Consider the following argument with premise \(βπ₯(π(π₯)β¨π(π₯))\) andβ¦
2020
Consider the following argument with premiseΒ \(βπ₯(π(π₯)β¨π(π₯))\)Β and conclusionΒ \((\forall _x P(x)) \wedge (\forall _x Q(x))\)
\(\begin{array}{|ll|l|} \hline (A) & \forall _x (P(x) \vee Q(x)) & \text{Premise} \\ \hline (B) & P(c) \vee Q(c) & \text{Universal instantiation from (A)} \\ \hline (C) & P(c) & \text{Simplification from (B)} \\ \hline (D) & \forall _x P(x) & \text{Universal Generalization of (C)} \\ \hline (E) & Q(c) & \text{Simplification from (B)} \\ \hline (F) & \forall _x Q(x) & \text{Universal Generalization of (E)} \\ \hline (G) & (\forall _x P(x)) \wedge (\forall _xQ(x)) & \text{Conjuction of (D) and (F)} \\ \hline \end{array}\)
- A.
This is a valid argument
- B.
StepsΒ (πΆ)Β andΒ (πΈ)Β are not correct inferences
- C.
StepsΒ (π·)Β andΒ (πΉ)Β are not correct inferences
- D.
StepΒ (πΊ)Β is not a correct inference
Attempted by 26 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 βx (P(x) β¨ Q(x)) you may validly perform universal instantiation to get P(c) β¨ Q(c) for any constant c.
Invalid move:
You cannot infer P(c) (or Q(c)) from P(c) β¨ 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 βx P(x) and βx 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 βx (P(x) β¨ Q(x)) is true, but βx P(x) is false and βx Q(x) is false. Thus the conclusion (βx P(x)) β§ (βx 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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