> 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, (01)
No, actually, the vicious circularity of which Russell (mostly) and
Whitehead spoke didn't have anything whatever to do with whatever it
was that Duane was talking about. The so-called Vicious Circle
Principle, which was implemented in the ramified type theory of
Principia Mathematica, put constraints on the acceptable range of the
quantifiers in a class definition, and its purpose was the avoidance
of semantic paradoxes like the Liar Paradox and Richard's Paradox.
(Basically, the VCP placed the blame for the paradoxes on the use of
nonconstructive class definitions in which quantifiers ranged over
the very classes of which the classes being defined were themselves
members. Such definitions are in fact ubiquitous (and generally
quite harmless) in classical mathematics.) (02)
> until Kurt Gödel came along and demolished their shiny, perfect,
> world. (03)
I will have to disagree. In fact, Gödel's results didn't really have
any particular bearing on the central philosophical motivations of
Principia Mathematica, viz., the avoidance of semantic paradox, which
PM managed to do quite well. It was in fact Ramsay who had already
shown several years before Gödel's 1931 paper that PM was an
inadequate foundation for classical mathematics unless it adopted a
very unintuitive principle known as the Axiom of Reducibility. And
even then, to my knowledge, Russell and Whitehead never claimed that
PM would provide a logically *complete* foundation for mathematics.
Gödel's theorems had a far greater impact on Hilbert's program, which
(somewhat anachronistically put) explicitly sought a complete,
consistent, computationally decidable foundation for all of
mathematics. Gödel's work did effectively show that goal to be
unattainable in principle even for elementary arithmetic, let alone
all of mathematics, and its implications are directly relevant to the
goals of modern computational ontology. (04)
-chris (05)
> 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.
>
>
> 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 (06)
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