A statement is said to be a tautology if its truth value is always T irrespective of the truth values of its component statements. It is denoted by T.

**Mathematical Logic**

**Tautology,
Contradiction, and Contingency**

A statement is said to be a
**tautology** if its truth value is always *T* irrespective of the truth values of its component statements. It
is denoted by **T**.

A statement is said to be a
**contradiction**
if its truth value is always *F* irrespective of the truth values of its component statements. It
is denoted by **F**.

A statement which is neither a tautology nor a contradiction is
called **contingency**

** **

1. For a
tautology, all the entries in the column corresponding to the statement formula
will contain *T*.

2. For a
contradiction, all the entries in the column corresponding to the statement
formula will contain *F*.

3. The
negation of a tautology is a contradiction and the negation of a contradiction
is a tautology.

4**.** The disjunction of a statement with
its negation is a tautology and the conjunction of a statement with its
negation is a contradiction. That is *p*
∨¬*p* is a **tautology** and *p* ∧¬*p* is a **contradiction. **This
can be easily seen by constructing their truth tables as given below.

** **

Since
the last column of *p *∨* *¬*p *contains only T,* p *∨* *¬*p *is a tautology.

** **

Since
the last column contains only *F*, *p* ∧ ¬*p* is a contradiction.

All the
entries in the last column of Table 12.10 are *F* and hence ( p ⊽
q) ∧ ( p ⊽ ¬q) is a contradiction.

** **

In the
above truth table, the entries in the last column are a combination of *T* and *F*. The given statement is neither a tautology nor a contradiction.
It is a contingency.

Tags : Discrete Mathematics | Mathematics , 12th Maths : UNIT 12 : Discrete Mathematics

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12th Maths : UNIT 12 : Discrete Mathematics : Mathematical Logic: Tautology, Contradiction, and Contingency | Discrete Mathematics | Mathematics

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