Consider the following argument with premise \(∀𝑥(𝑃(𝑥)∨𝑄(𝑥))\) and…
2020
Consider the following argument with premise \(∀𝑥(𝑃(𝑥)∨𝑄(𝑥))\) and conclusion \((\forall _x P(x)) \wedge (\forall _x Q(x))\)
\(\begin{array}{|ll|l|} \hline (A) & \forall _x (P(x) \vee Q(x)) & \text{Premise} \\ \hline (B) & P(c) \vee Q(c) & \text{Universal instantiation from (A)} \\ \hline (C) & P(c) & \text{Simplification from (B)} \\ \hline (D) & \forall _x P(x) & \text{Universal Generalization of (C)} \\ \hline (E) & Q(c) & \text{Simplification from (B)} \\ \hline (F) & \forall _x Q(x) & \text{Universal Generalization of (E)} \\ \hline (G) & (\forall _x P(x)) \wedge (\forall _xQ(x)) & \text{Conjuction of (D) and (F)} \\ \hline \end{array}\)
- A.
This is a valid argument
- B.
Steps (𝐶) and (𝐸) are not correct inferences
- C.
Steps (𝐷) and (𝐹) are not correct inferences
- D.
Step (𝐺) is not a correct inference
Attempted by 27 students.
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Correct answer: B
Diagnosis: The argument is invalid; the specific faulty inferences are the moves that try to derive individual predicates from a disjunction.
Valid step:
From the premise ∀x (P(x) ∨ Q(x)) you may validly perform universal instantiation to get P(c) ∨ Q(c) for any constant c.
Invalid move:
You cannot infer P(c) (or Q(c)) from P(c) ∨ Q(c). This is not a valid rule of inference (it is the fallacy of affirming a disjunct). Steps that claim P(c) and Q(c) separately from the disjunction are therefore incorrect.
Because those inferences fail, the subsequent universal generalizations (deriving ∀x P(x) and ∀x Q(x)) are also unjustified, and so forming their conjunction is unsupported.
Counterexample showing invalidity:
Let the domain be {a, b} with P(a) true, Q(a) false; P(b) false, Q(b) true. Then for each element at least one of P or Q holds, so ∀x (P(x) ∨ Q(x)) is true, but ∀x P(x) is false and ∀x Q(x) is false. Thus the conclusion (∀x P(x)) ∧ (∀x Q(x)) is false while the premise is true.
Conclusion: Steps that infer P(c) or Q(c) from the disjunction are the immediate errors. Identifying and explaining that faulty inference (and giving a counterexample) resolves why the overall argument fails.
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