Not every proposition is categorical. A hypothetical proposition joins two whole statements with "if": "If it rains, then the game is canceled." The part after "if" is the antecedent; the part after "then" is the consequent. You met these in Affirming the Consequent and Denying the Antecedent.
Key idea
"If P, then Q" says that P is enough for Q. "Q if P", "Q provided that P", "Whenever P, Q" and "P only if Q" all say the same thing: P is the antecedent and Q the consequent.
Why it works
"P only if Q" says that P cannot happen without Q: Q is required. So whenever P happens, Q must have happened too, and that is exactly "If P, then Q". "You may drive only if you have a license" means: if you may drive, you have a license. It does not promise that having a license lets you drive (you might also need insurance). The word "only" points to the requirement, which becomes the consequent.
| Wording | If …, then … |
|---|---|
| The game is canceled if it rains. | If it rains, then the game is canceled. |
| You may drive only if you have a license. | If you may drive, then you have a license. |
| The cake burns whenever it bakes too long. | If the cake bakes too long, then it burns. |
| A license is necessary for driving. | If you drive, then you have a license. |
| Rain is sufficient for canceling the game. | If it rains, then the game is canceled. |
Worked examples
Worked example: Only if
Rewrite with "If …, then …": "Sam goes to the movies only if Sam finishes his chores."
- "Only if" introduces the consequent: Sam finishes his chores.
- So the antecedent is "Sam goes to the movies".
Answer: If Sam goes to the movies, then Sam finished his chores.
Worked example: Necessary condition
Rewrite: "Practice is necessary for making the team."
- A necessary condition is a requirement, so it is the consequent.
- Antecedent: you make the team.
Answer: If you make the team, then you practiced.
Common mistake
Reversing "only if". "P only if Q" means "If P then Q", not "If Q then P". Check with the license example: having a license does not by itself let you drive, so "only if" cannot mean "if you have a license, you may drive".