Aspire Faculty ID #18950 · Topic: UGC NET Computer Science Nov 2020 (Paper II) · Just now
UGC NET Computer Science Nov 2020 (Paper II)

Consider the following argument with premise $\forall x(P(x)\vee Q(x))$ and conclusion $(\forall xP(x))\wedge(\forall xQ(x))$
(A) $\forall x(P(x)\vee Q(x))$Premise
(B) $P(c)\vee Q(c)$Universal instantiation from (A)
(C) $P(c)$Simplification from (B)
(D) $\forall xP(x)$Universal generalization of (C)
(E) $Q(c)$Simplification from (B)
(F) $\forall xQ(x)$Universal generalization of (E)
(G) $(\forall xP(x))\wedge(\forall xQ(x))$Conjunction of (D) and (F)

Solution

From $\forall x(P(x)\vee Q(x))$, we can get:

$P(c)\vee Q(c)$

So, step (B) is correct.

But from $P(c)\vee Q(c)$, we cannot conclude $P(c)$ alone.

Also, we cannot conclude $Q(c)$ alone.

So, steps (C) and (E) are not correct inferences.

Step (G) would be correct only if (D) and (F) were validly obtained.

Hence, the main error is in steps (C) and (E).

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