# Can anything be deduced from the statement “If I am a truth teller, the other two are liars”?

There is a scenario which goes along the following lines:

There are three friends. The friends are either truth tellers, or liars. One friend makes the statement:

"If I am a truth teller, the other two are liars".

What can I deduce from this statement?

It depends on how you interpret the statement. From a more 'casual' point of view:

• If the speaker is a truth-teller, then his statement is true, so his two friends are liars.
• If the speaker is a liar, then his statement is false, so him being a truth-teller doesn't imply his two friends are liars ... but he's not a truth-teller anyway, so this information is useless.

But from a strictly logical point of view, the negation of the statement $A\Rightarrow B$ is $A\land¬B$, i.e. $A$ is true and $B$ is false (since this is the only way the statement "A implies B" could be strictly untrue as opposed to just irrelevant). So if everyone is a strict logician, then:

• If the speaker is a liar, then his statement is false, so he is a truth-teller and the other two are not both liars, contradiction.

Thus, under the strictest logical interpretation, we know that:

the speaker is a truth-teller and the other two are liars.

And this is the interpretation usually taken in these kinds of logic puzzle: that everyone who's a "liar" is always a perfect logician speaking the exact logical dual of the truth. See also this interesting discussion of types of liars.

• Great catch on forcing any liar to always lie, Rand al'Thor. $~\small P\Rightarrow Q$, where $\small P$ = "this gets accepted" and $\small Q$ = "I'll upvote the puzzle." Also $\small R$, where $\small R$ = "I probably will anyway," – humn Aug 27 '17 at 17:12
• @humn Nah, upvote the puzzle anyway - for inspiring a fun solution if nothing else :-) Besides, one more upvote on the puzzle and the OP will have enough rep to upvote in turn! (Edit to match your edit: $R=JG$, where $J$ = "jolly" and $G$ = "good".) – Rand al'Thor Aug 27 '17 at 17:28
• Right you are, Rand Al'T, Any puzzle that produces a good solution is a good puzzle, regardless of intent. And you just provided a good reason to not delay the vote - early reputation points. – humn Aug 27 '17 at 17:32
• Rand al'Thor, I would say, the conclusion is logically incorrect. 1. Statement A=>B does not say anything about the case "not A". 2. "The other two are liers" is a compound statement that consists of two statements: man 2 is a lier and man 3 is a lier. I don't see these facts has been taken into account. – Cordfield Aug 28 '17 at 7:24
• @cordfield suppose that A is a liar. Then the statement is true, since A is not a truth-teller and false implies anything (look up the principle of explosion). But the statement is false, since A is a liar, and we have a contradiction. So A must be a truth teller, and the other two liars. This is what Rand is saying. – boboquack Aug 28 '17 at 8:02

Yes, we can deduct that at least one of the is a truth-teller and at least one liar.

Simple explanation: We can't deduct who is who particularly, but we can be sure that there are at least 1 truth-teller and at least 1 liar in this group of three people, because if a liar says there are two other liars, at least one of the is not a liar. As well as truth-teller can't say about two other people being liars and there are no liar.

• ^vote with a note: This solution is more or less what I thought, too, before seeing Rand al'Thor's more complete solution. – humn Aug 28 '17 at 7:35
• Not technically true: Not being a truth-teller, in this context, does not automatically a consistent liar make. They could well all be liars. – Weckar E. Aug 28 '17 at 12:54

We can deduct that the first is a truth teller and the other two are liars.

If I make any statement "if A then B" and the statement A is false, then the whole statement "if A then B" is true. For example: "If the moon is made of green cheese, then I have a million pound in my bank account" is a true statement.

If I am a liar, and I make a statement "if A then B" then that statement must be false. However, if A is "I am a truth teller", which is false, then the whole statement "if A then B" is true. Therefore, a liar couldn't make this statement. Therefore, the first friend is a truth-teller, and the two other are liars.

• ^vote with a note: This would be the best solution so far, in my opinion, if only Rand al'Thor hadn't found it almost a day earlier. – humn Aug 28 '17 at 15:26

Well, it is another form of the classical paradox "This statement is false", for, if he tells the truth that - he is a truth teller and the other two are liars - contradicts the given data that the friends are either truth tellers, or liars, which essentially means:

all the THREE are liars or all the THREE are truth tellers, which cannot be.

• I don't think that's how we're meant to interpret the statement that "The friends are either truth tellers, or liars" - I think it means (as is usual in these liars puzzles) that each friend is either a truth teller or a liar. – Rand al'Thor Aug 27 '17 at 14:46
• I am not sure if truther/liar should even be considered binary here. (if you don't consider "this statement is false" to be binary, it ceases to be a paradox) – Weckar E. Aug 28 '17 at 12:57