The Pump Controller Manual
A science club at a PU college in Mangaluru has built a model automatic pump controller with four indicator lamps. The variables are: T - the overhead tank is full; M - the motor is overheating; P - mains power is available; S - the pump is switched on. The club's manual lists three rules: R1: The pump switches on if the tank is full and the motor is not overheating. R2: The pump switches on only if mains power is available. R3: If the motor overheats, the pump does not switch on. On Sunday the sensors read T = true, M = false and P = true, but the pump lamp stayed off (S = false). A technician says, 'Since mains power is available and the motor is not overheating, the pump must be on; the sensor must be faulty.' A second student says the manual is contradictory and no working pump can satisfy all three rules at once. The club has one class period to settle both claims symbolically.
1. Symbolise R1, R2 and R3, then state what kind of condition (T and not-M) is for S, and what kind of condition P is for S.
Read the worked answer
Using T, M, P, S as given:
- R1: (T ∧ ¬M) → S
- R2: S → P
- R3: M → ¬S
(T ∧ ¬M) is a sufficient condition for S. R1 guarantees S whenever the tank is full and the motor is cool, but it does not say this is the only way the pump starts - a manual override or a top-up cycle could produce S without contradicting R1. Sufficiency runs one way only.
P is a necessary condition for S. 'S only if P' means ¬P → ¬S by contraposition: no power, no pumping. But P alone is not sufficient - on Sunday P was true and S was false with no breach of R2.
Two equivalences are worth recording. R3 is equivalent by contraposition to S → ¬M, and to ¬(M ∧ S); so M and S can never both be true. R1 would become a biconditional, (T ∧ ¬M) ↔ S, only if the manual had said 'if and only if' - the club should check the design intention, because the whole dispute turns on whether the rule was meant as a definition of the on-condition or merely as one sufficient trigger.
2. The technician argues: 'P is true and M is false, therefore the pump must be on.' Test the argument by counterexample and name the error.
Read the worked answer
The argument has the form P, ¬M ∴ S, and it is invalid.
Counterexample row: take T = F, P = T, M = F, S = F.
- R1: (F ∧ T) → F, i.e. F → F = true
- R2: F → T = true
- R3: F → T = true
All three rules hold, both premises hold, and the conclusion S is false. One such row is enough to prove the form invalid.
Diagnosis: the technician has taken R2, S → P, and used it as though it were P → S. That is the fallacy of affirming the consequent: he observes that the necessary condition P is present and concludes that a sufficient condition for S has been met. What he actually needs is R1 plus both conjuncts, T ∧ ¬M, and then Modus Ponens on R1.
Note the coincidence that T is also true on Sunday, so the real readings do conflict with R1. That is a separate fault in the plant, not a vindication of the technician: his reasoning would be invalid on any day, because it never uses T at all.
3. Are R1, R2 and R3 mutually consistent? Do Sunday's readings violate them? Answer with a truth-table row and a short formal proof.
Read the worked answer
Consistency - yes. A set of formulae is consistent if some single assignment makes all of them true. Take T = F, M = T, P = T, S = F:
- R1: (F ∧ F) → F = T
- R2: F → T = T
- R3: T → T = T
A model exists, so the manual is not self-contradictory and the second student's claim fails.
Sunday's readings - R1 is violated. With T = T and M = F, the antecedent T ∧ ¬M is true, so R1 demands S = T; the observed S = F makes R1 false. Formal proof of the conflict:
- (T ∧ ¬M) → S [R1]
- T ∧ ¬M [sensor data]
- S [1, 2, Modus Ponens]
- ¬S [lamp observed off]
- S ∧ ¬S [3, 4, Conjunction - contradiction]
Equivalently, by indirect proof: assume the sensor data together with ¬S; Modus Ponens yields S; contradiction, so those data cannot all be true together with R1. Since R2 and R3 are never used in the derivation, the fault is local - the antecedent condition, the override circuit, or the sensors - rather than a conflict among the three rules.