Re: [ontolog-forum] Current Semantic Web Layer Cake

> 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,

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.)

> until Kurt Gödel came along and demolished their shiny, perfect,  
> world.

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.

-chris

> 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@ontolog.cim3.net [mailto:ontolog-forum- 
> bounces@ontolog.cim3.net] 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@cytanet.com.cy> 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

Received on Wednesday, 1 August 2007 16:06:14 UTC