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Logic

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Introduction:

The word logic is derived from ''LOGUS'', which means reason.Logic is the science of formal principles of reasoning or correct inference. Historically, logic originated with the ancient Greek philosopher Aristotle. Logic was further developed and systematized by the Stoics and by the medieval scholastic philosophers. In the late 19th and 20th centuries, logic saw explosive growth, which has continued up to the present. Logic is thwe science of correct reasoning. Actually,reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference. However, when logic is applied to specific subject matter, it is important to note that not all logical inference constitutes a scientifically valid demonstration. This is because a piece of formally correct reasoning is not scientifically valid unless it is based on a true and primary starting point. Furthermore, any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences. All reasoning, both scientific and non-scientific, must take place within the logical framework, but it is only a framework, nothing more. This is what is meant by saying that logic is a formal science.

Concept of Logic:

The science of reasoning is called the logic.In mathematics, we deal with various theorems and formulae. The different procedures used in giving the proof of various theorems and formulae are based on sound reasoning. The study of such procedures based on sound reasoning is known as a logic. Logic tells us the truth and the falsity of the particular statement. Logic is the process by which we arrive at a conclusion from the given statement with a valid reason.

In Logic, We use symbols and notations to denote words or sentence, hence it is also called symbolic logic or mathematical logic

Statements:

The declarative sentence which is either true or false but not both at the same time is called a statement.for example:-

  1. Ram is a doctor.
  2. 7+4=9
  3. 3-11<5

Here, a,b and c are statements as a and c are true and b is false.

The interrogative, imperative and exclamatory sentence are not statements because they do not declare the truth or falsity.examples:

  1. what is your name?open the door.
  2. How beautiful the scene is!

A sentence which is true or false after filling up the gap or after substituting the value of variable are called open sentences. Examples:

  1. ............ is father of sita.
  2. x+3=5

These sentences are not statements

Types of statements

Statement is of two types. They are simple statement and compound statement.

Simple statement: The statement which declares one thing at a time is called a simple statement. A simple statement cannot be divided in two or more than two statements. A simple statement is denoted by p,q,r,s, etc

Examples-

  1. p: Ram is a doctor.
  2. q: 4+3<11

Compound statement:when two or more than two simple statements are combined to form a statement, it is called compound statement. Simple statements are combined to form compound statement by using words or phrases, called connectives or logical connectives.

The simple statements are called components of the compound statement. Truth value of a compound statement depends on the truth value of the component statements. Examples:

  1. Ram is a doctor and sita is a nurse.
  2. 3+7<4 or 4+11>20

Truth value and truth table

The truthness or falsity of a statement is called its truth value. A true statement is denoted by 'T' and a false statement is denoted by 'F'. The table showing the truth value of a compound statement together with the truth value of its component statements is called a truth table.

Logical connectives

The word or phrases, ''and '', ''or'', ''if ..... they'' and ''if and only if'' are the logical connectives used to combine simple statements to form compound statements.

1. Conjunction

When two simple statements are combined with ''and'' to form a compound statement; it is called a conjunction. It is denoted by '∧'. example;

p:shyam is handsome. q:sita is beautiful.

Then, p∧q: shyam is handsome and sita is beautiful.

Truth table of p∧q

p q p∧q
T T T
T F F
F T F
F F F

2. Disjunction:

When two simple statements are combined with ''or'' to form a compound statement; it is called a disjunction. It is denoted by '∨'. example;

p:3+5=11, q:4+3=7

then, p∨q:3+5=11 or4+3=7

Truth table of p∨q

p q p∨q
T T T
T F T
F T T
F F F

3. Conditional:

When two simple statements are combined with ''if.........then'' to form a compound statement; it is called a conditional and denoted by '⇒'

Examples:

p: ABC is a triangle.

q: The sum of three angles is 180°.

p⇒q: IfABC is a triangle thenThe sum of three angles is 180°.

Truth table ofp⇒q

p q p⇒q
T T T
T F F
F T T
F F T

4. Biconditional:

When two simple statements are combined with ''if and only if'' to form a compound statement; it is called a biconditional and denoted by '⇔'

Examples:

p: Two triangles are similar.

q:The sides of triangles are proportional.

p⇔q:Two triangles are similar if and only if the sides of triangles are proportional.

Truth value ofp⇔q

p q p⇔q
T T T
T F F
F T F
F F T

Negation:

Negation of the statement denies the given statement. The negation of the statement is formed by using the word ''not'', "it is not true that", "it is not that case that", "it is false that",etc

The negation of a statement p is denoted by, $$\sim p$$

example: p: Ram is an intelligent boy.

∼p:Ram is not an intelligent boy.

Negation of

All= some......... not

some= no

some.......not = all

no= some

Truth table for negation

p ∼p
T F
F T

Law of Logic

The following are the laws of logic:

Let p,q and r be any three statements.

a) Law of excluded middle

only one statement p or∼pis true.

A statement cannot both be true and false at the same time.

b) Law of tautology

The statement p v ∼p is a tautology

The disjunction of a statement and its negation is a tautology.

c) Law of contradiction

The statement p ∧∼p is a contradiction.

The conjunction of a statement and its negation is a contradiction.

d) Law of involution

$$\sim (\sim p)\equiv p$$

The negation of negation of a statement is logically equivalent to a given statement. The law is also known as the law ofdouble negation.

e) Law of syllogism

If p⇒ q and q⇒ r then p⇒ r. That is

$$(p\Rightarrow q) ∧(q\Rightarrow r)\Rightarrow (p\Rightarrow r)$$

This can be verified to be a tautology.

f) Law of contraposition

$$(p\Rightarrow q)\equiv ((\sim q)\Rightarrow (\sim p)) $$

The conditional and its contrapositive are logically equivalent.

g) Inverse law

$$(\sim p)\Rightarrow(\sim q)\equiv q\Rightarrow p$$

The inverse and the converse of a conditional are logically equivalent.



  • The truthness or falsity of a statement is called its truth value. A true statement is denoted by 'T' and a false statement is denoted by 'F'. The table showing the truth value of a compound statement together with the truth value of its component statements is called a truth table.
  • When two simple statements are combined with ''and'' to form a compound statement; it is called a conjunction. It is denoted by '∧'. 
  • When two simple statements are combined with ''or'' to form a compound statement; it is called a disjunction. It is denoted by '∨'.
  • When two simple statements are combined with ''if.........then'' to form a compound statement; it is called a conditional and denoted by '⇒'

  • When two simple statements are combined with ''if and only if'' to form a compound statement; it is called a biconditional and denoted by '⇔'

 

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Questions and Answers

Click on the questions below to reveal the answers

Truth table

p q p (p)q
T T F F
T F F F
F T T T
F F T F

Truth table

p q p (pq) (pq)
(pq)(pq)
T T F F T T
T F F F T T
F T T T T T
F F T F F T

Solution:

Truth table

p p p(p)
T F F
T F F
F T F
F T F

Hence, $$p∧(\sim p)\equiv C$$

Solution:

Truth table

p p p(p) ∼(p(p))
T F F T
T F F T
F T F T
F T F T


Hence, $$\sim(p∧(\sim p))\equiv T$$

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