Worked Examples: Logical Deductions Under Time Pressure

This lesson brings together everything from the previous nine lessons with a set of fully worked examples, each solved using the systematic approaches you have learned. Every example includes a time budget, the solving method, and common pitfalls to avoid. Use this lesson as both a practice resource and a revision reference.

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Example 1: Syllogism (Yes/No Format)

Time budget: 50 seconds

Consider the following premises:

• All pharmacists are university graduates.
• Some university graduates work in hospitals.
• No hospital workers are under 18.

For each of the following, decide whether it follows logically from the premises. Answer Yes or No.

Statement	Your answer
A. Some pharmacists work in hospitals.
B. No pharmacists are under 18.
C. Some university graduates are under 18.
D. All hospital workers are university graduates.

Solution

Draw the Venn diagram:

• Pharmacists circle entirely inside University Graduates circle.
• University Graduates and Hospital Workers overlap (X in the overlap).
• Hospital Workers and Under 18 are completely separate (no overlap).

Statement A: "Some pharmacists work in hospitals." The X (some university graduates who work in hospitals) could be inside or outside the Pharmacists circle. It is not forced to be inside. → No.

Statement B: "No pharmacists are under 18." Pharmacists ⊂ University Graduates. Can pharmacists be under 18? The premises only say no hospital workers are under 18. Pharmacists who do not work in hospitals could potentially be under 18 — nothing in the premises prevents this. → No.

Wait — let me reconsider. The premises say: (1) All pharmacists are university graduates. (2) Some university graduates work in hospitals. (3) No hospital workers are under 18. There is no direct link between pharmacists and being under 18. A pharmacist who does not work in a hospital could be under 18. → No.

Statement C: "Some university graduates are under 18." The premises do not state this. University graduates could all be 18 or over. Nothing forces any to be under 18. → No.

Statement D: "All hospital workers are university graduates." Premise 2 says some university graduates work in hospitals. This does not mean all hospital workers are university graduates — there could be hospital workers who are not university graduates (e.g., administrative staff without degrees). → No.

Answers: A = No, B = No, C = No, D = No.

Common Pitfall: Statement A is the most tempting trap. "Some pharmacists might work in hospitals" is plausible, but "might" is not the same as "must." In logic, "Some A are B" means it is necessarily true that at least one A is B — not that it is merely possible.

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Example 2: Conditional Logic

Time budget: 45 seconds

The following rules apply at a medical school:

• If a student fails their clinical placement, they must retake the year.
• If a student retakes the year, they are not eligible for the prize.

Which of the following must be true?

A. If a student is eligible for the prize, they passed their clinical placement. B. If a student passed their clinical placement, they are eligible for the prize. C. If a student is not eligible for the prize, they failed their clinical placement. D. If a student retakes the year, they failed their clinical placement.

Solution

Chain the conditionals:

• Fails placement → Retakes year → Not eligible for prize.

By contrapositive: Eligible for prize → Did not retake year → Did not fail placement (i.e., passed placement).

Evaluate each option:

A. "If eligible for prize → passed placement." This is the contrapositive chain (eligible → didn't retake → didn't fail → passed). Must be true. ✓

B. "If passed placement → eligible for prize." This is the inverse. A student could pass placement but retake the year for another reason (e.g., failing a different module). Not necessarily true. ✗

C. "If not eligible → failed placement." This is the inverse of the contrapositive. A student might be ineligible for reasons other than failing placement. Not necessarily true. ✗

D. "If retakes year → failed placement." The premise says failing placement leads to retaking, but retaking could also happen for other reasons. Not necessarily true. ✗

Answer: A

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Example 3: Ordering Puzzle

Time budget: 70 seconds

Five presentations — V, W, X, Y, Z — are scheduled in five consecutive time slots (1st through 5th). The following is known:

• W is scheduled immediately before X
• Y is scheduled before W
• Z is not scheduled 1st or 5th
• V is scheduled 5th

Question: What is the complete order of presentations?

Step 1: Place fixed items. V = 5th.

1	2	3	4	5
_	_	_	_	V

Step 2: W is immediately before X → [W, X] block. Possible positions: (1,2), (2,3), (3,4). Not (4,5) because V is in 5th.

Step 3: Y is before W. And Z is not 1st or 5th.

Try [W,X] at (1,2): W=1, X=2. Y must be before W, but W is 1st. There is no position before 1st. Impossible. ✗

Try [W,X] at (2,3): W=2, X=3. Y must be before W → Y=1. Remaining: Z for position 4. Check: Z ≠ 1st ✓, Z ≠ 5th ✓ (Z=4). Valid.

1	2	3	4	5
Y	W	X	Z	V

Try [W,X] at (3,4): W=3, X=4. Y before W → Y=1 or Y=2. Remaining: Z and Y for positions 1 and 2. Z ≠ 1st → Z=2, Y=1. Check: Y(1) before W(3) ✓. Z ≠ 5th ✓.

1	2	3	4	5
Y	Z	W	X	V

Two valid orderings:

1. Y, W, X, Z, V
2. Y, Z, W, X, V