## Subtleties of Lies

In a puzzle book by Mari Berrondo (in Russian), I found the following logic problem:

Alfred, Bertran and Charles are asked about their profession. One of them always lies; another one always tells the truth; and the third one [who I will refer to as a “half-liar”] sometimes lies and sometime tells the truth. Here are their answers:

Bertran: I am a painter, Alfred is a piano-tuner, Charles is a decorator.
Alfred: I am a doctor, Charles is an insurance agent. Concerning Bertran, if you ask him, he will tell you that he is a painter.
Charles: Alfred is a piano-tuner. Bertran is a decorator, and I am an insurance agent.
What is the profession of the half-liar?

The solution in the book is the following. As Alfred is right about what Bertran would say, Alfred can’t be a liar. If Alfred is a half-liar then the other two people would give the opposite statements, since one will be a truth-teller and the other a liar. But they both say that Alfred is a piano-tuner, therefore Alfred must be a truth-teller. Hence, Alfred’s statement about everyone’s profession must be the truth. Now we know that Charles is an insurance agent. As Charles confirms that, thus telling the truth in this instance, we recognize that he must be a half-liar. The answer to the problem is that the half-liar is an insurance agent.

But I have a problem with this problem. You see, a liar can say many things. He can say that he is a conductor, a mathematician, a beekeeper or whatever. So there is no way of knowing what a person who decides to lie can say. Let’s just analyze the statement by Alfred: “Concerning Bertran, if you ask him, he will tell you that he is a painter.”

If Alfred tells the truth about what Bertran would say, he needs to know for sure that Bertran will say that he is a painter. Hence, Bertran must be a truth-teller and a painter. If Alfred lies, he needs to be sure that Bertran won’t say that he is a painter. So Bertran must be either a truth-teller and not a painter, or a liar and a painter. Bertran can’t be a half-liar, because a half-liar can say that he is a painter as well as he can say something else, no matter what his real profession.

There is one interesting aspect of this that many people overlook. There are different types of people who are half-liars. In some books half-liars are introduced as people who, before making a statement, flip a coin to decide whether to lie or to tell the truth. Such a person needs to know in advance exactly what other people are saying, in order to construct a statement about what those people might say that corresponds to the coin flip. On the other hand, other types of half-liars exist. One half-liar can say something and then see later whether it is true. If Alfred is a half-liar who doesn’t care in advance about the truth of his statement, he can say that Bertran will claim that he is a painter.

I leave it to my readers to finish my analysis and see that the problem doesn’t have a solution. To end my essay on a positive note, I decided to slightly change the problem, so that there is no contradiction. In the same setting:

Bertran: I am a painter, Alfred is a piano-tuner, Charles is a decorator.
Alfred: I am a doctor, Charles is an insurance agent. Concerning Bertran, if you ask him, he will tell you that he is not a painter.
Charles: Alfred is a piano-tuner. Bertran is a decorator, and I am an insurance agent.
What is the profession of the half-liar?

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1. #### Igor:

I disagree. B can be a consistant lier. Actually i think most liers will stick to one story in order to prevent being descovered as a lier. The conclusion, that if A knows what B will say, doesn’t necessary mean that B is telling the truth. Examples can be found for example in politics, where a politicion sticks to a lie although everyone knows that he is lying. Since everyone knows that he is lying (consistantly, sticking to a story), everyone can make the prediction based on experience that he will repeat the lie again.
So the only thing we know, that A somehow knows what B will say. Therefore i don’t see a mistake in the riddle. And even if something needs to be improved, than it is that the truth teller is defined as someone who tries to tell the truth and won’t tell knowingly a lie.

2. #### Greg:

I couldn’t resist quoting this:

https://xkcd.com/246/

3. #### Sheldon:

Following a different, simpler approach, I derive the same solution as the book, Charles as the half-liar.
If you chart out the possibilities, a minimum of three, then the solution becomes obvious.

How?
Chart the possibly cases, assuming each is a Truth-teller and another a Liar. Do not worry about the Half-Liar because that would be the logically third person not selected.
Chart each job versus the person. Where there is a contradiction, that chart tells the assumption is incorrect.