Re: rec.puzzles archive logic/hundred

You first stated that if it is possible to find a consistent set of
truth values, then that is the solution. The point of the boxes
paradox is to exhibit a counterexample to that claim. Of course it is
true that the statements on the box cannot force the gold to be in one
box or another. But if you assume that every statement is either true
or false, then you are forced to conclude that the gold is in box B,
which is ridiculous. So there must be something wrong with the
assumption that every statement is either true or false.

It doesn't do any good to say "well, OK, but in the hundred sentences
case the wording of the problem implies that every statement ie either
true or false." Because even if I explicitly tell you that a certain
statement either is true or false, I might be saying something that is
impossible. The most trivial example of this is "This sentence is
false." It's impossible for that statement to be true or false; it
doesn't help for me to tell you that it is.

Another way to see that there is something strange going on is to
switch the predicate "is true" to the predicate "is blue." If you try
this you'll see that puzzles like the hundred sentences puzzle don't
work. The self-referential nature of the predicate "is true" is
essential to the puzzle.

If I can draw an analogy, consider the various bogus proofs that 0 = 1.
One common trick is to divide by zero somewhere in the proof. Now
division by zero is not a defined operation. It's meaningless. Even
though you can sometimes get away with it, it's still meaningless. The
fact that it's meaningless is why you can prove ridiculous things like
0 = 1. But it's not meaningful sometimes, like when you get away with
it, and meaningless other times, like when you prove that 0 = 1. It's
always meaningless.

Similarly, statements that refer to their own truth value are
meaningless. This problem has been extensively studies by logicians,
and there is a way to talk about truth of sentences (using
metalanguage), but I won't go into that here. The relevant point is
that statements that refer to their own truth value cannot be true or
false. While it's true that the hundred sentences puzzle doesn't lead
to a ridiculous result because it doesn't refer to anything other than
sentences on the list, the puzzle could be trivially modified (a la the
boxes puzzle) so that the exact same reasoning that "solves" it also
leads to a ridiculous result.

That is why the archive entry is correct.

.

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