A hypothetical syllogism has a hypothetical proposition as a premise. The simplest is called modus ponens (Latin: "the way that affirms"), or affirming the antecedent.
| Form | Example | |
|---|---|---|
| Premise | If P, then Q. | If the dog hears the doorbell, the dog barks. |
| Premise | P. | The dog hears the doorbell. |
| Conclusion | So Q. | So the dog barks. |
Key idea
From "If P, then Q" and "P", conclude "Q". Modus ponens is valid: whenever both premises are true, the conclusion is true.
Why it works
"If P, then Q" rules out exactly one situation: P true and Q false. The second premise says P is true. The only situation left is the one where Q is true too. A truth table lists every case at once. Write p for "the dog hears the doorbell" and q for "the dog barks"; the arrow → means "if … then". The only row where both premises are true has q true.
Worked example
Worked example: What follows?
"If Maya finishes her chores, then Maya goes to the movies. Maya finishes her chores." What follows?
- The second premise affirms the antecedent.
- By modus ponens, the consequent follows.
Answer: Maya goes to the movies.
Worked example: Order does not matter
Is this modus ponens? "The puddles freeze, since the temperature dropped below freezing, and if the temperature drops below freezing, the puddles freeze."
- The conclusion comes first ("since" marks the premises), but the form is the same.
- Premises: If P then Q; P. Conclusion: Q.
Answer: yes: valid modus ponens
Common mistake
Rejecting a modus ponens because you doubt the first premise. "If the moon is made of cheese, then mice can live on the moon; the moon is made of cheese; so mice can live on the moon" is valid. Its premises are false, so it is unsound, but the form is fine. Doubting a premise is a question about soundness, not validity.