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Article 1213 of comp.ai.philosophy:
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>From: zeleny@brauer.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Re: Is there any such thing as informal logic?
Message-ID: <1991Nov6.122415.5160@husc3.harvard.edu>
Date: 6 Nov 91 17:24:13 GMT
References: <1991Oct22.041210.5931@watserv1.waterloo.edu> <JMC.91Nov5212441@SAIL.Stanford.EDU>
Organization: Dept. of Math, Harvard Univ.
Lines: 86
Nntp-Posting-Host: brauer.harvard.edu

In article <JMC.91Nov5212441@SAIL.Stanford.EDU> 
jmc@SAIL.Stanford.EDU (John McCarthy) writes:

>In article <1991Nov4.182613.5073@husc3.harvard.edu> 
>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

>   In article <JMC.91Nov3225619@SAIL.Stanford.EDU> 
>   jmc@SAIL.Stanford.EDU (John McCarthy) writes:

JMC:
>   >Since the remaining points of the argument are either terminological
>   >or concern the prospects of the logicist approach to AI, there is
>   >no point in continuing unless Zeleny or Yodaiken would like to offer
>   >some arguments against the prospects of AI in general or the logicist
>   >approach in particular.

MZ:
>   You mean you haven't been convinced by the Nasruddin story?  I guess this
>   means that I must try harder... please let me know if I get too abstruse.
>
>   Consider Davidson's argument in ``Theories of Meaning and Learnable
>   Languages'', in the context of his program of extensional semantics.  Now,
>   various devices like Frege's paradox of the name relation, or Putnam's
>   model-theoretic argument, demonstrate the inadequacy of the extensional
>   approach to the task of characterizing the linguistic relation of denoting;
>   hence I conclude that an adequate semantical theory must be intensional.
>   Furthermore, on the assumption of intensional semantical entities, an
>   infinite hierarchy thereof has to be admitted (all relevant details can be
>   found in Church's papers on the Logic of Sense and Denotation); moreover,
>   for each level of intensions, our cognitive grasp of the lower-level
>   semantical entities can be seen as dependent on that of higher, more finely
>   differentiated intensional level.  Thus it can be seen that the semantics
>   of natural languages fails to satisfy Davidson's finite learnability
>   criteria, the ones required by any equivalent of a finite-state automaton
>   with finite memory, Q.E.D.

JMC:
>The amusing thing about this one is the Q.E.D. at the end.  Does this
>purport to be a proof?

No, this is a mere philosophical argument; unlike a formal proof, it's
replete with rhetorical devices, and open to controversy at each step.

JMC:
>Intensional entities are indeed needed for intelligence.  Moreover,
>the levels can indeed get higher and higher.  In fact they can go
>up into transfinite ordinals.  However, one needs higher and
>higher levels of intension in the same sense as one needs larger
>and larger natural numbers.  Just as the natural numbers actually
>mentioned in any single mathematical argument are bounded, so are
>the levels needed in any given intensional argument.  I rather doubt
>that you can find any that go beyond 3 levels.

Funny, this is just the sort of comment I used to make apropos Davidson's
anti-Church argument about 3 years ago.  Unfortunately, the analogy between
the integers and the intensional hierarchy breaks down at the point where
one is forced to observe that, unlike the number x, which (setting aside
certain difficulties connected with the second-order postulate of
induction, on which more anon) doesn't in any way depend on its successor
S(x) for its definition, on a Fregean view, the success of our reference to
any entity, whether intensional or extensional, depends on our grasp of its
concept, which in turn depends on our grasp on the concept of its concept,
and so on.  Getting back to the arithmetical analogy, observe that our
understanding of any mathematical argument depends not only on the natural
numbers actually mentioned in it, but on every single entity, intensional
or extensional, on which our grasp of these numbers actually depends. Once
we realize that the second-order PA, the weakest theory capable of
describing the set of all integers up to isomorphism, in the full-strength
induction postulate makes an explicit reference to all properties of the
integers, and ipso facto an implicit one to all subsets thereof, it becomes
evident that our reference to each integer indeed depends on an implicit
reference to the *completed* totality of all integers (recall that the
Axiom of Infinity is required for the logicist proof of the postulate that
each integer has at most one predecessor).  In other words, once your
analogy is made to work, it starts working against your position.


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| ``If there are no Platonic ideals, then what did we fight for?''   |
|                                (A Spanish anarchist, after 1938)   |
| Mikhail Zeleny                                           Harvard   |
| 872 Massachusetts Ave., Apt. 707                         doesn't   |
| Cambridge, Massachusetts 02139                            think    |
| (617) 661-8151                                              so     |
| email zeleny@math.harvard.edu or zeleny@zariski.harvard.edu        |
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