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## Re: [ontolog-forum] Foundation ontology, CYC, and Mapping

 To: "[ontolog-forum]" "John F. Sowa" Fri, 05 Mar 2010 10:38:42 -0500 <4B912582.3000800@xxxxxxxxxxx>
 ```Chris, Ali, and Gary,    (01) These discussions help clarify many of the issues. In particular, they show why we need to distinguish the formal operations rather than vague "intuitions" about "intended meaning". In particular, I would prefer to avoid using the term 'meaning' except in discussions of how the formalism relates to informal discussions in natural languages.    (02) I would also like to add some examples that illustrate how these discussions can be rephrased in terms of the lattice of theories. And it is also important to show the implications for the finite, implemented subset, which may be called the *hierarchy* of theories.    (03) CM> To say that an extension T' of a theory T is conservative is > to say only that you can't prove anything new about the original > primitives in the language of the original theory...    (04) Your discussion of that point was a very good and clear summary and illustration in terms of Presburger arithmetic. I'd like to give another example in terms of Boolean algebra.    (05) Boole's original operators (or relations) were '+', 'x', and '-', which he interpreted variously as operators among propositions, monadic predicates, and sets. Although Boole used the same symbols that were used for ordinary arithmetic, he created a very different algebra by adding axioms that were false about the integers, such as    (06) p + p = p, for all p.    (07) This is definitely not a conservative extension. In fact, it is not an extension at all, since it is inconsistent with ordinary arithmetic.    (08) Peirce made a conservative extension to Boolean algebra by introducing a less-than-or-equal operator, which he wrote as the claw symbol '-<'. He showed that p- true > false > impossible.    (023) Cyc currently uses a 5-value algebra:    (024) true > true by default > unknown > false by default > false.    (025) Both versions are nonconservative extensions (specializations) of the original Boolean algebra.    (026) In terms of the lattice of theories, a conservative extension creates a larger theory (more theorems) but each of the new theorems has a translation to an equivalent theorem that uses only the original vocabulary. But in terms of MPTs (meaning preserving translations), the new theorems would be considered distinct from the old ones because they use different vocabulary.    (027) John    (028) _________________________________________________________________ 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    (029) ```
 Current Thread Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, (continued) Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Adrian Walker Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Kingsley Idehen Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Ali Hashemi Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Patrick Cassidy Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Ali Hashemi Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Gary Berg-Cross Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Ali Hashemi Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, John F. Sowa Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Christopher Menzel Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Ali Hashemi Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, John F. Sowa <= Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Matthew West Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Pat Hayes Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Chris Partridge Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Patrick Cassidy Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Chris Partridge Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, John F. Sowa Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Gary Berg-Cross Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Chris Partridge Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, John F. Sowa Re: [ontolog-forum] Foundation ontology, CYC, and Mapping, Matthew West