ECAI 2002 Workshop on Ontologies and Semantic Interoperability Ontology Theory Christopher Menzel Department of Philosophy Texas A&M University

ECAI 2002 Workshop on Ontologies and Semantic Interoperability 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 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 Intuitions I: Ontologies 1. Ontologies consist of propositions. 2. 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. 3. Ontologies are objects I.e, things we can talk about and quantify over.

ECAI 2002 Workshop on Ontologies and Semantic Interoperability Intuitions II: Propositions 4. Propositions are not sentences, they are what sentences express. Different sentences in different languages (or possibly the same language) can express the same proposition. 5. Propositions are structured Propositions “consist” of concepts Hence, propositions can be logically equivalent without being identical. 6. Propositions are objects

ECAI 2002 Workshop on Ontologies and Semantic Interoperability 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 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 A Language for Ontology Theory A modal second-order base language Individual and predicate constants/variables Boolean operators Quantifiers modal operators ,  Complex predicates [ x 1 … x n  ], for individual variables x i No modal operators or bound predicate variables in  No x i occurring in any complex predicates in  All predicates can also occur as terms in atomic formulas

ECAI 2002 Workshop on Ontologies and Semantic Interoperability 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 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([ x PrefersTo(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 Structured Intensions The syntax of complex predicates reflects the logical form of their referents The LF of [(  x) ( Planet(x)  Larger(sun,x))] Pred 1 2 (Larger,sun) = [ y Larger(sun,y))] Impl(Planet, [ y Larger(sun,y))] = [ xy Planet(x)  Larger(sun,y))] Refl 1 2 ([ xy Planet(x)  Larger(sun,y))]) = [ x Planet(x)  Larger(sun,x))] Univ 1 ([ x Planet(x)  Larger(sun,))]) = [(  x) ( Planet(x)  Larger(sun,x))] In sum: Univ 1 (Refl 1 2 (Impl(Planet, Pred 1 2 (Larger,sun))))

ECAI 2002 Workshop on Ontologies and Semantic Interoperability 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 Univ 1 (Refl 1 2 (Impl(Planet, Pred 1 2 (Larger,sun)))) are being a planet, the larger-than relation, and the sun.

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

ECAI 2002 Workshop on Ontologies and Semantic Interoperability Content I: Ontologies An ontology is a nonempty property (class) of propositions Ontology(O) = df Property(O)  ~Empty(O)   p(O(p)  Proposition(p)) A constituent of an ontology is a constituent of one of its instances OntConst(x,O) = df Ontology(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 Content II: Strong Entailment An ontology O entails 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)) O strongly 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 Some useful comparative notions Ontology O is a subontology of O iff every instance of O is an instance of O. SubOnt(O,O) = df (  p)(O(p)  O(p)) O subsumes O 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 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)])