Is it possible to say that a variable occurs free in a given formula, even though the formula is within the scope of a quantifier that binds the variable? For example, considering the statement $\exists y \forall x P(x, y)$, is it correct to say that the variable $y$ occurs free in the formula $\forall x P(x, y)$ and bound in the entire proposition $\exists y \forall x P(x, y)$? If not, is there a proper definition for the variable $y$ in $\forall x P(x, y)$? For instance, is it a constant in the formula or something like that?
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1I think your "is it correct ... ?" is correct. – Henry Oct 29 '20 at 23:08
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See the post Is there a definition for free and bound variables in logic? – Mauro ALLEGRANZA Oct 30 '20 at 07:35
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Yes, correct: In $\forall x P(x,y)$ as a subformula of $\exists y \forall x P(x,y)$, $y$ is free.
This is in fact sometimes used in definitions; for instance, you might run into something like "... if $\phi$ is of the form $\exists x \psi$, where $x$ occurs free in $\psi$..." to say that the quantifier is non-vacuous and actively binds occurrences of $x$ in $\psi$.
Natalie Clarius
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