We can classify propositions in terms of their truth values:
- Tautology
- Contradiction
- Contingency
Tautology
These are propositions whose truth value is always true. For example, the compound proposition, $(p\land q)\implies q$ is a tautology:
| $p$ | $q$ | $p\land q$ | $(p\land q)\implies q$ |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Contradiction
These are propositions whose truth value is always false. An example is $(p\lor q) \land (\neg p\land \neg q)$:
| $p$ | $q$ | $\neg p$ | $\neg q$ | $p\lor q$ | $\neg p\land \neg q$ | $(p\lor q) \land (\neg p\land \neg q)$ |
|---|---|---|---|---|---|---|
| T | T | F | F | T | F | F |
| T | F | F | T | T | F | F |
| F | T | T | F | T | F | F |
| F | F | T | T | F | T | F |
Contingency
These are propositions whose truth values are a mixture of true and false. An example is the implication, $p\implies q$:
| $p$ | $q$ | $p\implies q$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| T | T | T |
Logical equivalence
Two propositions are logically equivalent when all of their corresponding truth values are the same.
Example: implication
The implication $p\implies q$ is logically equivalent to $\neg p \lor q$:
| $p$ | $q$ | $\neg p$ | $\neg p \lor q$ | $p\implies q$ |
|---|---|---|---|---|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |