In a propositional language $\mathcal{L}$, give an example of an $\mathcal{L}$-expression which is not an $\mathcal{L}$-formula, but is not ruled out by the results in either Problems 3 or 4. State, and prove by induction on formulas, a result which rules it out.
Prob 3: every $\mathcal{L}$-formula has the same number of left and right parenthesis.
Prob 4: in every $\mathcal{L}$-formula $s=c+1$, where $s$ is the number of proposition variables and $c$ is the number of binary connectives.
Let $p$ be a proposition variable in the language $\mathcal{L}$. Then, the $\mathcal{L}$-expressions $)p($ is not a $\mathcal{L}$-formula but it has the same number of left and right parenthesis, and moreover $s = c + 1$ because for such an $\mathcal{L}$-expression the number $s$ of variables is 1 and the number $c$ of binary connectives is $0$.