I just started reading Mathematical Logic
MY QUESTION
This part is from Language, Proof and Logic by John Etchemendy and Jon Barwise
They mention "if the variable x is not free in wff P then ∀x(P) ⇔ P", then we can take P= ∃x f(x).
But I saw in Kenneth H. Rosen's Book
They mention a "quantified variable does not appear in a part of statement"
Both Statement are very different. But both statement are for P, So should be same.
Which Statement is correct?
P.S. :- Sorry in advance because my English is not upto that mark. Edits are welcome :)
Kenneth is refering to $A$ as the statement, not $P(x)$. Like this: $$\underset{\text{quantfied variable}}{\forall \underset\uparrow x~(P(x))}\vee \underset{\text{statement}}{\underset\uparrow A} \equiv \forall x~(P(x)\vee A)$$
So, as long as the quantified variable $x$ does not occur free within statement $A$, and the domain is not empty, we can establish rules for distribution.