I'm trying to get my head around the expression "term $t$ is free for variable $v$ in formula $\phi$". This is defined in many sources as:
(Def 1)"$t$ is free for $v$ in $\phi$ if there is no variable $v_1$ in $t$ such that $v$ has a free occurrence in the scope of a quantifier $\forall v_1$ or $\exists v_1$.
In other sources it is introduced (informally) as:
(Def 2)"If no variable capture occurs during the substitution of $t$ into free occurrences of $v$ in $\phi$, then $t$ is free for $v$ in $\phi$."
Question 1: After substituting $t$ into every free occurrence of $v$ in $\phi$, if every variable of $t$ is a free variable in the resulting formula, then $t$ is free for $v$ in $\phi$. Is this correct? I don't understand why.
Question 2: Consider the formula $\phi : \forall v, \ (v \doteq v_1)$. Is $v_1$ free for $v$ in $\phi$? According to definition 1, it is not, as $(v \doteq v_1)$ has a free occurrence of $v$. However, according to definition 2, it is, as $\phi[v_1/v] = \forall v, \ (v \doteq v_1)$ (as we are only concerned with free occurrences of $v$ when doing substitution), and no variable capture has occurred. Where have I gone wrong?
Is (non-appearing in $\phi$) variable $v_2$ free for variable $v_3$ in $\phi$? There is no variable capture after substitution, because substitution $\phi[v_2/v_3]$ in this case does nothing.
Question 3: Is the following true? $v$ is free for $v$ in $\psi$, no matter what the formula $\psi$ is.
Question 4: $v$ is not a free variable of sentence $\theta$ and $v$ is a free variable in $\phi(v)$. Then $v$ is free for $v$ in the formula $\theta\to \phi(v)$.
I'm finding this concept so confusing. Can anyone help clarify this? Thanks.