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Re: [ontolog-forum] First-Order Semantics

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 15 Jun 2007 11:39:01 -0500
Message-id: <74334EB5-9C99-494B-A6C7-2BAA9FD16E7F@xxxxxxxx>
On Jun 14, 2007, at 9:57 PM, John F. Sowa wrote:
> 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.
> 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.    (01)

Indeed the mathematical facts require that it *not* contain them.   
There are (as of course John and Pat know) 2^card(D) relations over  
any set (taking relations here to be sets of n-tuples).  If (per  
impossibile) D were to contain all possible relations over D then  
(even ignoring issues of non-well-foundedness) we'd have card(D) =  
2^card(D), which of course contradicts Cantor's Theorem.    (02)

-chris    (03)

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