Here are two sentences in Predicate Logic:

  1. For all x, there exists a y (If x is F, then x > y)

  2. For all x (If x is F, then there exists a y (x > y))

The difference is that in the first example, all quantifiers are in the front and that in the second the quantifier appears within the scope of the other quantifier. Is there a difference? If so, where?

1 Answer 1


No, there is no difference, because :

if x does not occur free in α, then (α → ∃xβ) ↔ ∃x(α → β).

In your example, we have :

∀x(F(x) → ∃y(x > y))

and thus, being y not free in F(x), it is equivalent to :

∀x∃y(F(x) → (x > y)).

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