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Re: [ontolog-forum] formal systems, common logic and lbase

To: rick@xxxxxxxxxxxxxx
Cc: ontolog-forum@xxxxxxxxxxxxxxxx
From: Pat Hayes <phayes@xxxxxxx>
Date: Mon, 19 Nov 2007 12:29:05 -0600
Message-id: <p06230920c36783270693@[]>
>Folks might enjoy the Soloman Feferman lecture 
>Nagel, Minds and Machines. After recounting an 
>exchange between Godel and Nagel circa 1956, 
> Feferman describes the implications of the 
>minds vs. machines dichotomy ensuing from the 
>exchange.    (01)

There has been a LOT of material written on this 
general topic since 1956. At that date, AI 
systems in the modern sense did not yet exist, so 
the discussion was not particularly well-informed.    (02)

>  To avoid the impass resulting from the 
>dichotomy, Feferman proposes the redefinition of 
>a formal system to an ³open ended schematic 
>axiomatic system.² He claims this redefinition 
>is a constructive step towards an ³informative, 
>systematic account at a theoretical level of how 
>the mathematical mind works that squares with 
>experience.²    (03)

Yes. His idea was subsequently formalized and 
then quite rapidly shown to suffer from the same 
limitations as the simpler Goedel notion. So 
considered as an end-run past Goedel, it did not 
succeed. It is now generally accepted that all 
such attempts to sneak past the Goedel result are 
doomed to failure, and this topic is considered 
to be closed. However, none of this has any 
bearing on AI, contrary to what Penrose and 
others (including Goedel) believe. For a fairly 
thorough refutation of the Goedel-based attack on 
AI, see:   LaForte, Hayes & Ford (1998) "Why 
Godel's theorem cannot refute computationalism" 
Artificial Intelligence 104 (1998) 265-286    (04)

>  He stresses the importance of a subject neutral 
>theory of operations with basic schemata for 
>language, arithmetic, set theory that would 
>amount to a version of an untyped lambda 
>calculus. Feferman concludes by rejecting any 
>effective machine representation of mind as 
>contemplated by Nagel, Penrose & others.  
>So, what does this mean to Common Logic and LBase ?     (05)

Absolutely nothing.    (06)

>Seems to me that efforts 
>like <http://cl.tamu.edu/>Common Logic and 
><http://www.w3.org/TR/lbase/>LBase would either 
>have to a) be defined within this type of an 
>open ended system, let¹s say as the natural 
>language description of the constraints to which 
>the axioms that make up the theory of such a 
>system would  adhere; or b) evolve into an open 
>ended system that exhibits characteristics 
>of transformation across languages, logics, 
>models and theories.    (07)

No, neither. Common logic is simply a modernized 
Net-savvy restatement of first-order logic, in an 
attempt to get past the interoperability problems 
arising from the huge variety of surface 
notations in use. LBase is (was, better, as it 
seems to have been widely ignored) an earlier 
attempt to do this for the W3C family of semantic 
web languages. Goedel's incompleteness result, 
which gave such a shock to foundations of 
mathematics, has no relevance to the completeness 
of first-order logic (which was also first proved 
by Goedel, by the way.)    (08)

Pat    (09)

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