This is an interesting thought experiment/puzzle I read about in the introduction to the latest book I’m reading, “Reasoning about Knowledge” by Ronald Fagin and others (Buy it from Amazon
The case of two children is most interesting. Suppose a father tells his two children (a boy and a girl) not to get dirty. They go outside and play, and they both get mud on their foreheads. The children cannot detect the mud on their own foreheads, as it is above their eyes. But each child can see the mud on the other child’s forehead. Since each wants to see the other get in trouble, neither says anything.
These children happen to be very intelligent, as a matter of fact, they never make logical mistakes nor fail to deduce something which is logically deducible. Prodigies of logic, if you will. Also, they never lie.
Eventually, the father comes along. “At least one of you has mud on their forehead,” the father tells them. He asks the girl: “Is there mud on your forehead?” She answers: “I don’t know.” Then the father asks the boy, “Is there mud on your forehead?” He sheepishly answers “yes”.
How was the boy able to tell he had mud on his forehead? He reasoned as follows. If he didn’t have mud on his forehead, then his sister would have seen as much. His sister knew that at least one of them had a muddy forehead, because the father said so. She would therefore have deduced, by process of elimination, the mud was on her own forehead, and she would have said so. Since she didn’t know, that must mean there was mud on her brother’s head, preventing her from making that deduction.
Now here’s the semi-paradox. Before the father came along, both children already knew that at least one of them had muddy foreheads. This is because they could see each other. Therefore, it seems like the father didn’t actually tell them anything new. It seems like, when the father said “At least one of you has a muddy forehead”, that information was totally superfluous.
But suppose the father hadn’t said that. Imagine, instead, the father walked up and, without telling them anything, asked the girl if her face had mud on it. She’d answer “I don’t know”. Then he’d ask the boy the same question, and he’d have to answer “I don’t know” as well. After all, in the alternate universe where the boy was clean, the scenario up to his being asked the question, would unfold exactly the same. Nothing would have changed depending on his cleanliness or dirtiness, so he couldn’t possibly determine which was the truth.
Although by telling them “one of you is dirty” the father didn’t present any new information, he nevertheless changed the state of knowledge: he made the boy know that the girl knew that at least one of them was dirty. Before the father’s announcement, the boy could not know that the girl knew that one of them was dirty, because he didn’t know whether he was dirty himself, and if he wasn’t, the girl couldn’t see her own forehead.
If, rather than making the announcement out loud, the father had instead pulled each kid aside and privately told them that at least one of them was dirty, then no deductions could be made. The boy would not know what the father had whispered to the girl, so could not know that the girl knew that at least one of them was dirty.
In all of this, we’re implicitly making many extra assumptions. We’re assuming not only that the children were honest and logical and intelligent (I made that assumption explicit), we’re also assuming that the boy knows the girl is honest/logical/intelligent. If the boy didn’t know this, then he would have to consider the possibility that when the girl said “I don’t know”, she was lying, or that her ignorance resulted from missing a logical deduction. Unable to rule those possibilities out, the boy would not be able to deduce with 100% certainty that his own forehead was dirty. We’re also assuming the girl isn’t blind or deaf and, moreover, the boy knows she isn’t blind or deaf.
Finally, we’re assuming the father is honest. But that’s not enough. We’re assuming the girl knows the father is honest. But even that is too little. We’re actually assuming that the boy knows the girl knows the father is honest. After all, if the boy didn’t know the girl knew the father was honest, then he couldn’t tell whether the girl’s ignorance was based on seeing his muddy forehead or whether it was based on not knowing whether to believe her father!
This illustrates a little of the subtlety of reasoning about knowledge. Nontrivial deductions can require not just knowledge of hard cold facts, but knowledge about knowledge itself. Compare this to my earlier article, where I showed how to fix Fitch’s Paradox of Knowability by rejecting an axiom about knowledge about knowledge. And again, knowledge about knowledge leads to paradox in my Truth-Knowledge Paradox.
This is just the case of two children. There’s a more general case with n children of whom k have muddy faces. Then, one needs to worry about whether a child knows that another child knows that another child knows that another child knows somebody’s honest, and so on.
F’ing knowledge… how does it work?!
FURTHER READING
Fitch’s Paradox of Knowability
A Truth-Knowledge Paradox
General to Specific and Specific to General
How Children Understand Language