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Tautology

Tautology


A tautology is a compound statement that is true for all values of the individual statements.

The word tautology is derived from a Greek word where "tauto" means "same" and "logia" means "logic".
A compound statement is made with two simpler statements using some conditional words like 'and', 'or', 'not', 'if', 'then', and 'if and only if'.
For example, for any two given statements, such as x and y, (x ⇒ y) ∨ (y ⇒ x) is a tautology.


Simple examples of tautology are;

  • Either Subham goes home or Subham doesn't go home.
  • He is healthy or he is not healthy.
  • A number is odd or a number is not odd.

Tautology in Math

A tautology is a compound statement in Mathematics that always results in a value of Truth. No matter what the individual part consists of, the result in the tautology is always true. The opposite of tautology is the contradiction or fallacy that we will learn here.
It is easy to translate the tautologies of ordinary language into mathematical expressions with the help of logical symbols.
For example, I will give you 10 rupees or I will not give you 10 rupees.
Here, let us take:
P = I will give you 10 rupees
~P = I will not give you 10 rupees (Since it is the opposite statement of P)

These two individual statements are connected by the logical operator "OR", which is usually denoted by the symbol "∨".

Therefore, the above statement can be written as P ∨ ~P.

Now, we will check if the given statement produces a valid response.

Case 1: I will give 10 Rupees. In this case, the first statement is true and the second statement is false. Since the given statement is connected using the OR operator, it returns the true statement.

Case 2: I will not give 10 Rupees. In this case, the first statement is false and the second statement is true. Therefore, it produces a true statement.


What are the symbols used in tautology?

The important logic symbol used in tautology are:
  • AND (∧)
  • OR (∨)
  • NOT (~)
  • Negation (¬)
  • Implies (→)
  • If and only if (⇔)

Tautology Logic Symbols

Tautology uses different logical symbols to present compound statements.
These are the symbols and their meaning used in mathematical logic:

Symbols Meaning Representation
∧ AND A ∧ B
∨ OR A ∨ B
¬ Negation ¬A
~ NOT ~A
→ Implies or If-then A→B
⇔ If and only if A⇔B
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