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
>with
>
> 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)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J (010)
|