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). (01)
Only if n=1. (02)
vQ (03)
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