Thanks a lot !!

Question

State the scope of each quantifier in the following expression:

“∀y, [R(x, y) ^ ∃x (P(x, y) ^ ~Q(x, y))]

and state whether the expression above is a statement. Explain your answer.

Answer 1

In the well-formed formula (wff):

∀y [R(x, y) ^ ∃x (P(x, y) ^ ~Q(x, y))]

the scope of the ∃x quantifier is (P(x, y) ^ ~Q(x, y)) and the score of the ∀y quantifier is [R(x, y) ^ ∃x (P(x, y) ^ ~Q(x, y))]. Note that the x in (P(x, y) ^ ~Q(x, y)) is bound by the existential quantifier but the x in R(x, y) is not bound by any quantifier, that is, this particular occurrence of x is free.

If by "statement" you mean "proposition", or a wff with no free variables, then given the above free occurrence of x, the wff is NOT a statement.