Gotta start reading my mail queue LIFO style. :-) Pat, as usual, is
right on the money here, though his attribution of ramified type
theory to Ramsey is not quite correct. RTT is the baroque type
theory of Russell and Whitehead's Principia Mathematica.
Ramification was introduced to deal with the semantic paradoxes.
Russell's initial "simple" type theory dealt only with the set
theoretic paradoxes like the one he himself discovered as well as
others having to do with cardinal and ordinal numbers like the so-
called Cantor paradox (not actually discovered by Cantor) and the
Burali-Forti paradox (sometimes also falsely attributed to Cantor).
Ramsey showed that one is unable to prove the least upper bound
principle for real analysis in RTT without the axiom of reducibility,
whose effect (its unintuitive character aside) is to make RTT
logically equivalent to simple type theory. (01)
-chris (02)
ps: For a truly beautiful account of ramified type theory, see Alonzo
Church, "A Comparison of Russell's resolution of the semantical
antinomies with that of Tarski," _Journal of Symbolic Logic_ 41
(1976) pp. 747-760, reprinted in R. M. Martin, _Recent Essays on
Truth and the Liar Paradox_, Oxford Univ. Press, Oxford, 1984, pp.
289-306. (03)
On Jul 31, 2007, at 4:54 PM, Pat Hayes wrote: (04)
>> Duane:
>> You are right. This goes to the heart of the
>> issue of "vicious circularity" that Whitehead
>> and Russell had thought was sorted with
>> Principia Mathematica, until Kurt Gödel came
>> along and demolished their shiny, perfect, world.
>
> Um, please don't think I am being too picky here,
> but this assertion contains so many
> misunderstandings that I may not have time to
> list them all. Goedel's result has nothing to do
> with Russell and Whitehead's set theory, and
> neither of them have anything much to do with
> ontology or ontologies. The shiny, perfect world
> that Goedel refuted was Hilbert's dream of a
> self-verifying formalization of mathematics which
> could prove itself consistent. Russell and
> Whitehead's Principia was a response to Russell's
> demolishing of Frege's naive set theory. Set
> theory is also a mathematician's dream, but a
> different one; and one that has not been
> demolished but in fact is, usually in the form of
> Zermelo-Fraenkel (Z-F) set theory rather than
> Russell & Whitehead's consistent but unworkable
> type theory (or even Ramsey's ramified type
> theory from six years later) still the widely
> accepted mathematical foundational language that
> is as near to consistent as anything can be. But
> in any case, ALL of this has to do with
> mathematics; and as Duane says, ontologies are
> usually taken to be about part of the real world,
> not the Platonic world of mathematics; and the
> question of how to ground such theories in the
> actual world (either of physical entities or of
> experiences) is not even remotely relevant to, or
> influenced by, such matters as axiomatic set
> theory or Goedel's second theorem.
>
> By the way, Im not sure what you are referring to
> by "vicious circularity", but what links
> Russell's paradox and Goedel's undecideability
> result (and Turing's uncomputability theorem) is
> that they all derive fairly directly from the
> classical Liar paradox "This sentence is false",
> which itself is made possible only by
> self-reference. If this is what you mean, it also
> has nothing much to do with ontologies and their
> grounding.
>
>> An ontology is not just some self-referencing
>> and self-sustaining model that is somehow
>> "complete"; it points out to the real world, as
>> you rightly say.
>>
>> Before there is a flame war on this, I should
>> underline that we discussed this extensively at
>> the Ontology Summit, and there was an
>> (uncomfortable for some) consensus that there is
>> "Ontology" as *the* study of being; and there
>> are "ontologies" that are domain-specific
>> encapsulations of some aspect of the real world.
>
> This distinction between the original,
> philosophical, meaning of 'ontology' and its more
> recent, computer-science/IT/AI/KR/engineering
> meaning has been noted every since the second
> usage was coined, and should be familiar to
> everyone who has any opinion to express in either
> field. There is a Wikipedia disambiguation page
> devoted to it. If anyone feels "uncomfortable"
> about it, they need to get more comfortable as
> soon as possible, and preferably before sending
> any more emails on the topic.
>
> Pat
>
>>
>> Peter
>>
>> -----Original Message-----
>> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
>> [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx]
>> On Behalf Of Duane Nickull
>> Sent: 31 July 2007 16:05
>> To: [ontolog-forum]; John F. Sowa
>> Cc: 'SW-forum'
>> Subject: Re: [ontolog-forum] Current Semantic Web Layer Cake
>>
>>
>>
>>
>> On 7/31/07 12:46 PM, "Azamat" <abdoul@xxxxxxxxxxxxxx> wrote:
>>
>>> The real semantics or meanings of any symbolism or notation is
>>> defined by
>>> ontology; for this is the only knowledge domain studying the
>>> Being of
>>> Everything which is, happens and relates.
>>
>> Not trying to start a nit picky argument, but I had always thought
>> that real
>> semantics are defined by how a term is used and what it is linked
>> to in a
>> physical world (which of course can be captured and expressed in an
>> ontology). Otherwise any ontology is just a huge circular
>> reference (like
>> the english dictionary when void of any grounding.
>>
>> How can one define and convey the true meaning of spicy food,
>> heat, pain etc
>> without the corresponding grounding experience?
>>
>> Duane
>>
>> --
>> *********************************************************************
>> *
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>> *
>>
>>
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