We can classify propositions in terms of their truth values:
- Tautology
- Contradiction
- Contingency
Tautology
These are propositions whose truth values are 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 values are 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, $p\implies q$:
| $p$ | $q$ | $p\implies q$ |
|---|---|---|
| $T$ | $T$ | $T$ |
| $T$ | $F$ | $F$ |
| $F$ | $T$ | $T$ |
| $T$ | $T$ | $T$ |
Contingencies tend to be the most common type of proposition we work with.
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 $\neg p \lor q$:
| $p$ | $q$ | $\neg p$ | $\neg p \lor q$ | $p\implies q$ |
|---|---|---|---|---|
| $T$ | $T$ | $F$ | $T$ | $T$ |
| $T$ | $F$ | $F$ | $F$ | $F$ |
| $F$ | $T$ | $T$ | $T$ | $T$ |
| $F$ | $F$ | $T$ | $T$ | $T$ |
Challenge
Show that the following are logically equivalent:
- An implication and its contrapositive
- The converse of an implication and the inverse of the same implication
Created using natural intelligence