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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 18:04:34 -0500
Message-id: <BF3E761C-B999-40D9-8620-C6A270947B02@xxxxxxxx>

On Jun 15, 2007, at 5:37 PM, Waclaw Kusnierczyk wrote:    (01)

> Christopher Menzel wrote:
>> 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.
>> 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).
> Only if n=1.    (02)

Only if D is finite.  The POINT, of course -- that D, no matter its  
size, cannot contain all of the relations over D -- holds regardless.    (03)

I must admit that, like any good platonist, I was thinking of D as  
infinite, in which case what I say is true for all n.  To cover the  
general case of D any size, insert "at least" above after "There are".    (04)

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