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On Sep 24, 2:25 pm, herbzet <herb...@gmail.com> wrote:
> Newberry wrote:
>
> > Issues with van Fraassen's Paper
>
> > 1. Presuppositions
>
> > Bas C. van Fraassen used the Logic of Presuppositions to solve the
> > Liar paradox. van Fraassen (1968: Presupposition, Implication, and
> > Self-Reference,http://www.jstor.org/pss/2024557, The  Journal of
> > Philosophy, pp. 136-152)
>
> > Let
> > X = This sentence is false
> > Y = This sentence is not true
>
> > Van Frassen concludes that the Plain Liar can is solved when ~T(X) &
> > ~T(~X). He further claims that ~T(Y) necessitates Y, and as result the
> > Strengthened Liar is unsolvable except by dropping the requirement
> > that T() be bivalent. But why does ~T(Y) necessitate Y? We will re-
> > examine this claim.
>
> > Van Frassen's definitions are below. The negation of sentence A is
> > true if (respectively, false) iff A is false (respectively, true.)
>
> > 1. A presupposes B if and only if
> > (a) if A is true then B is true
> > (b) if ~A is true then B is true
>
> > 2. A necessitates B iff whenever A is true then B is true
> > From 1 and 2 we obtain:
> > 3. A presupposes B if and only if
> > (a) A necessitates B
> > (b) ~A necessitates B
>
> > Truth:
> > a) if P is true then T(P) is true, otherwise T(P) is false
> > b) If the second level sentence is A is false, then ~A is true, and if
> > A is true then ~A is false.
> > c) If either A is true or B is true, then (A v B) is true, otherwise
> > (A v B) is false.
>
> > 2. The Plain Liar Paradox
>
> > Van Fraassen observes that X and T(~X) necessitate each other.
> > a) if ~X is true then T(~X) is true [d3]
> > b) if X is true then T(~X) is true  [e1]
> > Therefore X presupposes T(~X)
> > But T(~X) cannot be true. Hence X does not have a truth value.
>
> > 3. The Strengthened Liar Paradox
>
> > Van Fraassen claims that ~T(Y) necessitates Y. We can indeed show
> > this, as well as that T(Y) necessitates ~Y, by a relatively simple
> > argument using the principle that if two different sentences mean the
> > same thing and one is true then the other one is also true:
>
> > Principle 1
> > Assumption         A means B
> > Premise            T(B)
> > Conclusion         T(A)
>
> > Let us assume that Y and ~T(Y) have the same meaning.
>
> > (1)  Assumption                        Y means ~T(Y)
> > (2)  Premise                           T~(T(Y))
> > (3)  Tarski's principle from 2        ~T(Y)
> > (4)  Principle 1 from 1,3              T(Y)
>
> > (1)  Assumption                        Y means ~T(Y)
> > (2)  Premise                           T(T(Y))
> > (3)  Tarski's principle from 2         T(Y)
> > (4)  Principle 1 from 1,3              T(~Y)
>
> > We have indeed proven that ~T(Y) necessitates Y, but we have also
> > proven a contradiction in the process. Neither ~T(Y) not T(Y) can be
> > the case. If Y and ~T(Y) mean the same thing then one cannot be
> > trivalent and the other bivalent.
>
> > By the definition of necessitation if Y is N then ~T(Y) must not be T.
> > Analogically if T(Y) necessitates ~Y and Y is N then T(Y) must not be
> > T.
>
> > Van Fraassen suggests that The Strengthened Liar paradox is averted
> > if we hold that T(Y) and T(~Y) are themselves neither true nor false.
> > From this it follows immediately that the sentence ~T(Y) & ~T(~Y) also
> > is neither true nor false. But this is implausible. If ~T(Y) & ~T(~Y)
> > is the case then by Tarski's principle T(~T(Y) & ~T(~Y)).
>
> > 4. The Plain Liar Again
>
> > The same argument can can be used against the Plain Liar.
>
> > (1)  Assumption                         X means T(~X)
> > (2)  Premise                            T(~T(~X))
> > (3)  Tarski's principle from 2         ~T(~X)
> > (4)  Principle 1 from 1,3               T(~X)
>
> > Neither T(~X) nor ~T(~X) can be the case.
>
> > Haim Gaifman pointed out that: The contradiction returns also if the
> > sentence says of itself that it is false: If it lacks a truth value,
> > then it is not false; but it says of itself that it is false, hence it
> > is false after all. Gaifman (Pointers to Propositions ,
> >http://www.columbia.edu/~hg17/gaifman6.pdf, Columbia University, p.
> > 16) There is no essential difference between the Plain Liar and the
> > Stregngthened Liar.
>
> > 4. Conclusion
>
> > What is wrong with the argument is of course the assumption that Y
> > means ~T(Y). We have in front of us is a proof by contradiction that Y
> > is not the same as ~T(Y). And if ~T(Y) is the only thing Y could
> > possibly mean then Y is meaningless, and ~T(Y) says that Y is not
> > true.
>
> >http://members.lycos.co.uk/neologic/Issue_vanFraasen.pdf
>
> I've read VF's paper, I've read HG's paper, I've read your post.

HG's paper is also quite interesting, isn't it?

> 1) I tend to agree with you.
> 2) This stuff about the Liar really makes my head hurt.
> 3) I fail to really see what interests you so much about it.

Do you really want to know? Because van Frassen's result that X
presupposes T(~X) still holds. And once we have resolved the paradox
we obtain analogically Y presupposes T(Y). Then we have an analogy

Let G = ~(Ex)(Ey)(Pxy & Qy), there is only one y=m such that Qy,
and m = #("~(Ex)(Ey)(Pxy & Qy)")

Y is like G
T(Y) is like (Ex)(Px#G)
But since ~T(Y) and ~(Ex)(Px#G) the presupposition fails. Hence Y is
neither T nor F, and G is neither T nor F. The beauty of it is that
the provability of ~(Ex)(Ey)(Pxy & Qy) no longer results in a
contradiction.

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