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

 To: "[ontolog-forum]" "John F. Sowa" Sat, 13 Feb 2010 12:30:32 -0500 <4B76E1B8.1070804@xxxxxxxxxxx>
 ```Cory,    (01) I'd like to give some examples that may clarify that issue:    (02) CC> ... foundational concepts are very similar to the "minimally > axiomatized micro theories" I remember John-S describing as > a workable foundation, yet John does not see primitives > as workable - why the difference?    (03) My objection to using "primitives" as a foundation is that the meaning of a primitive changes with each theory in which it occurs. For example, the term 'point' is a "primitive" in Euclidean geometry and various non-Euclidean geometries. But the meaning of the term 'point' is specified by axioms that are different in each of those theories.    (04) Note that there are two kinds specifications:    (05) 1. Some terms are defined by a *closed form* definition, such as    (06) '3' is defined as '2+1'.    (07) In a closed-form definition, any occurrence of the term on the left can be replaced by the expression on the right.    (08) 2. But every formal theory has terms that cannot be defined by a closed-form definition.    (09) For example, both Euclidean and non-Euclidean geometries use the term 'point' without giving a closed-form definition. But calling it undefined is misleading because its "meaning" is determined by the pattern of relationships in the axioms in which the term occurs.    (010) The axioms specify the "meaning". But the axioms change from one theory to another. Therefore, the same term may have different meanings in theories with different axioms.    (011) For example, Euclidean and non-Euclidean geometries share the same "primitives". The following web site summarizes Euclid's five "postulates" (AKA axioms):    (012) http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml    (013) The first four are true in Euclidean and most non-Euclidean geometries. By deleting the fifth postulate, you would get a theory of geometry that had exactly the same "primitives", but with fewer axioms. That theory would be a generalization of the following three:    (014) 1. Euclid specified a geometry in which the sum of the three angles of a triangle always sum to exactly 180 degrees.    (015) 2. By changing the fifth postulate, Riemann defined a geometry in which the sum is < 180 degrees.    (016) 3. By a different change to the fifth postulate, Lobachevsky defined a geometry in which the sum is > 180 degrees.    (017) This gives us a generalization hierarchy of theories. The theories are generalized by adding axioms, specialized by deleting axioms, and revised by changing axioms (or by deleting some and replacing them with others).    (018) I have no objection to using collections of vague words, such as WordNet or Longman's, as *guidelines*. But the meanings of those words are ultimately determined by the axioms, not by the choice of primitives.    (019) Note to RF: Yes, the patterns of words in NL text impose strong constraints on the meanings of the words. That is important for NLP, but more explicit spec's are important for computer software.    (020) John    (021) _________________________________________________________________ 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    (022) ```
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