 Solution

A useful consequence of the formation rules is that logical formulas form a prefix-free set of strings. To see this, describe a deterministic parser that reads one logical formula from its starting position. On a logical variable it consumes that one symbol. On a negation symbol it recursively reads one following logical formula. On an opening parenthesis it recursively reads a left logical formula, requires a logical implication symbol, recursively reads a right logical formula, and requires the closing parenthesis. Induction on formation proves that, if a valid logical formula begins at the current position, this procedure consumes exactly that logical formula and stops at its end.

Suppose now that both $\alpha$ and $\varphi$ are logical formulas and $\varphi =\alpha ϵ$. The same parser, started at their common first symbol, must stop both at the end of $\alpha$ and at the end of $\varphi$. Its deterministic stopping position cannot differ, so $ϵ$ is empty. HenceThis argument applies to symbol strings; it does not use truth values or logical equivalence.