The Liar Paradox
The liar paradox concerns a statement such as 'I am lying' when it refers to that very statement. Treating it as true makes it false, while treating it as false makes it true, exposing a problem in unrestricted self-reference about truth.
THE IDEA
THE LIAR PARADOX
It is remarkably easy to get yourself into a logical paradox: just declare “I’m lying!” I have seen logically inclined children do this and then collapse into giggles. They have created a paradox – if they’re telling the truth, then they’re lying. But if they’re lying, they’re telling the truth. This is a classical paradox known as the liar paradox. It also happens if I say “Don’t take my advice.” Should you then take that advice? If you take it, that means you shouldn’t take it. And if you don’t take it, that means you’ve taken it.
We can make a pair of statements into similarly loopy paradoxes like this:
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The following statement is true.
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The previous statement is false.
There is also the curious sentence discussed by Douglas Hofstadter in Metamagical Themas:
Cette phrase en français est difficile à traduire en anglais.
which can be translated literally as
This sentence in French is difficult to translate into English.
but it no longer makes sense.
At this point I should apologise to the French translator of this book, if there is one. Except of course, that sentence has just made it worse because it is somewhat difficult to translate into French.
When to use it
Examining whether a self-referential claim produces a contradiction or has a consistent interpretation.
Teaching why self-reference about truth needs careful rules, while distinguishing it from legitimate recursive definitions.
In programming, comparing circular dependencies with logical self-reference while checking whether a particular cycle actually causes nontermination.
Exploring philosophical questions about truth, meaning, and the nature of language.
FROM THEORY TO EVERYDAY LIFE
In practice
The Lying Statement
When 'I'm lying' means that this very statement is false, assigning it either truth value creates a contradiction. If the speaker instead refers to some other statement, the same paradox need not arise.
Don't Take My Advice
Saying 'Don't take my advice' leads to a paradox: if you take the advice, you are following the instruction not to take it, so you shouldn't take it; if you don't take it, you are ignoring the advice, which means you have taken it (by not taking it).
Pair of Self-Referential Statements
Statement 1: 'The following statement is true.' Statement 2: 'The previous statement is false.' If Statement 1 is true, then Statement 2 is true, which would make Statement 1 false, a contradiction. If Statement 1 is false, then Statement 2 is false, which would make Statement 1 true, also a contradiction.
Translation Paradox
The French sentence 'Cette phrase en français est difficile à traduire en anglais' refers to itself as a French sentence. Its literal English rendering, 'This sentence in French is difficult to translate into English', changes the language of the sentence doing the referring. The difficulty concerns preserving that reference, rather than a strict true-or-false contradiction.
A MOMENT TO REFLECT
Put it to work.
Think of a situation where someone uses a self-referential statement (e.g., 'I always lie'). How would you analyze its truth value?
How might the liar paradox relate to paradoxes in computer science, such as the halting problem?
Can you construct a real-world example where a loop of self-reference leads to a practical deadlock or confusion?
References
- Metamagical Themas — Douglas Hofstadter
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)