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Ontology Theory

Ontology Theory. Christopher Menzel Department of Philosophy Texas A&M University cmenzel@tamu.edu. Analysis: A Historical Paradigm. 18th Century Analysis: Intuition Intuitive theoretical foundations Conceptual confusions Inconsistencies 19th Century Analysis: Arithmetization

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Ontology Theory

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  1. Ontology Theory Christopher Menzel Department of Philosophy Texas A&M University cmenzel@tamu.edu ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  2. Analysis: A Historical Paradigm • 18th Century Analysis: Intuition • Intuitive theoretical foundations • Conceptual confusions • Inconsistencies • 19th Century Analysis: Arithmetization • Rigorous theoretical foundations • Shared understanding • Broader applicability ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  3. Ontology: The Current Situation • Similar to 18th Century analysis • Intuitive theoretical foundations • Conceptual confusions • High potential for inconsistency • Ontology needs its own “arithmetization” • Benefits • Shared understanding • Broader applicability • Sound foundation for integration ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  4. Intuitions I: Ontologies • Ontologies consist of propositions. • The content of an ontology O consists of the propositions entailed by O that involve only concepts in O. • Ontologies are comparable in terms of their content. • In particular, two ontologies are equivalent if they have the same content. • Ontologies are objects • I.e, things we can talk about and quantify over. ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  5. Intuitions II: Propositions • Propositions are not sentences, they are what sentences express. • Different sentences in different languages (or possibly the same language) can express the same proposition. • Propositions are structured • Propositions “consist” of concepts • Hence, propositions can be logically equivalent without being identical. • Propositions are objects ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  6. Desiderata I: Ontologies • Re 1, we need formal notions of ontology and proposition, and a notion of constituency relation that can hold between them. • Re 2, we need a notion of content; • Hence also a strong notion of entailment between ontologies and propositions. • Hence also a notion of comparability of ontologies. • Re 3, ontologies must be “first-class citizens” in ontology theory. ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  7. Desiderata II: Propositions • Re 4, we need a notion of proposition that is independent of any particular language. • Re 5, we need a robust notion of structured proposition • Hence a notion of the constituent concepts of a proposition • Re 6, propositions must be “first-class citizens” in ontology theory. ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  8. A Language for Ontology Theory • A modal second-order base language • Individual and predicate constants/variables • Boolean operators • Quantifiers • modal operators ,  • Complex predicates • [x1… xn ] , for individual variables xi • No modal operators or bound predicate variables in  • No xi occurring in any complex predicates in  • All predicates can also occur as terms in atomic formulas ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  9. Semantics: Type-free, Structured Intensionality • Type-freedom • There is a single universe of discourse closed under a variety of logical operations • Individual variables range over the entire domain • Structured Intensionality • n-place predicate variables range over subsets of the domain — the n-place relations • Complex predicates denote logically complex relations generated from “simpler” objects — their constituents — by the logical operations ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  10. Data for Type-freedom: Nominalization • Gerunds • “Being famous is all that Quentin thinks about.” • (x)(ThinksAbout(quentin,x)x = Famous) • Infinitives • “To prefer wine to beer is evidence of good taste.” • EvidenceOf([xPrefersTo(x,wine,beer)],GoodTaste) • That- clauses • “John believes that the sun is larger than every planet.” • Believes(john,[(x)(Planet(x)  Larger(sun,x))]) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  11. Structured Intensions • The syntax of complex predicates reflects the logical form of their referents • The LF of [(x)(Planet(x)  Larger(sun,x))] • Pred12(Larger,sun) = [yLarger(sun,y))] • Impl(Planet, [yLarger(sun,y))] =[xyPlanet(x)  Larger(sun,y))] • Refl12([xyPlanet(x)  Larger(sun,y))]) =[xPlanet(x)  Larger(sun,x))] • Univ1([xPlanet(x)  Larger(sun,))]) = [(x)(Planet(x)  Larger(sun,x))] • In sum: • Univ1(Refl12(Impl(Planet, Pred12(Larger,sun)))) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  12. Constituency and Logical Form • The constituents of an n-place relation are those entities involved in its logical form. • The primitive constituents of an n-place relation are those entities that have no constituents • The primitive constituents of Univ1(Refl12(Impl(Planet, Pred12(Larger,sun))))are being a planet, the larger-than relation, and the sun. ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  13. Axioms for Constituency • Const(,), where  is a term occurring free in  •  occurs free in if (i)  is a constant or (ii)  is a variable and some occurrence of in  is not in the scope of a quantifier occurrence in  of the form (Q) • Const is transitive and asymmetric • Hence also irreflexive • Primitiveness • Prim(x) =df(y)Const(y,x) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  14. Some definitions • Proposition(p) =df (F0)p = F0 • Property(r) =df (F1)r = F1 • True(p) =df (F0)p = F0F0 • TrueOf(r,x) =df (F1)r = F1F1(x) •  =dfTrue(), where a term  occurs like a 0-place predicate • () =dfTrueOf(,), where a term  occurs like a 1-place predicate • Empty(r) =dfProperty(r)  ~(x)r(x) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  15. Content I: Ontologies • An ontology is a nonempty property (class) of propositions • Ontology(O) =dfProperty(O)  ~Empty(O) p(O(p)  Proposition(p)) • A constituent of an ontology is a constituent of one of its instances • OntConst(x,O) =dfOntology(O)  (p)(O(p)Const(x,p)) • An ontology holds if all its constituent propositions are true. • Holds(O) =df (p)(O(p) p) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  16. Content II: Strong Entailment • An ontology Oentails a proposition p if, necessarily, p is true if O holds. • Entails(O,p) =df Ontology(O) (Holds(O) p) • O and p share primitives if every primitive constituent of p is a constituent of O. • ShPrim(O,p) =df Ontology(O) Proposition(p)  (x)(Prim(x)Const(x,p)  OntConst(x,O)) • Ostrongly entails p iff O entails p and O and p share primitives • O p =df Entails(O,p) ShPrim(O,p) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  17. Some useful comparative notions • Ontology O is a subontology ofO iff every instance of O is an instance of O. • SubOnt(O,O) =df (p)(O(p) O(p)) • OsubsumesO iff O strongly entails every instance of O. • Subsumes(O,O) =df (p)(O O p) • O and O are equivalent iff each subsumes the other. • Equiv(O,O) =df Subsumes(O,O) Subsumes(O,O) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

  18. More useful notions • O and O are overlap iff both strongly entail some proposition. • Overlap(O,O) =df (p)(O pO p) • Theorem: Overlap(O,O)  (x)(OntConst(x,O) OntConst(x,O)) • O is consistent iff there is some proposition it does not entail. • Consistent(O) =df Ontology(O) (p)~O p • O and O are compatible iff their union is consistent. • Compatible(O,O) =df Consistent([x O(x)  O(x)]) ECAI 2002 Workshop on Ontologies and Semantic Interoperability

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