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## Re: [ontolog-forum] cyclic and acyclic definitions

 To: "[ontolog-forum]" "John F. Sowa" Fri, 24 Apr 2009 08:28:24 -0400 <49F1B068.9060908@xxxxxxxxxxx>
 ```Bart,    (01) Definitions of the following kind are trivial substitutions:    (02) > Momo = Man who has Only Male Offspring > (forall x (Momo(x) iff exists y s.t. Man(x) ^ Momo(y) ^ hasChild(y,x)))    (03) Following is the more common form of definition in mathematics:    (04) A dyadic relation R is transitive iff the following axiom is true:    (05) For all x, y, z, R(x,y) and R(y,z) implies R(x,z).    (06) More generally, the basic form of a definition in mathematics names the type of entity (e.g. 'transitive relation') and states a symbol such as R that represents an arbitrary instance. Then that statement is followed by one or more axioms that specify the constraints on any such entity R.    (07) Mathematicians have been using such definitions since Euclid, and they have never caused any trouble at all.    (08) Perhaps you're thinking of the following paradoxical statement in set theory:    (09) The set of all sets that are not elements of themselves.    (010) But that is a result of a peculiarity caused by the 'element of' relation. There is no paradox with the following statement:    (011) The set of all sets that are not subsets of themselves.    (012) Since every set is a subset of itself, the set defined by that sentence happens to be empty. No paradox.    (013) Suggestion: I recommend that we following the well established practice of mathematicians for stating definitions.    (014) John Sowa    (015) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: 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 join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (016) ```
 Current Thread [ontolog-forum] cyclic and acyclic definitions, Bart Gajderowicz Re: [ontolog-forum] cyclic and acyclic definitions, John F. Sowa Re: [ontolog-forum] cyclic and acyclic definitions, John Bottoms Re: [ontolog-forum] cyclic and acyclic definitions, Jawit Kien Re: [ontolog-forum] cyclic and acyclic definitions, Bart Gajderowicz Re: [ontolog-forum] cyclic and acyclic definitions, Jawit Kien Re: [ontolog-forum] cyclic and acyclic definitions, Bart Gajderowicz Re: [ontolog-forum] cyclic and acyclic definitions, Christopher Menzel Re: [ontolog-forum] cyclic and acyclic definitions, Bart Gajderowicz Re: [ontolog-forum] cyclic and acyclic definitions, John F. Sowa <= Re: [ontolog-forum] cyclic and acyclic definitions, Bart Gajderowicz Re: [ontolog-forum] cyclic and acyclic definitions, Azamat