Re: RDF Abstract Syntax: a strawman

On 5/30/01 10:26, "pat hayes" <phayes@ai.uwf.edu> wrote:

> OK. In brief: FOL model theory says that the universe of
> quantification is a set. It does not say that the set cannot contain
> relations. So quantifying over relations is not ruled out by FOL.
> What makes a language higher-order is when its relational quantifiers
> are required to range over a rather large set of relations (exactly
> how large depends on the logic, eg classical HOL= *all* relations, ie
> the set 2|(D|n) where D is the base domain of individuals; Henkin
> logic = all lambda-definable relations.) If one does not impose any
> requirement on the size of the relational universe (other than it
> provide a denotation for every relational term) then there is nothing
> higher-order in the semantics and it is easy to allow quantification
> over relations and still be first-order. Those quantifiers have only
> a first-order kind of 'bite', of course, and the language has no
> rules of lambda-conversion.

For those of you interested in a reference, what Pat mentions is described
in Enderton's "Mathematical Introduction to Logic", Ch. 4.  See "General
Models of Second Order Logic"...

 .bill

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Bill Andersen
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