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Re: [ontolog-forum] First-Order Semantics

To: "John F. Sowa" <sowa@xxxxxxxxxxx>
Cc: ontolog-forum@xxxxxxxxxxxxxxxx
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 15 Jun 2007 11:28:39 -0500
Message-id: <p06230901c2986d4c56fb@[]>
>Kathy and Pat,
>I agree with Pat's explanation, but I think it could be made
>somewhat clearer by distinguishing the base domain D from
>the domain D' of *all* relations over D for second-order logic
>(and then a domain D'' of *all* relations over D', etc.).
>PH> The key semantic difference between the other logics is
>  > that they all impose conditions on the domain, requiring
>  > it to contain some entities as a result of containing others.
>  > So for example, classical second-order logic semantics
>  > requires that the domain is contain all relations
>  > over the base domain.
>I would rephrase the last sentence in the following way:
>     So for example, classical second-order logic semantics
>     starts with the given base domain D and introduces
>     another domain D' of *all* relations over D.    (01)

That is closer to the usual formulation, and 
probably clearer, yes. And it is that extended 
domain D' which the second-order quantifiers 
range over.    (02)

>I just wanted to give different names D, D', D'', etc.
>to distinguish the base domain D from any domains that
>may be introduced by implicit assumptions.
>CL allows the domain D to contain relations, but it doesn't
>require D to contain *all possible* relations.    (03)

or indeed any relations, if you want to exclude 
them: CL has the notion of a 'segregated dialect' 
to allow for this textbook first-order case.    (04)

Pat    (05)

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