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

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 27 Nov 2007 20:30:53 -0600
Message-id: <60B199DF-C9F1-4AFF-897F-037ADF1FFBAC@xxxxxxxx>
On Nov 27, 2007, at 11:03 AM, Pat Hayes wrote:
> [Rick Murphy wrote:]
>> ...I remain unconvinced that there's no connection between what  
>> Feferman's writing about and CL.
> I don't know how to convince you without you first taking a graduate  
> course in foundations of mathematics, but if you have the above  
> reference right, perhaps you could just take it on authority? They  
> really have nothing at all to do with one another.    (01)

Right.    (02)

> (The IKL project, now, may indeed have some possible relationship to  
> Feferman's stuff, since IKL can define its own truth-predicate.)    (03)

Yes, indeed.  If axioms are added to IKL that bestow a "fine-grained",  
sentence-like structure on IKL's propositions, one could probably  
reframe a lot of what Feferman is doing in terms of IKL.  Feferman  
works in an extension of Peano Arithmetic, after all, just so that he  
can arithmetize the syntax of his theory.  By quantifying over Gödel  
numbers of sentences (and the other syntactic objects) of his theory,  
the theory, in effect, is its own metalanguage, but without all the  
messiness and complications required in a more standard sort of  
metalanguage.    (04)

Rather than quantifying over Gödel numbers of sentences, IKL  
quantifies over propositions *expressed* by sentences.  And since  
there is, in effect, a truth predicate in IKL, many of the same issues  
that Feferman is addressing can arise in IKL.    (05)

BTW, Rick singles out McGee and Feferman, but they are but two stars  
in a vast firmament here.  Their (very fine) work is just a small  
portion of the huge body of research on "post-Tarskian" semantic  
theories for languages that contain their own truth predicate,  
stemming (most notably) from Kripke's enormously influential 1975  
paper "Outline of a Theory of Truth."    (06)

>>>> Interesting, thanks for the info, anything you could refer me to so
>>>> I can read up on this ?
>>> On LBase as a precursor to Common Logic?  On the incompleteness
>>> (better, perhaps, incompleteability) of arithmetic?  On the
>>> completeness of first-order logic?
>> Completeness of FOL.
> Read any logic textbook that has a chapter on semantics or model  
> theory.    (07)

Many good choices here, but two I often recommend are Mendelson's  
_Introduction to Mathematical Logic_ and Enderton's _A Mathematical  
Introduction to Logic_.  A somewhat less expensive and more  
approachable text is Christopher Leary's _A Friendly Introduction to  
Mathematical Logic_.  The latter is out of print, but you can find  
used copies on Amazon.    (08)

-chris    (09)

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