[FOM] On >>this sentence cannot be proven true<<

[Hartley Slater]

Laureano Luna, and Haim Gaifman are troubled about pronouns:

Laureano Luna wrote:
>The second 'its' in the Quinean sentence determines
>its reference from the doubly quoted part, which is
>indeed inside the sentence, so that all the sentence
>needs to fix its meaning is the linguistic code and
>the sentence itself; consequently, the sentence seems
>reasonably context-free.

I did not say the sentence was not context-free.  Only that the 
predicate is such, and for the reason Laureano gives.  So when Haim 
Gaifman writes:

>Thus, 'appended to its quote is grammatical' can be used to describe
>a property;

he makes the crucial error, since his predicate only 'can be used to 
describe' a *variable* or *context dependent* property.  In 
Grelling's case the error arises when people replace 'is not self 
applicable' with 'is heterological' since the latter masks the 
variable 'self' in the former.  We have:
a is not self applicable iff a is not a applicable,
b is not self applicable iff b is not b applicable,
so given the subject is a, there is one property - being not a 
applicable - and given the subject is b, there is another property 
(if a is not b) - being not b applicable.  There is no paradox 
substituting anything for 'a', or 'b', so long as it is also 
susbstituted for 'self': Substituting 'is not self-applicable' we get:
'is not self-applicable' is not self applicable iff 'is not 
self-applicable' is not 'is not self-applicable' applicable,
which is to say merely
'is not self-applicable' is not [self, i.e.] 'is not self applicable' 
applicable iff 'is not self-applicable' is not 'is not 
self-applicable' applicable.
If we started from
a is heterological iff a is not a applicable
then substitution of 'is heterological' for 'a' certainly would lead 
to a contradiction.  But the reason for that is now clear: there 
would then be no fourth place to substitute the 'is heterological' 
into.  (See also the papers previously referenced for further 
discussion.)

The above two writers also have some things to say about the 
(ir)relevance of possible worlds.  But they also have difficulty 
solving the paradoxes, so they should not be so dismissive so 
quickly.  They just have not yet seen the relevance.  it is not an 
accident, I think, that the paradoxes arose in logicians minds before 
possible worlds, and context dependence, were studied fully, and if 
one keeps to that early twentieth century mind-set  one will still 
find the paradoxes puzzling (particularly followers of Quine on 
Modality, and eternal sentences, of course).  Contrariwise, moving 
out of it (and particularly moving away from the influence of 
Tarski), the answers become extraordinarily simple (contrast with the 
strenuous efforts of a whole range of people recently, like McGinn, 
Soames, Simmons, and Priest).  Possible worlds in fact are highly 
relevant to the solution of The Liar, and I made further crucial 
points about that on the Phil-Logic list the other day, in connection 
with another remark of Laureano Luna's (see 
http://philo.at/pipermail/phil-logic/2006-August/008899.html):

>as I see it, it is not necessary a self-referential sentence s
>to specify in which world it is to be such; if there is a
>possible world w in which, according to some linguistic code c, s
>says that s is not true, then there is a possible paradox and this
>is really all we need; if w is the actual world, then we have an
>actual paradox. You probably denies that such code is possible.

There is no paradox if s says that s is not true: for if s is
ambiguous then the T-schema does not apply to it, and one cannot get
that Ts iff -Ts.  Moreover, as I said before, your linguistic code,
that now extends to give the various referents, in different possible
worlds, of things like 'The winner of the 2006 Tour de France', 'the
sentence at the top of page n in book B', shows that such items are
trans-world, if not 'contextually' indexical, and so 'ambiguous',
along with standard referential indexicals like demonstratives, and
pronouns.  So if one constructs the original type of syntactic
identity
s = 's is not true',
then, since just 's' is involved on the RHS, without reference to a
world, no specific referent can be involved.  But the referent of 's'
in this world is involved on the LHS, because we are speaking in this
world (just as we would be if we said that s is not true - that is
just the singular difference with indirect speech).  But, again, if
you try to improve on this by constructing something like
t = 't when uttered in the actual world/in this world is not true'
then you either have brought in an explicit indexical, or an
equivalent (remembering D.K.Lewis).

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