a possible solution to the Strengthened Liar's Paradox
YKY (Yan King Yin, 甄景贤)
S1: S1 is false.
The TV of S1 is "unsolvable", which is NOT a truth value per se.
S2: S2 is either false or meaningless.
S2 is not meaningless, but its TV is not solvable.
S3: S3's TV is unsolvable.
S3 is false.
S4: S4 is either false or S4's TV is unsolvable.
S4's TV is unsolvable.
S5: S4's TV is unsolvable.
S5 is true!
S4 is unsolvable because S4 is self-referential -- not all
self-referential statements are unsolvable, but some are; they have
to be determined on a case-by-case basis.
YKY
Luke Palmer
interpretation is so closely tied to the formalism.
>
> S1: S1 is false.
> The TV of S1 is "unsolvable", which is NOT a truth value per se.
>
> S2: S2 is either false or meaningless.
> S2 is not meaningless, but its TV is not solvable.
>
> S3: S3's TV is unsolvable.
> S3 is false.
>
> S4: S4 is either false or S4's TV is unsolvable.
> S4's TV is unsolvable.
YKY (Yan King Yin, 甄景贤)
> I have trouble thinking about such informal statements when their
> interpretation is so closely tied to the formalism.
They can be expressed formally using something such as Tarski's truth
predicate. I will do so in a short paper, if no one can point out
further holes in this idea...
>> S4: S4 is either false or S4's TV is unsolvable.
>> S4's TV is unsolvable.
>
> Doesn't that mean that S4 is true?
But, if we assume S4 is true, we can derive that S4 is false, thus
contradicting the assumption. Thus, the only conclusion we can reach
is that S4 is unsolvable.
YKY
Luke Palmer
Except once we conclude that, we can go back and prove that S4 is
true, by its definition, contradicting that it is unsolvable. It
seems like true, false, and unsolvable all lead to contradictions.
That doesn't mean that S4 is unsolvable, that means it is a witness to
inconsistency.
Luke
YKY (Yan King Yin, 甄景贤)
> Except once we conclude that, we can go back and prove that S4 is
> true, by its definition, contradicting that it is unsolvable.
Good point, but just because we can deduce that S4 is true (using the
assumption that S4 is unsolvable) does not necessary mean that we have
to conclude that S4 is indeed true. It's just that S4 can lead the
prover to conclude in multiple TVs; thus, it seems that we can still
conclude that S4 is unsolvable.
YKY
Abram Demski
What's your answer to these?
S6: S6 is true.
S7: S7 is true or false.
In other words, when a number of assessments could work, which one is
taken? (Not a critical point, but it's got to be decided.)
Now to more critical points...
It doesn't mean much to me to say that "unsolvable" is not a truth
value. It is still an assessment that can be made of a sentence, so
what does it matter?
The difference between what you are proposing here and a typical
fixed-point, I take it, is that "unsolvable" (or "neither true nor
false" or whatever) is a predicate in the language--- and even a
predicate that is evaluated to true, when such is the case *and*
admitting it doesn't lead to paradox.
(BTW, Tim Maudlin's version of the fixed point solution contains a
"neither true nor false" predicate, but it never evaluates to true.)
One question:
S8: S8 is either false or unsolvable.
S9: S8 is either false or unsolvable.
My thinking is that you'll want S8 to be unsolvable and therefore S9
true. If so, how do you avoid them being provably equivalent?
(Answering this requires laying out the entire axiom system and
showing it works, I admit.)
Finally:
The interesting thing is what happens if we try to "tarski" the
system, and create a metalanguage. Is it more powerful? The
referential holes in this system are the sentences that are unsolvable
but which cannot admit their own unsolveability. It seems to me that
*if* you answer S8 and S9 the way I suspect, *and* can provide a
working axiomatization, there is nothing to be gained by jumping to a
metalanguage: the language itself already can talk about the holes, so
long as it avoids using the same "wording" as the problematic
sentences use....
But perhaps I am hitting my limit of tarski-ing and someone else can
find a more powerful metalanguage.
--Abram
Abram Demski
http://dragonlogic-ai.blogspot.com/
http://groups.google.com/group/one-logic