I am trying wrap my mind around how to show that something does not belong to PROP. Just for clarification:
The set PROP of propositions is the smallest set $X$ with the properties
(i) $p_{i} \in X \quad (i \in \mathbb{N}), \bot \in X$
(ii) $\phi, \psi \in X \implies (\phi \square \psi) \in X$
(iii) $\phi \in X \implies \neg \phi \in X$
As an example, how would I go about proving $(( \to \not \in \text{PROP}$? My problem is that I do not know how to interpret the information, i.e. $(( \to \not \in \text{PROP}$, in the context of the definition. What I would want to do is to suppose that $(( \to$ satisfy all the three criteria and then show that it is $\textbf{not}$ the smallest set.
First look at $((\to$ and see it does not fit (i) as it is not of the form $p_i$. Nor does it fit (iii) since it does not begin with $\lnot$. So the only possibility is that it fits (ii) which is of the form $(\phi,\psi)$ where then $\phi$ must start with $(\to$, because the first left parenthesis of $((\to$ has been "peeled off" in making it match (ii).
Now apply the same idea with $(\to$ and see as before it can only match (ii) and another parenthesis gets peeled off and you're left with just $\to$, whose first (and only) symbol cannot match the beginning of any of (i),(ii),(iii), so the whole thing is not in PROP.