There are some noteworthy variations of the implication:
- Converse
- Inverse
- Contrapositive
Converse
This is the implication which results from us swapping the antecedent and the conclusion for the original implication. For example, given the implication $p\implies q$, the converse will be $q\implies p$. Below is the truth table showing the converse of an implication. The original implication is included in the table for you to compare and appreciate the differences in the truth values.
| $p$ | $q$ | $p\implies q$ | $q\implies p$ |
|---|---|---|---|
| $T$ | $T$ | $T$ | $T$ |
| $T$ | $F$ | $F$ | $T$ |
| $F$ | $T$ | $T$ | $F$ |
| $T$ | $T$ | $T$ | $T$ |
Notice how the proposition before the arrow, $q$ must be true and the proposition after the arrow, $p$ must be false in order for the implication, $q\implies p$ to be false. Otherwise we get true.
It should be noted that the converse of an implication is relative to the original implication. Thus if the original implication is $\neg r\implies p$ then the converse will be $p\implies \neg r$. We arrive at the converse by swapping the two propositions, $\neg r$ and $p$.
Inverse
The inverse of an implication is when we negate both the antecedent and the conclusion of the original proposition. For example, given the implication $p\implies q$, the inverse will be $\neg p \implies \neg q$.
| $p$ | $q$ | $\neg p$ | $\neg q$ | $p\implies q$ | $\neg p\implies \neg q$ |
|---|---|---|---|---|---|
| $T$ | $T$ | $F$ | $F$ | $T$ | $T$ |
| $T$ | $F$ | $F$ | $T$ | $F$ | $T$ |
| $F$ | $T$ | $T$ | $F$ | $T$ | $F$ |
| $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
Contrapositive
The contrapositive of an implication is the result of swapping and negating the antecedent and the conclusion. Again, it is relative to the original implication. For example, given the implication $p\implies q$, the contrapositive will be $\neg q\implies \neg p$.
| $p$ | $q$ | $\neg p$ | $\neg q$ | $p\implies q$ | $\neg q\implies \neg p$ |
|---|---|---|---|---|---|
| $T$ | $T$ | $F$ | $F$ | $T$ | $T$ |
| $T$ | $F$ | $F$ | $T$ | $F$ | $F$ |
| $F$ | $T$ | $T$ | $F$ | $T$ | $T$ |
| $T$ | $T$ | $T$ | $T$ | $T$ | $T$ |
Created using natural intelligence