Order of Quantifiers

The order in which quantifiers appear is very important; it must be maintained.

Consider the ambiguous sentence, Every man loves some woman. This sentence could be interpreted as:

1. For every man, there is some woman (depending on who the man is) whom the man loves. This would be written:

   ∀ x [Man(x) → ∃ y [Woman(y) ∧ Loves(x,y)]]

   and Skolemized:

   Man(x) →
   [Woman(lover(x)) ∧ Loves(x,lover(x))]

2. There is some woman (perhaps Marilyn Monroe) who is loved by every man. This would be written:

   ∃ y ∀ x [Man(x) → [Woman(y) ∧ Loves(x,y)]]

   and Skolemized:

   Man(x) → [Woman(a) ∧ Loves(x,a)]