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Re: [ontolog-forum] First-Order Semantics

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Waclaw Kusnierczyk <Waclaw.Marcin.Kusnierczyk@xxxxxxxxxxx>
Date: Sat, 16 Jun 2007 08:28:33 +0200
Message-id: <46738311.3090109@xxxxxxxxxxx>
Christopher Menzel wrote:
> On Jun 15, 2007, at 6:39 PM, Waclaw Kusnierczyk wrote:
>> Christopher Menzel wrote:
>>>>> 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.
>>> Only if D is finite.
>> My point was that a relation over a single set is a set of 1-tuples;
> I am not following you.  In standard, classical logic and set theory,  
> an n-place relation over a set is a set of n-tuples of members of S.   
> So a 1-place relation -- i.e., a property -- is a set of 1-tuples  
> over S (or, more typically, just a subset of S).  A 2-place relation  
> is a set of 2-tuples (i.e., ordered pairs) over S, i.e., a subset of  
> SxS; a 3-place relation is a set of 3-tuples, i.e., a subset of  
> SxSxS; and so on.    (01)

OK.  I remembered a definition of a relation over sets S, S', S'', ... 
as a set of tuples from the Cartesian product S x S' x S'' x ...
If there are 1 sets (there is one set), the relation is a set of 1-tuples.    (02)

But of course, this may be wrong.    (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.
>> A relation over an infinite set is still an (infinite) set of 1- 
>> tuples.
> You appear to be using "relation" to mean "property".  Is that right?    (04)

Yes.    (05)

>>> To cover the general case of D any size, insert "at least" above  
>>> after "There are".
>> At least, unless n=1.
> Huh?  That's not even true if "relation" is understood to mean  
> "property".      (06)

?  If n=1, then there are exactly 2^card(D) properties over D?    (07)

> The above claim holds for any finite n -- assuming  
> "relation" is understood as it is standardly used in logic and set  
> theory.    (08)

We read the same statements in different ways.
I shut up.    (09)

vQ    (010)

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