Why " provided $t$ is not in $\theta$ " at the end of the definition of $\forall $ Intro Rule ? ( Shapiro's Entry On Classical Logic in SEP)

Question

Here is the passage :

"The introduction clause for the universal quantifier is a bit more complicated. Suppose that a sentence θ contains a closed term t, and that θ has been deduced from a set of premises Γ. If the closed term t does not occur in any member of Γ, then θ will hold no matter which object t may denote. That is, ∀vθ

follows.

(∀I)

For any closed term t, if Γ⊢θ(v|t), then Γ⊢∀vθ provided that t is not in Γ or θ".

First point : Something I do not understand is that the preliminary remark suggests that $t$ is contained in $\theta$ while the official statement of the rule requires this not to be the case.

Second point : Also, should we not have :

If, for any closed term $t$, $\Gamma |-\theta(v,t) $, then $\Gamma |- \forall v\theta$

instead of

For any closed term t, if Γ⊢θ(v|t), then Γ⊢∀vθ ?

I mean, in order the universal statement to be deduced, should it not be required that the substitution works for any closed term? ( It seems to me that in the official rule, " for any closed term" applies to the whole conditional, not to the antecedent in particular.)

Does the alternative reading I suggest make any difference? What do I miss?

Answer 1

$\theta$ might be $v=t$. Then $\theta(v|t)$ is $t=t$ and certainly provable. That doesn't mean that $\forall v\, v=t$ is true.

We might have $t=0$ in $\Gamma$. Then if $\theta$ is $v=0$, we have that $\theta(v|t)$ is $t=0$ and we certainly have $\Gamma\vdash \theta(v|t)$, but not $\forall v\,v=0$.