ontolog-forum
[Top] [All Lists]

## Re: [ontolog-forum] First-Order Semantics

 To: "[ontolog-forum]" Waclaw Kusnierczyk Sat, 16 Jun 2007 08:28:33 +0200 <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) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (011) ```
 Current Thread Re: [ontolog-forum] First-Order Semantics, (continued) Re: [ontolog-forum] First-Order Semantics, Pat Hayes Re: [ontolog-forum] First-Order Semantics, John F. Sowa Re: [ontolog-forum] First-Order Semantics, Pat Hayes Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk Re: [ontolog-forum] First-Order Semantics, John F. Sowa Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Christopher Menzel Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk <= Re: [ontolog-forum] First-Order Semantics, Waclaw Kusnierczyk