Introduction to Mathematical Logic

  • The word Logic is derived from a Greek word ‘Logos’ which means reason. The logic deals with the methods of reasoning. Logic is a process by which we arrive at a conclusion from known statements or assertions with the help of valid assumptions. The valid assumptions are known as laws of logic. The Greek philosopher and thinker Aristotle laid the foundation of the study of logic in the systematic form. Logic associated with mathematics is called mathematical logic. The mathematical approach to logic is developed by English philosopher and mathematician George Boole. Hence logic is also referred as Boolean Logic or Symbolic Logic.
  • Logic helps in the development of systematic and logical reasoning skill. It helps in understanding the precise meaning of statements of theorems, the converse of theorems, and corollaries of the theorem. It is the basis of circuit designing, artificial intelligence, and computer programming.


Statement:

  • Language is the medium of communication of thoughts. We do so by using sentences simple or complex. They may be assertive or imperative or exclamatory or interrogative or suggestive or wishes.
  • A declarative sentence, which is either true or false, but not both simultaneously, is called a statement in logic.
  • Sentences which are incomplete or imperative or interrogative or exclamatory or suggestive or wishes or perceptions are not taken as statements in logic.
  • An open sentence is a sentence whose truth value can vary according to some conditions which are not stated in the sentence. e.g. It is white in colour.
  • In logic, the statements are denoted by small case letters particularly p, q, r, ….

Law of Excluded Middle:

  • A statement is either true or false. It can not be both true and false and also neither true nor false. This fact is known as the law of excluded middle.

Truth Value of a Statement:

  • The truth or falsity of a statement is called the truth value of the statement.
  • If the statement is true then its truth value is denoted by the letter ‘T’.  If the statement is false then its truth value is denoted by the letter ‘F’. In boolean algebra 1 is used for T and 0 is used for F.

To find Truth Value of a Statement:

Determine which of the following sentences are statements in logic. If not give a reason. If a statement then give its truth value.

Do your homework today.

  • It is command or suggestion. Hence it is not a statement.

x2 – 5x + 6 = 0, when x = 2

  • It is a statement. Its truth value is ‘T’

x2 – 5x + 6 = 0, x e R

  • For x = 2 or x = 3 it is true for x ≠ 2 and x ≠ 3, it is false. Hence it is open sentence. It is not a statement.

It is white in colour.

  • We cannot decide whether the statement is true or false. It is not a statement.

X + 3 = 5

  • If x = 2 it is true and if x ≠ 2 it is false. Hence it is open sentence. It is not a statement.

Oh! What a beautiful scene!

  • It is an exclamation. Hence it is not a statement.

Let us go for a walk

  • It is a suggestion. Hence it is not a statement.

I wish man had wings

  • It is a wish. Hence it is not a statement.

Please give me a glass of water.

  • It is a request i.e. imperative sentence. Hence it is not a statement.

Get out of the class immediately.

  • It is a Command i.e. imperative sentence. Hence it is not a statement.

When is your examination going to start?

  • It is an interrogative sentence. Hence it is not a statement.

The sum of two odd integers is always odd.

  • It is a statement. Its truth value is ‘F’

The product of two odd integers is always odd.

  • It is a statement. Its truth value is ‘T’

Please come here.

  • It is request i.e. imperative sentence. Hence it is not a statement.

The quadratic equation x2 – 3x + 2 = 0 has two real roots.

  • It is a statement. Its truth value is ‘T’

The square of any real number is always positive.

  • It is a statement. Its truth value is ‘F’. Because the square of 0 is 0 which is neither positive nor negative.

5 + 4 = 11

  • It is a statement. Its truth value is ‘F’

Every square of an odd number is always even.

  • It is a statement. Its truth value is ‘F’

1 is a prime number

  • It is a statement. Its truth value is ‘F’

Every natural number is a whole number.

  • It is a statement. Its truth value is ‘T’

Are you ready for a picnic?

  • It is an interrogative sentence. Hence it is not a statement.

Sin 2θ = 2 sin θ co θ, for all θ

  • It is a statement. Its truth value is ‘T’

829 is divisible by 9

  • It is a statement. Its truth value is ‘F’

What a great fall of Humpty Dumpty!

  • It is an exclamatory sentence. Hence it is not a statement.

New Delhi is capital of India

  • It is a statement. Its truth value is ‘T’

Shut the door,

  • It is an imperative sentence. Hence it is not a statement.

Please give me a pen

  • It is an imperative sentence. Hence it is not a statement.

Do you like fruits?

  • It is an interrogative sentence. Hence it is not a statement.

What a heavy downpour!

  • It is an exclamatory sentence. Hence it is not a statement.

Have you ever seen a rainbow?

  • It is an interrogative sentence. Hence it is not a statement.

X + 2 = 7

  • It is true when x = 5 and false when x ≠ 5. It is an open sentence. Hence it is not a statement.

He is a musician.

  • Its truth cannot be determined. Hence it is not a statement.

Statistics is an easy subject.

  • Its truth  cannot be determined because it is perception which may change from person to person. Hence it is not a statement.

The sun is a star.

  • It is a statement. Its truth value is ‘T’

May God bless you!

  • It is a wish. Hence it is not a statement.

The sum of interior angles of a triangle is 180°.

  • It is a statement. Its truth value is ‘T’

Every real number is a complex number.

  • It is a statement. Its truth value is ‘T’. because every real number can be written in the form a + bi

Why are you upset?

  • It is an interrogative sentence. Hence it is not a statement.

The square root of – 9 is a rational number.

  • It is a statement. Its truth value is ‘F’

The sum of cube roots of unity is 1.

  • It is a statement. Its truth value is ‘F’. Because the sum is zero.

X2 – 3x + 2 implies that x = -1 or x = -2.

  • It is a statement. Its truth value is ‘T’

He is a good person.

  • Its truth value cannot be determined because it is perception which may change from person to person. Hence it is not a statement.

Two is the only even prime number.

  • It is a statement. Its truth value is ‘T’

Do not disturb.

  • It is an imperative sentence. Hence it is not a statement.

X2 – 3x – 4 = 0 when x = -1

  • It is a statement. Its truth value is ‘F’

It is red in colour.

  • We cannot decide whether the sentence is true or false. Hence it is not a statement.

Every parallelogram is a rhombus.

  • It is a statement. Its truth value is ‘F’.

Every set is a finite set.

  • It is a statement. Its truth value is ‘F’

Indians are intelligent.

  • Its truth value cannot be determined because it is perception which may change from person to person. Hence it is not a statement.

Do your work properly.

  • It is a suggestion. Hence it is not a statement.

X + 3 = 10

  • It is true when x = 7 and false when x ≠ 7. It is an open sentence. Hence it is not a statement.

Will you help me?

  • It is an imperative sentence. Hence it is not a statement.


Square of an odd number is odd.

  • It is a statement. Its truth value is ‘T’

Zero is a complex number.

  • It is a statement. Its truth value is ‘T’. Because any real number is a complex number

All real numbers are rational numbers.

  • It is a statement. Its truth value is ‘F’

2 + 3 < 6

  • It is a statement. Its truth value is ‘T’

X2 – 5x + 6 = 0 when x = 2

  • It is a statement. Its truth value is ‘T’.

The door is open.

  • Its truth cannot be determined because it is a perception (How much?) which may change from person to person. Hence it is not a statement.

I am lying.

  • Its truth cannot be determined because it is dependent of the truthfulness of the person saying it. Hence it is not a statement. It is called liar’s paradox.

 

 

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