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Larks comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong
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Comments (130)
What about
'all sentances are either true or false'.
This sounds like the sort of sentance we'd want to assign a truth value to. Yet we can instanciate it into
'this sentance is either true or false'
Which is problematic - and yet it seems that it must have a truth value if the first sentance did.
I'm comfortable (mostly, it's a bit of a bullet bite) saying 'all sentences are either true or false' doesn't have a truth value, since to determine one you have to reference the sentence itself and that function doesn't terminate. You can say in English or a Meta-language that all well-formed formulas in some system are either true or false. But you can't say this in the object language.
Did you intend to note that "this sentence is either true or false" is a true sentence (for most methods of evaluation) that can't be evaluated by Yvain's fairly straightforward approach? Because that's definitely interesting (thanks Jack).
Just not messing with recursion, in general, is a fairly old solution and not very satisfying. I blame Yvain's writing ability for leading 9 people astray :D
Why is that a problem? It is a true sentence.
I take it the problem is that it doesn't unpack even though it does have a truth value. Or at least it isn't obvious how to unpack it. It's a false negative candidate.
So the point is that it is a sentence that demonstrates a problem with using unpackability as a requirement for qualifying as meaningful English? That seems reasonable.
That's what I got from it.
Note that the universal "All sentences are either true or false" also, doesn't appear to meet the unpackability requirement, though I'm not confident I know how to make a Tarski sentence out of that.