Truth or lie? "At least one [of the two] of us is a liar."

Question

This problem involves two people. Person A and person B. They can either always tell the truth, or always lie. When asked, person A replies that: "At least one of us is a liar." Does person A and B tell the truth or do they lie?

What I have so far is that person A can either tell the truth or lie. If he tells the truth, then his statement can be expressed by "A$\lor$B", if he lies then it must be given that they are both speaking the truth, which contradicts that he would be a liar. So person A must be speaking the truth.

But if person A is speaking the truth, then person B could be either lying or speaking the truth. Both would still make person A tell the truth.

So how can I figure out whether person B is speaking the truth or not?

Answer 1

$$\begin{array}{cccc} &&&\text{A says}\\ \text{A lies} &\text{B lies}&\text{A or B lies}&\text{A or B lies}\\ a&b&a\lor b&(a\lor b)\oplus a\\ F& F& F& \color{red}F\\ T& F& T& \color{red}F\\ F& T& T& \color{green}T\\ T& T& T& \color{red}F\\ \end{array}$$

Answer 2

If B would speak the truth then both of them would speak the truth - a contradiction to A's statement that at least one of them is a liar. So B will obviously not speak the truth.

Answer 3

I find it easier to always suppose that A, B tell the truth in any of these riddles with two types of people (that either always tells truth, or always lies). This is my approach:

$a$: A tells the truth

$b$: B tells the truth

Proposition "at least one of us is a liar" is $\lnot a \lor \lnot b$. Person A tells us this proposition, so $a$ and $\lnot a \lor \lnot b$ must have the same truth value (if A tells the truth, his proposition is true and if he lies, his proposition is false). Therefore, the answer to the problem is if there is one or more $T$ values of the proposition $a\leftrightarrow (\lnot a \lor \lnot b$). Let's see:

\begin{array}{cccccc} a& b& \color{grey}{\lnot a}& \color{grey}{\lnot b}& \lnot a \lor \lnot b& a\leftrightarrow (\lnot a \lor \lnot b)\\ F& F& \color{grey}T& \color{grey}T& T& \color{red}F\\ T& F& \color{grey}F& \color{grey}T& T& \color{green}T\\ F& T& \color{grey}T& \color{grey}F& T& \color{red}F\\ T& T& \color{grey}F& \color{grey}F& F& \color{red}F\\ \end{array}

So, we see that $a\leftrightarrow (\lnot a \lor \lnot b)$ is true only when A tells the truth and B tells lies.

The same approach can be applied to any of this kind of problems.