Predicate logic: free variable after ∀

Question

Consider the formula ∀xα (where x is some variable). Is the occurrence of x which comes just after ∀ considered as a free variable?

Many thanks for your help.

Fish

Answer 1

The statement:

"Now suppose that $α$ is open. Then the only free variable in $∀xα$ is $x$ because we showed that $∀xα$ is closed"

makes little sense. I think that you have a typo; it must be: "Then the only free variable in $α$ is $x$ because we showed that $∀xα$ is closed."

If $∀xα$ is closed, when we remove the leading quantifier $∀x$, what remains is $α$. If now $α$ is open, the only possibility is that it contains a free occurrence of $x$ that in $∀xα$ has been binded by the quantifier.

If insted we have a free occurrence of $y$ in $α$, also $∀xα$ will have it, and thus also $∀xα$ will be open.

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The formal definitions are :

all the variables occurring in a term or atomic formula are free;

$FV(\lnot \varphi) = FV(\varphi)$;

$FV (\varphi \square \psi) = FV (\varphi) \cup FV (\psi)$, for every binary connective $\square$.

$FV(\forall x \varphi)=FV(\exists x \varphi)= FV(\varphi) \setminus \{ x \}$.

$\varphi$ is called closed if $FV(\varphi)= \emptyset$. A closed formula is also called a sentence. A formula without quantifiers is called open.