Model theory, abstract syntax

At 01:56 PM 7/16/01 -0500, Aaron Swartz wrote:
>I'm unclear on the difference between model theory and abstract syntax. 
>Can someone clarify?

I'll take a shot;  I guess the real formal systems folks will put me right...

I think they are clearly different, but related, issues.

- Abstract syntax defines a language (i.e. a set of well formed formulae, 
or wff) in terms of some set of terminal symbols.  Given a formula, it 
allows us to say whether or not it is a well formed sentence (instance) of 
the language.  It also provides us with an annotation for the the structure 
of a wff that can be used as a reference point for defining semantics for 
the various allowed forms.  In summary:  abstract syntax is primarily about 
forms.

- Model theory defines semantics for the various allowed forms, by telling 
us how they can be interpreted in terms of some universe of discourse.  The 
elements of the language refer to members of the universe, and statements 
can be interpreted to be true or false of of such a universe.  An 
"interpretation" of a formula is an assignment of values from the domain of 
discourse to symbols in the formula, such that the formula can be said to 
true or false.  A "model" of a formula is an interpretation for which the 
formula evaluates to true.  Hence "model theory".

- A third related concept is "proof theory":  a deductive apparatus based 
on syntactic transformations of wffs that preserves truth.

#g


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Graham Klyne                    Baltimore Technologies
Strategic Research              Content Security Group
<Graham.Klyne@Baltimore.com>    <http://www.mimesweeper.com>
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Received on Monday, 16 July 2001 16:11:15 UTC