Here's a proof that if A is a well-formed formula with no negations (), then A has an even number of propositional variable symbols.

We prove this by induction on A. It is not possible to have A=(B) for a wff B, since A does not contain negations. If A=(BC) for wffs B and C and some connective which is one of ,,,, then by the induction hypothesis, B and C each contain an even number of propositional variable symbols. The number of propositional variable symbols which appear in A is the sum of the numbers for B and C, so it is also even. Explain what the error is in our attempted proof.