|From:||Ali Hashemi <ali.hashemi+ontolog@xxxxxxxxxxx>|
|Date:||Thu, 4 Mar 2010 15:06:02 -0500|
I agree wholly.
I believe I noted that (to use terminology that has appeared on the forum) the "intended meaning" might change in both cases. To trot out the other example, when you add colour to the notion of elephants, of course you can now prove all sorts of things about them. However, both elephants+colour and 0++, are a particular variety conservative extension.
There's another variety of Conservative extension where we definitely wouldn't think that we've changed the intended meaning of our primitives. Imagine I have a theory of numbers, Tn, and I decide I want to quantify certain properties about elephants, so I extend it with a theory of elephants Te. I think we all agree that Te conservatively extends Tn, yielding our new theory Tn*. Now I can talk about numbers of elephants, and ostensibly, I can prove new things about say the number 22 when talking about elephants - i.e. elephant gestation takes 22 months, or that elephants in the wild live up to 82 years of age or whatever.
In our first scenario, with successors and +, or elephants and colour, we introduce, for want of a better word, a new dimension relating to our previous concept, which we, as humans might consider to be important parts of their intended meanings.
I scenario two, numbers get elephants (or vice versa). Here, while we are introducing a new "dimension" for talking about numbers (or elephants), and we as humans would likely not consider the intended meaning of either to have changed.
From a logical vantage, both scenarios are conservative extensions.
Additionally, in a conservative extension, I can prove new properties about elephants, or 0, only in relation to the new vocabulary I introduced. In a non-conservative extension, my new axioms tell me more all in the context of my old vocabulary.
To be clear, what I am suggesting is that in Proposal 1, you are changing the notion of the primitive in an "orthogonal" manner. Moreover, only occasionally might we say that the intended meanings of our primitives have changed. In Proposal 2, you are always changing the intended meanings as well as changing the meaning of the primitives (in a logical way) quite directly.
So Proposal 1 - we occasionally change intended meaning of primitives, and we change the logical meaning, "orthogonally."
In Proposal 2 - we always change the intended meaning of primitives, and we change the logical meaning, directly.
Note - this view on change of meaning comes from examining the sets of permissible models given your axioms. In Proposal 2, we're restricting that set. In Proposal 1, we're linking the current set with a novel set of models (hence leading to what I previously alluded to as an n-dimensional model space). This is why i call conservative extensions an "orthogonal" change in meaning.
But you're right, meaning is a slippery notion.
I hope that not lost in all this quibbling and clarifying is that it is still important and very useful to know and keep track of what type of extensions (specializations) one is conducting, and if possible to do them as separate steps before combining the results.
To be honest, I don't really care what label we use. There are noticeable differences between the two both logically and intended meaning, and that's what's important to capture. It is also very important in ontology design to know this difference, as it helps clarify what one is doing.
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