Alice, Bob and Charlie - Who is lying?

Question

We were given this exercise in logic class:

Three friends, Alice, Bob and Charlie, are talking to each other. They make three statements:

1. Alice says: ‘If Bob is telling the truth, then Charlie is lying.’
2. Bob says: ‘Alice is telling the truth.’
3. Charlie says: ‘Bob is lying. The question: Who is telling the truth and who is lying?

• Use logical consequence to analyze the statements and determine whether or not they are are compatible with each other. Try to construct a solution where the statements check their own consistency. In this case, who is telling the truth and who is lying?

I understand that we should start this with:

$ A → (B → ¬C)$

$B → A $

$C → ¬ B$

but after that I'm kind of lost on what exactly I should do to find who lies and who tells the truth in proper formalism. How can I proceed with this exercise?

Answer 1

Each individual statement is an equivalence, not an implication (as lying or telling the truth refers only to this particular statement).

Thus, the second statement is A <-> B, ie we can replace B by A in the other two statements. The third statement becomes C <-> not B, so we can substitute (not C) by A in the first statement and get: A <-> (A -> A). Since A -> A is a tautology, we conclude that Alice and Bob are telling the truth and Charlie lies.