Lab Access Rules and the Affirming-the-Consequent Error
A university computer lab posts four access rules for the 2024–25 session: (1) If a student has paid the lab fee and completed safety training, then the student is granted lab access. (2) If a student is a graduating senior, then the student has completed safety training. (3) If a student has not paid the lab fee, then the student is not granted lab access. (4) Lab access is granted only to students who have completed safety training. A student, Asha, is a graduating senior and has not paid the lab fee. The lab manager argues: 'Asha has completed safety training. Students who are granted lab access have completed safety training. Therefore Asha will be granted lab access.' A tutor claims the rules are inconsistent because rule (1) requires payment for access while rule (3) denies access without payment. The lab director asks for a symbolic analysis before the case is escalated.
1. Symbolize the four rules and Asha's facts using P, Q, R, and S. Then determine, by natural deduction or truth-table reasoning, whether the rules entail that Asha is not granted lab access.
Read the worked answer
Let P = has paid the lab fee; Q = has completed safety training; R = is granted lab access; S = is a graduating senior.
Rules:
- (P and Q) implies R. Formal: (P ∧ Q) → R.
- S implies Q. Formal: S → Q.
- Not-P implies not-R. Formal: ¬P → ¬R.
- R implies Q. Formal: R → Q.
Asha's facts: S ∧ ¬P.
Natural deduction:
- S ∧ ¬P (given)
- S (from 1)
- S → Q (rule 2)
- Q (from 2 and 3, modus ponens)
- ¬P (from 1)
- ¬P → ¬R (rule 3)
- ¬R (from 5 and 6, modus ponens)
So the rules entail that Asha is not granted lab access. The derivation does not need rule 1 or rule 4. A valuation S = true, P = false, Q = true, R = false satisfies all four rules and makes ¬R true, so the set is consistent with Asha's being denied access.
2. Identify the fallacy in the lab manager's argument, symbolize it, and give a valuation that shows it is invalid.
Read the worked answer
The manager argues:
- Q (Asha has completed safety training).
- R → Q (anyone with lab access has completed safety training).
- Therefore R (Asha will be granted lab access).
This is the fallacy of affirming the consequent. The form is: Q; R → Q; therefore R.
It is invalid because Q is a necessary condition for R, not a sufficient condition. The rule R → Q says that if someone has access, then they have training; it does not say that if someone has training, then they have access.
Counterexample valuation: Q = true, R = false, P = false, S = true. Check the premises: Q is true; R → Q is false → true, which is true. Check the conclusion: R is false. So the premises can be true while the conclusion is false. This valuation also satisfies the four lab rules: (P ∧ Q) → R is false → false, true; S → Q is true → true, true; ¬P → ¬R is true → true, true; R → Q is false → true, true. So the manager's conclusion is not entailed.
3. The tutor claims rules (1) and (3) are inconsistent. Evaluate that claim. Does rule (3) entail R → P? If so, how?
Read the worked answer
The claim of inconsistency is false. Two rules are inconsistent only if no valuation makes both true. But P = false, Q = true, R = false makes both true:
- Rule 1: (P ∧ Q) → R becomes (false ∧ true) → false, i.e. false → false, which is true.
- Rule 3: ¬P → ¬R becomes true → true, which is true. So the rules can both hold.
The tutor may be reading rule 1 as 'payment is sufficient for access' and rule 3 as 'no payment means no access,' and then imagining a student who pays but is denied. But rule 1 actually says payment plus training is sufficient; it does not say payment alone is sufficient. Rule 3 says payment is necessary for access. Those two ideas are compatible: payment plus training is sufficient, and payment is necessary.
Rule (3) does entail R → P in classical propositional logic. The contrapositive of ¬P → ¬R is R → P. This says that anyone granted access has paid the fee. It does not say that anyone who has paid gets access. So rule 3 supports a necessary-condition reading of payment, not a conflict with rule 1.