I've been working through problems in a logic textbook and came across this theorem (paraphrased):
A well-formed formula (wff) that contains biconditional as its only connective is a tautology iff each propositional parameter in that wff occurs an even number of times.
How do I go about proving that this is so? Hints would be appreciated. Thanks a lot.
HINT
First use induction to prove that a statement built up using $\leftrightarrow$ as its only connective is true iff there are an even number of false terms (by terms, I mean instances, i.e. the statement $A \leftrightarrow A$ has $2$ terms, even though it has just $1$ variable).
Once you have that result, then if any such statement has an odd number of occurrences of a variable is not a tautology, since if you can set that variable to False, and all others to true, you have an odd number of false terms, and hence the statement as a whole is false, and hence is not a tautology.
But if all variables occur an even number of times, then no matter whether you set the variables to true or false, there will be an even number of false terms, and hence the statement will always be true, and hence be a tautology.