[FOM] Godel's Theorems
4mjmu at rogers.com
Sat May 3 09:03:50 EDT 2003
> >The paper's
> >authors (and LW) allow Godel the formal result, but do not accept the
> >proposed English translation of the formal result.
> Again,would they give up the English translation of the second
> incompleteness theorem for PA?
I am hardly the gold standard of interpretation, either for LW or the
authors. However, since the second theorem depends on the first and the
first depends on the construction of sentence for which they would argue we
have no legitimate English translation, then I don't think they would have
to accept the second theorem as stated in English.
> >Put this way, the "notorious remark" looks like a variant of LW's
> >that technical philosophical languages are "de-linked " from Ordinary
> >Language, to the detriment of technical philosophy. This seems to be the
> >point behind Hempel's statement, quoted in the paper, that all formal
> >languages are "explained in" ordinary language, and that formal language
> >should/ought to bear a "hereditary" relationship to Natural Language. In
> >the absence of such a relationship the move from the formal result to
> >Natural Language counterpart is a leap across the abyss, as it were, and
> need not be accepted.
> But why isn't the link between Ordinary Language and formal languages
> and systems such as the language of PA and the system PA, already
> solid enough to justify such "leaps"?
The problem is with the "definitions and transformations" that get you from
the natural language to the formal system and back. With paraphrase, in
other words (I think this is the point Putnam and Floyd are making in the
last part of the paper). For example, I have several introductory logic
texts that claim that paraphrase is "more an art then a science". What does
this imply? What is the relationship between a natural language sentence
and the various paraphrases it goes through before it "becomes" something in
the formal system. I would think and argue that it is not entailment. And
if it is not--if there is no rule constrained means of going from one to the
other-- then is the relationship magical? A typical semantic theory might
take a natural language sentence, give it a surface syntactic structure, and
then a deep syntactic structure, before assigning a final interpreted
logical form. Does each transformation require a certain amount of voodoo?
If so, why get led down this garden path?
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