How to Understand Difficult Texts Quickly
Hypothetical Syllogism, the Idempotent Law, and Contraposition
Which premise is required for the following inference to be valid?
One night, a report came in that someone had broken into an unmanned warehouse. The investigator knows the following facts.
(1) If someone broke into the warehouse, they must have entered through either the front door or a rear window.
(2) If they entered through the front door, CCTV footage would have been recorded and the alarm would have sounded.
(3) If they entered through the rear window, the glass-break sensor and the motion sensor would have been triggered.
(4) If the motion sensor was triggered, the alarm would have sounded.
(5) If the glass-break sensor was triggered, CCTV footage would automatically have been recorded.
Therefore, no one broke into the warehouse.
① CCTV footage from the warehouse was recorded.
② The motion sensor was triggered.
③ The glass-break sensor was triggered.
④ No one entered through the rear window.
⑤ The warehouse alarm did not sound.
When you first read the problem, the front door, the window, the CCTV system, the alarm, the glass-break sensor, and the motion sensor all appear at once. If you try to retain every sentence exactly as it is written, the problem quickly becomes complicated.
In a case like this, it is more efficient to keep only the key terms than to memorize each sentence in full.
Turn Long Sentences into Short Structures
Let us use the arrow → to mean, “If the event on the left occurs, the event on the right occurs.”
The symbol ∨ means “or,” while the dot · means “and.”
The five sentences can then be shortened as follows:
Break-in → Front door ∨ Window
Front door → CCTV · Alarm
Window → Glass-break sensor · Motion sensor
Motion sensor → Alarm
Glass-break sensor → CCTV
According to the first premise, if someone broke into the warehouse, there are two possible routes:
They entered through either the front door or the rear window.
Let us first consider the case in which the intruder entered through the front door.
According to Premise 2, if someone entered through the front door, CCTV footage was recorded and the alarm sounded.
Front door → CCTV · Alarm
The route taken by an intruder who used the front door can therefore be represented as follows:
Break-in → Front door → CCTV · Alarm
Now consider the case in which the intruder entered through the rear window.
According to Premise 3, if someone entered through the window, the glass-break sensor and the motion sensor were triggered.
Window → Glass-break sensor · Motion sensor
By connecting Premises 1, 2, and 3, we can therefore represent the situation as follows:
Break-in → (CCTV · Alarm) ∨ (Glass-break sensor · Motion sensor)
In ordinary language, this means:
If someone broke into the warehouse, they either entered through the front door, causing CCTV footage to be recorded and the alarm to sound, or they entered through the window, triggering the glass-break sensor and the motion sensor.
The outcomes of the two routes still appear different.
The front-door route ends with CCTV footage and the alarm, while the window route ends with the glass-break sensor and the motion sensor.
But the window route has not yet reached its final outcome.
Turn Intermediate Results into Final Results
Premises 4 and 5 tell us what happens after the sensors are triggered.
Motion sensor → Alarm
Glass-break sensor → CCTV
If the motion sensor is triggered, the alarm sounds. If the glass-break sensor is triggered, CCTV footage is recorded.
Therefore, the result of entering through the window—
Glass-break sensor · Motion sensor
—ultimately leads to:
CCTV · Alarm
We can now replace the result of the window route in the earlier expression:
Break-in → (CCTV · Alarm) ∨ (Alarm · CCTV)
If someone enters through the front door, CCTV footage is recorded and the alarm sounds.
If someone enters through the window, the alarm also sounds and CCTV footage is recorded.
The order is different, but the two outcomes are exactly the same:
CCTV footage is recorded and the alarm sounds.
The alarm sounds and CCTV footage is recorded.
This is no different from saying, “It is raining and the wind is blowing,” and, “The wind is blowing and it is raining.” In logic, this is called the commutative law.
We can therefore rewrite the statement as follows:
Break-in → (CCTV · Alarm) ∨ (CCTV · Alarm)
The same statement now appears twice on the right-hand side.
“A or A” ultimately means nothing more than “A.” Someone who says they will eat either pizza or pizza has already decided what to order. In logic, this is called the idempotent law.
We can therefore remove the repetition and leave only one instance:
Break-in → CCTV · Alarm
What originally appeared as five long sentences has now been compressed into a single short statement:
If someone broke into the warehouse, CCTV footage must have been recorded and the alarm must have sounded.
This is the entire logical structure of the problem.
Read the Statement in the Opposite Direction of the Desired Conclusion
However, the conclusion we are trying to establish is not:
Someone broke into the warehouse.
It is:
No one broke into the warehouse.
So far, we have established the following:
If someone broke in, CCTV footage was recorded and the alarm sounded.
To derive the conclusion that no one broke in, we must consider what follows if the result did not occur.
Let us consider a simple example:
If something is a cat, then it is an animal.
If this statement is true, the following statement must also be true:
If something is not an animal, then it is not a cat.
This is because something cannot be a cat without being an animal.
However, we cannot say:
If something is an animal, then it is a cat.
Dogs are animals, and humans are animals as well. Simply reversing the original statement produces a false conclusion.
The correct method is to reverse the two parts while negating both of them:
If A, then B.
If not B, then not A.
This is called contraposition.
Therefore, the contrapositive of
Break-in → CCTV · Alarm
is:
If it is not the case that CCTV footage was recorded and the alarm sounded, then no one broke in.
In symbolic form:
~(CCTV · Alarm) → ~Break-in
The symbol ~ means “not.”
What Does It Mean to Say That “Both Did Not Occur”?
We must now unpack the following expression:
It is not the case that CCTV footage was recorded and the alarm sounded.
Under what circumstances would this statement be true?
The CCTV footage may not have been recorded. Alternatively, the alarm may not have sounded. It is also possible that neither occurred.
Therefore, the statement
It is not the case that both CCTV recording and the alarm occurred
means:
The CCTV footage was not recorded, or the alarm did not sound.
In symbolic form:
~CCTV ∨ ~Alarm
When an entire statement joined by “and” is negated, each part is negated and the connective changes to “or.” This is called De Morgan’s law.
We can therefore derive the following final statement:
If the CCTV footage was not recorded or the alarm did not sound, then no one broke into the warehouse.
Find the Required Premise among the Options
Let us now examine the answer choices again.
① The fact that CCTV footage was recorded is compatible with the possibility that someone broke in. It does not guarantee the conclusion that no one broke in.
② If the motion sensor was triggered, the alarm would likely have sounded. This also provides no basis for concluding that no one broke in.
③ The fact that the glass-break sensor was triggered is consistent with the route involving entry through the window.
④ The fact that no one entered through the rear window is insufficient. Even if no one used the window, someone could still have entered through the front door.
⑤ states that the alarm did not sound.
We established earlier that if someone broke into the warehouse, the alarm must have sounded regardless of whether they entered through the front door or the window.
But if the alarm did not in fact sound, the assumption that someone broke into the warehouse cannot be true.
Therefore, the correct answer is:
⑤ The warehouse alarm did not sound.