Can't find a logical formulation to this problem

Question

In this problem are only truth tellers and liars. When meeting two people, A and B, you ask A: "Is any of you a truth teller?", to which A replies: "If B is a liars, then i'm a liar" What are A and B?

What I got so far is:

If A is telling the truth, then the statement must be true when B is telling the truth, and false when B is lying. If A is lying, then the statement must be true when B is telling the truth, and false when he is lying.

I can't seem to find a way to logically formulate this, and I'm thinking this might be because I'm missing something. Have I got it right?

Answer 1

If we say that $L(x)$ means "$x$ is a liar", then $A$ said the following statement: $$L(B)\implies L(A)$$

First, examine the two cases:

1. If $A$ is a liar, then the statement is false, meaning that the statement $(\neg L(B)\implies L(A))$ is true. This is equivalent to $\neg(\neg L(B) \vee L(A))$ which is $L(B)\wedge \neg L(A)$, so if $A$ is a liar, then $B$ is a liar and $A$ is not a liar, a contradiction.
2. If $A$ is not a liar, then $L(B)\implies L(A)$ is true. If $B$ would be a liar, then $A$ would be a liar which would be a contradiction, so $B$ is not a liar. If $B$ is not a liar, then nothing can be said of $A$, so there is no contradiction.

Conclusion: neither of them is a liar.

Answer 2

You can deduce from "If A is telling the truth, then the statement must be true when B is telling the truth, and false when B is lying" that if A is a truth teller then B is a truth teller.

You can deduce a contradiction from "If A is lying, then the statement must be true when B is telling the truth, and false when he is lying" but I would prefer to consider all eight possibilities:

 A     B   Statement  Comment 
TT    TT   True       Consistent: B is not a liar so A need not be a liar
TT    TT   False      Inconsistent: A is a truth teller so cannot state false
TT    Liar True       Inconsistent: B is a liar so A must be a liar
TT    Liar False      Inconsistent: A is a truth teller so cannot state false
Liar  TT   True       Inconsistent: A is a liar so cannot state true
Liar  TT   False      Inconsistent: Statement is vacuously true as B not liar
Liar  Liar True       Inconsistent: A is a liar so cannot state true
Liar  Liar False      Inconsistent: Statement is true as A and B are both liars