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

 To: "[ontolog-forum] " Christopher Menzel Fri, 15 Jun 2007 18:04:34 -0500
 ``` 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 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. >> >> 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). > > 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) _________________________________________________________________ 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    (05) ```
 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