HSPT: Logic: what an if-then rule guarantees
Use an if-then rule and one fact to find what must be true, and reject choices that run the rule backward or add a reason.
When a question gives an if-then rule and one fact, how do you know what must be true?
Short answer
Match the fact to a part of the rule. If the if-part happened, the then-part happened; if the then-part didn't happen, the if-part didn't either. Nothing else is guaranteed.
Know first: The difference between must and might, Reading a rule word by word
Step 1 of 5
A rule is a one-way promise
An if-then rule links two parts. Call the if-part P and the then-part Q. If it snows, school closes means that every time P happens, Q happens too.
The rule says what P forces. It says nothing about other reasons Q might happen. School could close for a power outage, and the rule would still be true.
| Fact given | What must be true | Rule: if it snows, school closes |
|---|---|---|
| P happened | Q happened | It snowed, so school closed. |
| Q did not happen | P did not happen | School stayed open, so it did not snow. |
| Q happened | Nothing about P | School closed. Snow is possible, but so is a power outage. |
| P did not happen | Nothing about Q | It didn't snow. School could still close for another reason. |
Only the first two rows lead to a conclusion. The last two are the traps.
Step 2 of 5
Match the fact to a row
Worked example: The bakery flag
If the bakery has fresh bread, a flag hangs by the door. No flag hangs by the door. Assuming the rule was followed, what must be true?
- AThe bakery has no fresh bread.
- BThe baker is sick today.
- CThe bakery will have bread tomorrow.
- DThe bakery has closed for good.
- 1
Label the rule
P = the bakery has fresh bread. Q = a flag hangs by the door.
- 2
Match the fact
No flag hangs by the door. That's Q did not happen, the second row.
- 3
Conclude
If there were fresh bread, the flag would be there. It isn't, so there is no fresh bread.
- 4
Why the others are there
The baker is sick gives a reason, and the rule says nothing about reasons. Tomorrow adds a future time. Closed for good goes far past anything the rule covers.
Answer
The bakery has no fresh bread.
Find the wrong step
Rule: If a library book is overdue, the borrower pays a fee. Fact: Lena paid a fee. A student decides what must be true.
- 1
Labels P = a book is overdue and Q = the borrower pays a fee.
- 2
Since Q happened, concludes that P happened: Lena's book was overdue.
What went wrong
That runs the rule backward. Lena could have paid a fee for a lost or damaged book. The rule only says overdue books lead to fees.
- 3
Picks the choice Lena's book was overdue.
The fix
A fact that matches Q is the third row: nothing about P is guaranteed. The only valid moves are P to Q and not-Q to not-P, so no choice about Lena's book must be true.
Step 3 of 5
Find what the rule forces
Check yourself · Question 1
If the soup is too hot, the cook adds an ice cube. The soup is too hot. Assuming the rule was followed, what must be true?
Answer: A
The fact matches P, so Q follows: the cook adds an ice cube.
Check yourself · Question 2
Visitors may enter the gallery only if they have a ticket. Jada entered the gallery. Assuming the rule was followed, what must be true?
Answer: B
Only if makes the ticket the then-part: if a visitor enters, the visitor has a ticket. Jada entered, so Jada has a ticket.
Trap answer: Everyone with a ticket entered the gallery.
Rule read backward
- Why it looks right
- It uses the same words as the rule, so it sounds like a restatement.
- Why it's wrong
- Only if makes the ticket a requirement. Meeting a requirement doesn't mean you used it, since some ticket holders may have stayed home.
- The right answer
- Jada has a ticket, because entering requires one.
Check yourself · Question 3
If the river rises above the marker, the path along the bank closes. The path along the bank is open. Assuming the rule was followed, what must be true?
Answer: C
The path is open, so Q did not happen. If the river had risen above the marker, the path would be closed. So the river has not risen above it.
Common mistake
Choosing the answer that explains why something happened.
- Why it's tempting
- A reason makes the story feel complete, so it sounds like a strong conclusion.
- Do this instead
- The question asks what must be true, not why. If a choice adds a cause, a time or a new person, cross it out.
Step 4 of 5
What to remember
Remember
- If P happened, Q happened. If Q didn't happen, P didn't happen.
- Q happening, or P not happening, proves nothing about the other part.
- Rewrite only if and every X who rules as if-then before you use them.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what were P and Q, and which row of the table did the fact match?