# Propositional logic problem about truth tellers and liars

I'm not sure if I'm right about this, so I'd appreciate a little guidance.

In a remote village, there exists two types of people

• Truth tellers who will always tell the truth

• Liars who will always lie

One day, a stranger visited the village. He met two of the inhabitants, Jack and Emily. The stranger asked them: "Is anyone of you a liar?". Jack replied: "At least one of us is a liar."

What are Jack and Emily? Truth tellers or liars?

I know I can get the answer with the help of truth tables, but I'm not sure what the intial formula should be for Jack's reply. What I came up with for "At least one of us is a liar" was:

J: Jack is a truth teller

E: Emily is a truth teller

$$\neg(J \lor E)$$

Would you agree?

• More simply...If Jack is a liar then his statement is True, a contradiction! Hence he must be a truth teller. But then... – lulu Sep 27 '16 at 22:55
• your inititial formula is wrong (you don't mention that Jack is the one who is speaking) the right formula is J <-> ~(J v E) – Willemien Sep 27 '16 at 22:58
• Are you sure? That formula translated would be something in the region of "Jack is a truth teller if and only if Jack or Emily are lying." @Willemien – Steve Sep 27 '16 at 23:08
• @Steve the formula ~(J v E) does not mean "Jack or Emily are lying", it's saying "it is not the case that at least one of them is a truth teller", which is equivalent to "both of them are lying" and so the formula written by Williemien is saying "Jack is a truth teller if and only if Jack and Emily are both liers", wich is a contradiction (anyway that formula doest't seem to help) – la flaca Sep 27 '16 at 23:33
• @Eliana and Steve, sorry my fault the right translation is J <-> (~J v ~E) (and this is equivalent to J & ~E ) (I just was thinking it must start with J <-> and that bit was missing ) just start this way and then go simplify ing – Willemien Sep 28 '16 at 18:34

I don't know if it was a typo and you tryed to write $\lnot(J \land E)$ or your intention was the formula you wrote, since it is equivalent to $\lnot J \land \lnot E$ (both are liars). I will assume it was a typo and in that case (as Willemien pointed out) your formula $\lnot(J\land E)$ is a translation of what Jack said but in principle it could be false (he could be a liar). If it were false then Jack would be a liar so (as lulu pointed out) what he said would be true and that's a contradiction and because of that it must be the case that Jack is not a liar and that validates your initial formula, now you can add the formula $$J$$ to derive from them $$\lnot(J \land E)\land J$$ and from that conclude $$\lnot E$$ So Jack is a truth teller and Emily is a liar