|Date:||Thu, 22 Nov 2007 06:10:45 +0000|
Chris & All:
I've been on travel for the past few days. It will take a few days to catch up, but thanks for the opportunity to explain in more detail.
I can see from the reply emails that the ascii art didn't work out as planned. The ascii art was supposed to look like this ...
The arcs in this graph were intended to communicate a degenerate representation of a logical environment. It just says that a logical environment is made up of four key abstrations. Languages, Logics, Models (or more accurately, a model theory) and Theories. And I've specialized Theories to include theories that are expressed as axioms and theories that are expressed in natural language.
In a more fully developed logical environement, the arcs are relevant to transformations across the abstrations. For an example, see the logical environment of the Information Flow Framework here:
Ok, well here's what I was thinking. First restating my claim: CL & LBase are natural language descriptions of the constraints to which the axioms of a Theory would have to adhere. I didn't say CL & LBase were theories. Next, just to baseline a common understanding. I'm working from a standard defintion of theory: A theory in the Language L is defined as a set of sentences in the language L. The set of true sentences in a particular structure M under an intepretation are the closed theory. It can be consistent, satisfiable, yada, yada ... I'll also add that Theories require denotational truth whereas models or model theories only require approximations of the truth. So given this explanation, I think I can still claim that CL and LBase are natural language specifiations stating the constraints to which a Theory as defined above would have to adhere.
The main point I intended to communicate, proposition b), is that CL & LBase should be evolving into open ended axiomatic systems, or logial environments. I'll make the wild-ass claim that Feferman's going in the direction that Kent, Goguen & others have gone, because approaches like logical environments, or open ended axiomatic systems will cause us to revisit some things that haven't been adequately developed. For example, LBase seems much more loosely specified than the IFF logial environment referenced above especially with regard to translations. Especially where LBase claims its strategy is a way to translate various languages to a common base. Your question regarding the arcs is the diagram is right on target and clearly the arcs, or translations, specified in the IFF logical environment are very well thought out. What's important here is that one of the assumptions (ie: openess) of the logical environment caused the translations to be more formally specified.
While I have a chance to complain, the language used in LBase is pretty loose. First, it claims to provide the semantics for SWELs, then its a framework, then its a language, then a logic ... That's probably a characteristic of the fluid nature of W3C's specification process, but it comes across pretty loose.
Again, the main point is Feferman's interest in open ended axiomatic systems is evidence of his support for an information flow, IFF, category theory-like approach to formal systems. I think CL & LBase should be going in this direction too.
Anyway, I appreciate you listening to my rant. Have a Happy Thanksgiving and I'll get back and answer some other emails once I get my dose a triptophan and football.
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