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Re: [ontolog-forum] Constructs, primitives, terms

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Thu, 8 Mar 2012 19:32:39 +0100
Message-id: <52403578-1B67-4C86-A5E9-E0DBAEC93DBF@xxxxxxxx>
Am Mar 8, 2012 um 6:23 PM schrieb William Frank:
> I think is the essence of this point is:
> "The number of possible conflicts is infinite, and no fixed set
> of universal definitions can anticipate and rule out all of them."
> This is true of *logical necessity*; so it is something that we always live 
> The reason for this truth is that 
>   there is no guarantee that all the models for
>   any theory G are consistent with one another.    (01)

I not sure what you and John mean by that -- consistency is a property of 
theories or, more generally, sets of sentences. It doesn't make any clear sense 
that I can see to say that two models are inconsistent. I suppose we can make 
something up, e.g., models M1 and M2 of theory G are inconsistent with each 
other if there is some sentence A in the language of G (obviously not a theorem 
of G) to which M1 and M2 assign different truth values.    (02)

> More strongly, unless the theory G is complete, which no theory as expressed 
>in an application will be    (03)

It's not obvious to me that that is so. There are well-known examples of 
complete first-order theories (e.g., the first-order real number theory and 
Euclidean geometry both have complete axiomatizations). Such theories obviously 
cannot contain a lot of arithmetic, but it's not obvious that an interesting 
practical ontology has to contain the arithmetic needed to guarantee 
incompleteness and hence can't be complete.    (04)

> (only theories as bounded and richly expressed as second order arithmetic 
>tend to be complete),    (05)

Eh? There is no complete axiomatization of second-order arithmetic.    (06)

> there will always be models, describe in theories G' that are extensions of 
>G, as expressed in different applications, that ARE inconsistent with each 
>other.  As in John's example, "an employee must be at least 21 years of age," 
>may be a rule expressed in one applicaton, while in another employees may be 
>16, or have no specified age limits.    (07)

OK, so if you've got a theory G that doesn't entail anything specific about age 
limits, then you could extend G to theories G1 and G2 that specify different 
age limits. Hence, a model M1 of G1 and a model M2 of G2 will both be models of 
G but will be "inconsistent" in the sense above. Fine, but it seems to me this 
is all more easily expressed proof theoretically -- it's the simple logical 
fact that every consistent incomplete theory has consistent extensions that are 
mutually inconsistent.    (08)

-chris    (09)

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