In modus ponens only one premise is hypothetical. In a pure hypothetical syllogism, every statement is: the premises link conditionals into a chain.
| Form | Example | |
|---|---|---|
| Premise | If P, then Q. | If the alarm rings, Sam wakes up. |
| Premise | If Q, then R. | If Sam wakes up, Sam catches the bus. |
| Conclusion | So if P, then R. | So if the alarm rings, Sam catches the bus. |
Key idea
From "If P, then Q" and "If Q, then R", conclude "If P, then R". The consequent of the first must be the antecedent of the second; the links must run in the same direction.
Why it works
Imagine P happens. By the first premise, Q happens. By the second, R happens. So whenever P happens, R does: that is "If P, then R". The chain works like the categorical chain in a sorites: Q is the middle term, and it must sit at the end of one link and the start of the next.
Worked examples
Worked example: A longer chain
What follows? "If it snows tonight, school opens late. If the math quiz moves to Friday, Kim has an extra day to study. If school opens late, the math quiz moves to Friday."
- Order the links: snow → school opens late → quiz moves to Friday → extra day.
- Each consequent is the next antecedent.
Answer: If it snows tonight, Kim has an extra day to study.
Worked example: Links that do not chain
What follows? "If the power goes out, the freezer stops. If the door is left open, the freezer stops."
- Both conditionals end at the same consequent.
- Nothing runs from "power out" to "door open" or back.
Answer: nothing follows about how the two antecedents are related
Common mistake
Chaining conditionals that do not share a middle in the right place, such as "If P then Q" and "If R then Q". Both arrows point into Q, so nothing connects P and R. Check: draw the arrows; you must be able to follow them from start to end without going backward.