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

 To: "[ontolog-forum] " Christopher Menzel Fri, 15 Jun 2007 11:39:01 -0500 <74334EB5-9C99-494B-A6C7-2BAA9FD16E7F@xxxxxxxx>
 ```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.    (01) 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). If (per impossibile) D were to contain all possible relations over D then (even ignoring issues of non-well-foundedness) we'd have card(D) = 2^card(D), which of course contradicts Cantor's Theorem.    (02) -chris    (03) _________________________________________________________________ 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    (04) ```
 Current Thread Re: [ontolog-forum] First-Order Semantics, Kathryn Blackmond Laskey 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