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Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x.

Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x. Correct Answer ∀x(p(x) → W) ≡ ∀x p(x) → W

Statement I:

∀x(p(x) ∨ W) ≡ ∀x p(x) ∨ W

This is logically valid. Because, W is free of any quantifier so, ∀x would be associated with p(x) only and hence L.H.S and R.H.S are equal.

Statement II:

Ǝx(p(x) ∧ W) ≡ Ǝx p(x) ∧ W

This also logically valid. Because, W is free of any quantifier so, Ǝx would be associated with p(x) only and hence L.H.S and R.H.S are equal.

Statement III:

∀x(p(x) → W) ≡ ∀x p(x) → W

This is logically NOT valid.

L.H.S: ∀x(p(x) → W

        = ∀x(¬ p(x) ∨ W)

        = ∀x (¬ p(x)) ∨ W

         = ¬ Ǝx p(x) ∨ W

         = Ǝx p(x) → W.

This is not equal to R.H.S

Statement IV:

Ǝx(p(x) → W) ≡ ∀x p(x) → W

This is logically valid.

L.H.S: Ǝx(p(x) → W)

       = Ǝx(¬ p(x) ∨ W)

      = Ǝx(¬ p(x)) ∨ W

      = ¬ ∀x p(x) ∨ W

      = ∀x p(x) → W = R.H.S

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