Given an if-then statement "if pp, then qq," we can create three related statements; the converse, inverse and contrapositive. If we know the validity of the original conditional statement, then we can know the validity of these related statements immediately.
A conditional statement consists of two parts, a hypothesis in the “if” clause (pp in the above) and a conclusion in the “then” clause (qq in the above).
For instance, “If it rains, then they cancel school.”
- "it rains" is the hypothesis
- "they cancel school" is the conclusion
To form the converse of the conditional statement, interchange the hypothesis and the conclusion. The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains."
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain."
In logical form summary:
| Statement | If pp, then qq. |
|
| Converse | If qq, then pp. |
reverse |
| Inverse | If not pp, then not qq. |
negate |
| Contrapositive | If not qq, then not pp. |
reverse and negate |
With these forms defined, there is an important logical conclusion:
- If the original statement is true, then the contrapositive is also logically true.
- If the converse is true, then the inverse is also logically true.
- Note that if you treat the converse as a statement, then the inverse is that statement's contrapositive.
Armed with this knowledge, it is possible to quickly evaluate the logical truthfulness of some statements.