When I was young, I enjoyed logic puzzles involving knights and knaves, fictional creatures that never and always lies respectively. The crux to solving those puzzles usually rely on applying liar’s paradox appropriately.
Liars paradox is about the statement that a liar makes declaring that he is lying. Evaluating the truthfulness of the sentence leads to a contradiction via circular reasoning. As a quick aside: a statement can be true or false. For example, “the sky is blue” is a statement that is true everywhere in the world but false in London.
As such, we can conclude that the statement “I am lying” cannot have a truth value and we don’t expect such statements to be proclaimed and to be taken seriously in real life. On its own it’s as meaningful as not saying anything; a sort of contra-positive statement of a tautology.
Consider a slight variant of the statement: “I am a liar”. In the world where liars lie all the time, such a statement similarly cannot be proclaimed even though this statement itself can have a truth value without leading to a contradiction.
However, if we relax the condition that a liar can choose when to lie, rather than always lies, and that a truthful person never lie, then the only way the person can truthfully make that statement is when the person is a liar. A truthful person that never lies will not make such a statement. Hence, if someone makes the statement “I am a liar” with the intention of making us evaluate its truthfulness, we must then believe that the statement is truthful and that the person is lying. What is surprising, and ironic, is that we ought to believe the liar in this case that he is telling the truth.
| Would I say “I am a liar” | Knight | Knave (sometime lies) |
|---|---|---|
| Lying | - (Not possible) | N |
| Truthful | N | Y |
What happens if we extend this to other sentences such as “I am a psychopath” or sentences where it will not be made by someone for which the statement is false. For example, someone who is not a psychopath would not declare that he/she is a psychopath; only a psychopath would employ this reverse psychology to deceive us. Hence he/she must be a psychopath, as claimed. Can we employ the same reasoning to deduce something about the truth?
| Would I say “I am a psychopath” | Psychopath | Not a psychopath |
|---|---|---|
| Psychopathic | Y (by telling the truth) | - (Not possible by definition) |
| Truthful | Y | N |
Generalizing this sounds pretty fishy. It is a slippery slope to absurd claims such as “I am a wizard” because someone who is not a wizard would not make such a statement. Upon some closer inspection, the above example assumes that all non-psychopaths are truthful. However, if a non-psychopathic liar comes along, he can make the declaration without any logical contradiction.
| Would I say “I am a psychopath” | Psychopathic knight | Non-psychopathic knight | Psychopathic knave | Non-psychopathic knave |
|---|---|---|---|---|
| Lying | - (Not possible by definition) | - | N | Y |
| Truthful | Y | N | Y | N |
The lying row will always be the negation of the truthful row. By adding a dimension for whether the speaker is lying or not, we see that it is impossible to differentiate a psychopath from one who is not. In fact, we can replace “psychopath” with any other attribute unrelated to the truthful nature of the speaker, and we would reach this impasse. This shows that this argument doesn’t generalize well.
In conclusion, the liar’s statement presents a concrete example where truthfulness of the statement cannot be evaluated. In general, when truthfulness of a statement can be defined, it is not possible to distinguish the knights from the knaves. For the special case sentence “I am a liar”, due to it’s self-referential nature, we can deduce that only a truthful liar will make such a statement.