Tautologies and Contradiction
Tautologies
A proposition p is a tautology when it is true in all situations. P tautology contains only the True in the field of the column of its truth table.
Example: (p⟶q) ↔(∼q⟶∼p) is a tautology
| p | q | p⟶q | ~p | ~q | ~q⟶~p | (p⟶q) ↔(∼q⟶∼p) |
| T | T | T | F | F | T | T |
| T | F | F | F | T | F | T |
| F | T | T | T | F | T | T |
| F | F | T | T | T | T | T |
So the (p⟶q) ↔(∼q⟶∼p) is containing True in all of its rows.
Contradiction
In Contradiction, a statement is always false. It is the opposite of Tautology.
Example: p ∧∼p is a contradiction
| p | ~p | p^~p |
| T | F | F |
| F | T | F |
Here in p^~p all statements are false, so it will be a contradiction.
Contingency
A statement can either be True or False depending on the truth values of its variables is called a contingency. It is neither tautology nor contradiction.
| p | q | p⟶q | p∧q | (p⟶q) ⟶ (p∧q) |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | F |
| F | F | T | F | F |
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