# no worries yet

What happens if you feed ChatGPT some logical puzzles? I took one, from Smullyan’s textbook on basics of mathematical logics. It is easy to solve. Predictably, ChatGPT failed to do so, but in rather instructive ways.

Here’s the puzzle:

[On an island, the inhabitants are either knights, and always tell the truth, or knaves, and always lie.] On the day of his arrival to that island, Abercrombie came across three inhabitants, whom we will call A, B and C. He asked A: “Are you a knight or a knave?” A answered, but so indistinctly that Abercrombie could not understand what he said. He then asked B: “What did he say?” B replied: “He said that he is a knave.” At this point, C piped up and said: “Don’t believe that; it’s a lie!”
Was C a knight or a knave?

After R. Smullyan, Logical Labyrinths.

And here are four solutions by the model. We can see that it correctly derived identity of C in 2 cases out of four. Yet even in those cases, it either relied on circular logic, or on a non sequitur. (OK, OK, it does not rely: more pedantically, if ChatGPT’s reasonings were made by a human, that’s what one would infer.)

In all these cases, one has a distinct whiff of a C-student who learned to sleekly emulate educated speech. An ideal Social Text author or editor, perhaps…

So, no risk for our trade, as of yet. Below, the “solutions”.

##### #1

To solve this logic puzzle, we need to determine whether A is a knight or a knave.
If A is a knight, then he would have truthfully answered “I am a knight” or “I am a knave” when asked about his identity. However, since Abercrombie could not understand his response, we cannot determine if A is a knight or a knave based on his answer.
If A is a knave, then he would have lied about his identity and said “I am a knight” or “I am a knave”. In this case, B’s response “He said that he is a knave” would be a truthful statement, since A would have lied about being a knight.
Now, if B is a knight, then he would have repeated A’s truthful statement and said “He said that he is a knave”, which means that A is indeed a knave. In this case, C’s statement “Don’t believe that; it’s a lie!” would be a lie, since B’s statement is true.
If B is a knave, then he would have repeated A’s lie and said “He said that he is a knave”, which means that A is actually a knight. In this case, C’s statement “Don’t believe that; it’s a lie!” would be a truthful statement, since B’s statement is a lie.
Therefore, we have two possible scenarios:

1. If A is a knave, then B is a knight and C is a knave.
2. If A is a knight, then we cannot determine the truthfulness of B and C.

Since we cannot determine whether A is a knight or a knave based on the information provided, we cannot definitively determine whether C is a knight or a knave.

##### #2

C was a knight.

To solve this puzzle, we need to analyze the statements made by A, B, and C. Let’s start with what we know about A. A’s statement was unclear, so we cannot determine whether A is a knight or a knave.

Next, we need to consider what B said. B claimed that A said he is a knave. If B is a knight, then A must be a knave because knights always tell the truth. However, this contradicts what C said, which was “Don’t believe that; it’s a lie!” Therefore, B must be a knave.

Now, we know that A is either a knight or a knave, and B is a knave. So, we can deduce that if A is a knight, then A would have said he is a knight. If A is a knave, then A would have said he is a knave (which is what B claimed). However, since we know B is lying, we can conclude that A must be a knight because A did not say he is a knave.

Finally, we can look at what C said. C claimed that B’s statement was a lie. We have already established that B is a knave, which means B’s statement was indeed a lie. Since C’s statement is true, C must be a knight.

In conclusion, A is a knight, B is a knave, and C is a knight.

##### #3

Let’s analyze the statements made by each inhabitant to determine if C is a knight or a knave.

A’s response is indistinct, so we cannot use it to determine their identity.

B’s response is that A said they were a knave. This means that either A is a knave and told the truth or A is a knight and lied. If A is a knave and told the truth, then they would indeed be a knave. If A is a knight and lied, then they would not be a knave but would be a knight. Therefore, we cannot determine A’s identity from B’s response alone.

C’s response is that B’s statement is a lie. This means that either C is a knave and lying or C is a knight and telling the truth. If C is a knave, then B’s statement that A is a knave would be true, which means A is indeed a knave. However, this would mean that C is telling the truth, which contradicts the assumption that C is a knave. Therefore, C cannot be a knave. If C is a knight, then B’s statement that A is a knave would be false, which means A is not a knave but is a knight. This means that C is lying, which is consistent with the assumption that C is a knight. Therefore, C must be a knight.

In conclusion, C is a knight.

##### #4

To solve this puzzle, we need to analyze the statements of A, B, and C and determine whether each person is a knight or a knave.

Let’s start with A. Since Abercrombie could not understand what A said, we cannot determine whether A is a knight or a knave from this interaction alone.

Next, we have B’s statement. B claims that A said he is a knave. If B is a knight, then A must be a knave since a knight would not lie about their own identity. If B is a knave, then A could be either a knight or a knave since a knave would lie about their own identity. So, we cannot determine the identity of A from B’s statement alone.

Finally, we have C’s statement. C claims that B is lying, which implies that A is not a knave (otherwise, B’s statement would be true). If C is a knight, then A must be a knight since a knight would always tell the truth. If C is a knave, then A could be either a knight or a knave since a knave would always lie.

Therefore, we cannot determine for sure whether A is a knight or a knave, but we can determine that C is a knave.

ChatGPT Mar 23 Version. retrieved April 23, 2023.