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 -- Bill Andersen Chief Scientist, Ontology Works 1132 Annapolis Road, Suite 104 Odenton, Maryland, 21113 Mobile 443-858-6444 Office 410-674-7600Received on Wednesday, 30 May 2001 17:28:05 UTC
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