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 secondorder 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 secondorder 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 secondorder 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 ntuples).
>
> 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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