From: Hal Finney (hal@finney.org)
Date: Wed May 29 2002 - 10:54:22 MDT
I can't resist entering this thread, although I don't think it is really
on topic for us, and I apologize in advance for that.
In Godel, Escher, Bach, author Douglas Hofstadter quotes logician WvO
Quine as having created an alternative form of self referential sentence,
closely related to the Godel numbering trick. The example he gave was a
variant on the puzzle "this sentence is false," a version of the ancient
paradox of Epimenides the Cretan, who said that all Cretans are liars.
"yields falsehood when preceded by its quotation" yields falsehood
when preceded by its quotation
This says that X, when preceded by its quotation (to form "X" X) is a
falsehood. But if X is the phrase "yields falsehood when preceded by
its quotation", and we form "X" X, then we get the sentence above. So
the sentence is asserting its own falsehood in an indirect manner.
Hofstadter goes on to analyze other cases of phrases preceded by their
quotations; some are true, some false, and some meaningless:
"is a sentence fragment" is a sentence fragment. TRUE.
"is composed of ten words" is composed of ten words. FALSE.
"wandering through the trees" wandering through the trees. MEANINGLESS.
Lee's paradox ("if this sentence is true, then God exists") could
be expressed using a similar trick and avoid explicit self-reference,
as follows:
"when preceded by its quotation, if true, implies that God exists"
when preceded by its quotation, if true, implies that God exists
The point is that self-reference is not always obvious.
Incidentally another way to analyze Lee's paradox is to assume it is
false. In classical logic, the only way for an if-then statement to be
false is if the antecedant (the "if" part) is true and the consequent (the
"then" part) to be false. Therefore the "if" part must be true, and that
part says "this sentence is true". So we have reached a contradiction by
our assumption that the sentence was false, and so (?) it must be true.
IMO the real solution to this paradox is to keep in mind that all
sentences have three possible truth values, not two, as shown above.
Until you know that a sentence is making a meaningful statement, you
should not view it as either true or false.
Hal
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