Festival Auditorium Rules: Truth-Table Entailment and a Repair Report
During the annual cultural festival, the auditorium committee must decide whether to open the main hall. Let P = 'The auditorium is booked.' Q = 'The sound system is repaired.' R = 'The cultural secretary is present.' The committee adopts three rules: (1) If the auditorium is booked, then the sound system is repaired or the cultural secretary is present. (2) If the sound system is not repaired, then the cultural secretary is present. (3) The auditorium is booked. A student volunteer claims that rules (1)–(3) alone logically guarantee R, so the secretary need not be separately confirmed. The committee also has a repair report showing Q is false, but the secretary's attendance is still unknown. They need a truth-table or short-method check, a clear counterexample if the claim fails, and a decision on what follows about R. They also want to know whether adding the rule 'If the cultural secretary is absent, the auditorium is not booked' would change the result.
1. Symbolize the three rules and the volunteer's claimed conclusion. Use a truth table or short method to test whether (1)–(3) entail R.
Read the worked answer
Symbolization: (1) P → (Q ∨ R); (2) ¬Q → R; (3) P; claimed conclusion R. The claim fails. Use the assignment P = T, Q = T, R = F. Then rule (1) becomes T → (T ∨ F) = T. Rule (2) becomes ¬T → F, i.e. F → F = T. Rule (3) is T. All three premises are true, but R is false. Therefore (1)–(3) do not entail R. The volunteer has confused a possible consequence with a logical guarantee; the cultural secretary's presence is not forced by the stated rules alone.
2. Now use the repair report Q = false. What follows? Does the volunteer's conclusion hold under the actual evidence?
Read the worked answer
With Q = false, ¬Q is true. Rule (2), ¬Q → R, then yields R by modus ponens. Rule (1) also yields Q ∨ R because P is true, and with ¬Q the disjunctive syllogism yields R. So under the actual evidence, R does follow. But the volunteer's original reasoning was incomplete: the conclusion depends on the extra fact that the sound system is not repaired. If Q were true, R could be false while all three original rules remained true, as the counterexample showed. The committee should record both results: not entailed by (1)–(3) alone, but entailed once Q = false is added.
3. Would adding 'If the cultural secretary is absent, the auditorium is not booked' make R follow from (1)–(3) without the repair report? Explain.
Read the worked answer
Yes. The added rule is ¬R → ¬P, which is the contrapositive of P → R, so it is equivalent to P → R. Since rule (3) gives P, modus ponens gives R regardless of Q. The added rule is consistent with the earlier rules: it does not contradict P → (Q ∨ R) or ¬Q → R. It makes the secretary's presence a necessary condition of booking, so once booking is affirmed, presence follows. The committee can use this rule as a direct guarantee, but it should still verify that the rule is actually policy rather than assuming it.