposition and compound proposition.
A simple proposition is the one which does not contain any other statement as a
component. For example, ‘Roses are red’ symbolized by R. A compound proposition is the
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one that contains another statement as a component. Compound propositions are mainly
conjunctive, disjunctive, implicative and biconditional. Their symbolic forms are the
following:
Conjunction
p.q
Disjunction
pvq
Implication
p‫ ﬤ‬q
Biconditional
p≡q
7. Name the different types of compound propositions. Or
Present the symbolic forms of compound propositions.
The main types of compound statements are Conjunction, disjunction, implication and
biconditional. Their symbolic forms are the following:
Conjunction
p.q
Disjunction
pvq
Implication
p‫ ﬤ‬q
Biconditional
p≡q
8. Write a short note on constant symbols.
In symbolic logic, we use definite signs to represent logical relations. Constant symbols do
not change their value throughout the domain of logic. Common constant symbols used in
logic are the following:
Negation
~
Conjunction
.
Disjunction
v
Implication
⊃
Biconditional
≡
9. What is the main difference between variable symbols and constant symbols?
In symbolic logic, variable symbols keep on changing their value from argument to
argument. They do not have fixed values. Constant symbols do not change their value
throughout the domain of logic.
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Variables are represented by letters like p, q etc. Constant symbols are the following:
Conjunction
∙
Disjunction
v
Implication
⊃
Biconditional
≡
Negation
~
10. Match the following:
a) If -- then
b) ~ M
c) Biconditional
d) Inclusive disjunction
Answers:
a) Implication
-
Equivalence
Weak
Implication
Negation
b) Negation
c) Equivalence
d) Weak
PART - C - Paragraph answer questions
1. What are the advantages of using symbols in logic?
Symbols are useful to logicians in many ways.
Firstly, the logical form of an argument becomes explicit by using symbols. By using
symbols to represent ordinary language expressions, the logical form of an argument
becomes explicit. It helps us to avoid ambiguous and confusing use of words.
By using symbols, we can determine the validity of an argument quickly and accurately.
The use of symbols in logic is an economical device. Lengthy and complicated arguments
become small and precise by symbolization. Symbolization offers specific methods of
testing the validity of arguments like truth table method and formal proof of validity.
Thus, symbolic logic has the advantages of clarity, brevity and accuracy over traditional
logic.
It drastically reduces the space, time and energy needed for logical operations.
2. Distinguish between variable symbols and constant symbols.
In the traditional logic, only three variable symbols are used, that is S, P and M for the
subject, predicate and middle term in a categorical syllogism. A, E, I, O are symbols for the
four types of categorical propositions. In modern logic, the use of symbols is more
extensive to make it symbolic logic. In symbolic logic, we use two types of symbols variables and constants. A variable symbol keeps on changing its value from argument to
argument. These symbols are ‘dummies’ that stand for terms and concepts. They do not
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have fixed values. For example, the symbol ‘P’ in one argument may stand for ‘You will
pass’, in another argument for ‘It is a pleasant day’, and yet in a third argument for ‘We
will go for picnic today.’ Hence, the symbol P may have different meanings in different
contexts. Many types of variables are used in modern logic such as propositional variables,
predicates variables, class variables etc.
Modern logicians began to use constant symbols that gave them the advantage of clarity,
brevity and accuracy over traditional logic. Constant symbols do not change their value
throughout the domain of logic. Common constant symbols used in logic are the following:
Negation
~
Conjunction
∙
Disjunction
v
N Implication
⊃
Biconditional
≡
3. What do you mean by truth functional connectives?
The truth-value of a truth functional compound statement is determined by the truth-value
of its components. Logical constants are the connecting symbols between the component
statements. They show how the component statements are related. The truth-values of the
component statements along with the nature of truth functional connective determine the
truth-value of the compound statement. There are four types of truth functional
connectives. They are conjunction, disjunction, implication and biconditional or
equivalence. Their symbolic forms are given below:
Conjunction
∙
Disjunction
v
Implication
⊃
Biconditional
≡
4. Symbolize the following using the letters given in brackets.
a) Beethoven and Mozart were great composers (B, M)
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b) Either we will reduce poverty, or we will provide subsidies. (R, P)
c) If hydrogen is combined with oxygen in a specific ratio, water is produced.
(H, W)
d) Salim will not attend the function. (S)
Answers:
a) B∙M
b) R v P
c) H ⊃ W
d) ~S
5. Symbolize the following.
a) If p then q and r.
b) Neither a nor b.
c) It is not the case that either k or t.
d) If both a and b then both c and d.
Answers: a) p ⊃ (q ∙ r), b) ~a v ~b,
c) ~ (k v t),
d) (a ∙ b) ⊃ (c ∙ d)
6. Write a note on the symbolization and truth function of conjunction.
When two statements are joined by placing ‘and’ between them, the resulting statement is a
conjunction. The component statements are called the conjuncts. The ‘∙’ (dot) symbol
represents conjunction. Where ‘p’ and ‘q’ are statement variables representing any two
statements, their conjunction is represented as ‘p ∙ q’.
If ‘p’ represents ‘The city is heavily populated’ and ‘q’ represents ‘There is an outbreak of
fever’, then ‘p ∙ q’ represents their conjunction ‘The city is heavily populated and there is
an outbreak of fever’.
A conjunctive statement is true only if both of its conjuncts are true. ‘p ∙ q’ is true only if
both ‘p’ is true and ‘q’ is true. Hence, the truth-value of ‘p ∙ q’ is determined by the truthvalues of both ‘p’ and ‘q’.
7. Write a note on disjunction and distinguish between inclusive and exclusive disjunction.
When two statements are combined by inserting ‘or’ between them, the resulting
compound statement is a disjunction or alternation. The two statements thus combined are
called disjuncts or alternatives. The symbol ‘v’ called a wedge or a vee is used for
indicating disjunction. The disjunctive proposition ‘Either a disease is hereditary or it is
due to infection’ is symbolized as HvI
The word ‘or’ may be used in the weak or strong sense. Weak disjunction is called
inclusive because if both or at least one of the disjuncts is true the disjunction is true and it
is false only if both the disjuncts are false. E.g., Leave will be granted either in the case of
sickness or bus strike. Here, leave is granted on either one of the conditions or both the
conditions. Hence, the disjunction is weak.
In strong disjunction, ‘or’ is used in the exclusive sense. It means ‘at least one and at most
one. That is only one of the disjuncts but not both will be true at a time. E.g., I shall take
either coffee or tea. Here, it is clear that I want to take only one of the two options but not
both. Hence, the disjunction is strong.
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Part - D - Essay questions
1. Explain the salient features of symbolic logic and bring out its differences from
traditional logic. Or
Bring out the main advantages of using symbols in logic. Add a note on the different types
of symbols used by modern logicians.
Logic is defined as the science of valid reasoning. An argument or a piece of reasoning is a
relational arrangement of premises and conclusion. Hence, the validity and correctness of
an argument is ensured when the premises are strong enough to support the conclusion.
Arguments formulated in English or any other natural language are often confusing
because of the vague and equivocal nature of the words in which they are expressed.
Misleading idioms and emotional expressions make them vague. To avoid such difficulties
connected with ordinary language, logicians have developed specialized technical symbols
to represent logical sentences and arguments. The works of philosophers like Russell have
introduced symbolic logic that enables us to overcome the limitations of traditional logic.
Aristotle made use of certain symbols that had been used in traditional logic. Modern
logicians have introduced more specialized symbols that make logical analysis more easy
and accurate. The differences between old and new logic is one of degree rather than kind.
The special symbols in modern logic enable us to attain more clarity and precision in
presenting arguments. Symbolic expressions help us to avoid the problems of vagueness
and confusion of meaning. According to A. N. Whitehead, “By the aid of symbolism, we
can make transitions in reasoning almost mechanically by eye”.
The use of symbols in logic gives us many advantages:
Firstly, the logical form of an argument becomes explicit by using symbols. By replacing
language by symbols, the logical form of an argument becomes clear.
When the logical form of an argument is clear, it is easy to determine its validity. The
symbolic form of an argument makes logical analysis more quick and accurate.
Symbolization offers specific methods of testing the validity of arguments like truth table
method and formal proof of validity. Thus, symbolic logic has the advantages of clarity,
brevity and accuracy over traditional logic. It drastically reduces the space, time and energy
needed for logical operations.
The use of symbols in logic is also an economy device. The long and big arguments
become precise and clear through symbolization. This will help us to reduce the chances of
committing error in deciding their validity.
In the traditional logic, only three variable symbols are used, that is S, P and M for the
subject, predicate and middle term in a categorical syllogism. A, E, I, O are symbols for the
four types of categorical propositions. In modern logic, the use of symbols is more
extensive to make it symbolic logic. In symbolic logic, we use two types of symbols variables and constants. A variable symbol keeps on changing its value from argument to
argument. These symbols are ‘dummies’ that stand for terms and concepts. They do not
have fixed values. For example, the symbol ‘P’ in one argument may stand for ‘You will
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pass’, in another argument for ‘It is a pleasant day’, and yet in a third argument for ‘We
will go for picnic today.’ Hence, the symbol P may have different meanings in different
contexts. Many types of variables are used in modern logic such as propositional variables,
predicates variables, class variables etc.
Modern logicians began to use constant symbols that gave them the advantage of clarity,
brevity and accuracy over traditional logic. Constant symbols do not change their value
throughout the domain of logic. Common constant symbols used in logic are the following:
Negation
~
Conjunction
∙
Disjunction
v
Implication
⊃
Biconditional
≡
The main differences between traditional and symbolic logic are summarized below:
i) Traditional logic is concerned more with the relation of the subject and predicate terms
of propositions. Symbolic logic is more concerned with propositions as a unit and
propositional relations.
ii) Traditional logic is concerned with both form and matter of thought whereas symbolic
logic is purely formal in nature.
iii) Traditional logic has only a limited use of symbols, whereas there is an extensive use of
special symbols in symbolic logic.
iv) Syllogisms are central in Aristotelian logic. Instead, the internal structure of
propositions and arguments is the focus of modern logic. Hence, the set of symbols include
not only variable symbols but also the constants that represent logical connections.
v) Traditional logicians use non-mathematical methods to determine the validity of
arguments. Modern logicians adopt decision procedures that ensure mathematical precision
in analyzing arguments.
In spite of all the above differences, modern logic is not opposed to traditional logic. It is a
much improved form of traditional logic. We can say that what was implicit in Aristotelian
logic has become explicit in modern logic. The aim of all logicians, traditional as well as
modern, is to provide methods or devices to differentiate between correct and incorrect
reasoning. The difference between classical logic and symbolic logic is only of degree
rather than of kind.
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MODULE 2
TRUTH FUNCTIONAL CONNECTIVES
PART - A - Multiple-choice questions
For model questions, see the Question Bank.
PART - B - Short answer questions
1. What is an atomic/simple proposition?
All propositions are either atomic or molecular. An atomic proposition is one which does
not contain any other proposition as its component. An atomic proposition cannot be
divided into further component propositions. For example, “Ramesh is honest” which is
symbolized as ‘H’. Atomic proposition is also known as simple proposition because it does
not contain any other statement as its component.
2. Define compound proposition.
A compound proposition is that which contains two or more propositions as its
components. Compound proposition is also known as molecular proposition. “Ramesh is
honest and Dinesh is intelligent” is a compound proposition. It contains two simple
propositions as its components. Compound propositions may be conjunctive, disjunctive,
or implicative.
3. Define compound statement and present the symbolic form of its different types.
A compound proposition is that which contains two or more propositions as its
components. Compound proposition is also known as molecular proposition. The main
types of compound statements are Conjunction, disjunction, implication and biconditional.
Their symbolic forms are the following:
Conjunction
p·q
Disjunction
pvq
Implication
p‫ ﬤ‬q
Biconditional
p≡q
4. Define ‘Implication’.
Implication is the truth-functional relation between two simple propositions connected by
the phrase ‘if -- then’. For example, “If it rains, then the road will be wet”. The part of
proposition that lies in between ‘if ‘and ‘then’ is called the antecedent. That which follows
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the word ‘then’ is called the consequent. The symbol “” called “horseshoe” is used to
form an implicative statement. The symbolic form of an implicative proposition is as
follows:
P⊃Q
5. Define conjunction.
Conjunction is a compound proposition in which the word “and” connects simple
statements. To connect statements conjunctively, the dot ‘∙’ symbol is used.
E.g. ‘John is intelligent and John is attentive’ is symbolized as ‘I ∙ A’
In conjunction, if both its conjuncts are true, the conjunction is true, otherwise, it is false.
6. Define negation.
Negation is the denial of a statement formed by inserting ‘not’ to show the denied part. A
negative statement means ‘it is not the case that’. The negation of P is ‘it is not the case
that P.’ The symbol ‘~’called ‘curl’ or ‘tilde’ is used to indicate negation. E.g. ~ P
Logicians treat negation as a truth functional operator rather than a connective because
negation applies directly to a single proposition.
7. Define disjunction.
Disjunction is a truth-functional connective that indicates an ‘or’ relationship between two
propositions. The component statements are called disjuncts. The symbol for disjunction is
a wedge ‘∨’. E.g. ‘Either John is careless or John is ignorant’ is symbolized as ‘C v I’.
Present the truth table for material equivalence.
Two sentences are said to be materially equivalent when they have the same truth-value.
The symbol ‘≡’ called the tribar stands for material equivalence. The truth table for
biconditional or material equivalence is as follows:
P
Q
P≡Q
T
T
T
T
F
F
F
T
F
F
F
T
8. Match the following:
a) Conjunction
b) Disjunction
c) Negation
d) Implication
e) Bi-conditional
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Wedge
Tribar
Horseshoe
Tilde
Dot
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Ans: a) Dot, b) Wedge, c) Tilde, d) Horseshoe, e) Tribar
9. Explain logical equivalence.
When two statements are logically equivalent, each has the same truth-value under the
same truth conditions. For example, consider any statement p and its double negation ~ ~ p.
The “principle of double negation”, p ≡ ~ ~ p is proved tautologous by the following truth
table:
P
~p
~~p
p ≡ ~~p
T
F
T
T
F
T
F
F
The table proves that the logical equivalence of p is ~ ~ p
PART - C - Paragraph answer questions
1. Distinguish between simple and compound proposition.
All propositions are either simple or compound or general. A simple proposition is the one
which does not contain any other proposition as its component. A simple proposition
cannot be split into further component propositions. For example, “Ramesh is honest”
symbolized as ‘H’. It does not contain any other statement as a component. Simple
proposition is also known as atomic proposition.
A compound proposition is that which contains two or more propositions as its
components. ‘Ramesh is honest and Dinesh is intelligent’ is a compound proposition
symbolized as ‘H ∙ I’. It contains two simple propositions as its components. Compound
proposition is also known as molecular proposition. The main types of compound
statements are Conjunction, disjunction, implication and biconditional. Their symbolic
forms are the following:
Conjunction
p∙q
Disjunction
pvq
Implication
p‫ ﬤ‬q
Biconditional
p≡q
2. Draw the Truth Table for conjunction and negation.
Conjunction:
Conjunction is a compound proposition in which the word “and” is used to connect simple
statements. To connect statements conjunctively, the dot ‘∙’ symbol is used. In
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conjunction, if both its conjuncts are true, the conjunction is true, otherwise, it is false. The
truth table for conjunction is as follows:
P
Q
P∙Q
T
T
T
T
F
F
F
T
F
F
F
F
Negation:
Negation is a compound proposition in which the word ‘not” or the phrase ‘it is not the
case that’ is used. The symbol ‘~’ called “curl” or “tilde” is used to form the negation of a
statement. The truth table for negation is as follows:
P
~p
T
F
F
T
3. Describe the distinction between ‘inclusive disjunction’ and ‘exclusive
disjunction’.
Logicians recognize two kinds of disjunctions, inclusive disjunction and exclusive
disjunction. A disjunction containing non-exclusive alternatives is called inclusive
disjunction. Example, ‘Ramesh is either intelligent or hard working’. The sense of ‘or’ in
inclusive disjunction is ‘at least one, both may be’. A disjunction containing exclusive
alternatives is called exclusive disjunction. For example, ‘Today is either Wednesday or
Thursday’. The sense of ‘or’ in exclusive disjunction is ‘at least one, but not both’.
The truth table for ‘inclusive disjunction’ is as follows:
P
Q
PvQ
T
T
T
T
F
T
F
T
T
F
F
F
The truth table for ‘exclusive disjunction’ is as follows:
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P
Q
P^Q
T
T
F
T
F
T
F
T
T
F
F
F
4. State and explain De Morgan’s Theorems.
There are two logical equivalences that express important interrelations among
conjunction, disjunction and negation. Since a conjunction of p and q assert that both its
conjuncts are true, to contradict p ∙ q is to assert that at least one is false. Thus asserting the
negation of the conjunction p ∙ q is logically equivalent to asserting the disjunction of the
negations of p and of q. This is expressed by the bi-conditional ~ ( p ∙ q ) ≡ (~ p v ~ q )
which is proved to be a tautology.
Similarly, the disjunction p v q asserts that at least one of its two disjuncts is true, it is
contradicted only by asserting that both are false. Thus asserting the negation of the
disjunction p v q is logically equivalent to asserting the conjunction of the negations of p
and of q. This is expressed by the bi-conditional ~ (p v q) ≡ (~ p ∙ ~ q).
These two equivalences are known as De Morgan’s theorems. De Morgan’s theorem is
formulated as:
a. The negation of the conjunction of two statements is logically equivalent to the
disjunction of their negations. ~ ( p . q ) ≡ (~ p v ~ q )
b. The negation of the disjunction of two statements is logically equivalent to the
conjunction of their negations. ~ ( p v q) ≡ (~ p . ~ q ) .
For example, “It will not both rain and snow tomorrow” can be translated as ~ (R ∙ S), or
it can be translated as ~R ∨ ~S. Likewise, “It will neither rain nor snow tomorrow” can be
translated as ~ (R V S) or ~R ∙ ~S.
5. Distinguish between material implication and material equivalence.
Implication is a compound proposition in which the simple statements are connected by the
phrase ‘if -- then’. For example, “If it rains, then the road will be wet”. The part of
proposition which lies in between ‘if’ and ‘then’ is called the antecedent. The part of
proposition which follows the word ‘then’ is called the consequent. The general form of an
implicative proposition is as follows: “If antecedent, then consequent”. The symbol “”
called “horseshoe” is used to form an implicative statement. The truth table for logical
equivalence is as follows:
P
Q
PQ
T
T
T
T
F
F
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F
T
T
F
F
T
Two sentences are said to be materially equivalent when they have the same truth-value. It
is a compound proposition in which the simple statements are connected by the phrase ‘if
and only if’. The symbol ‘≡’ called the tribar stands for material equivalence. Material
equivalence is also called ‘Bi-conditional proposition’. The truth table for material
equivalence is as follows:
P
Q
P≡Q
T
T
T
T
F
F
F
T
F
F
F
T
6. Distinguish between logical equivalence and material equivalence.
In logic, there are important equivalences between statements that are a necessary result of
how the logical operators function. When two statements are logically equivalent, each has
the same truth-value under the same truth conditions. For example, consider any statement
p and its double negation ~ ~ p. The “principle of double negation ‘p ≡ ~ ~ p’ is proved to
be tautology by the following truth-table:
P
~p
~~p
p ≡ ~~p
T
F
T
T
F
T
F
T
This proves that the logical equivalence of p is ~ ~ p
There is difference between logical equivalence and material equivalence. Two statements
are logically equivalent only when it is impossible for the two statements to have different
truth-values. Therefore, logically equivalent statements have the same meaning and may be
substituted for one another. However, two statements are materially equivalent if they
merely happen to have the same truth-value. Statements that are merely materially
equivalent certainly cannot be substituted for one another .
PART - D - Essay questions
1. Explain the truth functional compound statements using truth tables.
All propositions are either simple or compound or general. A simple proposition is that
which does not contain any other proposition as its component. A simple proposition
cannot be split into further component propositions. For example, the proposition “Ramesh
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is honest” does not contain any other statement as a component. Simple proposition is also
known as atomic proposition.
A compound proposition is that which contains two or more propositions as its
components. “Ramesh is honest and Dinesh is intelligent” is a compound proposition. It
contains two simple propositions as its components. Compound proposition is also known
as molecular proposition. Compound propositions are of different types such as Negative,
Conjunctive, Disjunctive, Implicative, and Material Equivalent.
Negation:
Negation is a compound proposition in which the word ‘not” or the phrase ‘it is not the
case that’ is used. Example, ’Ramesh is not honest’ or ‘It is not the case that Ramesh is
honest’. It contains ‘Ramesh is honest’ as component part. . The symbol “~” called “curl”
or “tilde” is used to form the negation of a statement. The truth table for negation is as
follows:
P
~p
T
F
F
T
Conjunction:
Conjunction is a compound proposition in which the word “and” is used to connect simple
statements.
To connect statements conjunctively, the dot “∙” symbol is used for
conjunction. In conjunction, if both its conjuncts are true, the conjunction is true,
otherwise, it is false. The truth table for conjunction is as follows:
P
Q
T
T
P∙Q
T
T
F
F
F
T
F
F
F
F
Disjunction:
Disjunction is a compound proposition in which the simple propositions are connected by
the word ‘or‘or the phrase ’either….or’. Example, ‘Ramesh is either intelligent or hard
working’. ‘Today is either Wednesday or Thursday’. The components of a disjunction are
called disjuncts.
Logicians recognize two kinds of disjunction - inclusive disjunction and
disjunction. A disjunction containing non-exclusive alternatives is called
disjunction. Example, ‘Ramesh is either intelligent or hard working’. The sense
inclusive disjunction is ‘at least one, both may be’. A disjunction containing
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inclusive
of ‘or’ in
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alternatives is called exclusive disjunction. Example, ‘Today is either Wednesday or
Thursday’. The sense of ‘or’ in exclusive disjunction is ‘at least one, but not both’. The
truth table for inclusive disjunction is as follows:
P
Q
PvQ
T
T
T
T
F
T
F
T
T
F
F
F
Implication:
Implication is a compound proposition in which the simple statements are connected by the
phrase ‘if -- then’. For example, “If it rains, then the road will be wet”. The part of
proposition which lies in between ‘if ‘and ‘then’ is called antecedent. The part of
proposition which follows the word ‘then’ is called consequent. The general form of an
implicative proposition is as follows: “If antecedent, then consequent”. The symbol “  ”
called “horseshoe” is used to form a implicative statement. The truth table for logical
equivalence is as follows:
P
Q
PQ
T
T
T
T
F
F
F
T
T
F
F
T
Bi-conditional:
Bi-conditional proposition is a compound proposition in which the simple statements are
connected by the phrase ‘if and only if’. For example, “I will go to the cinema if and only if
my friend comes with me”. Bi-conditional proposition is also called ‘material equivalence’.
The symbol ′ ≡’ called the tribar, to stand for material equivalence. The truth table for
material equivalence is as follows:
P
Q
P≡Q
T
T
T
T
F
F
F
T
F
F
F
T
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MODULE 3
STATEMENT FORMS AND
ARGUMENT FORMS
PART - A - Multiple-choice questions
For model questions, see the Question Bank.
PART - B - Short answer questions
10. Explain tautologous statement form.
A sentence is a tautology if the column under its main connective is ‘True’ on every row of
a complete truth table. Now consider the statement - ‘It is raining or it is not raining’,
which is symbolized, as ‘p v ~ p
The truth table for p v ~ p is represented as follows:
P
~p
pv~p
T
F
T
F
T
T
Since we get only T in every row, this proposition is tautology.
11. Explain contradictory statement form.
A sentence is a contradictory if the column under its main connective is ‘False’ on every
row of a complete truth table. Now consider: ‘It is raining and it is not raining’ which is
symbolized as ‘p . ~ p
The truth table for p v ~ p is represented as follows:
P
~p
p∙ ~ p
T
F
F
F
T
F
Since we get only F in every row, the given proposition is contradictory.
12. Explain contingent statements.
A sentence is contingent if it is neither a tautology nor a contradiction; i.e. if it is T on at
least one row and F on at least one row. Now consider the statement: ‘If it is raining then
the roads are wet’ which is symbolized as P  Q. The truth table for implication is as
follows:
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P
Q
PQ
T
T
T
T
F
F
F
T
T
F
F
T
Since we get at least one T and one F in the rows, the given proposition is contingent.
13. Define argument form.
An argument form can be defined as an array of symbols containing statement variables
but no statements, such that when statements are substituted for statement variables-the
same statement being substituted for the same statement variable throughout – the result is
an argument. An argument form is a group of sentence forms such that all its substitution
instances are arguments. For example, all substitution instances of the form are arguments,
and hence that form is an argument form.
PQ
P
/∴ Q
PART - C - Paragraph answer questions
1. Explain the differences between argument and argument form.
An argument is a set of sentences, one of which (the conclusion) is claimed to be supported
by the others (the premises). Argument form is the logical structure of an argument. An
argument can be proved invalid by constructing another argument of the same form with
true premises and false conclusion. To prove the invalidity of an argument, it is sufficient
to construct another argument of the same form with true premises and false conclusion.
An argument form can be defined as an array of symbols containing statement variables
but no statements, such that when statements are substituted for statement variables-the
same statement being substituted for the same statement variable throughout – the result is
an argument.
An argument form is a group of sentence forms such that all its substitution instances are
arguments. For example, all substitution instances of the form
PQ
P
/∴Q are arguments, and hence that form is an argument form. The order of the
premises in an argument is irrelevant. Thus,
PQ
P
P
/∴Q and P  Q /∴Q both can be thought of as substitution instances of the
preceding form. Every valid argument is a substitution instance of at least one valid form.
An invalid argument cannot be a substitution instance of a valid argument form.
2. Distinguish between validity and invalidity.
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Truth and falsehood characterize propositions or statements. Arguments, however, are not
considered as true or false but as valid or invalid. There is a connection between the
validity or invalidity of an argument and the truth or falsehood of its premises and
conclusion. An argument is valid if and only if it is impossible for the premises to be true
and the conclusion false. In the case of a valid argument, it is impossible for the premises
to be true and at the same time the conclusion false. Validity is not about the actual truth or
falsity of the sentences in the argument but instead, it is about the form of the argument.
3. Distinguish between tautology and contradiction.
A sentence is a tautology if the column under its main connective is ‘True’ on every row of
a complete truth table. Now consider the statement - ‘It is raining or it is not raining’,
which is symbolized as ‘p v ~ p’.
The truth table for p v ~ p is represented as follows:
P
~p
pv~p
T
F
T
F
T
T
Since we get only T in every row, this proposition is tautology.
A sentence is a Contradictory if the column under its main connective is ‘False’ on every
row of a complete truth table. Now consider: ‘it is raining and it is not raining’ which is
symbolized as ‘p . ~ p
The truth table for p v ~ p is represented as follows:
P
~p
p∙~p
T
F
F
F
T
F
Since we get only F in every row, this proposition is contradictory.
PART - D - Essay questions
1. Explain tautology, contradiction and contingent statement forms. Construct truth table
to find out which of the following compound statements are tautology, contradiction
and contingent.
a. (P  Q) v ~ Q
b. (P  Q) ∙~Q
c. (P  Q) ∙ ~ (P  Q)
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Ans: In the case of compound statement, some will be contingent propositions, some will
be tautologies, which are always true, and some will be contradictions, which are always
false. We determine whether a given proposition is tautology, contradiction or contingent
by constructing the truth tables.
a. Tautology:
A sentence is a tautology if the column under its main connective is ‘True’ on every row of
a complete truth table. Now consider: ‘It is raining or it is not raining’ which is symbolized
as ‘p ∨ ~ p
The truth table for p v ~ p is represented as follows:
P
~p
pv~p
T
F
T
F
T
T
Since we get only T in every row, this proposition is tautology.
b. Contradictory:
A sentence is a contradictory if the column under its main connective is ‘False’ on every
row of a complete truth table. Now consider: ‘It is raining and it is not raining’, which is
symbolized as ‘p ∙ ~ p
The truth table for p v ~ p is represented as follows:
P
~p
p∙~p
T
F
F
F
T
F
Since we get only F in every row, this proposition is contradictory.
c. Contingent:
A sentence is contingent if it is neither a tautology nor a contradiction; i.e. if it is T on at
least one row and F on at least one row. Now consider the statement: ‘If it is raining then
the roads are wet’, which is symbolized as P  Q. The truth table for implication is as
follows:
P
Q
PQ
T
T
T
T
F
F
F
T
T
F
F
T
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Since we get at least one T and one F in the rows, this proposition is contingent.
The division of sentences into tautologies, contradiction and contingent sentences is of
fundamental importance. There is an important relationship between tautologies,
contradictions, and valid arguments. To every valid argument, there corresponds a
toutologous conditional sentence whose antecedent is the conjunction of the premises and
whose consequent is the conclusion. The truth values of all tautologies and contradictions
can be determine by logic alone, without appeal to experience or to any kind of empirical
test, although this is not the case for contingent sentences. Thus, the division into
tautologies, contradiction and contingent sentences is permitted to basic philosophical
questions about the way in which knowledge can be acquired.
Truth table for (P  Q) ∨ ~ Q
P
Q
~Q
(P  Q)
(P  Q) v ~ Q
T
T
F
T
T
T
F
T
F
T
F
T
F
T
T
F
F
T
T
T
(P  Q)
T
F
T
T
(P  Q) ∙ ~ Q
F
F
F
T
Since we get only T in every row, (P  Q) v ~ Q is tautology.
a. Truth table for (P  Q) ∙ ~ Q
P
Q
~Q
T
T
F
T
F
T
F
T
F
F
F
T
∙
~
Q
is
contingent.
Since we get both T and F, (P  Q)
b. Truth table for (P  Q) ∙ ~ (P  Q)
P
T
T
F
F
Q
T
F
T
F
(P  Q)
T
F
T
T
~ (P  Q)
F
T
F
F
(P  Q) ∙ ~ (P  Q)
F
F
F
F
Since we get only F in every row, (P  Q) ∙ ~ (P  Q) is contradiction.
2. Distinguish between validity and invalidity.
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Truth and falsehood characterize proposition or statements. Arguments, however, are
not properly characterized as being either true or false but as valid or invalid. There is a
connection between the validity or invalidity of an argument and the truth or falsehood of
its premises and conclusion. An argument is valid if and only if it is impossible for the
premises to be true and the conclusion false. The crucial thing about a valid argument is
that it is impossible for the premises to be true at the same time that the conclusion is false.
The important thing to remember is that validity is not about the actual truth or falsity of
the sentences in the argument. Instead, it is about the form of the argument.
An argument is deductively valid if and only if it is impossible for the premises to be
true and the conclusion false. The crucial thing about a valid argument is that it is
impossible for the premises to be true at the same time that the conclusion is false.
Consider this example:
Oranges are either fruits or musical instruments.
Oranges are not fruits.
Therefore, oranges are musical instruments.
The conclusion of this argument is ridiculous. Nevertheless, it follows validly from
the premises. This is a valid argument. If both premises were true, then the conclusion
would necessarily be true.
This shows that a deductively valid argument does not need to have true premises or
a true conclusion. Conversely, having true premises and a true conclusion is not enough to
make an argument valid.
Consider this example:
London is in England.
Paris is in France.
Therefore, Beijing is in China.
The premises and conclusion of this argument are, in fact, all true. This is a terrible
argument, however, because the premises have nothing to do with the conclusion. Imagine
what would happen if Beijing declared independence from the rest of China. Then the
conclusion would be false, even though the premises would both still be true. Thus, it is
logically possible for the premises of this argument to be true and the conclusion false. The
argument is invalid.
The important thing to remember is that validity is not about the actual truth or
falsity of the sentences in the argument. Instead, it is about the form of the argument: The
truth of the premises is incompatible with the falsity of the conclusion.
3. Distinguish between statements and statement forms.
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A statement form is any sequence of symbols containing statement variables but no
statements, such that when statements are substituted for the statement variables-the same
statement being substituted for the same statement variable throughout- the result is a
statement. Thus, ~p is called a negation form or denial form, p v q is a statement form
called disjunctive statement form, p . q is called conjunctive statement form and p Ͻ q is
conditional statement form. Any statement of a certain form is said to be a substitution
instance of that statement form.
The specific form of a given statement is defined as that statement form from which the
statement results by substituting a different simple statement for each different statement
variable. For example, p Ͻ q is the specific form of the statement A Ͻ B.
Tautologous, Contradictory, and Contingent statement forms:
We determine whether a given proposition is tautology, contradictory or contingent by
looking at the truth tables.
d. Tautology:
A statement is a tautology if the column under its main connective is ‘True’ on every
row of a complete truth table. Now consider the statement - ‘it is raining or it is not
raining’, which is symbolized as
‘p v ~ p
The truth table for p v ~ p is represented as follows:
P
~p
pv~p
T
F
T
F
T
T
Since we get only T in every row, this statement is tautology.
e. Contradictory:
A statement is a contradictory if the column under its main connective is ‘False’ on
every row of a complete truth table. Now consider: ‘it is raining and it is not raining’ which
is symbolized as ‘p . ~ p
The truth table for p v ~ p is represented as follows:
P
~p
p.~p
T
F
F
F
T
F
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Since we get only F in every row, this statement is contradictory.
f. Contingent:
A statement is contingent if it is neither a tautology nor a contradiction; i.e. if it is T on
at least one row and F on at least one row. Now consider the statement: ‘if it is raining then
the roads are wet’ which is symbolized as P Ͻ Q. The truth table for implication is as
follows:
P
Q
PϽQ
T
T
T
T
F
F
F
T
T
F
F
T
Since we get at least one T and one F in the rows, this statement is contingent.
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MODULE 4
FORMAL PROOF OF VALIDITY
PART - A - Multiple-choice questions
For model questions, see the Question Bank.
PART - B - Short answer questions
1. Define formal proof of validity.
If an argument contains many component statements, it is difficult to use truth table to test
their validity. A more efficient method of establishing the validity of an argument is to
deduce their conclusion from their premise by a sequence of shorter elementary arguments
that are known to be valid. There are nine rules of inference, which help us to construct the
formal proof of validity. In formal proof of validity, some rules are given and based on this
rule the conclusion is derived. It starts with certain given statements, and with the help of
certain self-evident rules, we deduce the conclusion. This method is called deductive
method. Thus, in the decision procedure of the formal proof of validity, conclusion is
drawn by applying the nine Rules of inference.
2. Define Modus ponens and give its symbolic form.
Modus ponens is a form of hypothetical syllogism in which the minor premise affirms
the antecedent and the conclusion affirms the consequent.
Example:
If one is a Gandhian, then she is a vegetarian.
X is a Gandhian.
∴ X is a vegetarian
p⊃q
p
∴q
3. Define Modus Tollens and give its symbolic form.
Modus Tollens is the rule of inference, which means denying the consequent and hence
denying the antecedent. The symbolic form is
p⊃q
~q
∴~p
4. Define conjunction and give its symbolic form.
In the rule of inference called conjunction, the conclusion is formed by joining all premises
together. The symbolic form is
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p
q
∴p∙q
5. Define simplification and give its symbolic form.
Simplification is the opposite rule of conjunction. Here the first component of the
conjunctive premise is inferred as the conclusion. The symbolic form is
p∙q
∴p
6. Define Addition and give its symbolic form.
According to this rule of inference, we can add any number of variables to the given
premise by disjunction. The symbolic form is
p
∴pvq
7. Define Absorption and give its symbolic form.
According to this rule, if p implies q it also implies p and q because any proposition
implies itself. The symbolic form is
p⊃q
∴ p ⊃ (p ∙ q)
8. Present the symbolic form of Modus Ponens, Modus Tollens and Constructive Dilemma.
a) Modus Ponens (M.P)
b) Modus Tollens (M.T)
p⊃q
p⊃q
p
~q
∴q
∴ ~p
c) Constructive Dilemma (C. D.)
(p ⊃ q) ∙ (r ⊃ s)
pvr
∴qvs
9. Construct the formal proof of validity of the following:
A⊃B
B⊃C
C⊃D
~D/∴~A
1. A ⊃ B
2. B ⊃ C
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3. C ⊃ D
4. ~D /∴~A
5. B⊃D 2,3H.S
6. ~B
5,4 M.T
7. ~A
1,6 M.T
10. State the justification for each line that is not a premise.
1. A ∙ B
2. (A v C) ⊃D/∴A ∙ D
3. A
4. A v C
5. D
6. A ∙ D
1. A∙B
2. (A v C) ⊃ D/∴A ∙ D
3. A
1,Simplification
4. A v C 3,Addition
5. D
2,4 Modus ponens
6. A ∙ D
3,5 conjunction
11. Write a note on Hypothetical Syllogism and its symbolic form.
In pure hypothetical syllogism, all the propositions are hypothetical propositions.
For example,
If john catches the train then he will meet his family.
If he meets his family then the company will appoint new person.
Therefore, if john catches the train then the company will appoint new person.
The rules of pure hypothetical syllogism are as follows:
1) Both of the premises should have one common categorical proposition.
2) This common proposition is the antecedent in one premise and consequent in other
premise.
3) The conclusion should not have this common term, but instead it should contain the
antecedent of one premise as antecedent (other than the common term) and consequent
other premise as consequent (other than the common term)
Symbolic form
p⊃ q
q⊃ r
∴p ⊃ r
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12. Construct the formal proof validity of the following arguments.
A. If it rains in time, there will be good crops. If there are good crops, then prices of
essential commodities will not increase. It rains in time. Therefore, the prices of essential
commodities will not increase.
1. R ⊃ C
2. C ⊃ ~I
3. R /∴~I
Answer
1. R ⊃ C
2. C ⊃ ~I
3. R /∴~I
4. C 1,3 M.P
5. ~I
2, 4 M.P
B. If the students protest hike in fees then either the college will withdraw extra facilities or
the government will have to spend more. The students will protest hike in fees. The college
will not withdraw extra facilities. Therefore, the government will have to spend more.
1.
2.
3.
4.
5.
S⊃ (F v G)
S
~F /∴G
FvG
1,2 M.P
G
4,3 D.S
13. State the rule of inference for the following arguments.
1. (A⊃~B) ∙ (~C ⊃ D)
∴A⊃~B
Ans. Simplification
2. E ⊃ ~F
∴ (E ⊃ ~F) v (~G ⊃ H)
Ans. Addition
3. (F⊃~G) ⊃ (~H v ~I)
F⊃~G
∴~H v ~I
Ans. Modus Ponens
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PART - C - Paragraph answer questions
1. Describe constructive dilemma and give its symbolic form.
Every dilemma has two conditional propositions as its major premises. In constructive
dilemma, the major premise has two different antecedents, both leading to different
consequents.
E.g. If the salaries are increased, the economy is adversely affected and If the salaries are
not increased, there will be wide spread agitation.
Either the salaries are increased or not.
∴ Either the economy is adversely affected or there are wide spread agitation.
If A is B, C is D and E is F, G is H
Either A is B or E is F
 Either C is D or G is H
This is constructive because disjunctive minor premise affirms the antecedents.
The Symbolic form
(p ⊃ q) ∙ (r ⊃ s)
pv r
∴qv s
2. Construct the formal proof of validity of the following argument.
N⊃ M
M⊃ D
M⊃ P
~P
N v M /∴D
Ans.
1.
2.
3.
4.
5.
6.
7.
8.
N⊃ M
M⊃ D
M⊃ P
~P
N v M /∴D
~M 3,4 M.T
~N 1,6 M.T
M 5,7 D.S
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9. D 2,8 M.P
3. Write a note on Disjunctive Syllogism.
Disjunctive syllogism is a mixed syllogism in which major premise is a disjunctive
proposition (Alternatives are joined by either -- or), minor premise and conclusion are
categorical propositions.
E.g. Ram is either mad or drunk.
Ram is not mad.
∴Ram is drunk.
Rules
1) First premise is disjunctive proposition.
2) Second premise is negation of one of the disjuncts of major premise.
3) Conclusion is the remaining disjunct or disjuncts.
A valid disjunctive syllogism can be one of the following types:
1) Either p or q
pvq
Not p
~p
Therefore, q
∴q
2) Either p or q
pvq
Not q
~q
Therefore, p
∴p
4. State the justification for the given proof of validity.
1. F v (G v H)
2. (G ⊃ I) ∙ (H ⊃ I)
3. (I v J) ⊃ (F v H)
4. ~F /∴H
5. G v H
6. I v J
7. F v H
8. H
Ans.
1. F v (G v H)
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2. (G ⊃I) ∙ (H Ɔ I)
3. (I v J) ⊃ (F v H)
4. ~F /∴H
5. G v H 1, 4 D.S
6. I v J 2,5C.D
7. F v H 3, 6 M.P
8. H
4, 7 D.S
5. Construct the formal validity of the following argument.
P ⊃ ~q
~q ⊃ r
r⊃s
(p ∙ s)⊃ u
/∴p ⊃ u
Ans:
1. P⊃~q
2. ~q⊃ r
3. r ⊃ s
4. (p ∙ s) ⊃ u
/∴p ⊃ u
5. p ⊃ r
1,2,H.S
6. p ⊃ s
5,3H.S
7. p⊃ (p ∙ s) 6 Abs.
8. p ⊃ u
7,4 H.S
6. Briefly explain Modus ponens and Modus tollens and give its symbolic form.
Modus Ponens (Constructive hypothetical syllogism)
It is a form of hypothetical syllogism in which the minor premise affirms the
antecedent and the conclusion affirms the consequent.
Example: If a man is a Gandhian, then he is a vegetarian.
X is a Gandhian.
∴ X is a vegetarian.
Symbolic form
P
p⊃ q
Modus Tollens (Destructive Hypothetical Syllogism)
It is a form of hypothetical syllogism in which the minor premise denies the
consequent and the conclusion denies the antecedent of major premise.
Example: If he is a thief, he will hide the goods.
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He has not hidden the goods.
∴ He is not a thief.
Symbolic form
p⊃ q
~q
∴ ~p
7. State the rules of inference by which the conclusion follows from premise or premises.
1. A v (B ⊃ D)
2. ~C ⊃ (D ⊃ E)
3. A⊃C
4. ~C /∴B ⊃ E
Answer
1. A v (B ⊃ D)
2. ~C ⊃ (D ⊃ E)
3. A ⊃ C
4. ~C /∴B⊃E
5. ~A 3, 4 M.T
6. B ⊃ D 1, 5 D.S
7. D ⊃ E 2, 4 M.P
8. B ⊃ E
6, 7 H.S
8. Construct the formal proof of validity for the following Argument.
A⊃ B
C⊃ D
A ∙ C /∴B ∙ D
1. A ⊃ B
2. C⊃D
3. A ∙ C /∴B ∙ D
4. A 3 Simp.
5. B 1,4 M.P
6. A ⊃ D 1,2 H.S
7. D 4, 6 M.P
8. B ∙ D 5, 7 Conj.
9. Symbolize the given argument using the symbols given in brackets and construct the
formal proof of validity for the following.
If a tenth planet exists, then its orbit is perpendicular to that of other planets. Either the
tenth planet is responsible for the death of the dinosaurs or its orbit is not perpendicular to
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that of other planets. A tenth planet is not responsible for the death of the dinosaurs.
Therefore, a tenth planet does not exist. (E, P, R)
Symbolic form:
E⊃ P
R v ~P
~R / ∴ ~E
Formal proof:
1. E⊃P
2. R v ~P
3. ~R / ∴~E
4. ~P 2, 3 D.S
5. ~E 1, 4 M.T
10. Write a note on any two of the following and present their symbolic forms.
a)Hypothetical syllogism
b) Formal proof of validity
c) Modus ponens
(Prepare the answers by collecting and arranging the relevant answers from Part B and C)
Part - D - Essay questions
1. Define formal proof of validity and present the symbolic form of nine elementary valid
argument forms.
(Prepare the essay by collecting and arranging the relevant answers from Part B and C)
2. Present the nine rules of inference and construct the formal proof of validity for the
given argument.
(W v X)⊃Y
(W v X) ∨Z
~Y / ∴Z
(Prepare the first part of the essay by collecting and arranging the relevant answers from
Part B and C)
Answer to the second part:
1. (W v X) ⊃Y
2. (W v X) v Z
3. ~Y / ∴Z
4. ~ (W v X) 1, 3 M.T
5. Z
2, 4 D.S
3. Write a note on any two of the following and give its symbolic form.
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a. Constructive Dilemma
b. Hypothetical syllogism
c. Modus Ponens and Modus Tollens
d. Disjunctive syllogism
(Prepare the essay by collecting and arranging the relevant answers from Part B and C)
3. Define formal proof of validity and construct the formal proof of validity for the
following argument using the abbreviations suggested.
If either algebra is required or geometry is required, then all students will study
mathematics. Algebra is required and Trigonometry is required. Therefore, all students will
study Mathematics. (
A: Algebra, G: Geometry, T: Trigonometry, S: study, M:
Mathematics)
(Prepare the first part of the essay by collecting and arranging the relevant answers from
Part B and C).
Answer to the second part:
1. (A v G) ⊃ M
2. A ∙ T /∴M
3. A
2, Simp.
4. A v G
3, Add.
5. M
1, 4 M.P
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MODULE 5
INFORMATICS
(The informatics portions in this module are prepared on the basis of the Textbook
Informatics: Technology in Action authored by Alan Evans et al.)
PART - A - Multiple-choice questions
For model questions, see the Question Bank.
PART - B - Short answer questions
1. Define informatics.
Informatics is the science of computer information system. As an academic field, it
involves the practice of information processing. As a combination of ‘information’
and ‘automatic’, informatics is also defined as the science of automating
information interactions. Informatics includes different fields like Health
informatics, Bio- informatics, Business informatics and Engineering informatics.
2. Define Data.
In computer terms, data is a representation of a fact or idea. Data can be a number,
a word, a picture, or even a recorded sound. For example, the number 6125553297
and the names Derek and Washington are pieces of data.
3. Distinguish between data and information.
In computer terms, data is a representation of a fact or idea. Data can be a number,
a word, a picture, or even a recorded sound. For example, numbers and the names
are pieces of data. Data becomes information when organized in a meaningful way.
For example, when a set of ordered numbers represent the telephone number of a
person it is an information.
3. Explain the term privacy.
Privacy refers to the right of a person to maintain certain facts to oneself without
the knowledge of others. It is a basic human right like the right to be treated with
dignity. Unlimited privacy is the right to be left alone to do as one pleases. The idea
of privacy is often associated with hiding something. In computer terms privacy is
the right to protect the digital data and information without giving access to others.
4. Describe Cyber addiction.
Cyber addiction is the abnormal tendency to excessive use of computers and
internet. It is an addiction that affects the routine life of an individual who becomes
too much dependent on computer and internet. Cyber addiction is just like any other
form of addiction where there is a strong craving towards internet surfing. Cyber
addiction appears in different forms like computer addiction, cyber relational
addiction and net gaming addiction.
5. What is meant by information overload?
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Information overload refers to the addiction towards unlimited information
available on the internet. The addict in this case has a craving for searching, reading
and storing information of his interest on the internet. It often results in stress and
anxiety related disorders. The term information overload was first used by the great
futurist and writer Alvin Toffler in his book Future Shock. Toffler regards
information overload as a psychological disorder caused by an abundance of
information availability.
6. Explain the notions of Computer addiction and Cyber relational addiction.
Computer addiction may or may not include addiction towards the internet. It
particularly includes addiction towards playing computer games. Cyber relational
addiction refers to the addiction towards socializing through the internet. The addict
in this case finds a craving for creating online friends through social networking
sites; thus often neglecting the real social life.
7. Explain the role of Cyber laws in the context of Information explosion.
Cyber laws are a set of legal provisions to regulate cyber activities including
internet use. The unusual increase in the number of internet users is followed by an
increase in the number cyber crimes. Hence, all national governments are forced to
frame rigid cyber laws. They include not only the laws to prevent cyber crimes but
also the intellectual property right rules. In India, cyber laws have been defined
under the IT Act 2000.
8. Present the diagrammatic representation of NOT and OR gates.
NOT Gate
OR Gate
9. Present the diagrammatic representation and truth table for AND gate.
Diagram
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Truth table
PART - C - Paragraph answer questions
1. What is the difference between data and information?
In ordinary language, the terms data and information are used interchangeably.
However, in computer terms, the distinction between data and information is very
important. In computer terms, data is a representation of a fact or idea. Data can be a
number, a word, a picture, or even a recorded sound. For example, numbers and the
names are pieces of data. Data becomes information when organized in a meaningful
way. For example, when a set of ordered numbers represent the telephone number of a
person it is information. Hence, data become useful when it assumes the status of
information.
Computers are very good at processing data into information. For example, the
necessary personal data about a citizen become specific information when it is
processed into the ADHAAR Card. This organized output of data on your ID card is a
set of useful information.
2. How do computers process data into information?
Computers work with a binary language that consists of just two digits: 0 and 1. All
data on a computer is stored in the combinations of 0s and 1s. Each 0 and 1 is a binary
digit, or bit for short. Eight binary digits or bits combine to create one byte. In
computers, each letter of alphabet, each number, and each special character is a unique
combination of eight bits, or a string of eight 0s and 1s. For example, in computer
language, the letter K is represented as 01001011. This equals eight bits, or one byte. A
kilobyte (KB) is approximately 1000bytes, a megabyte (MB) is about a million bytes,
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and a gigabyte (GB) is about a billion bytes. In the computer world, the storage needs
are so high today that some computers can store more than one quadrillion bytes, that
is, a petabyte of data. In a computer, bits and bytes represent the data and information it
inputs and outputs. All data processing is based on this special digital language used in
a computer.
3. What is ethical computing?
We have many examples of the unethical use of computers. There are stories about
cyber crimes coming out in media every day. Unexpected virus attacks and illegal
sharing of copyright protected materials are not rare today. However, it is not easy to
define what constitutes ethical behaviour while using a computer.
Ethics is a system of moral principles, rules, and accepted standards of conduct. So
what are the accepted standards of conduct when using computers? The Computer
Ethics Institute developed the Ten Commandments of Computer Ethics, which may
guide our ethical standards in the cyber world. These Ethical Computing Guidelines are
stated below:
1. Avoiding causing harm to others when using computers.
2. Do not interfere with other people’s efforts at accomplishing work with computers.
3. Resist the temptation to snoop in other people’s computer files.
4. Do not use computers to commit theft.
5. Agree not to use computers to promote lies.
6. Do not use software (or make illegal copies for others) without paying the creator
for it.
7. Avoid using other people’s computer resources without appropriate authorization or
proper compensation.
8. Do not claim other people’s intellectual output as your own.
9. Consider the social consequences of the products of your computer labour.
10. Only use computers in way that show consideration and respect for others.
4. Summarize the issue of privacy in the cyber world.
Privacy refers to the right of a person to maintain certain facts to oneself without the
knowledge of others. It is a basic human right like the right to be treated with dignity.
Unlimited privacy is the right to be left alone to do as one pleases. The idea of privacy
is often associated with hiding something. In computer terms, privacy is the right to
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protect the digital data and information without giving access to others. It is indeed
necessary in the present day world of digitalized services in all areas like
communication, banking and marketing. We are using debit and credit cards for
purchasing and bank transactions. E-mail is the common means to correspondence. In
such cases, it is necessary to maintain privacy by preventing the chances of ‘identity
theft’ and phishing. Computer experts advise us to change our behaviour to protect
privacy. It is necessary to stop giving personal information to unknown internet queries.
We should take proper care to keep our cyber identity like passwords, phone numbers
and secret codes. Governments are framing cyber laws that include the rules to protect
our privacy in the cyber world.
5. Discuss the for and against arguments of the problem of privacy.
Arguments for privacy:
The advocates for protecting privacy argue that the right to privacy is a basic human
right and it should be ensured in the digital world also. The main reasons they give are
the following:
1. If I am not doing anything wrong, then you have no reason to watch me.
2. If the government is collecting information by watching citizens, it might misuse
or lose control of the data.
3. By allowing the government to determine what behaviours are right and wrong,
we open ourselves to uncertainty because the government may arbitrarily change
the definition of acceptable behaviours.
4. Requiring national ID cards is reminiscent of the former Nazi or Soviet regimes.
5. Implementing privacy controls (such as national ID cards) is extremely
expensive and a waste of taxpayer funds.
Arguments against privacy:
Advocates for stronger monitoring of private citizens emphasize national security
concerns like the prevention of terrorist activities. Personal inconvenience is just the
price for social security and peace. The main reasons they give are the following:
1. If you are not doing anything wrong, you need not hide anything.
2. Electronic identification documents are essential in the digital world to exchange
information and to detect suspected terrorists.
3. Laws protect citizens from the abuse and misuse of government officials who are
involved in monitoring activities.
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4. It is not possible to put a price on freedom or security and hence projects like a
national ID system are worth the cost of implementation.
6. What is cyber addiction and explain the various terms related to cyber addiction.
Cyber addiction refers to the tendency to too much use of the computer and internet to
the extent of affecting the routine life of an individual. One who becomes too much
dependent on computer and internet is called a cyber addict. Cyber addiction is just like
any other form of addiction such as that of television, alcohol, gambling, drugs etc.
Cyber addiction is specified by many other terms, like computer addiction, cyber
relational addiction and net gaming.
Computer addiction: It may or may not include addiction towards the internet. It
particularly includes addiction towards playing computer games.
Cyber relational addiction: It refers to the addiction towards socialising though
the internet. The addict in this case finds a craving for creating online friends
through social networking sites; thus often neglecting the real social life.
Net gaming: As the name suggests, the addict in this case finds a strong craving
towards playing online games. The addict particularly finds great sense of triumph
in beating other online gamers.
7. Analyze the issues related with Information overload.
Information overload refers to the addiction towards unlimited information available
on the internet. The addict in this case has a craving for searching, reading and
storing information of his interest on the internet. It often results in stress and
anxiety related disorders. The great futurist writer Alvin Toffler in his book Future
Shock first used the term information overload. Toffler regards information overload
as a psychological disorder caused by an abundance of information availability.
The amount of information on the internet has led to an information explosion.
More and more number of internet users is becoming addicts of information
overload and they gradually enter a virtual world of websites, emails, blogs,
reviews, messengers, social networking sites etc. Indifference to day-to-day affairs,
increasing anxiety and stress are symptoms of this addiction. In order to protect
oneself from information overload it is necessary to access the information in a
systematic manner just to meet the necessary requirements.
8. Present the symbols for AND, OR and NOT gates.
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AND
OR
NOT
9. Present the truth tables for OR and NOT gates.
OR Gate
NOT Gate
PART – D - Essay questions
1. Define ‘informatics’ and bring out of the relationship and differences between data,
information and knowledge in the digital world.
Informatics is the science of computer information system. As an academic field, it
involves the practice of information processing. As a combination of ‘information’
and ‘automatic’, informatics is also defined as the science of automating
information interactions. Informatics includes different fields like Health
informatics, Bio- informatics, Business informatics and Engineering informatics.
We are living in a world of digitalization and information overload. Computers and
accessories have become a part of our day-to-day life.
Computers are basically data processing devices that may be used in a variety of
ways. In ordinary language, we often use the terms data and information
interchangeably. However, in the world of computers, the distinction between data
and information is not that simple.
In computer terms, data is a representation of a fact or idea. Data can be a number,
a word, a picture, or even a recorded sound. For example, numbers and names are
pieces of data. Data becomes information when organized in a meaningful way. For
example, when a set of ordered numbers represent the telephone number of a
person it is information. The main functions of a computer as a data processing
machine are the following:
i) Collecting and storing data input by the user.
ii) Processing that data into information.
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iii) Outputs data and information.
iv) Stores data and information.
These functions in different combinations make a computer the fastest and the most
efficient companion in all fields of human life like communication, education,
medicine, commerce etc.
We have seen that data is a representation of a fact or idea. For example, the
number 203 and the name Sofia are pieces of data. Information is data that has been
organised or presented in a meaningful fashion. When we know that 203 is the ID
Card number of Sofia the data mentioned earlier suddenly becomes useful - that is,
information. Knowledge is defined as information, understanding and skills learned
through education or experience. Information is identified as knowledge when it is
classified into a particular field such as science, philosophy or medicine. Moreover,
knowledge may be theoretical or practical. A knowledgeable person is of course
well informed about a specific field of study like biology or mathematics or about
the stream of life in this world.
How do computers interact with data and information? Computers are very good at
processing data into information. When you first arrived on campus, you probably
were directed to a place where you could get an ID card. You most likely provided
a clerk with personal data that was entered into a computer. The clerk then took
your picture with a digital camera. This information was then processed
appropriately so that it could be printed on your ID card. This organized output of
data on your ID card is useful information. Finally, the information was probably
stored as a digital data on the computer for later use.
Computers work with a binary language that consists of just two digits: 0 and 1. All data on
a computer is stored in the combinations of 0s and 1s. Each 0 and 1 is a binary digit, or bit
for short. Eight binary digits or bits combine to create one byte. In computers, each letter of
alphabet, each number, and each special character is a unique combination of eight bits, or
a string of eight 0s and 1s. For example, in computer language, the letter K is represented
as 01001011. This equals eight bits, or one byte. A kilobyte (KB) is approximately
1000bytes, a megabyte (MB) is about a million bytes, and a gigabyte (GB) is about a
billion bytes. In the computer world, the storage needs are so high today that some
computers can store more than one quadrillion bytes, that is, a petabyte of data. In a
computer, bits and bytes represent the data and information it inputs and outputs. All data
processing is based on this special digital language used in a computer.
2. Briefly describe the ethical issues related with cyber world.
We are living in a world of digitalization and information overload. Computers and
accessories have become a part of our day-to-day life. Informatics is the science of
computer information system. As an academic field, it involves the practice of
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information processing. Informatics includes different fields like Health informatics,
Bio- informatics, Business informatics and Engineering informatics.
There is a vast range of ethical issues related to the cyber world. Issues of privacy, cyber
addiction and information overload are considered as the most important among them.
Cyber ethics is an emerging area of ethics dealing with the good and bad in human relation
to the digital devices and the acquisition, storage and exchange of data and information
through such means. The terms like privacy, identity theft etc have become the concern of
computer literate people. Cyber crimes like phishing and hacking are increasing day by
day.
Privacy:
Privacy refers to the right of a person to maintain certain facts to oneself without the
knowledge of others. It is a basic human right like the right to be treated with dignity.
Unlimited privacy is the right to be left alone to do as one pleases. The idea of privacy is
often associated with hiding something.
In computer terms, privacy is the right to protect the digital data and information without
giving access to others. It is indeed necessary in the present day world of digitalized
services in all areas like communication, banking and marketing. We are using debit and
credit cards for purchasing and bank transactions. E-mail is the common means to
correspondence. In such cases, it is necessary to maintain privacy by preventing the
chances of ‘identity theft’ and phishing. As the use of digital devices is increasing, the
number of unlawful methods to break into personal computers is also increasing. Computer
experts advise us to change our behaviour to protect privacy. It is necessary to stop giving
personal information to unknown internet queries. We should take proper care to keep our
cyber identity like passwords, phone numbers and secret codes. Governments are framing
cyber laws that include the rules to protect our privacy in the cyber world.
The advocates for protecting privacy argue that the right to privacy is a basic human right
and it should be ensured in the digital world also. They point out the chances of
governments misusing the data of citizens for the interests of the state. Making national ID
cards compulsory is a waste of public money. The main argument for privacy is that if I am
not doing anything wrong, then you have no reason to watch me.
Advocates for stronger monitoring of private citizens emphasize national security concerns
like the prevention of terrorist activities. Personal inconvenience is just the price for social
security and peace. Laws protect citizens from the abuse and misuse of government
officials who are involved in monitoring activities.
Cyber addiction
Cyber addiction refers to the tendency to too much use of the computer and internet to the
extent of affecting the routine life of an individual. One who becomes too much dependent
on computer and internet is called a cyber addict. Cyber addiction is just like any other
form of addiction such as that of television, alcohol, gambling, drugs etc. Cyber addiction
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is specified by many other terms, like computer addiction, cyber relational addiction and
net gaming.
Computer addiction: It may or may not include addiction towards the internet. It
particularly includes addiction towards playing computer games.
Cyber relational addiction: It refers to the addiction towards socializing though
the internet. The addict in this case finds a craving for creating online friends
through social networking sites; thus often neglecting the real social life.
Net gaming: As the name suggests, the addict in this case finds a strong craving
towards playing online games. The addict particularly finds great sense of triumph
in beating other online gamers.
Information overload
Information overload refers to the addiction towards unlimited information available
on the internet. The addict in this case has a craving for searching, reading and
storing information of his interest on the internet. It often results in stress and
anxiety related disorders. The great futurist writer Alvin Toffler in his book Future
Shock first used the term information overload. Toffler regards information overload
as a psychological disorder caused by an abundance of information availability.
The amount of information on the internet has led to an information explosion.
More and more number of internet users is becoming addicts of information
overload and they gradually enter a virtual world of websites, emails, blogs,
reviews, messengers, social networking sites etc. Indifference to day-to-day affairs
and increasing anxiety and stress mark the addiction to information overload. In
order to protect oneself from information overload it is necessary to access the
information in a systematic manner just to meet the necessary requirements.
Like any other technological achievement, digital technology also has the chances of both
use and misuse. Hence, Cyber ethics is the need of the day. It is necessary for cyber experts
and rulers to join hands to define the crimes and decide the punishments in the cyber
world.
3. Define logical gate and present the symbols and truth tables for AND, NOT and
OR gates.*
Logic gates are the basic building blocks of any digital system. They process
signals, which represent the binaries of true/false. Normally, the positive supply voltage
+5V represent true and 0V represents false. A logic gate has one or more input and only
one output. The input-output relationship is based on a definite logic and hence the
name logic gate. The basic logic gates are AND, NOT and OR. There are also universal
gates like NAND gate in which an AND gate is followed by a NOT gate and the NOTOR operation called NOR gate. Combinational gates are X-OR gate and X-NOR gate.
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SYMBOLS
AND Gate
NOT Gate
OR Gate
TRUTH TABLES
AND Gate
OR Gate
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NOT Gate
* For a detailed account of Logic Gates, see the following web sources:
 Logic-gate, http://www.saylor.org/site/wp-content/uploads/2011/05/logic-gate.pdf
 Basic logic gates, http://www.allaboutcircuits.com/worksheets/gates.html
 Chapter 9 – Logic Gates
http://education.cambridge.org/media/577240/cambridge_igcse_computer_studies__rev
ision_
guide___cambridge_education___cambridge_university_press_samples.pdf
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