Is $((A\to B)\to B)\to A$ a tautology?

Question

I'm really new to logic and I'm wondering whether the formula above is a tautology.

I know that a tautology is a statement that's always true. I'm stuck.

Answer 1

No. If $A$ is false and $B$ is true, then the formula is false.

Answer 2

It may be informative to convert your original expression to disjunctive normal form: $$ [(A\to B)\to B]\to A\equiv(\neg A\land\neg B)\lor A. $$ Thus, you can see the counterexample Shaun gave is actually the only counterexample for which your desired implication is not true; that is, any other choice of truth values for $A$ and $B$ will result in a true implication.