One of the logical axioms in a Hilbert system for first order logic is the quantifier axiom (Q7 in the wikipedia article):

$\phi\to\forall x\phi$

whenever $\phi$ is a formula and $x$ is a variable *not occurring free* in $\phi$.

What would be an example in some simple model where $x$ is free in $\phi$ and the resulting axiom would not be valid?

## Best Answer

In the supposed case, consider $\phi$ states '$x$ is an even number'. Then, the proposition purported to be axiom would state:

If $x$ is an even number, all $x$'s (e.g., all natural numbers) are even.

Hence, there would be illegitimate generalisation.

We take heed of the detail that these are propositions in the object language, not to be confused with the rule of universal generalisation.