Relation SUBCLASS-PARTITION

• Defined in theory: Frame-ontology
• Source code: frame-ontology.lisp

Slots on this relation:

Documentation:

A subclass-partition of a class C is a set of subclasses of C that are mutually disjoint.

Instance-Of: Relation

Arity: 2

Domain: Class

Range: Class-partition

Equivalence Axioms:

(<=> (Subclass-Partition ?C ?Class-Partition)
     (And (Class ?C)
          (Class-Partition ?Class-Partition)
          (Forall (?Subclass)
                  (=> (Member ?Subclass ?Class-Partition)
                      (Subclass-Of ?Subclass ?C)))))

Other Related Axioms:

(=> (Subclass-Partition $X $Y) (Class-Partition $Y))

(=> (Subclass-Partition $X $Y) (Class $X))

(<=> (Subclass-Partition ?C ?Class-Partition)
     (And (Class ?C)
          (Class-Partition ?Class-Partition)
          (Forall (?Subclass)
                  (=> (Member ?Subclass ?Class-Partition)
                      (Subclass-Of ?Subclass ?C)))))

(Subrelation-Of Exhaustive-Subclass-Partition Subclass-Partition)

(<=> (Exhaustive-Subclass-Partition ?C ?Class-Partition)
     (And (Subclass-Partition ?C ?Class-Partition)
          (Forall (?Instance)
                  (=> (Instance-Of ?Instance ?C)
                      (Exists (?Subclass)
                              (And (Member ?Subclass
                                           ?Class-Partition)
                                   (Member ?Instance ?Subclass)))))))

Notes:

• Why is the second argument a set, rather than a sequence variable?

  Interesting design choice. The ``notation'' question here is not new syntax for a language, it's just the definition of a particular relation called subclass-partition. In KIF you can define relations that take an arbitrary number of arguments, using a a

  sequence variable that acts like &rest in Lisp. So I could have made the subclass-partition relation take a variable number of arguments. I decided not to use a sequence variable because that is not a minimal ontological commitment; it imposes an extra logical constraint for the sake of syntactic convenience. A sequence (list) imposes an order. But a class-partition requires no order among the classes. And I wanted the second argument to subclass-partition to be a class-partition -- a thing that is defined independently as a set of classes.