In a formal language, how does one show that $\neg \neg \bot \neq( \phi \wedge \psi) $ Or how do one go about showing that the former is not a proposition.
I've just started reading Dalen's Logic and structure and there the set of propositions is defined as the smallest set $X $ satisfying,
i) $p _i \in X(i \in N ) $, $\bot \in X $
ii) $\phi , \psi \in X \implies ( \phi \wedge \psi),( \phi \vee \psi), ( \phi \to \psi),( \phi \leftrightarrow \psi) \in X$
iii) $\phi \in X \implies (\neg \phi) \in X $
To show that $\neg \neg \bot $ is not a proposition one can go about to show that assuming $X $ and then setting $Y=X \setminus \{\neg \neg \bot \} $ again satisfies i),ii) and iii) and is smaller that $X $. But this hinges on that one can show that $\neg \neg \bot \neq( \phi \wedge \psi) $, where nothing has been said about syntax.
Dalen states that one should "look at the brackets", can one determine difference from this?
Thanks in advance!
