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Re: [ontac-forum] Theories, Models, Reasoning, Language, and Truth

To: ONTAC-WG General Discussion <ontac-forum@xxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Sat, 17 Dec 2005 01:07:19 -0600
Message-id: <20051217070719.GV57447@xxxxxxxx>
On Sat, Dec 17, 2005 at 12:52:16AM -0500, Lichtblau, Dale wrote:
> Chris:
> Thanks for your very cogent response (vis-a-vis models vs. theories)
> (on behalf of John Sowa).
> My only remaining question is your claim that: "A *model* of a
> theory...in which "0" is interpreted to mean the number 0,
> [usw--apologies to Paul for the German--yes, the Janik/Toulmin book is
> wonderful, I have a 1973 1sr edition]". So, the number 0 is in the
> real world, there's a theory of the real world that denotes the
> entities of that world via a certain "sign" (i.e., "0"), and there's a
> "model" that is the "mapping" between the "theory" and the "real
> world"? Is this right? So a model is just the "mapping"?    (01)

Close.  The model -- better the *interpretation* -- is the mapping
*plus* a specification of the *set* that is used for interpreting the
basic lexical elements and which serves as the domain over which the
quantifiers range.  (A model of a theory is just an interpretation in
which the sentences of the theory are all true.)  The domain may (as in
our example) or may not comprise some "natural" set of "real world"
objects.    (02)

> I (vaguely) remember doing Henkin-style proofs (of completeness) in
> modal logic back in the early '70s. The black-board sketches of
> possible worlds and accessibility relationsips between them were the
> "model(s)" that served to prove the completeness of the underlying
> modal logic (T, S4, S5, etc.)?     (03)

The notions of interpretation and model in modal logic are natural
generalizations of the notion that is used for their nonmodal
counterparts.  Note the notion of interpretation I provided in my
previous post was for *predicate* logic.  T, S4, and S5 are systems of
modal *propositional* logic.  The notion of an interpretation in
nonmodal propositional logic is much simpler, as there are only two
semantic objects, Truth and Falsity.    (04)

-chris    (05)

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