Re: logical paradoxes

From: Jeffrey Ketland (ketland_at_ketland.fsnet.co.uk)
Date: 08/18/04 

Date: Wed, 18 Aug 2004 02:30:03 +0100

JXStern wrote in message ...
>On Tue, 17 Aug 2004 23:50:12 +0100, "Jeffrey Ketland"
><ketland@ketland.fsnet.co.uk> wrote:
>>I used to be sympathetic to the idea that the liar sentence fails to
express
>>a proposition. But now I'm even more sympathetic to the idea that there is
>>no univocal notion of truth---there is only truth relative to a fixed
>>interpretation. Of course, natural language appears to contains a univocal
>>notion of truth ....
>
>... but, do you see natural language commiting itself to fixed
>interpretation?

Even setting aside vagueness, ambiguity and indexicality, no. Not in the
sense of a fixed interpreted language, modelled as a pair (L, M), where M is
the structure over which L-symbols are interpreted. The problem is Tarski's
indefinability result that genuine semantic self-representation appears to
be mathematically impossible. Such languages cannot express their own
semantical concepts (which natural languages at least appear to do), and the
problem doesn't go away even if you consider many-valued logic. This appears
to force a metalanguage/object-language distinction upon us, with the
metalanguage being "essentially richer" than the object language, as Tarski
put it. This is a general case of the Revenge Problem. And this leads to a
hierarchy, which is counter-intuitive. As Kripke put it near the end of his
1975 paper, "the ghost of the Tarski hierarchy is still with us".

There is a descendant of Tarski's hierarchy view, due to Parsons and Burge,
which proposes that the natural language usage of the word "true" is
systematically ambiguous in a subtle way, referring to an implicit
hierarchy, "true_0", "true_1", etc. The problem with any such view of this
kind is not that we cannot make sense of such hierarchies. We can and there
is interesting work on such hierarchies. The main problem---which has been
stressed several times by Hilary Putnam---is that we appear to be able to
make sense of the whole hierarchy and talk coherently about it within
English. But we cannot locate English inside the hierarchy, because English
appears powerful enough to talk about the whole hierarchy.

Unfortunately, I have no idea how to respond to this argument. Maybe humans
aren't smart enough to solve the problem of semantic self-representation.
Maybe the apparent semantic self-representation of natural languages is an
illusion of some sort. Maybe there will arise some new approach, which no
one has thought of yet. Dialetheists like Priest and others believe they
have the answer (namely, that natural languages can be modelled as
inconsistent languages), but I am sceptical of that approach.

--- Jeff

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