ECAI 2002

1
Ontology Theory

• Christopher Menzel
• Department of Philosophy
• Texas AM University
• cmenzel_at_tamu.edu

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

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

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.

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

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.

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.

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

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

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(?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)))

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) ?y Larger(sun,y))
• Impl(Planet, ?y Larger(sun,y)) ?xy
  Planet(x) ? Larger(sun,y))
• Refl12(?xy Planet(x) ? Larger(sun,y))) ?x
  Planet(x) ? Larger(sun,x))
• Univ1(?x Planet(x) ? Larger(sun,)))
  (?x)(Planet(x) ? Larger(sun,x))
• In sum
• Univ1(Refl12(Impl(Planet, Pred12(Larger,sun))))

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(P
  lanet, Pred12(Larger,sun))))are being a planet,
  the larger-than relation, and the sun.

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)

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)
• ? 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)

15
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)

16
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)

17
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)

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))