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

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

Explain what the error is in our attempted proof.