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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 20:02:06 -0500
Message-id: <8B7AB56F-2355-4A43-9BB6-80AFF5A28ACA@xxxxxxxx>
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;    (01)

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.    (02)

> ...
>> 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.    (03)

You appear to be using "relation" to mean "property".  Is that right?    (04)

> (But see above.)
>> To cover the general case of D any size, insert "at least" above  
>> after "There are".
> At least, unless n=1.    (05)

Huh?  That's not even true if "relation" is understood to mean  
"property".  The above claim holds for any finite n -- assuming  
"relation" is understood as it is standardly used in logic and set  
theory.    (06)

-chris    (07)

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