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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