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The inverse of a conditional statement negates the hypothesis q and the conclusion q, and it can also be written in if-then
form.
rcc & Conditional & Inverse [0.5em] [-0.5em] Words: & Ifp, then q & If notp, then not q Symbols: & p ⇒ q & ¬ p ⇒ ¬ q
Negating both parts of a statement about squares gives the following pair.
| Example | Symbols | |
|---|---|---|
| Conditional | If a figure is a square_p,thenit has four sides_q. | p ⇒ q |
| Inverse | If a figure is not a square_(¬ p),thenit does not have four sides_(¬ q). | ¬ p ⇒ ¬ q |
Notice that the conditional is true while its inverse is false, since a rectangle is not a square yet still has four sides. Negating both parts can change the truth value of a statement.
| p | q | ¬ p | ¬ q | Conditional p ⇒ q | Inverse ¬ p ⇒ ¬ q |
|---|---|---|---|---|---|
| T | T | F | F | T | T |
| T | F | F | T | F | T |
| F | T | T | F | T | F |
| F | F | T | T | T | T |
The inverse is itself a conditional statement, so it is false only when its own hypothesis ¬ p is true and its own conclusion ¬ q is false. That happens in the third row, where the conditional is true and its inverse is not.