 Solution

Apply structural induction to the formation of a propositional formula. Let $L(\theta )$ and $R(\theta )$ count its opening and closing parentheses, and use the usual grammar generated by propositional variables, any allowed truth constants, negation, and parenthesized binary logical connectives.

An atomic variable or truth constant contains neither kind of parenthesis, so $L=R=0$. If $L(\alpha )=R(\alpha )$, prefixing negation leaves both counts unchanged. If the notation writes a negated formula with an extra pair of parentheses, it adds one to each and still preserves equality.

For a binary construction $(\alpha \circ \beta )$, where $\circ$ is an allowed binary logical connective,The two induction hypotheses make these equal. Every grammar construction preserves the property, so every propositional formula contains equally many left and right parentheses. This is balanced parentheses in propositional syntax.