Tautologies and Contradictions

Tautologies and Contradictions

1. Tautology

A tautology is a compound proposition that is always true, no matter what truth values are assigned to its individual components.

In a truth table, a tautology will have T (True) in every row of the final column.

Examples:

a) $p \lor \neg p$

This is true for all possible values of $p$. So it's a tautology.

b) $(p \to q) \lor (q \to p)$

No matter the values of $p$ and $q$, this compound statement always evaluates to true.

2. Contradiction

A contradiction is a compound proposition that is always false, regardless of the truth values of the individual components.

In a truth table, a contradiction will have F (False) in every row.

Examples:

a) $p \land \neg p$

Always false → this is a contradiction.

b) $(p \to q) \land (p \land \neg q)$

This is only true if both $p \to q$ and $p \land \neg q$ are true at the same time, which is impossible. Always false.

3. Contingency

A contingency is a compound proposition that is sometimes true and sometimes false, depending on the truth values of its variables.

Example:

$p \to q$

This is not always true or always false → it's a contingency.

4. Identifying with Truth Tables

To determine if a statement is a tautology, contradiction, or contingency:

1. Construct the truth table.
2. Check the final column:
   • All T → Tautology
   • All F → Contradiction
   • Mixed T and F → Contingency

5. Why Tautologies and Contradictions Matter

• Tautologies represent logically valid forms.
  • Used in proofs and argument validation.
• Contradictions signal inconsistent or illogical statements.
  • Useful for proof by contradiction.
• Recognizing these helps evaluate logical correctness in mathematics, programming, and circuit design.